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Sandbox Plain spheres

posted 5mo ago by trichoplax‭ · edited 2mo ago by trichoplax‭

Article code-golf math graphical-output image

#20: Post edited by

trichoplax‭ · 2024-10-12T22:00:57Z (2 months ago)
Add TODO for checking choice of sphere makes little difference

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Given a list of spheres, output an image.
Produce a greyscale [orthographic](https://en.wikipedia.org/wiki/Parallel_projection#Orthographic_projection) (parallel projection - no perspective) ray caster, that renders only plain grey spheres with no lighting.
## Input
- A list of spheres, each consisting of:
- The coordinates of its centre $(x,y,z)$.
- Its radius $r$.
- Its greyscale value $g$, an integer from $0$ to $255$ (both inclusive).
- Only the greyscale value will be restricted to integer values. The coordinates and radius may have fractional values.
## Output
- An image $1024$ by $1024$ pixels with greyscale values from $0$ to $255$.
- The image covers x and y values from $-1$ to $1$.
- A sphere's greyscale value is applied to every visible part of it. There is no variation over the surface of the sphere, so it appears flat like a disk.
- You may output whatever image format is convenient, including simple text formats such as [PGM](https://en.wikipedia.org/wiki/Netpbm#PGM_example) or a list of pixel greyscale values. If you want to include an example of your output as an image in your answer, the image formats that Codidact supports are PNG, JPEG, and GIF. You could output PGM and then convert to PNG to upload to your answer. If you output just the greyscale values as a list of numbers from $0$ to $255$, I've made a `[ TODO page to convert these to greyscale PNG]`.
- You are free to output an image reflected or rotated by a multiple of 90 degrees. However, the z axis is required to be increasing in the forwards direction (into the image) otherwise some of the test cases will look incorrect.
- You may assume that all ray intersection points (not just nearest intersections) will have a $z$ value greater than zero, and less than $1024$.
```text
TODO: Adjust test cases to fit within this specified z range
```
## Calculating a pixel value
Imagine a ray passing into the image through a pixel, perpendicular to the image plane. If it hits a sphere, it shows the greyscale value of the first sphere it hits. Otherwise, it shows a value of zero (black).
For example, for each pixel:
- The pixel $x,y$ coordinates will each be from $0$ to $1023$. Scale these to the range $-1$ to $1$ (for example, divide by $512$ and subtract $1$).
- If these scaled $x,y$ coordinates are within the radius $r$ of the $x,y$ coordinates of a sphere's centre, the ray intersects that sphere (the $z$ coordinate is not relevant to this test because the image is an orthographic projection - the rays are all parallel to the $z$ axis). The distance $a$ of the ray from the sphere's centre can be calculated as follows:
$$a=\sqrt{(x_{ray}-x_{sphere})^2+(y_{ray}-y_{sphere})^2}$$
```text
CONSIDER A DIAGRAM HERE
```
- For each sphere that the ray intersects, find the $z$ coordinate of the nearest intersection (there will often be $2$ per sphere, $1$ on the front of the sphere and $1$ on the back of the sphere, as the ray passes through - only the nearest is relevant). This can be calculated as follows:
- The distance $d$ to the nearest intersection point can be calculated from the $z$ coordinate of the sphere's centre, the radius $r$, and the distance $a$ from the $x,y$ coordinates of the sphere's centre to the scaled $x,y$ coordinates of the pixel:
$$d=z_{sphere}-\sqrt{r_{sphere}^2-a^2}$$
```text
CONSIDER A DIAGRAM HERE
```
- Of all the spheres that the ray intersects, the sphere with the intersection point with the smallest $z$ coordinate $d$ is the one to display. Set the pixel to the greyscale value of this sphere.
- If $2$ or more spheres have an intersection point with the smallest $z$ coordinate $d$ then you may set the pixel to the greyscale value of any of these spheres.
- If the ray did not intersect any spheres, set the pixel to zero (black).
## Example implementations
These are ungolfed code to supplement understanding of the requirements. You are free to golf these or implement your own approach that matches the test cases up to reflection & rotation.
The Python implementation uses 64 bit floating point. The Rust implementation uses 32 bit floating point to show the difference in output. I've tried to optimise the test cases for compatibility with 32 bit floating point, so the differences are restricted to being a pixel out in some places.
<details>
<summary>Python</summary>
The function `grey_values` would be sufficient as an entry to this challenge. The rest of the code saves the output as a PNG file. The function itself has no dependence on the `import png` so it would not be required in an entry.
Instructions for installing the `png` package can be found on its [project page on PyPI](https://pypi.org/project/pypng/).
```python
def grey_values(spheres):
values = []
for y_pixel in range(1024):
for x_pixel in range(1024):
x_ray, y_ray = x_pixel / 512 - 1, y_pixel / 512 - 1
min_distance, hit_grey_value = None, 0
for sphere in spheres:
x_sphere, y_sphere, z_sphere, radius, grey = sphere
a = (
(x_ray - x_sphere) ** 2 + (y_ray - y_sphere) ** 2
) ** 0.5
if a < radius:
distance = z_sphere - (radius ** 2 - a ** 2) ** 0.5
if min_distance is None or distance < min_distance:
min_distance = distance
hit_grey_value = grey
values.append(hit_grey_value)
return values
import png
test_case = [
[0, 0, 1000, 2, 192],
[-2, 2, 100, 3, 98],
[2, 2, 100, 3, 98],
[0.5, -0.5, 500, 0.2, 255],
[-0.349469, -0.1, 20.000279, 0.6, 64],
[-0.349775, -0.100273, 20.000485, 0.6, 85],
[-0.350214, -0.100386, 20.000407, 0.6, 106],
[-0.350527, -0.100273, 20.000091, 0.6, 127],
[-0.350531, -0.1, 19.999721, 0.6, 148],
[-0.350225, -0.099727, 19.999515, 0.6, 169],
[-0.349786, -0.099614, 19.999593, 0.6, 190],
[-0.349473, -0.099727, 19.999909, 0.6, 211],
[0.20032, 0.349824, 15.000456, 0.3, 0],
[0.20032, 0.349517, 14.999924, 0.3, 0],
[0.19968, 0.350483, 15.000076, 0.3, 0],
[0.19968, 0.350176, 14.999544, 0.3, 0],
[0.200578, 0.350073, 14.999958, 0.3, 0],
[0.199817, 0.349519, 15.000277, 0.3, 0],
[0.200183, 0.350481, 14.999723, 0.3, 0],
[0.199422, 0.349927, 15.000042, 0.3, 0],
[0.200235, 0.35042, 15.000332, 0.3, 0],
[0.199765, 0.350078, 15.000529, 0.3, 0],
[0.200235, 0.349922, 14.999471, 0.3, 0],
[0.199765, 0.34958, 14.999668, 0.3, 0],
[0.200429, 0.350419, 15.000005, 0.3, 255],
[0.199571, 0.349795, 15.000366, 0.3, 255],
[0.200429, 0.350205, 14.999634, 0.3, 255],
[0.199571, 0.349581, 14.999995, 0.3, 255],
[0.200524, 0.349747, 15.000146, 0.3, 255],
[0.200196, 0.349509, 15.000283, 0.3, 255],
[0.199804, 0.350491, 14.999717, 0.3, 255],
[0.199476, 0.350253, 14.999854, 0.3, 255],
[0.200138, 0.349578, 14.999597, 0.3, 255],
[0.200138, 0.350138, 15.000567, 0.3, 255],
[0.199862, 0.349862, 14.999433, 0.3, 255],
[0.199862, 0.350422, 15.000403, 0.3, 255],
[0.200488, 0.350136, 15.000321, 0.3, 255],
[0.199957, 0.349751, 15.000544, 0.3, 255],
[0.200488, 0.34979, 14.999721, 0.3, 255],
[0.199957, 0.349404, 14.999944, 0.3, 255],
[0.200043, 0.350596, 15.000056, 0.3, 255],
[0.199512, 0.35021, 15.000279, 0.3, 255],
[0.200043, 0.350249, 14.999456, 0.3, 255],
[0.199512, 0.349864, 14.999679, 0.3, 255],
[-0.5, 0.500442, 10.000442, 0.2, 0],
[-0.5, 0.5, 10, 0.1995, 255],
[-0.499966, 0.49971, 9.999563, 0.199, 0],
[-0.500034, 0.499563, 9.99971, 0.199, 0],
[-0.499934, 0.499432, 9.999144, 0.1985, 255],
[-0.500066, 0.499144, 9.999432, 0.1985, 255],
[0.6, 0.5, 10, 0.25, 210],
[0.600122, 0.4999, 10.000122, 0.25, 210],
[0.600061, 0.49995, 10.000061, 0.25003, 50],
[0.595856, 0.505256, 9.991271, 0.24, 100],
[0.597973, 0.507461, 9.992175, 0.24, 210],
[0.595362, 0.507926, 9.993945, 0.24, 100],
[0.594618, 0.509055, 9.996832, 0.24, 210],
[0.592382, 0.506906, 9.996092, 0.24, 100],
[0.590438, 0.504882, 9.997604, 0.24, 210],
[0.591035, 0.503605, 9.994744, 0.24, 100],
[0.59121, 0.500708, 9.993425, 0.24, 210],
[0.593181, 0.502585, 9.991764, 0.24, 100],
[0.595867, 0.502302, 9.990069, 0.24, 210],
]
values = grey_values(test_case)
rows = [values[row * 1024:row * 1024 + 1024] for row in range(1024)]
with open('combination_scene.png', 'wb') as file:
w = png.Writer(1024, 1024, greyscale=True)
w.write(file, rows)
```
</details>
<details>
<summary>Rust</summary>
The function `grey_values` would be sufficient as an entry to this challenge. The rest of the code saves the output as a PNG file.
The `png` crate can be installed using `cargo add png`. The `png` crate is not required for the `grey_values` function, only for saving as a PNG file.
```rust
fn grey_values(spheres: &Vec<[f32; 5]>) -> Vec<u8> {
let mut values = vec![];
for y_pixel in 0..1024 {
for x_pixel in 0..1024 {
let x_ray = x_pixel as f32 / 512.0 - 1.0;
let y_ray = y_pixel as f32 / 512.0 - 1.0;
let mut min_distance = f32::INFINITY;
let mut hit_grey_value = 0;
for sphere in spheres {
let [
x_sphere, y_sphere, z_sphere, radius, grey
] = *sphere;
let a = (
(x_ray - x_sphere).powf(2.0)
+ (y_ray - y_sphere).powf(2.0)
).sqrt();
if a < radius {
let distance = z_sphere - (
radius.powf(2.0) - a.powf(2.0)
).sqrt();
if distance < min_distance {
min_distance = distance;
hit_grey_value = grey as u8;
}
}
}
values.push(hit_grey_value);
}
}
values
}
fn main() {
let test_case: Vec<[f32; 5]> = vec![
[0.0, 0.0, 1000.0, 2.0, 192.0],
[-2.0, 2.0, 100.0, 3.0, 98.0],
[2.0, 2.0, 100.0, 3.0, 98.0],
[0.5, -0.5, 500.0, 0.2, 255.0],
[-0.349469, -0.1, 20.000279, 0.6, 64.0],
[-0.349775, -0.100273, 20.000485, 0.6, 85.0],
[-0.350214, -0.100386, 20.000407, 0.6, 106.0],
[-0.350527, -0.100273, 20.000091, 0.6, 127.0],
[-0.350531, -0.1, 19.999721, 0.6, 148.0],
[-0.350225, -0.099727, 19.999515, 0.6, 169.0],
[-0.349786, -0.099614, 19.999593, 0.6, 190.0],
[-0.349473, -0.099727, 19.999909, 0.6, 211.0],
[0.20032, 0.349824, 15.000456, 0.3, 0.0],
[0.20032, 0.349517, 14.999924, 0.3, 0.0],
[0.19968, 0.350483, 15.000076, 0.3, 0.0],
[0.19968, 0.350176, 14.999544, 0.3, 0.0],
[0.200578, 0.350073, 14.999958, 0.3, 0.0],
[0.199817, 0.349519, 15.000277, 0.3, 0.0],
[0.200183, 0.350481, 14.999723, 0.3, 0.0],
[0.199422, 0.349927, 15.000042, 0.3, 0.0],
[0.200235, 0.35042, 15.000332, 0.3, 0.0],
[0.199765, 0.350078, 15.000529, 0.3, 0.0],
[0.200235, 0.349922, 14.999471, 0.3, 0.0],
[0.199765, 0.34958, 14.999668, 0.3, 0.0],
[0.200429, 0.350419, 15.000005, 0.3, 255.0],
[0.199571, 0.349795, 15.000366, 0.3, 255.0],
[0.200429, 0.350205, 14.999634, 0.3, 255.0],
[0.199571, 0.349581, 14.999995, 0.3, 255.0],
[0.200524, 0.349747, 15.000146, 0.3, 255.0],
[0.200196, 0.349509, 15.000283, 0.3, 255.0],
[0.199804, 0.350491, 14.999717, 0.3, 255.0],
[0.199476, 0.350253, 14.999854, 0.3, 255.0],
[0.200138, 0.349578, 14.999597, 0.3, 255.0],
[0.200138, 0.350138, 15.000567, 0.3, 255.0],
[0.199862, 0.349862, 14.999433, 0.3, 255.0],
[0.199862, 0.350422, 15.000403, 0.3, 255.0],
[0.200488, 0.350136, 15.000321, 0.3, 255.0],
[0.199957, 0.349751, 15.000544, 0.3, 255.0],
[0.200488, 0.34979, 14.999721, 0.3, 255.0],
[0.199957, 0.349404, 14.999944, 0.3, 255.0],
[0.200043, 0.350596, 15.000056, 0.3, 255.0],
[0.199512, 0.35021, 15.000279, 0.3, 255.0],
[0.200043, 0.350249, 14.999456, 0.3, 255.0],
[0.199512, 0.349864, 14.999679, 0.3, 255.0],
[-0.5, 0.500442, 10.000442, 0.2, 0.0],
[-0.5, 0.5, 10.0, 0.1995, 255.0],
[-0.499966, 0.49971, 9.999563, 0.199, 0.0],
[-0.500034, 0.499563, 9.99971, 0.199, 0.0],
[-0.499934, 0.499432, 9.999144, 0.1985, 255.0],
[-0.500066, 0.499144, 9.999432, 0.1985, 255.0],
[0.6, 0.5, 10.0, 0.25, 210.0],
[0.600122, 0.4999, 10.000122, 0.25, 210.0],
[0.600061, 0.49995, 10.000061, 0.25003, 50.0],
[0.595856, 0.505256, 9.991271, 0.24, 100.0],
[0.597973, 0.507461, 9.992175, 0.24, 210.0],
[0.595362, 0.507926, 9.993945, 0.24, 100.0],
[0.594618, 0.509055, 9.996832, 0.24, 210.0],
[0.592382, 0.506906, 9.996092, 0.24, 100.0],
[0.590438, 0.504882, 9.997604, 0.24, 210.0],
[0.591035, 0.503605, 9.994744, 0.24, 100.0],
[0.59121, 0.500708, 9.993425, 0.24, 210.0],
[0.593181, 0.502585, 9.991764, 0.24, 100.0],
[0.595867, 0.502302, 9.990069, 0.24, 210.0],
];
let values = grey_values(&test_case);
let path = std::path::Path::new("combination_scene.png");
let file = std::fs::File::create(path).unwrap();
let w = &mut std::io::BufWriter::new(file);
let mut encoder = png::Encoder::new(w, 1024, 1024);
encoder.set_color(png::ColorType::Grayscale);
encoder.set_depth(png::BitDepth::Eight);
let mut writer = encoder.write_header().unwrap();
writer.write_image_data(&values).unwrap();
}
```
</details>
## Test cases
The output images may be scaled down to fit this page. Click on any image to open it full size.
### Single sphere
Input:
```text
[
[0, 0, 0, 1, 128],
]
```
Output:
[![Grey circle on a black background](https://codegolf.codidact.com/uploads/jlru163pa2pgm7aaq70b2tff16uf)](https://codegolf.codidact.com/uploads/jlru163pa2pgm7aaq70b2tff16uf)
### Two separated spheres
Input:
```text
[
[0, -0.55, 0, 0.4, 100],
[0, 0.55, 0, 0.4, 200],
]
```
Output:
[![Light and dark grey circles on a black background](https://codegolf.codidact.com/uploads/8yw7agli78y65kiomb1u5yebs7c5)](https://codegolf.codidact.com/uploads/8yw7agli78y65kiomb1u5yebs7c5)
### Ring of spheres
Input:
```text
[
[0.75, 0, 0, 0.25, 64],
[0.53033, 0.375, -0.375, 0.25, 185],
[0, 0.53033, -0.53033, 0.25, 64],
[-0.53033, 0.375, -0.375, 0.25, 185],
[-0.75, 0, 0, 0.25, 64],
[-0.53033, -0.375, 0.375, 0.25, 185],
[0, -0.53033, 0.53033, 0.25, 64],
[0.53033, -0.375, 0.375, 0.25, 185],
]
```
Output:
[![Eight circles in alternating light and dark grey in a ring some overlapping](https://codegolf.codidact.com/uploads/0lsenynig27au0sryf3ruozt1uwh)](https://codegolf.codidact.com/uploads/0lsenynig27au0sryf3ruozt1uwh)
### Beach ball
Input:
```text
[
[0.001, 0, 0, 1, 64],
[0.000707, 0.000707, 0, 1, 85],
[0, 0.001, 0, 1, 106],
[-0.000707, 0.000707, 0, 1, 127],
[-0.001, 0, 0, 1, 148],
[-0.000707, -0.000707, 0, 1, 169],
[0, -0.001, 0, 1, 190],
[0.000707, -0.000707, 0, 1, 211],
]
```
Output:
[![A circle with eight sectors in different shades of grey](https://codegolf.codidact.com/uploads/51a3ni0cfwsqcdw5b49gy6282qno)](https://codegolf.codidact.com/uploads/51a3ni0cfwsqcdw5b49gy6282qno)
### Football
Input:
```text
[
[0, 0.000058, 0.001947, 1, 0],
[0, -0.001715, 0.000923, 1, 0],
[0, 0.001715, -0.000923, 1, 0],
[0, -0.000058, -0.001947, 1, 0],
[0.001657, -0.000512, 0.000887, 1, 0],
[-0.001657, -0.000512, 0.000887, 1, 0],
[0.001657, 0.000512, -0.000887, 1, 0],
[-0.001657, 0.000512, -0.000887, 1, 0],
[0.001024, 0.001435, 0.000829, 1, 0],
[-0.001024, 0.001435, 0.000829, 1, 0],
[0.001024, -0.001435, -0.000829, 1, 0],
[-0.001024, -0.001435, -0.000829, 1, 0],
[0.001868, 0.000618, 0.000357, 1, 255],
[-0.001868, 0.000618, 0.000357, 1, 255],
[0.001868, -0.000618, -0.000357, 1, 255],
[-0.001868, -0.000618, -0.000357, 1, 255],
[0.000714, -0.000934, 0.001618, 1, 255],
[-0.000714, -0.000934, 0.001618, 1, 255],
[0.000714, 0.000934, -0.001618, 1, 255],
[-0.000714, 0.000934, -0.001618, 1, 255],
[0, -0.001975, -0.000316, 1, 255],
[0, 0.001261, 0.001552, 1, 255],
[0, -0.001261, -0.001552, 1, 255],
[0, 0.001975, 0.000316, 1, 255],
[0.001155, 0.000423, 0.001577, 1, 255],
[-0.001155, 0.000423, 0.001577, 1, 255],
[0.001155, -0.001577, 0.000423, 1, 255],
[-0.001155, -0.001577, 0.000423, 1, 255],
[0.001155, 0.001577, -0.000423, 1, 255],
[-0.001155, 0.001577, -0.000423, 1, 255],
[0.001155, -0.000423, -0.001577, 1, 255],
[-0.001155, -0.000423, -0.001577, 1, 255],
]
```
Output:
[![Black pentagons on a white ball](https://codegolf.codidact.com/uploads/bmrbsw65p59imvwu8ssd42uvvx1r)](https://codegolf.codidact.com/uploads/bmrbsw65p59imvwu8ssd42uvvx1r)
### Eight ball
Input:
```text
[
[0, 0, 1000, 2, 128],
[0, 0, 0.003125, 1, 0],
[0, 0, 0, 0.9975, 255],
[0, 0.000547, -0.002573, 0.995, 0],
[0, -0.000547, -0.002573, 0.995, 0],
[0, 0.001071, -0.005037, 0.9925, 255],
[0, -0.001071, -0.005037, 0.9925, 255],
]
```
Output:
[![A ball with the number 8 in a white circle](https://codegolf.codidact.com/uploads/dvl68crpmy62rqt94mwanditt4wc)](https://codegolf.codidact.com/uploads/dvl68crpmy62rqt94mwanditt4wc)
### Star ball
Input:
```text
[
[0, 0, 0, 1, 210],
[0, 0, 0.0008, 1, 210],
[0, 0, 0.0004, 1.00012, 50],
[0.012969, 0, -0.042045, 0.96, 100],
[0.016397, 0.011913, -0.039054, 0.96, 210],
[0.004008, 0.012334, -0.042045, 0.96, 100],
[-0.006263, 0.019276, -0.039054, 0.96, 210],
[-0.010492, 0.007623, -0.042045, 0.96, 100],
[-0.020268, 0, -0.039054, 0.96, 210],
[-0.010492, -0.007623, -0.042045, 0.96, 100],
[-0.006263, -0.019276, -0.039054, 0.96, 210],
[0.004008, -0.012334, -0.042045, 0.96, 100],
[0.016397, -0.011913, -0.039054, 0.96, 210],
]
```
Output:
[![A light grey ball with a dark grey five pointed star](https://codegolf.codidact.com/uploads/f3wdgwru6j35ue72x25mazxikfx2)](https://codegolf.codidact.com/uploads/f3wdgwru6j35ue72x25mazxikfx2)
### Combination scene
Input:
```text
[
[0, 0, 1000, 2, 192],
[-2, 2, 100, 3, 98],
[2, 2, 100, 3, 98],
[0.5, -0.5, 500, 0.2, 255],
[-0.349469, -0.1, 20.000279, 0.6, 64],
[-0.349775, -0.100273, 20.000485, 0.6, 85],
[-0.350214, -0.100386, 20.000407, 0.6, 106],
[-0.350527, -0.100273, 20.000091, 0.6, 127],
[-0.350531, -0.1, 19.999721, 0.6, 148],
[-0.350225, -0.099727, 19.999515, 0.6, 169],
[-0.349786, -0.099614, 19.999593, 0.6, 190],
[-0.349473, -0.099727, 19.999909, 0.6, 211],
[0.20032, 0.349824, 15.000456, 0.3, 0],
[0.20032, 0.349517, 14.999924, 0.3, 0],
[0.19968, 0.350483, 15.000076, 0.3, 0],
[0.19968, 0.350176, 14.999544, 0.3, 0],
[0.200578, 0.350073, 14.999958, 0.3, 0],
[0.199817, 0.349519, 15.000277, 0.3, 0],
[0.200183, 0.350481, 14.999723, 0.3, 0],
[0.199422, 0.349927, 15.000042, 0.3, 0],
[0.200235, 0.35042, 15.000332, 0.3, 0],
[0.199765, 0.350078, 15.000529, 0.3, 0],
[0.200235, 0.349922, 14.999471, 0.3, 0],
[0.199765, 0.34958, 14.999668, 0.3, 0],
[0.200429, 0.350419, 15.000005, 0.3, 255],
[0.199571, 0.349795, 15.000366, 0.3, 255],
[0.200429, 0.350205, 14.999634, 0.3, 255],
[0.199571, 0.349581, 14.999995, 0.3, 255],
[0.200524, 0.349747, 15.000146, 0.3, 255],
[0.200196, 0.349509, 15.000283, 0.3, 255],
[0.199804, 0.350491, 14.999717, 0.3, 255],
[0.199476, 0.350253, 14.999854, 0.3, 255],
[0.200138, 0.349578, 14.999597, 0.3, 255],
[0.200138, 0.350138, 15.000567, 0.3, 255],
[0.199862, 0.349862, 14.999433, 0.3, 255],
[0.199862, 0.350422, 15.000403, 0.3, 255],
[0.200488, 0.350136, 15.000321, 0.3, 255],
[0.199957, 0.349751, 15.000544, 0.3, 255],
[0.200488, 0.34979, 14.999721, 0.3, 255],
[0.199957, 0.349404, 14.999944, 0.3, 255],
[0.200043, 0.350596, 15.000056, 0.3, 255],
[0.199512, 0.35021, 15.000279, 0.3, 255],
[0.200043, 0.350249, 14.999456, 0.3, 255],
[0.199512, 0.349864, 14.999679, 0.3, 255],
[-0.5, 0.500442, 10.000442, 0.2, 0],
[-0.5, 0.5, 10, 0.1995, 255],
[-0.499966, 0.49971, 9.999563, 0.199, 0],
[-0.500034, 0.499563, 9.99971, 0.199, 0],
[-0.499934, 0.499432, 9.999144, 0.1985, 255],
[-0.500066, 0.499144, 9.999432, 0.1985, 255],
[0.6, 0.5, 10, 0.25, 210],
[0.600122, 0.4999, 10.000122, 0.25, 210],
[0.600061, 0.49995, 10.000061, 0.25003, 50],
[0.595856, 0.505256, 9.991271, 0.24, 100],
[0.597973, 0.507461, 9.992175, 0.24, 210],
[0.595362, 0.507926, 9.993945, 0.24, 100],
[0.594618, 0.509055, 9.996832, 0.24, 210],
[0.592382, 0.506906, 9.996092, 0.24, 100],
[0.590438, 0.504882, 9.997604, 0.24, 210],
[0.591035, 0.503605, 9.994744, 0.24, 100],
[0.59121, 0.500708, 9.993425, 0.24, 210],
[0.593181, 0.502585, 9.991764, 0.24, 100],
[0.595867, 0.502302, 9.990069, 0.24, 210],
]
```
Output:
[![A beachball, football, eightball, and starball with the sun behind hills in the background](https://codegolf.codidact.com/uploads/usv9npm1t5bjpcaknn6rbzgkqwmi)](https://codegolf.codidact.com/uploads/usv9npm1t5bjpcaknn6rbzgkqwmi)
## Accuracy
The outputs for the test cases were generated using 64 bit floating point numbers. The following is the last test case using 32 bit floating point rather than 64 bit floating point:
[![A beachball, football, eightball, and starball with the sun behind hills in the background](https://codegolf.codidact.com/uploads/7uuxjxq7a9oxc9q0jf2bi8l68qk3)](https://codegolf.codidact.com/uploads/7uuxjxq7a9oxc9q0jf2bi8l68qk3)
The inaccuracies in the 32 bit floating point version are subtle. You may need to click on the image to open it full size to see the occasional inaccurate pixel. For example, around the inside of the holes in the 8 on the eightball. These occasional inaccurate pixels are acceptable in your output (allowing languages which only support 32 bit floating point to compete).
More widespread speckling as in the following image would not be valid, so you may need to be careful if changing the order of calculations during golfing:
[![A scene with significant areas of inaccurate speckles obscuring the intended image](https://codegolf.codidact.com/uploads/1sa7i6z0cv2kgyrmw2tj2ytr1ps1)](https://codegolf.codidact.com/uploads/1sa7i6z0cv2kgyrmw2tj2ytr1ps1)
This exaggerated inaccuracy was achieved by adding 1000 to and then subtracting 1000 from the distance calculation, exceeding the available precision of 32 bit floating point and causing some distances to appear in the wrong order. Note that this same approach causes no inaccuracy with 64 bit floating point, so if your language supports both you may have more leeway for rearranging calculations if you use 64 bit floating point.
## Explanatory animations
The example images were produced from plain grey spheres with no surface pattern or texture. The following animations show how they break down into their component plain spheres.
<details>
<summary>Animations</summary>
TODO
</details>
## Scoring
This is a [code golf challenge](https://codegolf.codidact.com/categories/49/tags/4274). Your score is the number of bytes in your code. Lowest score for each language wins.
> Explanations are optional, but I'm more likely to upvote answers that have one.

Given a list of spheres, output an image.
Produce a greyscale [orthographic](https://en.wikipedia.org/wiki/Parallel_projection#Orthographic_projection) (parallel projection - no perspective) ray caster, that renders only plain grey spheres with no lighting.
## Input
- A list of spheres, each consisting of:
- The coordinates of its centre $(x,y,z)$.
- Its radius $r$.
- Its greyscale value $g$, an integer from $0$ to $255$ (both inclusive).
- Only the greyscale value will be restricted to integer values. The coordinates and radius may have fractional values.
## Output
- An image $1024$ by $1024$ pixels with greyscale values from $0$ to $255$.
- The image covers x and y values from $-1$ to $1$.
- A sphere's greyscale value is applied to every visible part of it. There is no variation over the surface of the sphere, so it appears flat like a disk.
- You may output whatever image format is convenient, including simple text formats such as [PGM](https://en.wikipedia.org/wiki/Netpbm#PGM_example) or a list of pixel greyscale values. If you want to include an example of your output as an image in your answer, the image formats that Codidact supports are PNG, JPEG, and GIF. You could output PGM and then convert to PNG to upload to your answer. If you output just the greyscale values as a list of numbers from $0$ to $255$, I've made a `[ TODO page to convert these to greyscale PNG]`.
- You are free to output an image reflected or rotated by a multiple of 90 degrees. However, the z axis is required to be increasing in the forwards direction (into the image) otherwise some of the test cases will look incorrect.
- You may assume that all ray intersection points (not just nearest intersections) will have a $z$ value greater than zero, and less than $1024$.
```text
TODO: Adjust test cases to fit within this specified z range
```
## Calculating a pixel value
Imagine a ray passing into the image through a pixel, perpendicular to the image plane. If it hits a sphere, it shows the greyscale value of the first sphere it hits. Otherwise, it shows a value of zero (black).
For example, for each pixel:
- The pixel $x,y$ coordinates will each be from $0$ to $1023$. Scale these to the range $-1$ to $1$ (for example, divide by $512$ and subtract $1$).
- If these scaled $x,y$ coordinates are within the radius $r$ of the $x,y$ coordinates of a sphere's centre, the ray intersects that sphere (the $z$ coordinate is not relevant to this test because the image is an orthographic projection - the rays are all parallel to the $z$ axis). The distance $a$ of the ray from the sphere's centre can be calculated as follows:
$$a=\sqrt{(x_{ray}-x_{sphere})^2+(y_{ray}-y_{sphere})^2}$$
```text
CONSIDER A DIAGRAM HERE
```
- For each sphere that the ray intersects, find the $z$ coordinate of the nearest intersection (there will often be $2$ per sphere, $1$ on the front of the sphere and $1$ on the back of the sphere, as the ray passes through - only the nearest is relevant). This can be calculated as follows:
- The distance $d$ to the nearest intersection point can be calculated from the $z$ coordinate of the sphere's centre, the radius $r$, and the distance $a$ from the $x,y$ coordinates of the sphere's centre to the scaled $x,y$ coordinates of the pixel:
$$d=z_{sphere}-\sqrt{r_{sphere}^2-a^2}$$
```text
CONSIDER A DIAGRAM HERE
```
- Of all the spheres that the ray intersects, the sphere with the intersection point with the smallest $z$ coordinate $d$ is the one to display. Set the pixel to the greyscale value of this sphere.
- If $2$ or more spheres have an intersection point with the smallest $z$ coordinate $d$ then you may set the pixel to the greyscale value of any of these spheres.
```text
TODO: generate output images for each test case
both ways to confirm that this makes less than
a pixel's width difference.
```
- If the ray did not intersect any spheres, set the pixel to zero (black).
## Example implementations
These are ungolfed code to supplement understanding of the requirements. You are free to golf these or implement your own approach that matches the test cases up to reflection & rotation.
The Python implementation uses 64 bit floating point. The Rust implementation uses 32 bit floating point to show the difference in output. I've tried to optimise the test cases for compatibility with 32 bit floating point, so the differences are restricted to being a pixel out in some places.
<details>
<summary>Python</summary>
The function `grey_values` would be sufficient as an entry to this challenge. The rest of the code saves the output as a PNG file. The function itself has no dependence on the `import png` so it would not be required in an entry.
Instructions for installing the `png` package can be found on its [project page on PyPI](https://pypi.org/project/pypng/).
```python
def grey_values(spheres):
values = []
for y_pixel in range(1024):
for x_pixel in range(1024):
x_ray, y_ray = x_pixel / 512 - 1, y_pixel / 512 - 1
min_distance, hit_grey_value = None, 0
for sphere in spheres:
x_sphere, y_sphere, z_sphere, radius, grey = sphere
a = (
(x_ray - x_sphere) ** 2 + (y_ray - y_sphere) ** 2
) ** 0.5
if a < radius:
distance = z_sphere - (radius ** 2 - a ** 2) ** 0.5
if min_distance is None or distance < min_distance:
min_distance = distance
hit_grey_value = grey
values.append(hit_grey_value)
return values
import png
test_case = [
[0, 0, 1000, 2, 192],
[-2, 2, 100, 3, 98],
[2, 2, 100, 3, 98],
[0.5, -0.5, 500, 0.2, 255],
[-0.349469, -0.1, 20.000279, 0.6, 64],
[-0.349775, -0.100273, 20.000485, 0.6, 85],
[-0.350214, -0.100386, 20.000407, 0.6, 106],
[-0.350527, -0.100273, 20.000091, 0.6, 127],
[-0.350531, -0.1, 19.999721, 0.6, 148],
[-0.350225, -0.099727, 19.999515, 0.6, 169],
[-0.349786, -0.099614, 19.999593, 0.6, 190],
[-0.349473, -0.099727, 19.999909, 0.6, 211],
[0.20032, 0.349824, 15.000456, 0.3, 0],
[0.20032, 0.349517, 14.999924, 0.3, 0],
[0.19968, 0.350483, 15.000076, 0.3, 0],
[0.19968, 0.350176, 14.999544, 0.3, 0],
[0.200578, 0.350073, 14.999958, 0.3, 0],
[0.199817, 0.349519, 15.000277, 0.3, 0],
[0.200183, 0.350481, 14.999723, 0.3, 0],
[0.199422, 0.349927, 15.000042, 0.3, 0],
[0.200235, 0.35042, 15.000332, 0.3, 0],
[0.199765, 0.350078, 15.000529, 0.3, 0],
[0.200235, 0.349922, 14.999471, 0.3, 0],
[0.199765, 0.34958, 14.999668, 0.3, 0],
[0.200429, 0.350419, 15.000005, 0.3, 255],
[0.199571, 0.349795, 15.000366, 0.3, 255],
[0.200429, 0.350205, 14.999634, 0.3, 255],
[0.199571, 0.349581, 14.999995, 0.3, 255],
[0.200524, 0.349747, 15.000146, 0.3, 255],
[0.200196, 0.349509, 15.000283, 0.3, 255],
[0.199804, 0.350491, 14.999717, 0.3, 255],
[0.199476, 0.350253, 14.999854, 0.3, 255],
[0.200138, 0.349578, 14.999597, 0.3, 255],
[0.200138, 0.350138, 15.000567, 0.3, 255],
[0.199862, 0.349862, 14.999433, 0.3, 255],
[0.199862, 0.350422, 15.000403, 0.3, 255],
[0.200488, 0.350136, 15.000321, 0.3, 255],
[0.199957, 0.349751, 15.000544, 0.3, 255],
[0.200488, 0.34979, 14.999721, 0.3, 255],
[0.199957, 0.349404, 14.999944, 0.3, 255],
[0.200043, 0.350596, 15.000056, 0.3, 255],
[0.199512, 0.35021, 15.000279, 0.3, 255],
[0.200043, 0.350249, 14.999456, 0.3, 255],
[0.199512, 0.349864, 14.999679, 0.3, 255],
[-0.5, 0.500442, 10.000442, 0.2, 0],
[-0.5, 0.5, 10, 0.1995, 255],
[-0.499966, 0.49971, 9.999563, 0.199, 0],
[-0.500034, 0.499563, 9.99971, 0.199, 0],
[-0.499934, 0.499432, 9.999144, 0.1985, 255],
[-0.500066, 0.499144, 9.999432, 0.1985, 255],
[0.6, 0.5, 10, 0.25, 210],
[0.600122, 0.4999, 10.000122, 0.25, 210],
[0.600061, 0.49995, 10.000061, 0.25003, 50],
[0.595856, 0.505256, 9.991271, 0.24, 100],
[0.597973, 0.507461, 9.992175, 0.24, 210],
[0.595362, 0.507926, 9.993945, 0.24, 100],
[0.594618, 0.509055, 9.996832, 0.24, 210],
[0.592382, 0.506906, 9.996092, 0.24, 100],
[0.590438, 0.504882, 9.997604, 0.24, 210],
[0.591035, 0.503605, 9.994744, 0.24, 100],
[0.59121, 0.500708, 9.993425, 0.24, 210],
[0.593181, 0.502585, 9.991764, 0.24, 100],
[0.595867, 0.502302, 9.990069, 0.24, 210],
]
values = grey_values(test_case)
rows = [values[row * 1024:row * 1024 + 1024] for row in range(1024)]
with open('combination_scene.png', 'wb') as file:
w = png.Writer(1024, 1024, greyscale=True)
w.write(file, rows)
```
</details>
<details>
<summary>Rust</summary>
The function `grey_values` would be sufficient as an entry to this challenge. The rest of the code saves the output as a PNG file.
The `png` crate can be installed using `cargo add png`. The `png` crate is not required for the `grey_values` function, only for saving as a PNG file.
```rust
fn grey_values(spheres: &Vec<[f32; 5]>) -> Vec<u8> {
let mut values = vec![];
for y_pixel in 0..1024 {
for x_pixel in 0..1024 {
let x_ray = x_pixel as f32 / 512.0 - 1.0;
let y_ray = y_pixel as f32 / 512.0 - 1.0;
let mut min_distance = f32::INFINITY;
let mut hit_grey_value = 0;
for sphere in spheres {
let [
x_sphere, y_sphere, z_sphere, radius, grey
] = *sphere;
let a = (
(x_ray - x_sphere).powf(2.0)
+ (y_ray - y_sphere).powf(2.0)
).sqrt();
if a < radius {
let distance = z_sphere - (
radius.powf(2.0) - a.powf(2.0)
).sqrt();
if distance < min_distance {
min_distance = distance;
hit_grey_value = grey as u8;
}
}
}
values.push(hit_grey_value);
}
}
values
}
fn main() {
let test_case: Vec<[f32; 5]> = vec![
[0.0, 0.0, 1000.0, 2.0, 192.0],
[-2.0, 2.0, 100.0, 3.0, 98.0],
[2.0, 2.0, 100.0, 3.0, 98.0],
[0.5, -0.5, 500.0, 0.2, 255.0],
[-0.349469, -0.1, 20.000279, 0.6, 64.0],
[-0.349775, -0.100273, 20.000485, 0.6, 85.0],
[-0.350214, -0.100386, 20.000407, 0.6, 106.0],
[-0.350527, -0.100273, 20.000091, 0.6, 127.0],
[-0.350531, -0.1, 19.999721, 0.6, 148.0],
[-0.350225, -0.099727, 19.999515, 0.6, 169.0],
[-0.349786, -0.099614, 19.999593, 0.6, 190.0],
[-0.349473, -0.099727, 19.999909, 0.6, 211.0],
[0.20032, 0.349824, 15.000456, 0.3, 0.0],
[0.20032, 0.349517, 14.999924, 0.3, 0.0],
[0.19968, 0.350483, 15.000076, 0.3, 0.0],
[0.19968, 0.350176, 14.999544, 0.3, 0.0],
[0.200578, 0.350073, 14.999958, 0.3, 0.0],
[0.199817, 0.349519, 15.000277, 0.3, 0.0],
[0.200183, 0.350481, 14.999723, 0.3, 0.0],
[0.199422, 0.349927, 15.000042, 0.3, 0.0],
[0.200235, 0.35042, 15.000332, 0.3, 0.0],
[0.199765, 0.350078, 15.000529, 0.3, 0.0],
[0.200235, 0.349922, 14.999471, 0.3, 0.0],
[0.199765, 0.34958, 14.999668, 0.3, 0.0],
[0.200429, 0.350419, 15.000005, 0.3, 255.0],
[0.199571, 0.349795, 15.000366, 0.3, 255.0],
[0.200429, 0.350205, 14.999634, 0.3, 255.0],
[0.199571, 0.349581, 14.999995, 0.3, 255.0],
[0.200524, 0.349747, 15.000146, 0.3, 255.0],
[0.200196, 0.349509, 15.000283, 0.3, 255.0],
[0.199804, 0.350491, 14.999717, 0.3, 255.0],
[0.199476, 0.350253, 14.999854, 0.3, 255.0],
[0.200138, 0.349578, 14.999597, 0.3, 255.0],
[0.200138, 0.350138, 15.000567, 0.3, 255.0],
[0.199862, 0.349862, 14.999433, 0.3, 255.0],
[0.199862, 0.350422, 15.000403, 0.3, 255.0],
[0.200488, 0.350136, 15.000321, 0.3, 255.0],
[0.199957, 0.349751, 15.000544, 0.3, 255.0],
[0.200488, 0.34979, 14.999721, 0.3, 255.0],
[0.199957, 0.349404, 14.999944, 0.3, 255.0],
[0.200043, 0.350596, 15.000056, 0.3, 255.0],
[0.199512, 0.35021, 15.000279, 0.3, 255.0],
[0.200043, 0.350249, 14.999456, 0.3, 255.0],
[0.199512, 0.349864, 14.999679, 0.3, 255.0],
[-0.5, 0.500442, 10.000442, 0.2, 0.0],
[-0.5, 0.5, 10.0, 0.1995, 255.0],
[-0.499966, 0.49971, 9.999563, 0.199, 0.0],
[-0.500034, 0.499563, 9.99971, 0.199, 0.0],
[-0.499934, 0.499432, 9.999144, 0.1985, 255.0],
[-0.500066, 0.499144, 9.999432, 0.1985, 255.0],
[0.6, 0.5, 10.0, 0.25, 210.0],
[0.600122, 0.4999, 10.000122, 0.25, 210.0],
[0.600061, 0.49995, 10.000061, 0.25003, 50.0],
[0.595856, 0.505256, 9.991271, 0.24, 100.0],
[0.597973, 0.507461, 9.992175, 0.24, 210.0],
[0.595362, 0.507926, 9.993945, 0.24, 100.0],
[0.594618, 0.509055, 9.996832, 0.24, 210.0],
[0.592382, 0.506906, 9.996092, 0.24, 100.0],
[0.590438, 0.504882, 9.997604, 0.24, 210.0],
[0.591035, 0.503605, 9.994744, 0.24, 100.0],
[0.59121, 0.500708, 9.993425, 0.24, 210.0],
[0.593181, 0.502585, 9.991764, 0.24, 100.0],
[0.595867, 0.502302, 9.990069, 0.24, 210.0],
];
let values = grey_values(&test_case);
let path = std::path::Path::new("combination_scene.png");
let file = std::fs::File::create(path).unwrap();
let w = &mut std::io::BufWriter::new(file);
let mut encoder = png::Encoder::new(w, 1024, 1024);
encoder.set_color(png::ColorType::Grayscale);
encoder.set_depth(png::BitDepth::Eight);
let mut writer = encoder.write_header().unwrap();
writer.write_image_data(&values).unwrap();
}
```
</details>
## Test cases
The output images may be scaled down to fit this page. Click on any image to open it full size.
### Single sphere
Input:
```text
[
[0, 0, 0, 1, 128],
]
```
Output:
[![Grey circle on a black background](https://codegolf.codidact.com/uploads/jlru163pa2pgm7aaq70b2tff16uf)](https://codegolf.codidact.com/uploads/jlru163pa2pgm7aaq70b2tff16uf)
### Two separated spheres
Input:
```text
[
[0, -0.55, 0, 0.4, 100],
[0, 0.55, 0, 0.4, 200],
]
```
Output:
[![Light and dark grey circles on a black background](https://codegolf.codidact.com/uploads/8yw7agli78y65kiomb1u5yebs7c5)](https://codegolf.codidact.com/uploads/8yw7agli78y65kiomb1u5yebs7c5)
### Ring of spheres
Input:
```text
[
[0.75, 0, 0, 0.25, 64],
[0.53033, 0.375, -0.375, 0.25, 185],
[0, 0.53033, -0.53033, 0.25, 64],
[-0.53033, 0.375, -0.375, 0.25, 185],
[-0.75, 0, 0, 0.25, 64],
[-0.53033, -0.375, 0.375, 0.25, 185],
[0, -0.53033, 0.53033, 0.25, 64],
[0.53033, -0.375, 0.375, 0.25, 185],
]
```
Output:
[![Eight circles in alternating light and dark grey in a ring some overlapping](https://codegolf.codidact.com/uploads/0lsenynig27au0sryf3ruozt1uwh)](https://codegolf.codidact.com/uploads/0lsenynig27au0sryf3ruozt1uwh)
### Beach ball
Input:
```text
[
[0.001, 0, 0, 1, 64],
[0.000707, 0.000707, 0, 1, 85],
[0, 0.001, 0, 1, 106],
[-0.000707, 0.000707, 0, 1, 127],
[-0.001, 0, 0, 1, 148],
[-0.000707, -0.000707, 0, 1, 169],
[0, -0.001, 0, 1, 190],
[0.000707, -0.000707, 0, 1, 211],
]
```
Output:
[![A circle with eight sectors in different shades of grey](https://codegolf.codidact.com/uploads/51a3ni0cfwsqcdw5b49gy6282qno)](https://codegolf.codidact.com/uploads/51a3ni0cfwsqcdw5b49gy6282qno)
### Football
Input:
```text
[
[0, 0.000058, 0.001947, 1, 0],
[0, -0.001715, 0.000923, 1, 0],
[0, 0.001715, -0.000923, 1, 0],
[0, -0.000058, -0.001947, 1, 0],
[0.001657, -0.000512, 0.000887, 1, 0],
[-0.001657, -0.000512, 0.000887, 1, 0],
[0.001657, 0.000512, -0.000887, 1, 0],
[-0.001657, 0.000512, -0.000887, 1, 0],
[0.001024, 0.001435, 0.000829, 1, 0],
[-0.001024, 0.001435, 0.000829, 1, 0],
[0.001024, -0.001435, -0.000829, 1, 0],
[-0.001024, -0.001435, -0.000829, 1, 0],
[0.001868, 0.000618, 0.000357, 1, 255],
[-0.001868, 0.000618, 0.000357, 1, 255],
[0.001868, -0.000618, -0.000357, 1, 255],
[-0.001868, -0.000618, -0.000357, 1, 255],
[0.000714, -0.000934, 0.001618, 1, 255],
[-0.000714, -0.000934, 0.001618, 1, 255],
[0.000714, 0.000934, -0.001618, 1, 255],
[-0.000714, 0.000934, -0.001618, 1, 255],
[0, -0.001975, -0.000316, 1, 255],
[0, 0.001261, 0.001552, 1, 255],
[0, -0.001261, -0.001552, 1, 255],
[0, 0.001975, 0.000316, 1, 255],
[0.001155, 0.000423, 0.001577, 1, 255],
[-0.001155, 0.000423, 0.001577, 1, 255],
[0.001155, -0.001577, 0.000423, 1, 255],
[-0.001155, -0.001577, 0.000423, 1, 255],
[0.001155, 0.001577, -0.000423, 1, 255],
[-0.001155, 0.001577, -0.000423, 1, 255],
[0.001155, -0.000423, -0.001577, 1, 255],
[-0.001155, -0.000423, -0.001577, 1, 255],
]
```
Output:
[![Black pentagons on a white ball](https://codegolf.codidact.com/uploads/bmrbsw65p59imvwu8ssd42uvvx1r)](https://codegolf.codidact.com/uploads/bmrbsw65p59imvwu8ssd42uvvx1r)
### Eight ball
Input:
```text
[
[0, 0, 1000, 2, 128],
[0, 0, 0.003125, 1, 0],
[0, 0, 0, 0.9975, 255],
[0, 0.000547, -0.002573, 0.995, 0],
[0, -0.000547, -0.002573, 0.995, 0],
[0, 0.001071, -0.005037, 0.9925, 255],
[0, -0.001071, -0.005037, 0.9925, 255],
]
```
Output:
[![A ball with the number 8 in a white circle](https://codegolf.codidact.com/uploads/dvl68crpmy62rqt94mwanditt4wc)](https://codegolf.codidact.com/uploads/dvl68crpmy62rqt94mwanditt4wc)
### Star ball
Input:
```text
[
[0, 0, 0, 1, 210],
[0, 0, 0.0008, 1, 210],
[0, 0, 0.0004, 1.00012, 50],
[0.012969, 0, -0.042045, 0.96, 100],
[0.016397, 0.011913, -0.039054, 0.96, 210],
[0.004008, 0.012334, -0.042045, 0.96, 100],
[-0.006263, 0.019276, -0.039054, 0.96, 210],
[-0.010492, 0.007623, -0.042045, 0.96, 100],
[-0.020268, 0, -0.039054, 0.96, 210],
[-0.010492, -0.007623, -0.042045, 0.96, 100],
[-0.006263, -0.019276, -0.039054, 0.96, 210],
[0.004008, -0.012334, -0.042045, 0.96, 100],
[0.016397, -0.011913, -0.039054, 0.96, 210],
]
```
Output:
[![A light grey ball with a dark grey five pointed star](https://codegolf.codidact.com/uploads/f3wdgwru6j35ue72x25mazxikfx2)](https://codegolf.codidact.com/uploads/f3wdgwru6j35ue72x25mazxikfx2)
### Combination scene
Input:
```text
[
[0, 0, 1000, 2, 192],
[-2, 2, 100, 3, 98],
[2, 2, 100, 3, 98],
[0.5, -0.5, 500, 0.2, 255],
[-0.349469, -0.1, 20.000279, 0.6, 64],
[-0.349775, -0.100273, 20.000485, 0.6, 85],
[-0.350214, -0.100386, 20.000407, 0.6, 106],
[-0.350527, -0.100273, 20.000091, 0.6, 127],
[-0.350531, -0.1, 19.999721, 0.6, 148],
[-0.350225, -0.099727, 19.999515, 0.6, 169],
[-0.349786, -0.099614, 19.999593, 0.6, 190],
[-0.349473, -0.099727, 19.999909, 0.6, 211],
[0.20032, 0.349824, 15.000456, 0.3, 0],
[0.20032, 0.349517, 14.999924, 0.3, 0],
[0.19968, 0.350483, 15.000076, 0.3, 0],
[0.19968, 0.350176, 14.999544, 0.3, 0],
[0.200578, 0.350073, 14.999958, 0.3, 0],
[0.199817, 0.349519, 15.000277, 0.3, 0],
[0.200183, 0.350481, 14.999723, 0.3, 0],
[0.199422, 0.349927, 15.000042, 0.3, 0],
[0.200235, 0.35042, 15.000332, 0.3, 0],
[0.199765, 0.350078, 15.000529, 0.3, 0],
[0.200235, 0.349922, 14.999471, 0.3, 0],
[0.199765, 0.34958, 14.999668, 0.3, 0],
[0.200429, 0.350419, 15.000005, 0.3, 255],
[0.199571, 0.349795, 15.000366, 0.3, 255],
[0.200429, 0.350205, 14.999634, 0.3, 255],
[0.199571, 0.349581, 14.999995, 0.3, 255],
[0.200524, 0.349747, 15.000146, 0.3, 255],
[0.200196, 0.349509, 15.000283, 0.3, 255],
[0.199804, 0.350491, 14.999717, 0.3, 255],
[0.199476, 0.350253, 14.999854, 0.3, 255],
[0.200138, 0.349578, 14.999597, 0.3, 255],
[0.200138, 0.350138, 15.000567, 0.3, 255],
[0.199862, 0.349862, 14.999433, 0.3, 255],
[0.199862, 0.350422, 15.000403, 0.3, 255],
[0.200488, 0.350136, 15.000321, 0.3, 255],
[0.199957, 0.349751, 15.000544, 0.3, 255],
[0.200488, 0.34979, 14.999721, 0.3, 255],
[0.199957, 0.349404, 14.999944, 0.3, 255],
[0.200043, 0.350596, 15.000056, 0.3, 255],
[0.199512, 0.35021, 15.000279, 0.3, 255],
[0.200043, 0.350249, 14.999456, 0.3, 255],
[0.199512, 0.349864, 14.999679, 0.3, 255],
[-0.5, 0.500442, 10.000442, 0.2, 0],
[-0.5, 0.5, 10, 0.1995, 255],
[-0.499966, 0.49971, 9.999563, 0.199, 0],
[-0.500034, 0.499563, 9.99971, 0.199, 0],
[-0.499934, 0.499432, 9.999144, 0.1985, 255],
[-0.500066, 0.499144, 9.999432, 0.1985, 255],
[0.6, 0.5, 10, 0.25, 210],
[0.600122, 0.4999, 10.000122, 0.25, 210],
[0.600061, 0.49995, 10.000061, 0.25003, 50],
[0.595856, 0.505256, 9.991271, 0.24, 100],
[0.597973, 0.507461, 9.992175, 0.24, 210],
[0.595362, 0.507926, 9.993945, 0.24, 100],
[0.594618, 0.509055, 9.996832, 0.24, 210],
[0.592382, 0.506906, 9.996092, 0.24, 100],
[0.590438, 0.504882, 9.997604, 0.24, 210],
[0.591035, 0.503605, 9.994744, 0.24, 100],
[0.59121, 0.500708, 9.993425, 0.24, 210],
[0.593181, 0.502585, 9.991764, 0.24, 100],
[0.595867, 0.502302, 9.990069, 0.24, 210],
]
```
Output:
[![A beachball, football, eightball, and starball with the sun behind hills in the background](https://codegolf.codidact.com/uploads/usv9npm1t5bjpcaknn6rbzgkqwmi)](https://codegolf.codidact.com/uploads/usv9npm1t5bjpcaknn6rbzgkqwmi)
## Accuracy
The outputs for the test cases were generated using 64 bit floating point numbers. The following is the last test case using 32 bit floating point rather than 64 bit floating point:
[![A beachball, football, eightball, and starball with the sun behind hills in the background](https://codegolf.codidact.com/uploads/7uuxjxq7a9oxc9q0jf2bi8l68qk3)](https://codegolf.codidact.com/uploads/7uuxjxq7a9oxc9q0jf2bi8l68qk3)
The inaccuracies in the 32 bit floating point version are subtle. You may need to click on the image to open it full size to see the occasional inaccurate pixel. For example, around the inside of the holes in the 8 on the eightball. These occasional inaccurate pixels are acceptable in your output (allowing languages which only support 32 bit floating point to compete).
More widespread speckling as in the following image would not be valid, so you may need to be careful if changing the order of calculations during golfing:
[![A scene with significant areas of inaccurate speckles obscuring the intended image](https://codegolf.codidact.com/uploads/1sa7i6z0cv2kgyrmw2tj2ytr1ps1)](https://codegolf.codidact.com/uploads/1sa7i6z0cv2kgyrmw2tj2ytr1ps1)
This exaggerated inaccuracy was achieved by adding 1000 to and then subtracting 1000 from the distance calculation, exceeding the available precision of 32 bit floating point and causing some distances to appear in the wrong order. Note that this same approach causes no inaccuracy with 64 bit floating point, so if your language supports both you may have more leeway for rearranging calculations if you use 64 bit floating point.
## Explanatory animations
The example images were produced from plain grey spheres with no surface pattern or texture. The following animations show how they break down into their component plain spheres.
<details>
<summary>Animations</summary>
TODO
</details>
## Scoring
This is a [code golf challenge](https://codegolf.codidact.com/categories/49/tags/4274). Your score is the number of bytes in your code. Lowest score for each language wins.
> Explanations are optional, but I'm more likely to upvote answers that have one.

#19: Post edited by

trichoplax‭ · 2024-10-12T18:51:03Z (2 months ago)
Specify z range

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Given a list of spheres, output an image.
Produce a greyscale [orthographic](https://en.wikipedia.org/wiki/Parallel_projection#Orthographic_projection) (parallel projection - no perspective) ray caster, that renders only plain grey spheres with no lighting.
## Input
- A list of spheres, each consisting of:
- The coordinates of its centre $(x,y,z)$.
- Its radius $r$.
- Its greyscale value $g$, an integer from $0$ to $255$ (both inclusive).
- Only the greyscale value will be restricted to integer values. The coordinates and radius may have fractional values.
## Output
- An image $1024$ by $1024$ pixels with greyscale values from $0$ to $255$.
- The image covers x and y values from $-1$ to $1$.
- A sphere's greyscale value is applied to every visible part of it. There is no variation over the surface of the sphere, so it appears flat like a disk.
- You may output whatever image format is convenient, including simple text formats such as [PGM](https://en.wikipedia.org/wiki/Netpbm#PGM_example) or a list of pixel greyscale values. If you want to include an example of your output as an image in your answer, the image formats that Codidact supports are PNG, JPEG, and GIF. You could output PGM and then convert to PNG to upload to your answer. If you output just the greyscale values as a list of numbers from $0$ to $255$, I've made a `[ TODO page to convert these to greyscale PNG]`.
- You are free to output an image reflected or rotated by a multiple of 90 degrees. However, the z axis is required to be increasing in the forwards direction (into the image) otherwise some of the test cases will look incorrect.
## Calculating a pixel value
Imagine a ray passing into the image through a pixel, perpendicular to the image plane. If it hits a sphere, it shows the greyscale value of the first sphere it hits. Otherwise, it shows a value of zero (black).
For example, for each pixel:
- The pixel $x,y$ coordinates will each be from $0$ to $1023$. Scale these to the range $-1$ to $1$ (for example, divide by $512$ and subtract $1$).
- If these scaled $x,y$ coordinates are within the radius $r$ of the $x,y$ coordinates of a sphere's centre, the ray intersects that sphere (the $z$ coordinate is not relevant to this test because the image is an orthographic projection - the rays are all parallel to the $z$ axis). The distance $a$ of the ray from the sphere's centre can be calculated as follows:
$$a=\sqrt{(x_{ray}-x_{sphere})^2+(y_{ray}-y_{sphere})^2}$$
```text
CONSIDER A DIAGRAM HERE
```
- For each sphere that the ray intersects, find the $z$ coordinate of the nearest intersection (there will often be $2$ per sphere, $1$ on the front of the sphere and $1$ on the back of the sphere, as the ray passes through - only the nearest is relevant). This can be calculated as follows:
- The distance $d$ to the nearest intersection point can be calculated from the $z$ coordinate of the sphere's centre, the radius $r$, and the distance $a$ from the $x,y$ coordinates of the sphere's centre to the scaled $x,y$ coordinates of the pixel:
$$d=z_{sphere}-\sqrt{r_{sphere}^2-a^2}$$
```text
CONSIDER A DIAGRAM HERE
```
- Of all the spheres that the ray intersects, the sphere with the intersection point with the smallest $z$ coordinate $d$ is the one to display. Set the pixel to the greyscale value of this sphere.
- If $2$ or more spheres have an intersection point with the smallest $z$ coordinate $d$ then you may set the pixel to the greyscale value of any of these spheres.
- If the ray did not intersect any spheres, set the pixel to zero (black).
## Example implementations
These are ungolfed code to supplement understanding of the requirements. You are free to golf these or implement your own approach that matches the test cases up to reflection & rotation.
The Python implementation uses 64 bit floating point. The Rust implementation uses 32 bit floating point to show the difference in output. I've tried to optimise the test cases for compatibility with 32 bit floating point, so the differences are restricted to being a pixel out in some places.
<details>
<summary>Python</summary>
The function `grey_values` would be sufficient as an entry to this challenge. The rest of the code saves the output as a PNG file. The function itself has no dependence on the `import png` so it would not be required in an entry.
Instructions for installing the `png` package can be found on its [project page on PyPI](https://pypi.org/project/pypng/).
```python
def grey_values(spheres):
values = []
for y_pixel in range(1024):
for x_pixel in range(1024):
x_ray, y_ray = x_pixel / 512 - 1, y_pixel / 512 - 1
min_distance, hit_grey_value = None, 0
for sphere in spheres:
x_sphere, y_sphere, z_sphere, radius, grey = sphere
a = (
(x_ray - x_sphere) ** 2 + (y_ray - y_sphere) ** 2
) ** 0.5
if a < radius:
distance = z_sphere - (radius ** 2 - a ** 2) ** 0.5
if min_distance is None or distance < min_distance:
min_distance = distance
hit_grey_value = grey
values.append(hit_grey_value)
return values
import png
test_case = [
[0, 0, 1000, 2, 192],
[-2, 2, 100, 3, 98],
[2, 2, 100, 3, 98],
[0.5, -0.5, 500, 0.2, 255],
[-0.349469, -0.1, 20.000279, 0.6, 64],
[-0.349775, -0.100273, 20.000485, 0.6, 85],
[-0.350214, -0.100386, 20.000407, 0.6, 106],
[-0.350527, -0.100273, 20.000091, 0.6, 127],
[-0.350531, -0.1, 19.999721, 0.6, 148],
[-0.350225, -0.099727, 19.999515, 0.6, 169],
[-0.349786, -0.099614, 19.999593, 0.6, 190],
[-0.349473, -0.099727, 19.999909, 0.6, 211],
[0.20032, 0.349824, 15.000456, 0.3, 0],
[0.20032, 0.349517, 14.999924, 0.3, 0],
[0.19968, 0.350483, 15.000076, 0.3, 0],
[0.19968, 0.350176, 14.999544, 0.3, 0],
[0.200578, 0.350073, 14.999958, 0.3, 0],
[0.199817, 0.349519, 15.000277, 0.3, 0],
[0.200183, 0.350481, 14.999723, 0.3, 0],
[0.199422, 0.349927, 15.000042, 0.3, 0],
[0.200235, 0.35042, 15.000332, 0.3, 0],
[0.199765, 0.350078, 15.000529, 0.3, 0],
[0.200235, 0.349922, 14.999471, 0.3, 0],
[0.199765, 0.34958, 14.999668, 0.3, 0],
[0.200429, 0.350419, 15.000005, 0.3, 255],
[0.199571, 0.349795, 15.000366, 0.3, 255],
[0.200429, 0.350205, 14.999634, 0.3, 255],
[0.199571, 0.349581, 14.999995, 0.3, 255],
[0.200524, 0.349747, 15.000146, 0.3, 255],
[0.200196, 0.349509, 15.000283, 0.3, 255],
[0.199804, 0.350491, 14.999717, 0.3, 255],
[0.199476, 0.350253, 14.999854, 0.3, 255],
[0.200138, 0.349578, 14.999597, 0.3, 255],
[0.200138, 0.350138, 15.000567, 0.3, 255],
[0.199862, 0.349862, 14.999433, 0.3, 255],
[0.199862, 0.350422, 15.000403, 0.3, 255],
[0.200488, 0.350136, 15.000321, 0.3, 255],
[0.199957, 0.349751, 15.000544, 0.3, 255],
[0.200488, 0.34979, 14.999721, 0.3, 255],
[0.199957, 0.349404, 14.999944, 0.3, 255],
[0.200043, 0.350596, 15.000056, 0.3, 255],
[0.199512, 0.35021, 15.000279, 0.3, 255],
[0.200043, 0.350249, 14.999456, 0.3, 255],
[0.199512, 0.349864, 14.999679, 0.3, 255],
[-0.5, 0.500442, 10.000442, 0.2, 0],
[-0.5, 0.5, 10, 0.1995, 255],
[-0.499966, 0.49971, 9.999563, 0.199, 0],
[-0.500034, 0.499563, 9.99971, 0.199, 0],
[-0.499934, 0.499432, 9.999144, 0.1985, 255],
[-0.500066, 0.499144, 9.999432, 0.1985, 255],
[0.6, 0.5, 10, 0.25, 210],
[0.600122, 0.4999, 10.000122, 0.25, 210],
[0.600061, 0.49995, 10.000061, 0.25003, 50],
[0.595856, 0.505256, 9.991271, 0.24, 100],
[0.597973, 0.507461, 9.992175, 0.24, 210],
[0.595362, 0.507926, 9.993945, 0.24, 100],
[0.594618, 0.509055, 9.996832, 0.24, 210],
[0.592382, 0.506906, 9.996092, 0.24, 100],
[0.590438, 0.504882, 9.997604, 0.24, 210],
[0.591035, 0.503605, 9.994744, 0.24, 100],
[0.59121, 0.500708, 9.993425, 0.24, 210],
[0.593181, 0.502585, 9.991764, 0.24, 100],
[0.595867, 0.502302, 9.990069, 0.24, 210],
]
values = grey_values(test_case)
rows = [values[row * 1024:row * 1024 + 1024] for row in range(1024)]
with open('combination_scene.png', 'wb') as file:
w = png.Writer(1024, 1024, greyscale=True)
w.write(file, rows)
```
</details>
<details>
<summary>Rust</summary>
The function `grey_values` would be sufficient as an entry to this challenge. The rest of the code saves the output as a PNG file.
The `png` crate can be installed using `cargo add png`. The `png` crate is not required for the `grey_values` function, only for saving as a PNG file.
```rust
fn grey_values(spheres: &Vec<[f32; 5]>) -> Vec<u8> {
let mut values = vec![];
for y_pixel in 0..1024 {
for x_pixel in 0..1024 {
let x_ray = x_pixel as f32 / 512.0 - 1.0;
let y_ray = y_pixel as f32 / 512.0 - 1.0;
let mut min_distance = f32::INFINITY;
let mut hit_grey_value = 0;
for sphere in spheres {
let [
x_sphere, y_sphere, z_sphere, radius, grey
] = *sphere;
let a = (
(x_ray - x_sphere).powf(2.0)
+ (y_ray - y_sphere).powf(2.0)
).sqrt();
if a < radius {
let distance = z_sphere - (
radius.powf(2.0) - a.powf(2.0)
).sqrt();
if distance < min_distance {
min_distance = distance;
hit_grey_value = grey as u8;
}
}
}
values.push(hit_grey_value);
}
}
values
}
fn main() {
let test_case: Vec<[f32; 5]> = vec![
[0.0, 0.0, 1000.0, 2.0, 192.0],
[-2.0, 2.0, 100.0, 3.0, 98.0],
[2.0, 2.0, 100.0, 3.0, 98.0],
[0.5, -0.5, 500.0, 0.2, 255.0],
[-0.349469, -0.1, 20.000279, 0.6, 64.0],
[-0.349775, -0.100273, 20.000485, 0.6, 85.0],
[-0.350214, -0.100386, 20.000407, 0.6, 106.0],
[-0.350527, -0.100273, 20.000091, 0.6, 127.0],
[-0.350531, -0.1, 19.999721, 0.6, 148.0],
[-0.350225, -0.099727, 19.999515, 0.6, 169.0],
[-0.349786, -0.099614, 19.999593, 0.6, 190.0],
[-0.349473, -0.099727, 19.999909, 0.6, 211.0],
[0.20032, 0.349824, 15.000456, 0.3, 0.0],
[0.20032, 0.349517, 14.999924, 0.3, 0.0],
[0.19968, 0.350483, 15.000076, 0.3, 0.0],
[0.19968, 0.350176, 14.999544, 0.3, 0.0],
[0.200578, 0.350073, 14.999958, 0.3, 0.0],
[0.199817, 0.349519, 15.000277, 0.3, 0.0],
[0.200183, 0.350481, 14.999723, 0.3, 0.0],
[0.199422, 0.349927, 15.000042, 0.3, 0.0],
[0.200235, 0.35042, 15.000332, 0.3, 0.0],
[0.199765, 0.350078, 15.000529, 0.3, 0.0],
[0.200235, 0.349922, 14.999471, 0.3, 0.0],
[0.199765, 0.34958, 14.999668, 0.3, 0.0],
[0.200429, 0.350419, 15.000005, 0.3, 255.0],
[0.199571, 0.349795, 15.000366, 0.3, 255.0],
[0.200429, 0.350205, 14.999634, 0.3, 255.0],
[0.199571, 0.349581, 14.999995, 0.3, 255.0],
[0.200524, 0.349747, 15.000146, 0.3, 255.0],
[0.200196, 0.349509, 15.000283, 0.3, 255.0],
[0.199804, 0.350491, 14.999717, 0.3, 255.0],
[0.199476, 0.350253, 14.999854, 0.3, 255.0],
[0.200138, 0.349578, 14.999597, 0.3, 255.0],
[0.200138, 0.350138, 15.000567, 0.3, 255.0],
[0.199862, 0.349862, 14.999433, 0.3, 255.0],
[0.199862, 0.350422, 15.000403, 0.3, 255.0],
[0.200488, 0.350136, 15.000321, 0.3, 255.0],
[0.199957, 0.349751, 15.000544, 0.3, 255.0],
[0.200488, 0.34979, 14.999721, 0.3, 255.0],
[0.199957, 0.349404, 14.999944, 0.3, 255.0],
[0.200043, 0.350596, 15.000056, 0.3, 255.0],
[0.199512, 0.35021, 15.000279, 0.3, 255.0],
[0.200043, 0.350249, 14.999456, 0.3, 255.0],
[0.199512, 0.349864, 14.999679, 0.3, 255.0],
[-0.5, 0.500442, 10.000442, 0.2, 0.0],
[-0.5, 0.5, 10.0, 0.1995, 255.0],
[-0.499966, 0.49971, 9.999563, 0.199, 0.0],
[-0.500034, 0.499563, 9.99971, 0.199, 0.0],
[-0.499934, 0.499432, 9.999144, 0.1985, 255.0],
[-0.500066, 0.499144, 9.999432, 0.1985, 255.0],
[0.6, 0.5, 10.0, 0.25, 210.0],
[0.600122, 0.4999, 10.000122, 0.25, 210.0],
[0.600061, 0.49995, 10.000061, 0.25003, 50.0],
[0.595856, 0.505256, 9.991271, 0.24, 100.0],
[0.597973, 0.507461, 9.992175, 0.24, 210.0],
[0.595362, 0.507926, 9.993945, 0.24, 100.0],
[0.594618, 0.509055, 9.996832, 0.24, 210.0],
[0.592382, 0.506906, 9.996092, 0.24, 100.0],
[0.590438, 0.504882, 9.997604, 0.24, 210.0],
[0.591035, 0.503605, 9.994744, 0.24, 100.0],
[0.59121, 0.500708, 9.993425, 0.24, 210.0],
[0.593181, 0.502585, 9.991764, 0.24, 100.0],
[0.595867, 0.502302, 9.990069, 0.24, 210.0],
];
let values = grey_values(&test_case);
let path = std::path::Path::new("combination_scene.png");
let file = std::fs::File::create(path).unwrap();
let w = &mut std::io::BufWriter::new(file);
let mut encoder = png::Encoder::new(w, 1024, 1024);
encoder.set_color(png::ColorType::Grayscale);
encoder.set_depth(png::BitDepth::Eight);
let mut writer = encoder.write_header().unwrap();
writer.write_image_data(&values).unwrap();
}
```
</details>
## Test cases
The output images may be scaled down to fit this page. Click on any image to open it full size.
### Single sphere
Input:
```text
[
[0, 0, 0, 1, 128],
]
```
Output:
[![Grey circle on a black background](https://codegolf.codidact.com/uploads/jlru163pa2pgm7aaq70b2tff16uf)](https://codegolf.codidact.com/uploads/jlru163pa2pgm7aaq70b2tff16uf)
### Two separated spheres
Input:
```text
[
[0, -0.55, 0, 0.4, 100],
[0, 0.55, 0, 0.4, 200],
]
```
Output:
[![Light and dark grey circles on a black background](https://codegolf.codidact.com/uploads/8yw7agli78y65kiomb1u5yebs7c5)](https://codegolf.codidact.com/uploads/8yw7agli78y65kiomb1u5yebs7c5)
### Ring of spheres
Input:
```text
[
[0.75, 0, 0, 0.25, 64],
[0.53033, 0.375, -0.375, 0.25, 185],
[0, 0.53033, -0.53033, 0.25, 64],
[-0.53033, 0.375, -0.375, 0.25, 185],
[-0.75, 0, 0, 0.25, 64],
[-0.53033, -0.375, 0.375, 0.25, 185],
[0, -0.53033, 0.53033, 0.25, 64],
[0.53033, -0.375, 0.375, 0.25, 185],
]
```
Output:
[![Eight circles in alternating light and dark grey in a ring some overlapping](https://codegolf.codidact.com/uploads/0lsenynig27au0sryf3ruozt1uwh)](https://codegolf.codidact.com/uploads/0lsenynig27au0sryf3ruozt1uwh)
### Beach ball
Input:
```text
[
[0.001, 0, 0, 1, 64],
[0.000707, 0.000707, 0, 1, 85],
[0, 0.001, 0, 1, 106],
[-0.000707, 0.000707, 0, 1, 127],
[-0.001, 0, 0, 1, 148],
[-0.000707, -0.000707, 0, 1, 169],
[0, -0.001, 0, 1, 190],
[0.000707, -0.000707, 0, 1, 211],
]
```
Output:
[![A circle with eight sectors in different shades of grey](https://codegolf.codidact.com/uploads/51a3ni0cfwsqcdw5b49gy6282qno)](https://codegolf.codidact.com/uploads/51a3ni0cfwsqcdw5b49gy6282qno)
### Football
Input:
```text
[
[0, 0.000058, 0.001947, 1, 0],
[0, -0.001715, 0.000923, 1, 0],
[0, 0.001715, -0.000923, 1, 0],
[0, -0.000058, -0.001947, 1, 0],
[0.001657, -0.000512, 0.000887, 1, 0],
[-0.001657, -0.000512, 0.000887, 1, 0],
[0.001657, 0.000512, -0.000887, 1, 0],
[-0.001657, 0.000512, -0.000887, 1, 0],
[0.001024, 0.001435, 0.000829, 1, 0],
[-0.001024, 0.001435, 0.000829, 1, 0],
[0.001024, -0.001435, -0.000829, 1, 0],
[-0.001024, -0.001435, -0.000829, 1, 0],
[0.001868, 0.000618, 0.000357, 1, 255],
[-0.001868, 0.000618, 0.000357, 1, 255],
[0.001868, -0.000618, -0.000357, 1, 255],
[-0.001868, -0.000618, -0.000357, 1, 255],
[0.000714, -0.000934, 0.001618, 1, 255],
[-0.000714, -0.000934, 0.001618, 1, 255],
[0.000714, 0.000934, -0.001618, 1, 255],
[-0.000714, 0.000934, -0.001618, 1, 255],
[0, -0.001975, -0.000316, 1, 255],
[0, 0.001261, 0.001552, 1, 255],
[0, -0.001261, -0.001552, 1, 255],
[0, 0.001975, 0.000316, 1, 255],
[0.001155, 0.000423, 0.001577, 1, 255],
[-0.001155, 0.000423, 0.001577, 1, 255],
[0.001155, -0.001577, 0.000423, 1, 255],
[-0.001155, -0.001577, 0.000423, 1, 255],
[0.001155, 0.001577, -0.000423, 1, 255],
[-0.001155, 0.001577, -0.000423, 1, 255],
[0.001155, -0.000423, -0.001577, 1, 255],
[-0.001155, -0.000423, -0.001577, 1, 255],
]
```
Output:
[![Black pentagons on a white ball](https://codegolf.codidact.com/uploads/bmrbsw65p59imvwu8ssd42uvvx1r)](https://codegolf.codidact.com/uploads/bmrbsw65p59imvwu8ssd42uvvx1r)
### Eight ball
Input:
```text
[
[0, 0, 1000, 2, 128],
[0, 0, 0.003125, 1, 0],
[0, 0, 0, 0.9975, 255],
[0, 0.000547, -0.002573, 0.995, 0],
[0, -0.000547, -0.002573, 0.995, 0],
[0, 0.001071, -0.005037, 0.9925, 255],
[0, -0.001071, -0.005037, 0.9925, 255],
]
```
Output:
[![A ball with the number 8 in a white circle](https://codegolf.codidact.com/uploads/dvl68crpmy62rqt94mwanditt4wc)](https://codegolf.codidact.com/uploads/dvl68crpmy62rqt94mwanditt4wc)
### Star ball
Input:
```text
[
[0, 0, 0, 1, 210],
[0, 0, 0.0008, 1, 210],
[0, 0, 0.0004, 1.00012, 50],
[0.012969, 0, -0.042045, 0.96, 100],
[0.016397, 0.011913, -0.039054, 0.96, 210],
[0.004008, 0.012334, -0.042045, 0.96, 100],
[-0.006263, 0.019276, -0.039054, 0.96, 210],
[-0.010492, 0.007623, -0.042045, 0.96, 100],
[-0.020268, 0, -0.039054, 0.96, 210],
[-0.010492, -0.007623, -0.042045, 0.96, 100],
[-0.006263, -0.019276, -0.039054, 0.96, 210],
[0.004008, -0.012334, -0.042045, 0.96, 100],
[0.016397, -0.011913, -0.039054, 0.96, 210],
]
```
Output:
[![A light grey ball with a dark grey five pointed star](https://codegolf.codidact.com/uploads/f3wdgwru6j35ue72x25mazxikfx2)](https://codegolf.codidact.com/uploads/f3wdgwru6j35ue72x25mazxikfx2)
### Combination scene
Input:
```text
[
[0, 0, 1000, 2, 192],
[-2, 2, 100, 3, 98],
[2, 2, 100, 3, 98],
[0.5, -0.5, 500, 0.2, 255],
[-0.349469, -0.1, 20.000279, 0.6, 64],
[-0.349775, -0.100273, 20.000485, 0.6, 85],
[-0.350214, -0.100386, 20.000407, 0.6, 106],
[-0.350527, -0.100273, 20.000091, 0.6, 127],
[-0.350531, -0.1, 19.999721, 0.6, 148],
[-0.350225, -0.099727, 19.999515, 0.6, 169],
[-0.349786, -0.099614, 19.999593, 0.6, 190],
[-0.349473, -0.099727, 19.999909, 0.6, 211],
[0.20032, 0.349824, 15.000456, 0.3, 0],
[0.20032, 0.349517, 14.999924, 0.3, 0],
[0.19968, 0.350483, 15.000076, 0.3, 0],
[0.19968, 0.350176, 14.999544, 0.3, 0],
[0.200578, 0.350073, 14.999958, 0.3, 0],
[0.199817, 0.349519, 15.000277, 0.3, 0],
[0.200183, 0.350481, 14.999723, 0.3, 0],
[0.199422, 0.349927, 15.000042, 0.3, 0],
[0.200235, 0.35042, 15.000332, 0.3, 0],
[0.199765, 0.350078, 15.000529, 0.3, 0],
[0.200235, 0.349922, 14.999471, 0.3, 0],
[0.199765, 0.34958, 14.999668, 0.3, 0],
[0.200429, 0.350419, 15.000005, 0.3, 255],
[0.199571, 0.349795, 15.000366, 0.3, 255],
[0.200429, 0.350205, 14.999634, 0.3, 255],
[0.199571, 0.349581, 14.999995, 0.3, 255],
[0.200524, 0.349747, 15.000146, 0.3, 255],
[0.200196, 0.349509, 15.000283, 0.3, 255],
[0.199804, 0.350491, 14.999717, 0.3, 255],
[0.199476, 0.350253, 14.999854, 0.3, 255],
[0.200138, 0.349578, 14.999597, 0.3, 255],
[0.200138, 0.350138, 15.000567, 0.3, 255],
[0.199862, 0.349862, 14.999433, 0.3, 255],
[0.199862, 0.350422, 15.000403, 0.3, 255],
[0.200488, 0.350136, 15.000321, 0.3, 255],
[0.199957, 0.349751, 15.000544, 0.3, 255],
[0.200488, 0.34979, 14.999721, 0.3, 255],
[0.199957, 0.349404, 14.999944, 0.3, 255],
[0.200043, 0.350596, 15.000056, 0.3, 255],
[0.199512, 0.35021, 15.000279, 0.3, 255],
[0.200043, 0.350249, 14.999456, 0.3, 255],
[0.199512, 0.349864, 14.999679, 0.3, 255],
[-0.5, 0.500442, 10.000442, 0.2, 0],
[-0.5, 0.5, 10, 0.1995, 255],
[-0.499966, 0.49971, 9.999563, 0.199, 0],
[-0.500034, 0.499563, 9.99971, 0.199, 0],
[-0.499934, 0.499432, 9.999144, 0.1985, 255],
[-0.500066, 0.499144, 9.999432, 0.1985, 255],
[0.6, 0.5, 10, 0.25, 210],
[0.600122, 0.4999, 10.000122, 0.25, 210],
[0.600061, 0.49995, 10.000061, 0.25003, 50],
[0.595856, 0.505256, 9.991271, 0.24, 100],
[0.597973, 0.507461, 9.992175, 0.24, 210],
[0.595362, 0.507926, 9.993945, 0.24, 100],
[0.594618, 0.509055, 9.996832, 0.24, 210],
[0.592382, 0.506906, 9.996092, 0.24, 100],
[0.590438, 0.504882, 9.997604, 0.24, 210],
[0.591035, 0.503605, 9.994744, 0.24, 100],
[0.59121, 0.500708, 9.993425, 0.24, 210],
[0.593181, 0.502585, 9.991764, 0.24, 100],
[0.595867, 0.502302, 9.990069, 0.24, 210],
]
```
Output:
[![A beachball, football, eightball, and starball with the sun behind hills in the background](https://codegolf.codidact.com/uploads/usv9npm1t5bjpcaknn6rbzgkqwmi)](https://codegolf.codidact.com/uploads/usv9npm1t5bjpcaknn6rbzgkqwmi)
## Accuracy
The outputs for the test cases were generated using 64 bit floating point numbers. The following is the last test case using 32 bit floating point rather than 64 bit floating point:
[![A beachball, football, eightball, and starball with the sun behind hills in the background](https://codegolf.codidact.com/uploads/7uuxjxq7a9oxc9q0jf2bi8l68qk3)](https://codegolf.codidact.com/uploads/7uuxjxq7a9oxc9q0jf2bi8l68qk3)
The inaccuracies in the 32 bit floating point version are subtle. You may need to click on the image to open it full size to see the occasional inaccurate pixel. For example, around the inside of the holes in the 8 on the eightball. These occasional inaccurate pixels are acceptable in your output (allowing languages which only support 32 bit floating point to compete).
More widespread speckling as in the following image would not be valid, so you may need to be careful if changing the order of calculations during golfing:
[![A scene with significant areas of inaccurate speckles obscuring the intended image](https://codegolf.codidact.com/uploads/1sa7i6z0cv2kgyrmw2tj2ytr1ps1)](https://codegolf.codidact.com/uploads/1sa7i6z0cv2kgyrmw2tj2ytr1ps1)
This exaggerated inaccuracy was achieved by adding 1000 to and then subtracting 1000 from the distance calculation, exceeding the available precision of 32 bit floating point and causing some distances to appear in the wrong order. Note that this same approach causes no inaccuracy with 64 bit floating point, so if your language supports both you may have more leeway for rearranging calculations if you use 64 bit floating point.
## Explanatory animations
The example images were produced from plain grey spheres with no surface pattern or texture. The following animations show how they break down into their component plain spheres.
<details>
<summary>Animations</summary>
TODO
</details>
## Scoring
This is a [code golf challenge](https://codegolf.codidact.com/categories/49/tags/4274). Your score is the number of bytes in your code. Lowest score for each language wins.
> Explanations are optional, but I'm more likely to upvote answers that have one.

Given a list of spheres, output an image.
Produce a greyscale [orthographic](https://en.wikipedia.org/wiki/Parallel_projection#Orthographic_projection) (parallel projection - no perspective) ray caster, that renders only plain grey spheres with no lighting.
## Input
- A list of spheres, each consisting of:
- The coordinates of its centre $(x,y,z)$.
- Its radius $r$.
- Its greyscale value $g$, an integer from $0$ to $255$ (both inclusive).
- Only the greyscale value will be restricted to integer values. The coordinates and radius may have fractional values.
## Output
- An image $1024$ by $1024$ pixels with greyscale values from $0$ to $255$.
- The image covers x and y values from $-1$ to $1$.
- A sphere's greyscale value is applied to every visible part of it. There is no variation over the surface of the sphere, so it appears flat like a disk.
- You may output whatever image format is convenient, including simple text formats such as [PGM](https://en.wikipedia.org/wiki/Netpbm#PGM_example) or a list of pixel greyscale values. If you want to include an example of your output as an image in your answer, the image formats that Codidact supports are PNG, JPEG, and GIF. You could output PGM and then convert to PNG to upload to your answer. If you output just the greyscale values as a list of numbers from $0$ to $255$, I've made a `[ TODO page to convert these to greyscale PNG]`.
- You are free to output an image reflected or rotated by a multiple of 90 degrees. However, the z axis is required to be increasing in the forwards direction (into the image) otherwise some of the test cases will look incorrect.
- You may assume that all ray intersection points (not just nearest intersections) will have a $z$ value greater than zero, and less than $1024$.
```text
TODO: Adjust test cases to fit within this specified z range
```
## Calculating a pixel value
Imagine a ray passing into the image through a pixel, perpendicular to the image plane. If it hits a sphere, it shows the greyscale value of the first sphere it hits. Otherwise, it shows a value of zero (black).
For example, for each pixel:
- The pixel $x,y$ coordinates will each be from $0$ to $1023$. Scale these to the range $-1$ to $1$ (for example, divide by $512$ and subtract $1$).
- If these scaled $x,y$ coordinates are within the radius $r$ of the $x,y$ coordinates of a sphere's centre, the ray intersects that sphere (the $z$ coordinate is not relevant to this test because the image is an orthographic projection - the rays are all parallel to the $z$ axis). The distance $a$ of the ray from the sphere's centre can be calculated as follows:
$$a=\sqrt{(x_{ray}-x_{sphere})^2+(y_{ray}-y_{sphere})^2}$$
```text
CONSIDER A DIAGRAM HERE
```
- For each sphere that the ray intersects, find the $z$ coordinate of the nearest intersection (there will often be $2$ per sphere, $1$ on the front of the sphere and $1$ on the back of the sphere, as the ray passes through - only the nearest is relevant). This can be calculated as follows:
- The distance $d$ to the nearest intersection point can be calculated from the $z$ coordinate of the sphere's centre, the radius $r$, and the distance $a$ from the $x,y$ coordinates of the sphere's centre to the scaled $x,y$ coordinates of the pixel:
$$d=z_{sphere}-\sqrt{r_{sphere}^2-a^2}$$
```text
CONSIDER A DIAGRAM HERE
```
- Of all the spheres that the ray intersects, the sphere with the intersection point with the smallest $z$ coordinate $d$ is the one to display. Set the pixel to the greyscale value of this sphere.
- If $2$ or more spheres have an intersection point with the smallest $z$ coordinate $d$ then you may set the pixel to the greyscale value of any of these spheres.
- If the ray did not intersect any spheres, set the pixel to zero (black).
## Example implementations
These are ungolfed code to supplement understanding of the requirements. You are free to golf these or implement your own approach that matches the test cases up to reflection & rotation.
The Python implementation uses 64 bit floating point. The Rust implementation uses 32 bit floating point to show the difference in output. I've tried to optimise the test cases for compatibility with 32 bit floating point, so the differences are restricted to being a pixel out in some places.
<details>
<summary>Python</summary>
The function `grey_values` would be sufficient as an entry to this challenge. The rest of the code saves the output as a PNG file. The function itself has no dependence on the `import png` so it would not be required in an entry.
Instructions for installing the `png` package can be found on its [project page on PyPI](https://pypi.org/project/pypng/).
```python
def grey_values(spheres):
values = []
for y_pixel in range(1024):
for x_pixel in range(1024):
x_ray, y_ray = x_pixel / 512 - 1, y_pixel / 512 - 1
min_distance, hit_grey_value = None, 0
for sphere in spheres:
x_sphere, y_sphere, z_sphere, radius, grey = sphere
a = (
(x_ray - x_sphere) ** 2 + (y_ray - y_sphere) ** 2
) ** 0.5
if a < radius:
distance = z_sphere - (radius ** 2 - a ** 2) ** 0.5
if min_distance is None or distance < min_distance:
min_distance = distance
hit_grey_value = grey
values.append(hit_grey_value)
return values
import png
test_case = [
[0, 0, 1000, 2, 192],
[-2, 2, 100, 3, 98],
[2, 2, 100, 3, 98],
[0.5, -0.5, 500, 0.2, 255],
[-0.349469, -0.1, 20.000279, 0.6, 64],
[-0.349775, -0.100273, 20.000485, 0.6, 85],
[-0.350214, -0.100386, 20.000407, 0.6, 106],
[-0.350527, -0.100273, 20.000091, 0.6, 127],
[-0.350531, -0.1, 19.999721, 0.6, 148],
[-0.350225, -0.099727, 19.999515, 0.6, 169],
[-0.349786, -0.099614, 19.999593, 0.6, 190],
[-0.349473, -0.099727, 19.999909, 0.6, 211],
[0.20032, 0.349824, 15.000456, 0.3, 0],
[0.20032, 0.349517, 14.999924, 0.3, 0],
[0.19968, 0.350483, 15.000076, 0.3, 0],
[0.19968, 0.350176, 14.999544, 0.3, 0],
[0.200578, 0.350073, 14.999958, 0.3, 0],
[0.199817, 0.349519, 15.000277, 0.3, 0],
[0.200183, 0.350481, 14.999723, 0.3, 0],
[0.199422, 0.349927, 15.000042, 0.3, 0],
[0.200235, 0.35042, 15.000332, 0.3, 0],
[0.199765, 0.350078, 15.000529, 0.3, 0],
[0.200235, 0.349922, 14.999471, 0.3, 0],
[0.199765, 0.34958, 14.999668, 0.3, 0],
[0.200429, 0.350419, 15.000005, 0.3, 255],
[0.199571, 0.349795, 15.000366, 0.3, 255],
[0.200429, 0.350205, 14.999634, 0.3, 255],
[0.199571, 0.349581, 14.999995, 0.3, 255],
[0.200524, 0.349747, 15.000146, 0.3, 255],
[0.200196, 0.349509, 15.000283, 0.3, 255],
[0.199804, 0.350491, 14.999717, 0.3, 255],
[0.199476, 0.350253, 14.999854, 0.3, 255],
[0.200138, 0.349578, 14.999597, 0.3, 255],
[0.200138, 0.350138, 15.000567, 0.3, 255],
[0.199862, 0.349862, 14.999433, 0.3, 255],
[0.199862, 0.350422, 15.000403, 0.3, 255],
[0.200488, 0.350136, 15.000321, 0.3, 255],
[0.199957, 0.349751, 15.000544, 0.3, 255],
[0.200488, 0.34979, 14.999721, 0.3, 255],
[0.199957, 0.349404, 14.999944, 0.3, 255],
[0.200043, 0.350596, 15.000056, 0.3, 255],
[0.199512, 0.35021, 15.000279, 0.3, 255],
[0.200043, 0.350249, 14.999456, 0.3, 255],
[0.199512, 0.349864, 14.999679, 0.3, 255],
[-0.5, 0.500442, 10.000442, 0.2, 0],
[-0.5, 0.5, 10, 0.1995, 255],
[-0.499966, 0.49971, 9.999563, 0.199, 0],
[-0.500034, 0.499563, 9.99971, 0.199, 0],
[-0.499934, 0.499432, 9.999144, 0.1985, 255],
[-0.500066, 0.499144, 9.999432, 0.1985, 255],
[0.6, 0.5, 10, 0.25, 210],
[0.600122, 0.4999, 10.000122, 0.25, 210],
[0.600061, 0.49995, 10.000061, 0.25003, 50],
[0.595856, 0.505256, 9.991271, 0.24, 100],
[0.597973, 0.507461, 9.992175, 0.24, 210],
[0.595362, 0.507926, 9.993945, 0.24, 100],
[0.594618, 0.509055, 9.996832, 0.24, 210],
[0.592382, 0.506906, 9.996092, 0.24, 100],
[0.590438, 0.504882, 9.997604, 0.24, 210],
[0.591035, 0.503605, 9.994744, 0.24, 100],
[0.59121, 0.500708, 9.993425, 0.24, 210],
[0.593181, 0.502585, 9.991764, 0.24, 100],
[0.595867, 0.502302, 9.990069, 0.24, 210],
]
values = grey_values(test_case)
rows = [values[row * 1024:row * 1024 + 1024] for row in range(1024)]
with open('combination_scene.png', 'wb') as file:
w = png.Writer(1024, 1024, greyscale=True)
w.write(file, rows)
```
</details>
<details>
<summary>Rust</summary>
The function `grey_values` would be sufficient as an entry to this challenge. The rest of the code saves the output as a PNG file.
The `png` crate can be installed using `cargo add png`. The `png` crate is not required for the `grey_values` function, only for saving as a PNG file.
```rust
fn grey_values(spheres: &Vec<[f32; 5]>) -> Vec<u8> {
let mut values = vec![];
for y_pixel in 0..1024 {
for x_pixel in 0..1024 {
let x_ray = x_pixel as f32 / 512.0 - 1.0;
let y_ray = y_pixel as f32 / 512.0 - 1.0;
let mut min_distance = f32::INFINITY;
let mut hit_grey_value = 0;
for sphere in spheres {
let [
x_sphere, y_sphere, z_sphere, radius, grey
] = *sphere;
let a = (
(x_ray - x_sphere).powf(2.0)
+ (y_ray - y_sphere).powf(2.0)
).sqrt();
if a < radius {
let distance = z_sphere - (
radius.powf(2.0) - a.powf(2.0)
).sqrt();
if distance < min_distance {
min_distance = distance;
hit_grey_value = grey as u8;
}
}
}
values.push(hit_grey_value);
}
}
values
}
fn main() {
let test_case: Vec<[f32; 5]> = vec![
[0.0, 0.0, 1000.0, 2.0, 192.0],
[-2.0, 2.0, 100.0, 3.0, 98.0],
[2.0, 2.0, 100.0, 3.0, 98.0],
[0.5, -0.5, 500.0, 0.2, 255.0],
[-0.349469, -0.1, 20.000279, 0.6, 64.0],
[-0.349775, -0.100273, 20.000485, 0.6, 85.0],
[-0.350214, -0.100386, 20.000407, 0.6, 106.0],
[-0.350527, -0.100273, 20.000091, 0.6, 127.0],
[-0.350531, -0.1, 19.999721, 0.6, 148.0],
[-0.350225, -0.099727, 19.999515, 0.6, 169.0],
[-0.349786, -0.099614, 19.999593, 0.6, 190.0],
[-0.349473, -0.099727, 19.999909, 0.6, 211.0],
[0.20032, 0.349824, 15.000456, 0.3, 0.0],
[0.20032, 0.349517, 14.999924, 0.3, 0.0],
[0.19968, 0.350483, 15.000076, 0.3, 0.0],
[0.19968, 0.350176, 14.999544, 0.3, 0.0],
[0.200578, 0.350073, 14.999958, 0.3, 0.0],
[0.199817, 0.349519, 15.000277, 0.3, 0.0],
[0.200183, 0.350481, 14.999723, 0.3, 0.0],
[0.199422, 0.349927, 15.000042, 0.3, 0.0],
[0.200235, 0.35042, 15.000332, 0.3, 0.0],
[0.199765, 0.350078, 15.000529, 0.3, 0.0],
[0.200235, 0.349922, 14.999471, 0.3, 0.0],
[0.199765, 0.34958, 14.999668, 0.3, 0.0],
[0.200429, 0.350419, 15.000005, 0.3, 255.0],
[0.199571, 0.349795, 15.000366, 0.3, 255.0],
[0.200429, 0.350205, 14.999634, 0.3, 255.0],
[0.199571, 0.349581, 14.999995, 0.3, 255.0],
[0.200524, 0.349747, 15.000146, 0.3, 255.0],
[0.200196, 0.349509, 15.000283, 0.3, 255.0],
[0.199804, 0.350491, 14.999717, 0.3, 255.0],
[0.199476, 0.350253, 14.999854, 0.3, 255.0],
[0.200138, 0.349578, 14.999597, 0.3, 255.0],
[0.200138, 0.350138, 15.000567, 0.3, 255.0],
[0.199862, 0.349862, 14.999433, 0.3, 255.0],
[0.199862, 0.350422, 15.000403, 0.3, 255.0],
[0.200488, 0.350136, 15.000321, 0.3, 255.0],
[0.199957, 0.349751, 15.000544, 0.3, 255.0],
[0.200488, 0.34979, 14.999721, 0.3, 255.0],
[0.199957, 0.349404, 14.999944, 0.3, 255.0],
[0.200043, 0.350596, 15.000056, 0.3, 255.0],
[0.199512, 0.35021, 15.000279, 0.3, 255.0],
[0.200043, 0.350249, 14.999456, 0.3, 255.0],
[0.199512, 0.349864, 14.999679, 0.3, 255.0],
[-0.5, 0.500442, 10.000442, 0.2, 0.0],
[-0.5, 0.5, 10.0, 0.1995, 255.0],
[-0.499966, 0.49971, 9.999563, 0.199, 0.0],
[-0.500034, 0.499563, 9.99971, 0.199, 0.0],
[-0.499934, 0.499432, 9.999144, 0.1985, 255.0],
[-0.500066, 0.499144, 9.999432, 0.1985, 255.0],
[0.6, 0.5, 10.0, 0.25, 210.0],
[0.600122, 0.4999, 10.000122, 0.25, 210.0],
[0.600061, 0.49995, 10.000061, 0.25003, 50.0],
[0.595856, 0.505256, 9.991271, 0.24, 100.0],
[0.597973, 0.507461, 9.992175, 0.24, 210.0],
[0.595362, 0.507926, 9.993945, 0.24, 100.0],
[0.594618, 0.509055, 9.996832, 0.24, 210.0],
[0.592382, 0.506906, 9.996092, 0.24, 100.0],
[0.590438, 0.504882, 9.997604, 0.24, 210.0],
[0.591035, 0.503605, 9.994744, 0.24, 100.0],
[0.59121, 0.500708, 9.993425, 0.24, 210.0],
[0.593181, 0.502585, 9.991764, 0.24, 100.0],
[0.595867, 0.502302, 9.990069, 0.24, 210.0],
];
let values = grey_values(&test_case);
let path = std::path::Path::new("combination_scene.png");
let file = std::fs::File::create(path).unwrap();
let w = &mut std::io::BufWriter::new(file);
let mut encoder = png::Encoder::new(w, 1024, 1024);
encoder.set_color(png::ColorType::Grayscale);
encoder.set_depth(png::BitDepth::Eight);
let mut writer = encoder.write_header().unwrap();
writer.write_image_data(&values).unwrap();
}
```
</details>
## Test cases
The output images may be scaled down to fit this page. Click on any image to open it full size.
### Single sphere
Input:
```text
[
[0, 0, 0, 1, 128],
]
```
Output:
[![Grey circle on a black background](https://codegolf.codidact.com/uploads/jlru163pa2pgm7aaq70b2tff16uf)](https://codegolf.codidact.com/uploads/jlru163pa2pgm7aaq70b2tff16uf)
### Two separated spheres
Input:
```text
[
[0, -0.55, 0, 0.4, 100],
[0, 0.55, 0, 0.4, 200],
]
```
Output:
[![Light and dark grey circles on a black background](https://codegolf.codidact.com/uploads/8yw7agli78y65kiomb1u5yebs7c5)](https://codegolf.codidact.com/uploads/8yw7agli78y65kiomb1u5yebs7c5)
### Ring of spheres
Input:
```text
[
[0.75, 0, 0, 0.25, 64],
[0.53033, 0.375, -0.375, 0.25, 185],
[0, 0.53033, -0.53033, 0.25, 64],
[-0.53033, 0.375, -0.375, 0.25, 185],
[-0.75, 0, 0, 0.25, 64],
[-0.53033, -0.375, 0.375, 0.25, 185],
[0, -0.53033, 0.53033, 0.25, 64],
[0.53033, -0.375, 0.375, 0.25, 185],
]
```
Output:
[![Eight circles in alternating light and dark grey in a ring some overlapping](https://codegolf.codidact.com/uploads/0lsenynig27au0sryf3ruozt1uwh)](https://codegolf.codidact.com/uploads/0lsenynig27au0sryf3ruozt1uwh)
### Beach ball
Input:
```text
[
[0.001, 0, 0, 1, 64],
[0.000707, 0.000707, 0, 1, 85],
[0, 0.001, 0, 1, 106],
[-0.000707, 0.000707, 0, 1, 127],
[-0.001, 0, 0, 1, 148],
[-0.000707, -0.000707, 0, 1, 169],
[0, -0.001, 0, 1, 190],
[0.000707, -0.000707, 0, 1, 211],
]
```
Output:
[![A circle with eight sectors in different shades of grey](https://codegolf.codidact.com/uploads/51a3ni0cfwsqcdw5b49gy6282qno)](https://codegolf.codidact.com/uploads/51a3ni0cfwsqcdw5b49gy6282qno)
### Football
Input:
```text
[
[0, 0.000058, 0.001947, 1, 0],
[0, -0.001715, 0.000923, 1, 0],
[0, 0.001715, -0.000923, 1, 0],
[0, -0.000058, -0.001947, 1, 0],
[0.001657, -0.000512, 0.000887, 1, 0],
[-0.001657, -0.000512, 0.000887, 1, 0],
[0.001657, 0.000512, -0.000887, 1, 0],
[-0.001657, 0.000512, -0.000887, 1, 0],
[0.001024, 0.001435, 0.000829, 1, 0],
[-0.001024, 0.001435, 0.000829, 1, 0],
[0.001024, -0.001435, -0.000829, 1, 0],
[-0.001024, -0.001435, -0.000829, 1, 0],
[0.001868, 0.000618, 0.000357, 1, 255],
[-0.001868, 0.000618, 0.000357, 1, 255],
[0.001868, -0.000618, -0.000357, 1, 255],
[-0.001868, -0.000618, -0.000357, 1, 255],
[0.000714, -0.000934, 0.001618, 1, 255],
[-0.000714, -0.000934, 0.001618, 1, 255],
[0.000714, 0.000934, -0.001618, 1, 255],
[-0.000714, 0.000934, -0.001618, 1, 255],
[0, -0.001975, -0.000316, 1, 255],
[0, 0.001261, 0.001552, 1, 255],
[0, -0.001261, -0.001552, 1, 255],
[0, 0.001975, 0.000316, 1, 255],
[0.001155, 0.000423, 0.001577, 1, 255],
[-0.001155, 0.000423, 0.001577, 1, 255],
[0.001155, -0.001577, 0.000423, 1, 255],
[-0.001155, -0.001577, 0.000423, 1, 255],
[0.001155, 0.001577, -0.000423, 1, 255],
[-0.001155, 0.001577, -0.000423, 1, 255],
[0.001155, -0.000423, -0.001577, 1, 255],
[-0.001155, -0.000423, -0.001577, 1, 255],
]
```
Output:
[![Black pentagons on a white ball](https://codegolf.codidact.com/uploads/bmrbsw65p59imvwu8ssd42uvvx1r)](https://codegolf.codidact.com/uploads/bmrbsw65p59imvwu8ssd42uvvx1r)
### Eight ball
Input:
```text
[
[0, 0, 1000, 2, 128],
[0, 0, 0.003125, 1, 0],
[0, 0, 0, 0.9975, 255],
[0, 0.000547, -0.002573, 0.995, 0],
[0, -0.000547, -0.002573, 0.995, 0],
[0, 0.001071, -0.005037, 0.9925, 255],
[0, -0.001071, -0.005037, 0.9925, 255],
]
```
Output:
[![A ball with the number 8 in a white circle](https://codegolf.codidact.com/uploads/dvl68crpmy62rqt94mwanditt4wc)](https://codegolf.codidact.com/uploads/dvl68crpmy62rqt94mwanditt4wc)
### Star ball
Input:
```text
[
[0, 0, 0, 1, 210],
[0, 0, 0.0008, 1, 210],
[0, 0, 0.0004, 1.00012, 50],
[0.012969, 0, -0.042045, 0.96, 100],
[0.016397, 0.011913, -0.039054, 0.96, 210],
[0.004008, 0.012334, -0.042045, 0.96, 100],
[-0.006263, 0.019276, -0.039054, 0.96, 210],
[-0.010492, 0.007623, -0.042045, 0.96, 100],
[-0.020268, 0, -0.039054, 0.96, 210],
[-0.010492, -0.007623, -0.042045, 0.96, 100],
[-0.006263, -0.019276, -0.039054, 0.96, 210],
[0.004008, -0.012334, -0.042045, 0.96, 100],
[0.016397, -0.011913, -0.039054, 0.96, 210],
]
```
Output:
[![A light grey ball with a dark grey five pointed star](https://codegolf.codidact.com/uploads/f3wdgwru6j35ue72x25mazxikfx2)](https://codegolf.codidact.com/uploads/f3wdgwru6j35ue72x25mazxikfx2)
### Combination scene
Input:
```text
[
[0, 0, 1000, 2, 192],
[-2, 2, 100, 3, 98],
[2, 2, 100, 3, 98],
[0.5, -0.5, 500, 0.2, 255],
[-0.349469, -0.1, 20.000279, 0.6, 64],
[-0.349775, -0.100273, 20.000485, 0.6, 85],
[-0.350214, -0.100386, 20.000407, 0.6, 106],
[-0.350527, -0.100273, 20.000091, 0.6, 127],
[-0.350531, -0.1, 19.999721, 0.6, 148],
[-0.350225, -0.099727, 19.999515, 0.6, 169],
[-0.349786, -0.099614, 19.999593, 0.6, 190],
[-0.349473, -0.099727, 19.999909, 0.6, 211],
[0.20032, 0.349824, 15.000456, 0.3, 0],
[0.20032, 0.349517, 14.999924, 0.3, 0],
[0.19968, 0.350483, 15.000076, 0.3, 0],
[0.19968, 0.350176, 14.999544, 0.3, 0],
[0.200578, 0.350073, 14.999958, 0.3, 0],
[0.199817, 0.349519, 15.000277, 0.3, 0],
[0.200183, 0.350481, 14.999723, 0.3, 0],
[0.199422, 0.349927, 15.000042, 0.3, 0],
[0.200235, 0.35042, 15.000332, 0.3, 0],
[0.199765, 0.350078, 15.000529, 0.3, 0],
[0.200235, 0.349922, 14.999471, 0.3, 0],
[0.199765, 0.34958, 14.999668, 0.3, 0],
[0.200429, 0.350419, 15.000005, 0.3, 255],
[0.199571, 0.349795, 15.000366, 0.3, 255],
[0.200429, 0.350205, 14.999634, 0.3, 255],
[0.199571, 0.349581, 14.999995, 0.3, 255],
[0.200524, 0.349747, 15.000146, 0.3, 255],
[0.200196, 0.349509, 15.000283, 0.3, 255],
[0.199804, 0.350491, 14.999717, 0.3, 255],
[0.199476, 0.350253, 14.999854, 0.3, 255],
[0.200138, 0.349578, 14.999597, 0.3, 255],
[0.200138, 0.350138, 15.000567, 0.3, 255],
[0.199862, 0.349862, 14.999433, 0.3, 255],
[0.199862, 0.350422, 15.000403, 0.3, 255],
[0.200488, 0.350136, 15.000321, 0.3, 255],
[0.199957, 0.349751, 15.000544, 0.3, 255],
[0.200488, 0.34979, 14.999721, 0.3, 255],
[0.199957, 0.349404, 14.999944, 0.3, 255],
[0.200043, 0.350596, 15.000056, 0.3, 255],
[0.199512, 0.35021, 15.000279, 0.3, 255],
[0.200043, 0.350249, 14.999456, 0.3, 255],
[0.199512, 0.349864, 14.999679, 0.3, 255],
[-0.5, 0.500442, 10.000442, 0.2, 0],
[-0.5, 0.5, 10, 0.1995, 255],
[-0.499966, 0.49971, 9.999563, 0.199, 0],
[-0.500034, 0.499563, 9.99971, 0.199, 0],
[-0.499934, 0.499432, 9.999144, 0.1985, 255],
[-0.500066, 0.499144, 9.999432, 0.1985, 255],
[0.6, 0.5, 10, 0.25, 210],
[0.600122, 0.4999, 10.000122, 0.25, 210],
[0.600061, 0.49995, 10.000061, 0.25003, 50],
[0.595856, 0.505256, 9.991271, 0.24, 100],
[0.597973, 0.507461, 9.992175, 0.24, 210],
[0.595362, 0.507926, 9.993945, 0.24, 100],
[0.594618, 0.509055, 9.996832, 0.24, 210],
[0.592382, 0.506906, 9.996092, 0.24, 100],
[0.590438, 0.504882, 9.997604, 0.24, 210],
[0.591035, 0.503605, 9.994744, 0.24, 100],
[0.59121, 0.500708, 9.993425, 0.24, 210],
[0.593181, 0.502585, 9.991764, 0.24, 100],
[0.595867, 0.502302, 9.990069, 0.24, 210],
]
```
Output:
[![A beachball, football, eightball, and starball with the sun behind hills in the background](https://codegolf.codidact.com/uploads/usv9npm1t5bjpcaknn6rbzgkqwmi)](https://codegolf.codidact.com/uploads/usv9npm1t5bjpcaknn6rbzgkqwmi)
## Accuracy
The outputs for the test cases were generated using 64 bit floating point numbers. The following is the last test case using 32 bit floating point rather than 64 bit floating point:
[![A beachball, football, eightball, and starball with the sun behind hills in the background](https://codegolf.codidact.com/uploads/7uuxjxq7a9oxc9q0jf2bi8l68qk3)](https://codegolf.codidact.com/uploads/7uuxjxq7a9oxc9q0jf2bi8l68qk3)
The inaccuracies in the 32 bit floating point version are subtle. You may need to click on the image to open it full size to see the occasional inaccurate pixel. For example, around the inside of the holes in the 8 on the eightball. These occasional inaccurate pixels are acceptable in your output (allowing languages which only support 32 bit floating point to compete).
More widespread speckling as in the following image would not be valid, so you may need to be careful if changing the order of calculations during golfing:
[![A scene with significant areas of inaccurate speckles obscuring the intended image](https://codegolf.codidact.com/uploads/1sa7i6z0cv2kgyrmw2tj2ytr1ps1)](https://codegolf.codidact.com/uploads/1sa7i6z0cv2kgyrmw2tj2ytr1ps1)
This exaggerated inaccuracy was achieved by adding 1000 to and then subtracting 1000 from the distance calculation, exceeding the available precision of 32 bit floating point and causing some distances to appear in the wrong order. Note that this same approach causes no inaccuracy with 64 bit floating point, so if your language supports both you may have more leeway for rearranging calculations if you use 64 bit floating point.
## Explanatory animations
The example images were produced from plain grey spheres with no surface pattern or texture. The following animations show how they break down into their component plain spheres.
<details>
<summary>Animations</summary>
TODO
</details>
## Scoring
This is a [code golf challenge](https://codegolf.codidact.com/categories/49/tags/4274). Your score is the number of bytes in your code. Lowest score for each language wins.
> Explanations are optional, but I'm more likely to upvote answers that have one.

#18: Post edited by

trichoplax‭ · 2024-10-12T18:37:54Z (2 months ago)
Specify what happens when 2 or more intersections are nearest

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Given a list of spheres, output an image.
Produce a greyscale [orthographic](https://en.wikipedia.org/wiki/Parallel_projection#Orthographic_projection) (parallel projection - no perspective) ray caster, that renders only plain grey spheres with no lighting.
## Input
- A list of spheres, each consisting of:
- The coordinates of its centre $(x,y,z)$.
- Its radius $r$.
- Its greyscale value $g$, an integer from $0$ to $255$ (both inclusive).
- Only the greyscale value will be restricted to integer values. The coordinates and radius may have fractional values.
## Output
- An image $1024$ by $1024$ pixels with greyscale values from $0$ to $255$.
- The image covers x and y values from $-1$ to $1$.
- A sphere's greyscale value is applied to every visible part of it. There is no variation over the surface of the sphere, so it appears flat like a disk.
- You may output whatever image format is convenient, including simple text formats such as [PGM](https://en.wikipedia.org/wiki/Netpbm#PGM_example) or a list of pixel greyscale values. If you want to include an example of your output as an image in your answer, the image formats that Codidact supports are PNG, JPEG, and GIF. You could output PGM and then convert to PNG to upload to your answer. If you output just the greyscale values as a list of numbers from $0$ to $255$, I've made a `[ TODO page to convert these to greyscale PNG]`.
- You are free to output an image reflected or rotated by a multiple of 90 degrees. However, the z axis is required to be increasing in the forwards direction (into the image) otherwise some of the test cases will look incorrect.
## Calculating a pixel value
Imagine a ray passing into the image through a pixel, perpendicular to the image plane. If it hits a sphere, it shows the greyscale value of the first sphere it hits. Otherwise, it shows a value of zero (black).
For example, for each pixel:
- The pixel $x,y$ coordinates will each be from $0$ to $1023$. Scale these to the range $-1$ to $1$ (for example, divide by $512$ and subtract $1$).
- If these scaled $x,y$ coordinates are within the radius $r$ of the $x,y$ coordinates of a sphere's centre, the ray intersects that sphere (the $z$ coordinate is not relevant to this test because the image is an orthographic projection - the rays are all parallel to the $z$ axis). The distance $a$ of the ray from the sphere's centre can be calculated as follows:
$$a=\sqrt{(x_{ray}-x_{sphere})^2+(y_{ray}-y_{sphere})^2}$$
```text
CONSIDER A DIAGRAM HERE
```
- For each sphere that the ray intersects, find the $z$ coordinate of the nearest intersection (there will often be $2$ per sphere, $1$ on the front of the sphere and $1$ on the back of the sphere, as the ray passes through - only the nearest is relevant). This can be calculated as follows:
- The distance $d$ to the nearest intersection point can be calculated from the $z$ coordinate of the sphere's centre, the radius $r$, and the distance $a$ from the $x,y$ coordinates of the sphere's centre to the scaled $x,y$ coordinates of the pixel:
$$d=z_{sphere}-\sqrt{r_{sphere}^2-a^2}$$
```text
CONSIDER A DIAGRAM HERE
```
- Of all the spheres that the ray intersects, the sphere with the intersection point with the smallest $z$ coordinate $d$ is the one to display. Set the pixel to the greyscale value of this sphere.
- If the ray did not intersect any spheres, set the pixel to zero (black).
## Example implementations
These are ungolfed code to supplement understanding of the requirements. You are free to golf these or implement your own approach that matches the test cases up to reflection & rotation.
The Python implementation uses 64 bit floating point. The Rust implementation uses 32 bit floating point to show the difference in output. I've tried to optimise the test cases for compatibility with 32 bit floating point, so the differences are restricted to being a pixel out in some places.
<details>
<summary>Python</summary>
The function `grey_values` would be sufficient as an entry to this challenge. The rest of the code saves the output as a PNG file. The function itself has no dependence on the `import png` so it would not be required in an entry.
Instructions for installing the `png` package can be found on its [project page on PyPI](https://pypi.org/project/pypng/).
```python
def grey_values(spheres):
values = []
for y_pixel in range(1024):
for x_pixel in range(1024):
x_ray, y_ray = x_pixel / 512 - 1, y_pixel / 512 - 1
min_distance, hit_grey_value = None, 0
for sphere in spheres:
x_sphere, y_sphere, z_sphere, radius, grey = sphere
a = (
(x_ray - x_sphere) ** 2 + (y_ray - y_sphere) ** 2
) ** 0.5
if a < radius:
distance = z_sphere - (radius ** 2 - a ** 2) ** 0.5
if min_distance is None or distance < min_distance:
min_distance = distance
hit_grey_value = grey
values.append(hit_grey_value)
return values
import png
test_case = [
[0, 0, 1000, 2, 192],
[-2, 2, 100, 3, 98],
[2, 2, 100, 3, 98],
[0.5, -0.5, 500, 0.2, 255],
[-0.349469, -0.1, 20.000279, 0.6, 64],
[-0.349775, -0.100273, 20.000485, 0.6, 85],
[-0.350214, -0.100386, 20.000407, 0.6, 106],
[-0.350527, -0.100273, 20.000091, 0.6, 127],
[-0.350531, -0.1, 19.999721, 0.6, 148],
[-0.350225, -0.099727, 19.999515, 0.6, 169],
[-0.349786, -0.099614, 19.999593, 0.6, 190],
[-0.349473, -0.099727, 19.999909, 0.6, 211],
[0.20032, 0.349824, 15.000456, 0.3, 0],
[0.20032, 0.349517, 14.999924, 0.3, 0],
[0.19968, 0.350483, 15.000076, 0.3, 0],
[0.19968, 0.350176, 14.999544, 0.3, 0],
[0.200578, 0.350073, 14.999958, 0.3, 0],
[0.199817, 0.349519, 15.000277, 0.3, 0],
[0.200183, 0.350481, 14.999723, 0.3, 0],
[0.199422, 0.349927, 15.000042, 0.3, 0],
[0.200235, 0.35042, 15.000332, 0.3, 0],
[0.199765, 0.350078, 15.000529, 0.3, 0],
[0.200235, 0.349922, 14.999471, 0.3, 0],
[0.199765, 0.34958, 14.999668, 0.3, 0],
[0.200429, 0.350419, 15.000005, 0.3, 255],
[0.199571, 0.349795, 15.000366, 0.3, 255],
[0.200429, 0.350205, 14.999634, 0.3, 255],
[0.199571, 0.349581, 14.999995, 0.3, 255],
[0.200524, 0.349747, 15.000146, 0.3, 255],
[0.200196, 0.349509, 15.000283, 0.3, 255],
[0.199804, 0.350491, 14.999717, 0.3, 255],
[0.199476, 0.350253, 14.999854, 0.3, 255],
[0.200138, 0.349578, 14.999597, 0.3, 255],
[0.200138, 0.350138, 15.000567, 0.3, 255],
[0.199862, 0.349862, 14.999433, 0.3, 255],
[0.199862, 0.350422, 15.000403, 0.3, 255],
[0.200488, 0.350136, 15.000321, 0.3, 255],
[0.199957, 0.349751, 15.000544, 0.3, 255],
[0.200488, 0.34979, 14.999721, 0.3, 255],
[0.199957, 0.349404, 14.999944, 0.3, 255],
[0.200043, 0.350596, 15.000056, 0.3, 255],
[0.199512, 0.35021, 15.000279, 0.3, 255],
[0.200043, 0.350249, 14.999456, 0.3, 255],
[0.199512, 0.349864, 14.999679, 0.3, 255],
[-0.5, 0.500442, 10.000442, 0.2, 0],
[-0.5, 0.5, 10, 0.1995, 255],
[-0.499966, 0.49971, 9.999563, 0.199, 0],
[-0.500034, 0.499563, 9.99971, 0.199, 0],
[-0.499934, 0.499432, 9.999144, 0.1985, 255],
[-0.500066, 0.499144, 9.999432, 0.1985, 255],
[0.6, 0.5, 10, 0.25, 210],
[0.600122, 0.4999, 10.000122, 0.25, 210],
[0.600061, 0.49995, 10.000061, 0.25003, 50],
[0.595856, 0.505256, 9.991271, 0.24, 100],
[0.597973, 0.507461, 9.992175, 0.24, 210],
[0.595362, 0.507926, 9.993945, 0.24, 100],
[0.594618, 0.509055, 9.996832, 0.24, 210],
[0.592382, 0.506906, 9.996092, 0.24, 100],
[0.590438, 0.504882, 9.997604, 0.24, 210],
[0.591035, 0.503605, 9.994744, 0.24, 100],
[0.59121, 0.500708, 9.993425, 0.24, 210],
[0.593181, 0.502585, 9.991764, 0.24, 100],
[0.595867, 0.502302, 9.990069, 0.24, 210],
]
values = grey_values(test_case)
rows = [values[row * 1024:row * 1024 + 1024] for row in range(1024)]
with open('combination_scene.png', 'wb') as file:
w = png.Writer(1024, 1024, greyscale=True)
w.write(file, rows)
```
</details>
<details>
<summary>Rust</summary>
The function `grey_values` would be sufficient as an entry to this challenge. The rest of the code saves the output as a PNG file.
The `png` crate can be installed using `cargo add png`. The `png` crate is not required for the `grey_values` function, only for saving as a PNG file.
```rust
fn grey_values(spheres: &Vec<[f32; 5]>) -> Vec<u8> {
let mut values = vec![];
for y_pixel in 0..1024 {
for x_pixel in 0..1024 {
let x_ray = x_pixel as f32 / 512.0 - 1.0;
let y_ray = y_pixel as f32 / 512.0 - 1.0;
let mut min_distance = f32::INFINITY;
let mut hit_grey_value = 0;
for sphere in spheres {
let [
x_sphere, y_sphere, z_sphere, radius, grey
] = *sphere;
let a = (
(x_ray - x_sphere).powf(2.0)
+ (y_ray - y_sphere).powf(2.0)
).sqrt();
if a < radius {
let distance = z_sphere - (
radius.powf(2.0) - a.powf(2.0)
).sqrt();
if distance < min_distance {
min_distance = distance;
hit_grey_value = grey as u8;
}
}
}
values.push(hit_grey_value);
}
}
values
}
fn main() {
let test_case: Vec<[f32; 5]> = vec![
[0.0, 0.0, 1000.0, 2.0, 192.0],
[-2.0, 2.0, 100.0, 3.0, 98.0],
[2.0, 2.0, 100.0, 3.0, 98.0],
[0.5, -0.5, 500.0, 0.2, 255.0],
[-0.349469, -0.1, 20.000279, 0.6, 64.0],
[-0.349775, -0.100273, 20.000485, 0.6, 85.0],
[-0.350214, -0.100386, 20.000407, 0.6, 106.0],
[-0.350527, -0.100273, 20.000091, 0.6, 127.0],
[-0.350531, -0.1, 19.999721, 0.6, 148.0],
[-0.350225, -0.099727, 19.999515, 0.6, 169.0],
[-0.349786, -0.099614, 19.999593, 0.6, 190.0],
[-0.349473, -0.099727, 19.999909, 0.6, 211.0],
[0.20032, 0.349824, 15.000456, 0.3, 0.0],
[0.20032, 0.349517, 14.999924, 0.3, 0.0],
[0.19968, 0.350483, 15.000076, 0.3, 0.0],
[0.19968, 0.350176, 14.999544, 0.3, 0.0],
[0.200578, 0.350073, 14.999958, 0.3, 0.0],
[0.199817, 0.349519, 15.000277, 0.3, 0.0],
[0.200183, 0.350481, 14.999723, 0.3, 0.0],
[0.199422, 0.349927, 15.000042, 0.3, 0.0],
[0.200235, 0.35042, 15.000332, 0.3, 0.0],
[0.199765, 0.350078, 15.000529, 0.3, 0.0],
[0.200235, 0.349922, 14.999471, 0.3, 0.0],
[0.199765, 0.34958, 14.999668, 0.3, 0.0],
[0.200429, 0.350419, 15.000005, 0.3, 255.0],
[0.199571, 0.349795, 15.000366, 0.3, 255.0],
[0.200429, 0.350205, 14.999634, 0.3, 255.0],
[0.199571, 0.349581, 14.999995, 0.3, 255.0],
[0.200524, 0.349747, 15.000146, 0.3, 255.0],
[0.200196, 0.349509, 15.000283, 0.3, 255.0],
[0.199804, 0.350491, 14.999717, 0.3, 255.0],
[0.199476, 0.350253, 14.999854, 0.3, 255.0],
[0.200138, 0.349578, 14.999597, 0.3, 255.0],
[0.200138, 0.350138, 15.000567, 0.3, 255.0],
[0.199862, 0.349862, 14.999433, 0.3, 255.0],
[0.199862, 0.350422, 15.000403, 0.3, 255.0],
[0.200488, 0.350136, 15.000321, 0.3, 255.0],
[0.199957, 0.349751, 15.000544, 0.3, 255.0],
[0.200488, 0.34979, 14.999721, 0.3, 255.0],
[0.199957, 0.349404, 14.999944, 0.3, 255.0],
[0.200043, 0.350596, 15.000056, 0.3, 255.0],
[0.199512, 0.35021, 15.000279, 0.3, 255.0],
[0.200043, 0.350249, 14.999456, 0.3, 255.0],
[0.199512, 0.349864, 14.999679, 0.3, 255.0],
[-0.5, 0.500442, 10.000442, 0.2, 0.0],
[-0.5, 0.5, 10.0, 0.1995, 255.0],
[-0.499966, 0.49971, 9.999563, 0.199, 0.0],
[-0.500034, 0.499563, 9.99971, 0.199, 0.0],
[-0.499934, 0.499432, 9.999144, 0.1985, 255.0],
[-0.500066, 0.499144, 9.999432, 0.1985, 255.0],
[0.6, 0.5, 10.0, 0.25, 210.0],
[0.600122, 0.4999, 10.000122, 0.25, 210.0],
[0.600061, 0.49995, 10.000061, 0.25003, 50.0],
[0.595856, 0.505256, 9.991271, 0.24, 100.0],
[0.597973, 0.507461, 9.992175, 0.24, 210.0],
[0.595362, 0.507926, 9.993945, 0.24, 100.0],
[0.594618, 0.509055, 9.996832, 0.24, 210.0],
[0.592382, 0.506906, 9.996092, 0.24, 100.0],
[0.590438, 0.504882, 9.997604, 0.24, 210.0],
[0.591035, 0.503605, 9.994744, 0.24, 100.0],
[0.59121, 0.500708, 9.993425, 0.24, 210.0],
[0.593181, 0.502585, 9.991764, 0.24, 100.0],
[0.595867, 0.502302, 9.990069, 0.24, 210.0],
];
let values = grey_values(&test_case);
let path = std::path::Path::new("combination_scene.png");
let file = std::fs::File::create(path).unwrap();
let w = &mut std::io::BufWriter::new(file);
let mut encoder = png::Encoder::new(w, 1024, 1024);
encoder.set_color(png::ColorType::Grayscale);
encoder.set_depth(png::BitDepth::Eight);
let mut writer = encoder.write_header().unwrap();
writer.write_image_data(&values).unwrap();
}
```
</details>
## Test cases
The output images may be scaled down to fit this page. Click on any image to open it full size.
### Single sphere
Input:
```text
[
[0, 0, 0, 1, 128],
]
```
Output:
[![Grey circle on a black background](https://codegolf.codidact.com/uploads/jlru163pa2pgm7aaq70b2tff16uf)](https://codegolf.codidact.com/uploads/jlru163pa2pgm7aaq70b2tff16uf)
### Two separated spheres
Input:
```text
[
[0, -0.55, 0, 0.4, 100],
[0, 0.55, 0, 0.4, 200],
]
```
Output:
[![Light and dark grey circles on a black background](https://codegolf.codidact.com/uploads/8yw7agli78y65kiomb1u5yebs7c5)](https://codegolf.codidact.com/uploads/8yw7agli78y65kiomb1u5yebs7c5)
### Ring of spheres
Input:
```text
[
[0.75, 0, 0, 0.25, 64],
[0.53033, 0.375, -0.375, 0.25, 185],
[0, 0.53033, -0.53033, 0.25, 64],
[-0.53033, 0.375, -0.375, 0.25, 185],
[-0.75, 0, 0, 0.25, 64],
[-0.53033, -0.375, 0.375, 0.25, 185],
[0, -0.53033, 0.53033, 0.25, 64],
[0.53033, -0.375, 0.375, 0.25, 185],
]
```
Output:
[![Eight circles in alternating light and dark grey in a ring some overlapping](https://codegolf.codidact.com/uploads/0lsenynig27au0sryf3ruozt1uwh)](https://codegolf.codidact.com/uploads/0lsenynig27au0sryf3ruozt1uwh)
### Beach ball
Input:
```text
[
[0.001, 0, 0, 1, 64],
[0.000707, 0.000707, 0, 1, 85],
[0, 0.001, 0, 1, 106],
[-0.000707, 0.000707, 0, 1, 127],
[-0.001, 0, 0, 1, 148],
[-0.000707, -0.000707, 0, 1, 169],
[0, -0.001, 0, 1, 190],
[0.000707, -0.000707, 0, 1, 211],
]
```
Output:
[![A circle with eight sectors in different shades of grey](https://codegolf.codidact.com/uploads/51a3ni0cfwsqcdw5b49gy6282qno)](https://codegolf.codidact.com/uploads/51a3ni0cfwsqcdw5b49gy6282qno)
### Football
Input:
```text
[
[0, 0.000058, 0.001947, 1, 0],
[0, -0.001715, 0.000923, 1, 0],
[0, 0.001715, -0.000923, 1, 0],
[0, -0.000058, -0.001947, 1, 0],
[0.001657, -0.000512, 0.000887, 1, 0],
[-0.001657, -0.000512, 0.000887, 1, 0],
[0.001657, 0.000512, -0.000887, 1, 0],
[-0.001657, 0.000512, -0.000887, 1, 0],
[0.001024, 0.001435, 0.000829, 1, 0],
[-0.001024, 0.001435, 0.000829, 1, 0],
[0.001024, -0.001435, -0.000829, 1, 0],
[-0.001024, -0.001435, -0.000829, 1, 0],
[0.001868, 0.000618, 0.000357, 1, 255],
[-0.001868, 0.000618, 0.000357, 1, 255],
[0.001868, -0.000618, -0.000357, 1, 255],
[-0.001868, -0.000618, -0.000357, 1, 255],
[0.000714, -0.000934, 0.001618, 1, 255],
[-0.000714, -0.000934, 0.001618, 1, 255],
[0.000714, 0.000934, -0.001618, 1, 255],
[-0.000714, 0.000934, -0.001618, 1, 255],
[0, -0.001975, -0.000316, 1, 255],
[0, 0.001261, 0.001552, 1, 255],
[0, -0.001261, -0.001552, 1, 255],
[0, 0.001975, 0.000316, 1, 255],
[0.001155, 0.000423, 0.001577, 1, 255],
[-0.001155, 0.000423, 0.001577, 1, 255],
[0.001155, -0.001577, 0.000423, 1, 255],
[-0.001155, -0.001577, 0.000423, 1, 255],
[0.001155, 0.001577, -0.000423, 1, 255],
[-0.001155, 0.001577, -0.000423, 1, 255],
[0.001155, -0.000423, -0.001577, 1, 255],
[-0.001155, -0.000423, -0.001577, 1, 255],
]
```
Output:
[![Black pentagons on a white ball](https://codegolf.codidact.com/uploads/bmrbsw65p59imvwu8ssd42uvvx1r)](https://codegolf.codidact.com/uploads/bmrbsw65p59imvwu8ssd42uvvx1r)
### Eight ball
Input:
```text
[
[0, 0, 1000, 2, 128],
[0, 0, 0.003125, 1, 0],
[0, 0, 0, 0.9975, 255],
[0, 0.000547, -0.002573, 0.995, 0],
[0, -0.000547, -0.002573, 0.995, 0],
[0, 0.001071, -0.005037, 0.9925, 255],
[0, -0.001071, -0.005037, 0.9925, 255],
]
```
Output:
[![A ball with the number 8 in a white circle](https://codegolf.codidact.com/uploads/dvl68crpmy62rqt94mwanditt4wc)](https://codegolf.codidact.com/uploads/dvl68crpmy62rqt94mwanditt4wc)
### Star ball
Input:
```text
[
[0, 0, 0, 1, 210],
[0, 0, 0.0008, 1, 210],
[0, 0, 0.0004, 1.00012, 50],
[0.012969, 0, -0.042045, 0.96, 100],
[0.016397, 0.011913, -0.039054, 0.96, 210],
[0.004008, 0.012334, -0.042045, 0.96, 100],
[-0.006263, 0.019276, -0.039054, 0.96, 210],
[-0.010492, 0.007623, -0.042045, 0.96, 100],
[-0.020268, 0, -0.039054, 0.96, 210],
[-0.010492, -0.007623, -0.042045, 0.96, 100],
[-0.006263, -0.019276, -0.039054, 0.96, 210],
[0.004008, -0.012334, -0.042045, 0.96, 100],
[0.016397, -0.011913, -0.039054, 0.96, 210],
]
```
Output:
[![A light grey ball with a dark grey five pointed star](https://codegolf.codidact.com/uploads/f3wdgwru6j35ue72x25mazxikfx2)](https://codegolf.codidact.com/uploads/f3wdgwru6j35ue72x25mazxikfx2)
### Combination scene
Input:
```text
[
[0, 0, 1000, 2, 192],
[-2, 2, 100, 3, 98],
[2, 2, 100, 3, 98],
[0.5, -0.5, 500, 0.2, 255],
[-0.349469, -0.1, 20.000279, 0.6, 64],
[-0.349775, -0.100273, 20.000485, 0.6, 85],
[-0.350214, -0.100386, 20.000407, 0.6, 106],
[-0.350527, -0.100273, 20.000091, 0.6, 127],
[-0.350531, -0.1, 19.999721, 0.6, 148],
[-0.350225, -0.099727, 19.999515, 0.6, 169],
[-0.349786, -0.099614, 19.999593, 0.6, 190],
[-0.349473, -0.099727, 19.999909, 0.6, 211],
[0.20032, 0.349824, 15.000456, 0.3, 0],
[0.20032, 0.349517, 14.999924, 0.3, 0],
[0.19968, 0.350483, 15.000076, 0.3, 0],
[0.19968, 0.350176, 14.999544, 0.3, 0],
[0.200578, 0.350073, 14.999958, 0.3, 0],
[0.199817, 0.349519, 15.000277, 0.3, 0],
[0.200183, 0.350481, 14.999723, 0.3, 0],
[0.199422, 0.349927, 15.000042, 0.3, 0],
[0.200235, 0.35042, 15.000332, 0.3, 0],
[0.199765, 0.350078, 15.000529, 0.3, 0],
[0.200235, 0.349922, 14.999471, 0.3, 0],
[0.199765, 0.34958, 14.999668, 0.3, 0],
[0.200429, 0.350419, 15.000005, 0.3, 255],
[0.199571, 0.349795, 15.000366, 0.3, 255],
[0.200429, 0.350205, 14.999634, 0.3, 255],
[0.199571, 0.349581, 14.999995, 0.3, 255],
[0.200524, 0.349747, 15.000146, 0.3, 255],
[0.200196, 0.349509, 15.000283, 0.3, 255],
[0.199804, 0.350491, 14.999717, 0.3, 255],
[0.199476, 0.350253, 14.999854, 0.3, 255],
[0.200138, 0.349578, 14.999597, 0.3, 255],
[0.200138, 0.350138, 15.000567, 0.3, 255],
[0.199862, 0.349862, 14.999433, 0.3, 255],
[0.199862, 0.350422, 15.000403, 0.3, 255],
[0.200488, 0.350136, 15.000321, 0.3, 255],
[0.199957, 0.349751, 15.000544, 0.3, 255],
[0.200488, 0.34979, 14.999721, 0.3, 255],
[0.199957, 0.349404, 14.999944, 0.3, 255],
[0.200043, 0.350596, 15.000056, 0.3, 255],
[0.199512, 0.35021, 15.000279, 0.3, 255],
[0.200043, 0.350249, 14.999456, 0.3, 255],
[0.199512, 0.349864, 14.999679, 0.3, 255],
[-0.5, 0.500442, 10.000442, 0.2, 0],
[-0.5, 0.5, 10, 0.1995, 255],
[-0.499966, 0.49971, 9.999563, 0.199, 0],
[-0.500034, 0.499563, 9.99971, 0.199, 0],
[-0.499934, 0.499432, 9.999144, 0.1985, 255],
[-0.500066, 0.499144, 9.999432, 0.1985, 255],
[0.6, 0.5, 10, 0.25, 210],
[0.600122, 0.4999, 10.000122, 0.25, 210],
[0.600061, 0.49995, 10.000061, 0.25003, 50],
[0.595856, 0.505256, 9.991271, 0.24, 100],
[0.597973, 0.507461, 9.992175, 0.24, 210],
[0.595362, 0.507926, 9.993945, 0.24, 100],
[0.594618, 0.509055, 9.996832, 0.24, 210],
[0.592382, 0.506906, 9.996092, 0.24, 100],
[0.590438, 0.504882, 9.997604, 0.24, 210],
[0.591035, 0.503605, 9.994744, 0.24, 100],
[0.59121, 0.500708, 9.993425, 0.24, 210],
[0.593181, 0.502585, 9.991764, 0.24, 100],
[0.595867, 0.502302, 9.990069, 0.24, 210],
]
```
Output:
[![A beachball, football, eightball, and starball with the sun behind hills in the background](https://codegolf.codidact.com/uploads/usv9npm1t5bjpcaknn6rbzgkqwmi)](https://codegolf.codidact.com/uploads/usv9npm1t5bjpcaknn6rbzgkqwmi)
## Accuracy
The outputs for the test cases were generated using 64 bit floating point numbers. The following is the last test case using 32 bit floating point rather than 64 bit floating point:
[![A beachball, football, eightball, and starball with the sun behind hills in the background](https://codegolf.codidact.com/uploads/7uuxjxq7a9oxc9q0jf2bi8l68qk3)](https://codegolf.codidact.com/uploads/7uuxjxq7a9oxc9q0jf2bi8l68qk3)
The inaccuracies in the 32 bit floating point version are subtle. You may need to click on the image to open it full size to see the occasional inaccurate pixel. For example, around the inside of the holes in the 8 on the eightball. These occasional inaccurate pixels are acceptable in your output (allowing languages which only support 32 bit floating point to compete).
More widespread speckling as in the following image would not be valid, so you may need to be careful if changing the order of calculations during golfing:
[![A scene with significant areas of inaccurate speckles obscuring the intended image](https://codegolf.codidact.com/uploads/1sa7i6z0cv2kgyrmw2tj2ytr1ps1)](https://codegolf.codidact.com/uploads/1sa7i6z0cv2kgyrmw2tj2ytr1ps1)
This exaggerated inaccuracy was achieved by adding 1000 to and then subtracting 1000 from the distance calculation, exceeding the available precision of 32 bit floating point and causing some distances to appear in the wrong order. Note that this same approach causes no inaccuracy with 64 bit floating point, so if your language supports both you may have more leeway for rearranging calculations if you use 64 bit floating point.
## Explanatory animations
The example images were produced from plain grey spheres with no surface pattern or texture. The following animations show how they break down into their component plain spheres.
<details>
<summary>Animations</summary>
TODO
</details>
## Scoring
This is a [code golf challenge](https://codegolf.codidact.com/categories/49/tags/4274). Your score is the number of bytes in your code. Lowest score for each language wins.
> Explanations are optional, but I'm more likely to upvote answers that have one.

Given a list of spheres, output an image.
Produce a greyscale [orthographic](https://en.wikipedia.org/wiki/Parallel_projection#Orthographic_projection) (parallel projection - no perspective) ray caster, that renders only plain grey spheres with no lighting.
## Input
- A list of spheres, each consisting of:
- The coordinates of its centre $(x,y,z)$.
- Its radius $r$.
- Its greyscale value $g$, an integer from $0$ to $255$ (both inclusive).
- Only the greyscale value will be restricted to integer values. The coordinates and radius may have fractional values.
## Output
- An image $1024$ by $1024$ pixels with greyscale values from $0$ to $255$.
- The image covers x and y values from $-1$ to $1$.
- A sphere's greyscale value is applied to every visible part of it. There is no variation over the surface of the sphere, so it appears flat like a disk.
- You may output whatever image format is convenient, including simple text formats such as [PGM](https://en.wikipedia.org/wiki/Netpbm#PGM_example) or a list of pixel greyscale values. If you want to include an example of your output as an image in your answer, the image formats that Codidact supports are PNG, JPEG, and GIF. You could output PGM and then convert to PNG to upload to your answer. If you output just the greyscale values as a list of numbers from $0$ to $255$, I've made a `[ TODO page to convert these to greyscale PNG]`.
- You are free to output an image reflected or rotated by a multiple of 90 degrees. However, the z axis is required to be increasing in the forwards direction (into the image) otherwise some of the test cases will look incorrect.
## Calculating a pixel value
Imagine a ray passing into the image through a pixel, perpendicular to the image plane. If it hits a sphere, it shows the greyscale value of the first sphere it hits. Otherwise, it shows a value of zero (black).
For example, for each pixel:
- The pixel $x,y$ coordinates will each be from $0$ to $1023$. Scale these to the range $-1$ to $1$ (for example, divide by $512$ and subtract $1$).
- If these scaled $x,y$ coordinates are within the radius $r$ of the $x,y$ coordinates of a sphere's centre, the ray intersects that sphere (the $z$ coordinate is not relevant to this test because the image is an orthographic projection - the rays are all parallel to the $z$ axis). The distance $a$ of the ray from the sphere's centre can be calculated as follows:
$$a=\sqrt{(x_{ray}-x_{sphere})^2+(y_{ray}-y_{sphere})^2}$$
```text
CONSIDER A DIAGRAM HERE
```
- For each sphere that the ray intersects, find the $z$ coordinate of the nearest intersection (there will often be $2$ per sphere, $1$ on the front of the sphere and $1$ on the back of the sphere, as the ray passes through - only the nearest is relevant). This can be calculated as follows:
- The distance $d$ to the nearest intersection point can be calculated from the $z$ coordinate of the sphere's centre, the radius $r$, and the distance $a$ from the $x,y$ coordinates of the sphere's centre to the scaled $x,y$ coordinates of the pixel:
$$d=z_{sphere}-\sqrt{r_{sphere}^2-a^2}$$
```text
CONSIDER A DIAGRAM HERE
```
- Of all the spheres that the ray intersects, the sphere with the intersection point with the smallest $z$ coordinate $d$ is the one to display. Set the pixel to the greyscale value of this sphere.
- If $2$ or more spheres have an intersection point with the smallest $z$ coordinate $d$ then you may set the pixel to the greyscale value of any of these spheres.
- If the ray did not intersect any spheres, set the pixel to zero (black).
## Example implementations
These are ungolfed code to supplement understanding of the requirements. You are free to golf these or implement your own approach that matches the test cases up to reflection & rotation.
The Python implementation uses 64 bit floating point. The Rust implementation uses 32 bit floating point to show the difference in output. I've tried to optimise the test cases for compatibility with 32 bit floating point, so the differences are restricted to being a pixel out in some places.
<details>
<summary>Python</summary>
The function `grey_values` would be sufficient as an entry to this challenge. The rest of the code saves the output as a PNG file. The function itself has no dependence on the `import png` so it would not be required in an entry.
Instructions for installing the `png` package can be found on its [project page on PyPI](https://pypi.org/project/pypng/).
```python
def grey_values(spheres):
values = []
for y_pixel in range(1024):
for x_pixel in range(1024):
x_ray, y_ray = x_pixel / 512 - 1, y_pixel / 512 - 1
min_distance, hit_grey_value = None, 0
for sphere in spheres:
x_sphere, y_sphere, z_sphere, radius, grey = sphere
a = (
(x_ray - x_sphere) ** 2 + (y_ray - y_sphere) ** 2
) ** 0.5
if a < radius:
distance = z_sphere - (radius ** 2 - a ** 2) ** 0.5
if min_distance is None or distance < min_distance:
min_distance = distance
hit_grey_value = grey
values.append(hit_grey_value)
return values
import png
test_case = [
[0, 0, 1000, 2, 192],
[-2, 2, 100, 3, 98],
[2, 2, 100, 3, 98],
[0.5, -0.5, 500, 0.2, 255],
[-0.349469, -0.1, 20.000279, 0.6, 64],
[-0.349775, -0.100273, 20.000485, 0.6, 85],
[-0.350214, -0.100386, 20.000407, 0.6, 106],
[-0.350527, -0.100273, 20.000091, 0.6, 127],
[-0.350531, -0.1, 19.999721, 0.6, 148],
[-0.350225, -0.099727, 19.999515, 0.6, 169],
[-0.349786, -0.099614, 19.999593, 0.6, 190],
[-0.349473, -0.099727, 19.999909, 0.6, 211],
[0.20032, 0.349824, 15.000456, 0.3, 0],
[0.20032, 0.349517, 14.999924, 0.3, 0],
[0.19968, 0.350483, 15.000076, 0.3, 0],
[0.19968, 0.350176, 14.999544, 0.3, 0],
[0.200578, 0.350073, 14.999958, 0.3, 0],
[0.199817, 0.349519, 15.000277, 0.3, 0],
[0.200183, 0.350481, 14.999723, 0.3, 0],
[0.199422, 0.349927, 15.000042, 0.3, 0],
[0.200235, 0.35042, 15.000332, 0.3, 0],
[0.199765, 0.350078, 15.000529, 0.3, 0],
[0.200235, 0.349922, 14.999471, 0.3, 0],
[0.199765, 0.34958, 14.999668, 0.3, 0],
[0.200429, 0.350419, 15.000005, 0.3, 255],
[0.199571, 0.349795, 15.000366, 0.3, 255],
[0.200429, 0.350205, 14.999634, 0.3, 255],
[0.199571, 0.349581, 14.999995, 0.3, 255],
[0.200524, 0.349747, 15.000146, 0.3, 255],
[0.200196, 0.349509, 15.000283, 0.3, 255],
[0.199804, 0.350491, 14.999717, 0.3, 255],
[0.199476, 0.350253, 14.999854, 0.3, 255],
[0.200138, 0.349578, 14.999597, 0.3, 255],
[0.200138, 0.350138, 15.000567, 0.3, 255],
[0.199862, 0.349862, 14.999433, 0.3, 255],
[0.199862, 0.350422, 15.000403, 0.3, 255],
[0.200488, 0.350136, 15.000321, 0.3, 255],
[0.199957, 0.349751, 15.000544, 0.3, 255],
[0.200488, 0.34979, 14.999721, 0.3, 255],
[0.199957, 0.349404, 14.999944, 0.3, 255],
[0.200043, 0.350596, 15.000056, 0.3, 255],
[0.199512, 0.35021, 15.000279, 0.3, 255],
[0.200043, 0.350249, 14.999456, 0.3, 255],
[0.199512, 0.349864, 14.999679, 0.3, 255],
[-0.5, 0.500442, 10.000442, 0.2, 0],
[-0.5, 0.5, 10, 0.1995, 255],
[-0.499966, 0.49971, 9.999563, 0.199, 0],
[-0.500034, 0.499563, 9.99971, 0.199, 0],
[-0.499934, 0.499432, 9.999144, 0.1985, 255],
[-0.500066, 0.499144, 9.999432, 0.1985, 255],
[0.6, 0.5, 10, 0.25, 210],
[0.600122, 0.4999, 10.000122, 0.25, 210],
[0.600061, 0.49995, 10.000061, 0.25003, 50],
[0.595856, 0.505256, 9.991271, 0.24, 100],
[0.597973, 0.507461, 9.992175, 0.24, 210],
[0.595362, 0.507926, 9.993945, 0.24, 100],
[0.594618, 0.509055, 9.996832, 0.24, 210],
[0.592382, 0.506906, 9.996092, 0.24, 100],
[0.590438, 0.504882, 9.997604, 0.24, 210],
[0.591035, 0.503605, 9.994744, 0.24, 100],
[0.59121, 0.500708, 9.993425, 0.24, 210],
[0.593181, 0.502585, 9.991764, 0.24, 100],
[0.595867, 0.502302, 9.990069, 0.24, 210],
]
values = grey_values(test_case)
rows = [values[row * 1024:row * 1024 + 1024] for row in range(1024)]
with open('combination_scene.png', 'wb') as file:
w = png.Writer(1024, 1024, greyscale=True)
w.write(file, rows)
```
</details>
<details>
<summary>Rust</summary>
The function `grey_values` would be sufficient as an entry to this challenge. The rest of the code saves the output as a PNG file.
The `png` crate can be installed using `cargo add png`. The `png` crate is not required for the `grey_values` function, only for saving as a PNG file.
```rust
fn grey_values(spheres: &Vec<[f32; 5]>) -> Vec<u8> {
let mut values = vec![];
for y_pixel in 0..1024 {
for x_pixel in 0..1024 {
let x_ray = x_pixel as f32 / 512.0 - 1.0;
let y_ray = y_pixel as f32 / 512.0 - 1.0;
let mut min_distance = f32::INFINITY;
let mut hit_grey_value = 0;
for sphere in spheres {
let [
x_sphere, y_sphere, z_sphere, radius, grey
] = *sphere;
let a = (
(x_ray - x_sphere).powf(2.0)
+ (y_ray - y_sphere).powf(2.0)
).sqrt();
if a < radius {
let distance = z_sphere - (
radius.powf(2.0) - a.powf(2.0)
).sqrt();
if distance < min_distance {
min_distance = distance;
hit_grey_value = grey as u8;
}
}
}
values.push(hit_grey_value);
}
}
values
}
fn main() {
let test_case: Vec<[f32; 5]> = vec![
[0.0, 0.0, 1000.0, 2.0, 192.0],
[-2.0, 2.0, 100.0, 3.0, 98.0],
[2.0, 2.0, 100.0, 3.0, 98.0],
[0.5, -0.5, 500.0, 0.2, 255.0],
[-0.349469, -0.1, 20.000279, 0.6, 64.0],
[-0.349775, -0.100273, 20.000485, 0.6, 85.0],
[-0.350214, -0.100386, 20.000407, 0.6, 106.0],
[-0.350527, -0.100273, 20.000091, 0.6, 127.0],
[-0.350531, -0.1, 19.999721, 0.6, 148.0],
[-0.350225, -0.099727, 19.999515, 0.6, 169.0],
[-0.349786, -0.099614, 19.999593, 0.6, 190.0],
[-0.349473, -0.099727, 19.999909, 0.6, 211.0],
[0.20032, 0.349824, 15.000456, 0.3, 0.0],
[0.20032, 0.349517, 14.999924, 0.3, 0.0],
[0.19968, 0.350483, 15.000076, 0.3, 0.0],
[0.19968, 0.350176, 14.999544, 0.3, 0.0],
[0.200578, 0.350073, 14.999958, 0.3, 0.0],
[0.199817, 0.349519, 15.000277, 0.3, 0.0],
[0.200183, 0.350481, 14.999723, 0.3, 0.0],
[0.199422, 0.349927, 15.000042, 0.3, 0.0],
[0.200235, 0.35042, 15.000332, 0.3, 0.0],
[0.199765, 0.350078, 15.000529, 0.3, 0.0],
[0.200235, 0.349922, 14.999471, 0.3, 0.0],
[0.199765, 0.34958, 14.999668, 0.3, 0.0],
[0.200429, 0.350419, 15.000005, 0.3, 255.0],
[0.199571, 0.349795, 15.000366, 0.3, 255.0],
[0.200429, 0.350205, 14.999634, 0.3, 255.0],
[0.199571, 0.349581, 14.999995, 0.3, 255.0],
[0.200524, 0.349747, 15.000146, 0.3, 255.0],
[0.200196, 0.349509, 15.000283, 0.3, 255.0],
[0.199804, 0.350491, 14.999717, 0.3, 255.0],
[0.199476, 0.350253, 14.999854, 0.3, 255.0],
[0.200138, 0.349578, 14.999597, 0.3, 255.0],
[0.200138, 0.350138, 15.000567, 0.3, 255.0],
[0.199862, 0.349862, 14.999433, 0.3, 255.0],
[0.199862, 0.350422, 15.000403, 0.3, 255.0],
[0.200488, 0.350136, 15.000321, 0.3, 255.0],
[0.199957, 0.349751, 15.000544, 0.3, 255.0],
[0.200488, 0.34979, 14.999721, 0.3, 255.0],
[0.199957, 0.349404, 14.999944, 0.3, 255.0],
[0.200043, 0.350596, 15.000056, 0.3, 255.0],
[0.199512, 0.35021, 15.000279, 0.3, 255.0],
[0.200043, 0.350249, 14.999456, 0.3, 255.0],
[0.199512, 0.349864, 14.999679, 0.3, 255.0],
[-0.5, 0.500442, 10.000442, 0.2, 0.0],
[-0.5, 0.5, 10.0, 0.1995, 255.0],
[-0.499966, 0.49971, 9.999563, 0.199, 0.0],
[-0.500034, 0.499563, 9.99971, 0.199, 0.0],
[-0.499934, 0.499432, 9.999144, 0.1985, 255.0],
[-0.500066, 0.499144, 9.999432, 0.1985, 255.0],
[0.6, 0.5, 10.0, 0.25, 210.0],
[0.600122, 0.4999, 10.000122, 0.25, 210.0],
[0.600061, 0.49995, 10.000061, 0.25003, 50.0],
[0.595856, 0.505256, 9.991271, 0.24, 100.0],
[0.597973, 0.507461, 9.992175, 0.24, 210.0],
[0.595362, 0.507926, 9.993945, 0.24, 100.0],
[0.594618, 0.509055, 9.996832, 0.24, 210.0],
[0.592382, 0.506906, 9.996092, 0.24, 100.0],
[0.590438, 0.504882, 9.997604, 0.24, 210.0],
[0.591035, 0.503605, 9.994744, 0.24, 100.0],
[0.59121, 0.500708, 9.993425, 0.24, 210.0],
[0.593181, 0.502585, 9.991764, 0.24, 100.0],
[0.595867, 0.502302, 9.990069, 0.24, 210.0],
];
let values = grey_values(&test_case);
let path = std::path::Path::new("combination_scene.png");
let file = std::fs::File::create(path).unwrap();
let w = &mut std::io::BufWriter::new(file);
let mut encoder = png::Encoder::new(w, 1024, 1024);
encoder.set_color(png::ColorType::Grayscale);
encoder.set_depth(png::BitDepth::Eight);
let mut writer = encoder.write_header().unwrap();
writer.write_image_data(&values).unwrap();
}
```
</details>
## Test cases
The output images may be scaled down to fit this page. Click on any image to open it full size.
### Single sphere
Input:
```text
[
[0, 0, 0, 1, 128],
]
```
Output:
[![Grey circle on a black background](https://codegolf.codidact.com/uploads/jlru163pa2pgm7aaq70b2tff16uf)](https://codegolf.codidact.com/uploads/jlru163pa2pgm7aaq70b2tff16uf)
### Two separated spheres
Input:
```text
[
[0, -0.55, 0, 0.4, 100],
[0, 0.55, 0, 0.4, 200],
]
```
Output:
[![Light and dark grey circles on a black background](https://codegolf.codidact.com/uploads/8yw7agli78y65kiomb1u5yebs7c5)](https://codegolf.codidact.com/uploads/8yw7agli78y65kiomb1u5yebs7c5)
### Ring of spheres
Input:
```text
[
[0.75, 0, 0, 0.25, 64],
[0.53033, 0.375, -0.375, 0.25, 185],
[0, 0.53033, -0.53033, 0.25, 64],
[-0.53033, 0.375, -0.375, 0.25, 185],
[-0.75, 0, 0, 0.25, 64],
[-0.53033, -0.375, 0.375, 0.25, 185],
[0, -0.53033, 0.53033, 0.25, 64],
[0.53033, -0.375, 0.375, 0.25, 185],
]
```
Output:
[![Eight circles in alternating light and dark grey in a ring some overlapping](https://codegolf.codidact.com/uploads/0lsenynig27au0sryf3ruozt1uwh)](https://codegolf.codidact.com/uploads/0lsenynig27au0sryf3ruozt1uwh)
### Beach ball
Input:
```text
[
[0.001, 0, 0, 1, 64],
[0.000707, 0.000707, 0, 1, 85],
[0, 0.001, 0, 1, 106],
[-0.000707, 0.000707, 0, 1, 127],
[-0.001, 0, 0, 1, 148],
[-0.000707, -0.000707, 0, 1, 169],
[0, -0.001, 0, 1, 190],
[0.000707, -0.000707, 0, 1, 211],
]
```
Output:
[![A circle with eight sectors in different shades of grey](https://codegolf.codidact.com/uploads/51a3ni0cfwsqcdw5b49gy6282qno)](https://codegolf.codidact.com/uploads/51a3ni0cfwsqcdw5b49gy6282qno)
### Football
Input:
```text
[
[0, 0.000058, 0.001947, 1, 0],
[0, -0.001715, 0.000923, 1, 0],
[0, 0.001715, -0.000923, 1, 0],
[0, -0.000058, -0.001947, 1, 0],
[0.001657, -0.000512, 0.000887, 1, 0],
[-0.001657, -0.000512, 0.000887, 1, 0],
[0.001657, 0.000512, -0.000887, 1, 0],
[-0.001657, 0.000512, -0.000887, 1, 0],
[0.001024, 0.001435, 0.000829, 1, 0],
[-0.001024, 0.001435, 0.000829, 1, 0],
[0.001024, -0.001435, -0.000829, 1, 0],
[-0.001024, -0.001435, -0.000829, 1, 0],
[0.001868, 0.000618, 0.000357, 1, 255],
[-0.001868, 0.000618, 0.000357, 1, 255],
[0.001868, -0.000618, -0.000357, 1, 255],
[-0.001868, -0.000618, -0.000357, 1, 255],
[0.000714, -0.000934, 0.001618, 1, 255],
[-0.000714, -0.000934, 0.001618, 1, 255],
[0.000714, 0.000934, -0.001618, 1, 255],
[-0.000714, 0.000934, -0.001618, 1, 255],
[0, -0.001975, -0.000316, 1, 255],
[0, 0.001261, 0.001552, 1, 255],
[0, -0.001261, -0.001552, 1, 255],
[0, 0.001975, 0.000316, 1, 255],
[0.001155, 0.000423, 0.001577, 1, 255],
[-0.001155, 0.000423, 0.001577, 1, 255],
[0.001155, -0.001577, 0.000423, 1, 255],
[-0.001155, -0.001577, 0.000423, 1, 255],
[0.001155, 0.001577, -0.000423, 1, 255],
[-0.001155, 0.001577, -0.000423, 1, 255],
[0.001155, -0.000423, -0.001577, 1, 255],
[-0.001155, -0.000423, -0.001577, 1, 255],
]
```
Output:
[![Black pentagons on a white ball](https://codegolf.codidact.com/uploads/bmrbsw65p59imvwu8ssd42uvvx1r)](https://codegolf.codidact.com/uploads/bmrbsw65p59imvwu8ssd42uvvx1r)
### Eight ball
Input:
```text
[
[0, 0, 1000, 2, 128],
[0, 0, 0.003125, 1, 0],
[0, 0, 0, 0.9975, 255],
[0, 0.000547, -0.002573, 0.995, 0],
[0, -0.000547, -0.002573, 0.995, 0],
[0, 0.001071, -0.005037, 0.9925, 255],
[0, -0.001071, -0.005037, 0.9925, 255],
]
```
Output:
[![A ball with the number 8 in a white circle](https://codegolf.codidact.com/uploads/dvl68crpmy62rqt94mwanditt4wc)](https://codegolf.codidact.com/uploads/dvl68crpmy62rqt94mwanditt4wc)
### Star ball
Input:
```text
[
[0, 0, 0, 1, 210],
[0, 0, 0.0008, 1, 210],
[0, 0, 0.0004, 1.00012, 50],
[0.012969, 0, -0.042045, 0.96, 100],
[0.016397, 0.011913, -0.039054, 0.96, 210],
[0.004008, 0.012334, -0.042045, 0.96, 100],
[-0.006263, 0.019276, -0.039054, 0.96, 210],
[-0.010492, 0.007623, -0.042045, 0.96, 100],
[-0.020268, 0, -0.039054, 0.96, 210],
[-0.010492, -0.007623, -0.042045, 0.96, 100],
[-0.006263, -0.019276, -0.039054, 0.96, 210],
[0.004008, -0.012334, -0.042045, 0.96, 100],
[0.016397, -0.011913, -0.039054, 0.96, 210],
]
```
Output:
[![A light grey ball with a dark grey five pointed star](https://codegolf.codidact.com/uploads/f3wdgwru6j35ue72x25mazxikfx2)](https://codegolf.codidact.com/uploads/f3wdgwru6j35ue72x25mazxikfx2)
### Combination scene
Input:
```text
[
[0, 0, 1000, 2, 192],
[-2, 2, 100, 3, 98],
[2, 2, 100, 3, 98],
[0.5, -0.5, 500, 0.2, 255],
[-0.349469, -0.1, 20.000279, 0.6, 64],
[-0.349775, -0.100273, 20.000485, 0.6, 85],
[-0.350214, -0.100386, 20.000407, 0.6, 106],
[-0.350527, -0.100273, 20.000091, 0.6, 127],
[-0.350531, -0.1, 19.999721, 0.6, 148],
[-0.350225, -0.099727, 19.999515, 0.6, 169],
[-0.349786, -0.099614, 19.999593, 0.6, 190],
[-0.349473, -0.099727, 19.999909, 0.6, 211],
[0.20032, 0.349824, 15.000456, 0.3, 0],
[0.20032, 0.349517, 14.999924, 0.3, 0],
[0.19968, 0.350483, 15.000076, 0.3, 0],
[0.19968, 0.350176, 14.999544, 0.3, 0],
[0.200578, 0.350073, 14.999958, 0.3, 0],
[0.199817, 0.349519, 15.000277, 0.3, 0],
[0.200183, 0.350481, 14.999723, 0.3, 0],
[0.199422, 0.349927, 15.000042, 0.3, 0],
[0.200235, 0.35042, 15.000332, 0.3, 0],
[0.199765, 0.350078, 15.000529, 0.3, 0],
[0.200235, 0.349922, 14.999471, 0.3, 0],
[0.199765, 0.34958, 14.999668, 0.3, 0],
[0.200429, 0.350419, 15.000005, 0.3, 255],
[0.199571, 0.349795, 15.000366, 0.3, 255],
[0.200429, 0.350205, 14.999634, 0.3, 255],
[0.199571, 0.349581, 14.999995, 0.3, 255],
[0.200524, 0.349747, 15.000146, 0.3, 255],
[0.200196, 0.349509, 15.000283, 0.3, 255],
[0.199804, 0.350491, 14.999717, 0.3, 255],
[0.199476, 0.350253, 14.999854, 0.3, 255],
[0.200138, 0.349578, 14.999597, 0.3, 255],
[0.200138, 0.350138, 15.000567, 0.3, 255],
[0.199862, 0.349862, 14.999433, 0.3, 255],
[0.199862, 0.350422, 15.000403, 0.3, 255],
[0.200488, 0.350136, 15.000321, 0.3, 255],
[0.199957, 0.349751, 15.000544, 0.3, 255],
[0.200488, 0.34979, 14.999721, 0.3, 255],
[0.199957, 0.349404, 14.999944, 0.3, 255],
[0.200043, 0.350596, 15.000056, 0.3, 255],
[0.199512, 0.35021, 15.000279, 0.3, 255],
[0.200043, 0.350249, 14.999456, 0.3, 255],
[0.199512, 0.349864, 14.999679, 0.3, 255],
[-0.5, 0.500442, 10.000442, 0.2, 0],
[-0.5, 0.5, 10, 0.1995, 255],
[-0.499966, 0.49971, 9.999563, 0.199, 0],
[-0.500034, 0.499563, 9.99971, 0.199, 0],
[-0.499934, 0.499432, 9.999144, 0.1985, 255],
[-0.500066, 0.499144, 9.999432, 0.1985, 255],
[0.6, 0.5, 10, 0.25, 210],
[0.600122, 0.4999, 10.000122, 0.25, 210],
[0.600061, 0.49995, 10.000061, 0.25003, 50],
[0.595856, 0.505256, 9.991271, 0.24, 100],
[0.597973, 0.507461, 9.992175, 0.24, 210],
[0.595362, 0.507926, 9.993945, 0.24, 100],
[0.594618, 0.509055, 9.996832, 0.24, 210],
[0.592382, 0.506906, 9.996092, 0.24, 100],
[0.590438, 0.504882, 9.997604, 0.24, 210],
[0.591035, 0.503605, 9.994744, 0.24, 100],
[0.59121, 0.500708, 9.993425, 0.24, 210],
[0.593181, 0.502585, 9.991764, 0.24, 100],
[0.595867, 0.502302, 9.990069, 0.24, 210],
]
```
Output:
[![A beachball, football, eightball, and starball with the sun behind hills in the background](https://codegolf.codidact.com/uploads/usv9npm1t5bjpcaknn6rbzgkqwmi)](https://codegolf.codidact.com/uploads/usv9npm1t5bjpcaknn6rbzgkqwmi)
## Accuracy
The outputs for the test cases were generated using 64 bit floating point numbers. The following is the last test case using 32 bit floating point rather than 64 bit floating point:
[![A beachball, football, eightball, and starball with the sun behind hills in the background](https://codegolf.codidact.com/uploads/7uuxjxq7a9oxc9q0jf2bi8l68qk3)](https://codegolf.codidact.com/uploads/7uuxjxq7a9oxc9q0jf2bi8l68qk3)
The inaccuracies in the 32 bit floating point version are subtle. You may need to click on the image to open it full size to see the occasional inaccurate pixel. For example, around the inside of the holes in the 8 on the eightball. These occasional inaccurate pixels are acceptable in your output (allowing languages which only support 32 bit floating point to compete).
More widespread speckling as in the following image would not be valid, so you may need to be careful if changing the order of calculations during golfing:
[![A scene with significant areas of inaccurate speckles obscuring the intended image](https://codegolf.codidact.com/uploads/1sa7i6z0cv2kgyrmw2tj2ytr1ps1)](https://codegolf.codidact.com/uploads/1sa7i6z0cv2kgyrmw2tj2ytr1ps1)
This exaggerated inaccuracy was achieved by adding 1000 to and then subtracting 1000 from the distance calculation, exceeding the available precision of 32 bit floating point and causing some distances to appear in the wrong order. Note that this same approach causes no inaccuracy with 64 bit floating point, so if your language supports both you may have more leeway for rearranging calculations if you use 64 bit floating point.
## Explanatory animations
The example images were produced from plain grey spheres with no surface pattern or texture. The following animations show how they break down into their component plain spheres.
<details>
<summary>Animations</summary>
TODO
</details>
## Scoring
This is a [code golf challenge](https://codegolf.codidact.com/categories/49/tags/4274). Your score is the number of bytes in your code. Lowest score for each language wins.
> Explanations are optional, but I'm more likely to upvote answers that have one.

#17: Post edited by

trichoplax‭ · 2024-08-09T18:47:25Z (4 months ago)
Collapse example implementations because they are optional

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Given a list of spheres, output an image.
Produce a greyscale [orthographic](https://en.wikipedia.org/wiki/Parallel_projection#Orthographic_projection) (parallel projection - no perspective) ray caster, that renders only plain grey spheres with no lighting.
## Input
- A list of spheres, each consisting of:
- The coordinates of its centre $(x,y,z)$.
- Its radius $r$.
- Its greyscale value $g$, an integer from $0$ to $255$ (both inclusive).
- Only the greyscale value will be restricted to integer values. The coordinates and radius may have fractional values.
## Output
- An image $1024$ by $1024$ pixels with greyscale values from $0$ to $255$.
- The image covers x and y values from $-1$ to $1$.
- A sphere's greyscale value is applied to every visible part of it. There is no variation over the surface of the sphere, so it appears flat like a disk.
- You may output whatever image format is convenient, including simple text formats such as [PGM](https://en.wikipedia.org/wiki/Netpbm#PGM_example) or a list of pixel greyscale values. If you want to include an example of your output as an image in your answer, the image formats that Codidact supports are PNG, JPEG, and GIF. You could output PGM and then convert to PNG to upload to your answer. If you output just the greyscale values as a list of numbers from $0$ to $255$, I've made a `[ TODO page to convert these to greyscale PNG]`.
- You are free to output an image reflected or rotated by a multiple of 90 degrees. However, the z axis is required to be increasing in the forwards direction (into the image) otherwise some of the test cases will look incorrect.
## Calculating a pixel value
Imagine a ray passing into the image through a pixel, perpendicular to the image plane. If it hits a sphere, it shows the greyscale value of the first sphere it hits. Otherwise, it shows a value of zero (black).
For example, for each pixel:
- The pixel $x,y$ coordinates will each be from $0$ to $1023$. Scale these to the range $-1$ to $1$ (for example, divide by $512$ and subtract $1$).
- If these scaled $x,y$ coordinates are within the radius $r$ of the $x,y$ coordinates of a sphere's centre, the ray intersects that sphere (the $z$ coordinate is not relevant to this test because the image is an orthographic projection - the rays are all parallel to the $z$ axis). The distance $a$ of the ray from the sphere's centre can be calculated as follows:
$$a=\sqrt{(x_{ray}-x_{sphere})^2+(y_{ray}-y_{sphere})^2}$$
```text
CONSIDER A DIAGRAM HERE
```
- For each sphere that the ray intersects, find the $z$ coordinate of the nearest intersection (there will often be $2$ per sphere, $1$ on the front of the sphere and $1$ on the back of the sphere, as the ray passes through - only the nearest is relevant). This can be calculated as follows:
- The distance $d$ to the nearest intersection point can be calculated from the $z$ coordinate of the sphere's centre, the radius $r$, and the distance $a$ from the $x,y$ coordinates of the sphere's centre to the scaled $x,y$ coordinates of the pixel:
$$d=z_{sphere}-\sqrt{r_{sphere}^2-a^2}$$
```text
CONSIDER A DIAGRAM HERE
```
- Of all the spheres that the ray intersects, the sphere with the intersection point with the smallest $z$ coordinate $d$ is the one to display. Set the pixel to the greyscale value of this sphere.
- If the ray did not intersect any spheres, set the pixel to zero (black).
## Example implementations
These are ungolfed code to supplement understanding of the requirements. You are free to golf these or implement your own approach that matches the test cases up to reflection & rotation.
The Python implementation uses 64 bit floating point. The Rust implementation uses 32 bit floating point to show the difference in output. I've tried to optimise the test cases for compatibility with 32 bit floating point, so the differences are restricted to being a pixel out in some places.
### Python
The function `grey_values` would be sufficient as an entry to this challenge. The rest of the code saves the output as a PNG file. The function itself has no dependence on the `import png` so it would not be required in an entry.
Instructions for installing the `png` package can be found on its [project page on PyPI](https://pypi.org/project/pypng/).
```python
def grey_values(spheres):
values = []
for y_pixel in range(1024):
for x_pixel in range(1024):
x_ray, y_ray = x_pixel / 512 - 1, y_pixel / 512 - 1
min_distance, hit_grey_value = None, 0
for sphere in spheres:
x_sphere, y_sphere, z_sphere, radius, grey = sphere
a = (
(x_ray - x_sphere) ** 2 + (y_ray - y_sphere) ** 2
) ** 0.5
if a < radius:
distance = z_sphere - (radius ** 2 - a ** 2) ** 0.5
if min_distance is None or distance < min_distance:
min_distance = distance
hit_grey_value = grey
values.append(hit_grey_value)
return values
import png
test_case = [
[0, 0, 1000, 2, 192],
[-2, 2, 100, 3, 98],
[2, 2, 100, 3, 98],
[0.5, -0.5, 500, 0.2, 255],
[-0.349469, -0.1, 20.000279, 0.6, 64],
[-0.349775, -0.100273, 20.000485, 0.6, 85],
[-0.350214, -0.100386, 20.000407, 0.6, 106],
[-0.350527, -0.100273, 20.000091, 0.6, 127],
[-0.350531, -0.1, 19.999721, 0.6, 148],
[-0.350225, -0.099727, 19.999515, 0.6, 169],
[-0.349786, -0.099614, 19.999593, 0.6, 190],
[-0.349473, -0.099727, 19.999909, 0.6, 211],
[0.20032, 0.349824, 15.000456, 0.3, 0],
[0.20032, 0.349517, 14.999924, 0.3, 0],
[0.19968, 0.350483, 15.000076, 0.3, 0],
[0.19968, 0.350176, 14.999544, 0.3, 0],
[0.200578, 0.350073, 14.999958, 0.3, 0],
[0.199817, 0.349519, 15.000277, 0.3, 0],
[0.200183, 0.350481, 14.999723, 0.3, 0],
[0.199422, 0.349927, 15.000042, 0.3, 0],
[0.200235, 0.35042, 15.000332, 0.3, 0],
[0.199765, 0.350078, 15.000529, 0.3, 0],
[0.200235, 0.349922, 14.999471, 0.3, 0],
[0.199765, 0.34958, 14.999668, 0.3, 0],
[0.200429, 0.350419, 15.000005, 0.3, 255],
[0.199571, 0.349795, 15.000366, 0.3, 255],
[0.200429, 0.350205, 14.999634, 0.3, 255],
[0.199571, 0.349581, 14.999995, 0.3, 255],
[0.200524, 0.349747, 15.000146, 0.3, 255],
[0.200196, 0.349509, 15.000283, 0.3, 255],
[0.199804, 0.350491, 14.999717, 0.3, 255],
[0.199476, 0.350253, 14.999854, 0.3, 255],
[0.200138, 0.349578, 14.999597, 0.3, 255],
[0.200138, 0.350138, 15.000567, 0.3, 255],
[0.199862, 0.349862, 14.999433, 0.3, 255],
[0.199862, 0.350422, 15.000403, 0.3, 255],
[0.200488, 0.350136, 15.000321, 0.3, 255],
[0.199957, 0.349751, 15.000544, 0.3, 255],
[0.200488, 0.34979, 14.999721, 0.3, 255],
[0.199957, 0.349404, 14.999944, 0.3, 255],
[0.200043, 0.350596, 15.000056, 0.3, 255],
[0.199512, 0.35021, 15.000279, 0.3, 255],
[0.200043, 0.350249, 14.999456, 0.3, 255],
[0.199512, 0.349864, 14.999679, 0.3, 255],
[-0.5, 0.500442, 10.000442, 0.2, 0],
[-0.5, 0.5, 10, 0.1995, 255],
[-0.499966, 0.49971, 9.999563, 0.199, 0],
[-0.500034, 0.499563, 9.99971, 0.199, 0],
[-0.499934, 0.499432, 9.999144, 0.1985, 255],
[-0.500066, 0.499144, 9.999432, 0.1985, 255],
[0.6, 0.5, 10, 0.25, 210],
[0.600122, 0.4999, 10.000122, 0.25, 210],
[0.600061, 0.49995, 10.000061, 0.25003, 50],
[0.595856, 0.505256, 9.991271, 0.24, 100],
[0.597973, 0.507461, 9.992175, 0.24, 210],
[0.595362, 0.507926, 9.993945, 0.24, 100],
[0.594618, 0.509055, 9.996832, 0.24, 210],
[0.592382, 0.506906, 9.996092, 0.24, 100],
[0.590438, 0.504882, 9.997604, 0.24, 210],
[0.591035, 0.503605, 9.994744, 0.24, 100],
[0.59121, 0.500708, 9.993425, 0.24, 210],
[0.593181, 0.502585, 9.991764, 0.24, 100],
[0.595867, 0.502302, 9.990069, 0.24, 210],
]
values = grey_values(test_case)
rows = [values[row * 1024:row * 1024 + 1024] for row in range(1024)]
with open('combination_scene.png', 'wb') as file:
w = png.Writer(1024, 1024, greyscale=True)
w.write(file, rows)
```
~~### Rust~~
The function `grey_values` would be sufficient as an entry to this challenge. The rest of the code saves the output as a PNG file.
The `png` crate can be installed using `cargo add png`. The `png` crate is not required for the `grey_values` function, only for saving as a PNG file.
```rust
fn grey_values(spheres: &Vec<[f32; 5]>) -> Vec<u8> {
let mut values = vec![];
for y_pixel in 0..1024 {
for x_pixel in 0..1024 {
let x_ray = x_pixel as f32 / 512.0 - 1.0;
let y_ray = y_pixel as f32 / 512.0 - 1.0;
let mut min_distance = f32::INFINITY;
let mut hit_grey_value = 0;
for sphere in spheres {
let [
x_sphere, y_sphere, z_sphere, radius, grey
] = *sphere;
let a = (
(x_ray - x_sphere).powf(2.0)
+ (y_ray - y_sphere).powf(2.0)
).sqrt();
if a < radius {
let distance = z_sphere - (
radius.powf(2.0) - a.powf(2.0)
).sqrt();
if distance < min_distance {
min_distance = distance;
hit_grey_value = grey as u8;
}
}
}
values.push(hit_grey_value);
}
}
values
}
fn main() {
let test_case: Vec<[f32; 5]> = vec![
[0.0, 0.0, 1000.0, 2.0, 192.0],
[-2.0, 2.0, 100.0, 3.0, 98.0],
[2.0, 2.0, 100.0, 3.0, 98.0],
[0.5, -0.5, 500.0, 0.2, 255.0],
[-0.349469, -0.1, 20.000279, 0.6, 64.0],
[-0.349775, -0.100273, 20.000485, 0.6, 85.0],
[-0.350214, -0.100386, 20.000407, 0.6, 106.0],
[-0.350527, -0.100273, 20.000091, 0.6, 127.0],
[-0.350531, -0.1, 19.999721, 0.6, 148.0],
[-0.350225, -0.099727, 19.999515, 0.6, 169.0],
[-0.349786, -0.099614, 19.999593, 0.6, 190.0],
[-0.349473, -0.099727, 19.999909, 0.6, 211.0],
[0.20032, 0.349824, 15.000456, 0.3, 0.0],
[0.20032, 0.349517, 14.999924, 0.3, 0.0],
[0.19968, 0.350483, 15.000076, 0.3, 0.0],
[0.19968, 0.350176, 14.999544, 0.3, 0.0],
[0.200578, 0.350073, 14.999958, 0.3, 0.0],
[0.199817, 0.349519, 15.000277, 0.3, 0.0],
[0.200183, 0.350481, 14.999723, 0.3, 0.0],
[0.199422, 0.349927, 15.000042, 0.3, 0.0],
[0.200235, 0.35042, 15.000332, 0.3, 0.0],
[0.199765, 0.350078, 15.000529, 0.3, 0.0],
[0.200235, 0.349922, 14.999471, 0.3, 0.0],
[0.199765, 0.34958, 14.999668, 0.3, 0.0],
[0.200429, 0.350419, 15.000005, 0.3, 255.0],
[0.199571, 0.349795, 15.000366, 0.3, 255.0],
[0.200429, 0.350205, 14.999634, 0.3, 255.0],
[0.199571, 0.349581, 14.999995, 0.3, 255.0],
[0.200524, 0.349747, 15.000146, 0.3, 255.0],
[0.200196, 0.349509, 15.000283, 0.3, 255.0],
[0.199804, 0.350491, 14.999717, 0.3, 255.0],
[0.199476, 0.350253, 14.999854, 0.3, 255.0],
[0.200138, 0.349578, 14.999597, 0.3, 255.0],
[0.200138, 0.350138, 15.000567, 0.3, 255.0],
[0.199862, 0.349862, 14.999433, 0.3, 255.0],
[0.199862, 0.350422, 15.000403, 0.3, 255.0],
[0.200488, 0.350136, 15.000321, 0.3, 255.0],
[0.199957, 0.349751, 15.000544, 0.3, 255.0],
[0.200488, 0.34979, 14.999721, 0.3, 255.0],
[0.199957, 0.349404, 14.999944, 0.3, 255.0],
[0.200043, 0.350596, 15.000056, 0.3, 255.0],
[0.199512, 0.35021, 15.000279, 0.3, 255.0],
[0.200043, 0.350249, 14.999456, 0.3, 255.0],
[0.199512, 0.349864, 14.999679, 0.3, 255.0],
[-0.5, 0.500442, 10.000442, 0.2, 0.0],
[-0.5, 0.5, 10.0, 0.1995, 255.0],
[-0.499966, 0.49971, 9.999563, 0.199, 0.0],
[-0.500034, 0.499563, 9.99971, 0.199, 0.0],
[-0.499934, 0.499432, 9.999144, 0.1985, 255.0],
[-0.500066, 0.499144, 9.999432, 0.1985, 255.0],
[0.6, 0.5, 10.0, 0.25, 210.0],
[0.600122, 0.4999, 10.000122, 0.25, 210.0],
[0.600061, 0.49995, 10.000061, 0.25003, 50.0],
[0.595856, 0.505256, 9.991271, 0.24, 100.0],
[0.597973, 0.507461, 9.992175, 0.24, 210.0],
[0.595362, 0.507926, 9.993945, 0.24, 100.0],
[0.594618, 0.509055, 9.996832, 0.24, 210.0],
[0.592382, 0.506906, 9.996092, 0.24, 100.0],
[0.590438, 0.504882, 9.997604, 0.24, 210.0],
[0.591035, 0.503605, 9.994744, 0.24, 100.0],
[0.59121, 0.500708, 9.993425, 0.24, 210.0],
[0.593181, 0.502585, 9.991764, 0.24, 100.0],
[0.595867, 0.502302, 9.990069, 0.24, 210.0],
];
let values = grey_values(&test_case);
let path = std::path::Path::new("combination_scene.png");
let file = std::fs::File::create(path).unwrap();
let w = &mut std::io::BufWriter::new(file);
let mut encoder = png::Encoder::new(w, 1024, 1024);
encoder.set_color(png::ColorType::Grayscale);
encoder.set_depth(png::BitDepth::Eight);
let mut writer = encoder.write_header().unwrap();
writer.write_image_data(&values).unwrap();
}
```
## Test cases
The output images may be scaled down to fit this page. Click on any image to open it full size.
### Single sphere
Input:
```text
[
[0, 0, 0, 1, 128],
]
```
Output:
[![Grey circle on a black background](https://codegolf.codidact.com/uploads/jlru163pa2pgm7aaq70b2tff16uf)](https://codegolf.codidact.com/uploads/jlru163pa2pgm7aaq70b2tff16uf)
### Two separated spheres
Input:
```text
[
[0, -0.55, 0, 0.4, 100],
[0, 0.55, 0, 0.4, 200],
]
```
Output:
[![Light and dark grey circles on a black background](https://codegolf.codidact.com/uploads/8yw7agli78y65kiomb1u5yebs7c5)](https://codegolf.codidact.com/uploads/8yw7agli78y65kiomb1u5yebs7c5)
### Ring of spheres
Input:
```text
[
[0.75, 0, 0, 0.25, 64],
[0.53033, 0.375, -0.375, 0.25, 185],
[0, 0.53033, -0.53033, 0.25, 64],
[-0.53033, 0.375, -0.375, 0.25, 185],
[-0.75, 0, 0, 0.25, 64],
[-0.53033, -0.375, 0.375, 0.25, 185],
[0, -0.53033, 0.53033, 0.25, 64],
[0.53033, -0.375, 0.375, 0.25, 185],
]
```
Output:
[![Eight circles in alternating light and dark grey in a ring some overlapping](https://codegolf.codidact.com/uploads/0lsenynig27au0sryf3ruozt1uwh)](https://codegolf.codidact.com/uploads/0lsenynig27au0sryf3ruozt1uwh)
### Beach ball
Input:
```text
[
[0.001, 0, 0, 1, 64],
[0.000707, 0.000707, 0, 1, 85],
[0, 0.001, 0, 1, 106],
[-0.000707, 0.000707, 0, 1, 127],
[-0.001, 0, 0, 1, 148],
[-0.000707, -0.000707, 0, 1, 169],
[0, -0.001, 0, 1, 190],
[0.000707, -0.000707, 0, 1, 211],
]
```
Output:
[![A circle with eight sectors in different shades of grey](https://codegolf.codidact.com/uploads/51a3ni0cfwsqcdw5b49gy6282qno)](https://codegolf.codidact.com/uploads/51a3ni0cfwsqcdw5b49gy6282qno)
### Football
Input:
```text
[
[0, 0.000058, 0.001947, 1, 0],
[0, -0.001715, 0.000923, 1, 0],
[0, 0.001715, -0.000923, 1, 0],
[0, -0.000058, -0.001947, 1, 0],
[0.001657, -0.000512, 0.000887, 1, 0],
[-0.001657, -0.000512, 0.000887, 1, 0],
[0.001657, 0.000512, -0.000887, 1, 0],
[-0.001657, 0.000512, -0.000887, 1, 0],
[0.001024, 0.001435, 0.000829, 1, 0],
[-0.001024, 0.001435, 0.000829, 1, 0],
[0.001024, -0.001435, -0.000829, 1, 0],
[-0.001024, -0.001435, -0.000829, 1, 0],
[0.001868, 0.000618, 0.000357, 1, 255],
[-0.001868, 0.000618, 0.000357, 1, 255],
[0.001868, -0.000618, -0.000357, 1, 255],
[-0.001868, -0.000618, -0.000357, 1, 255],
[0.000714, -0.000934, 0.001618, 1, 255],
[-0.000714, -0.000934, 0.001618, 1, 255],
[0.000714, 0.000934, -0.001618, 1, 255],
[-0.000714, 0.000934, -0.001618, 1, 255],
[0, -0.001975, -0.000316, 1, 255],
[0, 0.001261, 0.001552, 1, 255],
[0, -0.001261, -0.001552, 1, 255],
[0, 0.001975, 0.000316, 1, 255],
[0.001155, 0.000423, 0.001577, 1, 255],
[-0.001155, 0.000423, 0.001577, 1, 255],
[0.001155, -0.001577, 0.000423, 1, 255],
[-0.001155, -0.001577, 0.000423, 1, 255],
[0.001155, 0.001577, -0.000423, 1, 255],
[-0.001155, 0.001577, -0.000423, 1, 255],
[0.001155, -0.000423, -0.001577, 1, 255],
[-0.001155, -0.000423, -0.001577, 1, 255],
]
```
Output:
[![Black pentagons on a white ball](https://codegolf.codidact.com/uploads/bmrbsw65p59imvwu8ssd42uvvx1r)](https://codegolf.codidact.com/uploads/bmrbsw65p59imvwu8ssd42uvvx1r)
### Eight ball
Input:
```text
[
[0, 0, 1000, 2, 128],
[0, 0, 0.003125, 1, 0],
[0, 0, 0, 0.9975, 255],
[0, 0.000547, -0.002573, 0.995, 0],
[0, -0.000547, -0.002573, 0.995, 0],
[0, 0.001071, -0.005037, 0.9925, 255],
[0, -0.001071, -0.005037, 0.9925, 255],
]
```
Output:
[![A ball with the number 8 in a white circle](https://codegolf.codidact.com/uploads/dvl68crpmy62rqt94mwanditt4wc)](https://codegolf.codidact.com/uploads/dvl68crpmy62rqt94mwanditt4wc)
### Star ball
Input:
```text
[
[0, 0, 0, 1, 210],
[0, 0, 0.0008, 1, 210],
[0, 0, 0.0004, 1.00012, 50],
[0.012969, 0, -0.042045, 0.96, 100],
[0.016397, 0.011913, -0.039054, 0.96, 210],
[0.004008, 0.012334, -0.042045, 0.96, 100],
[-0.006263, 0.019276, -0.039054, 0.96, 210],
[-0.010492, 0.007623, -0.042045, 0.96, 100],
[-0.020268, 0, -0.039054, 0.96, 210],
[-0.010492, -0.007623, -0.042045, 0.96, 100],
[-0.006263, -0.019276, -0.039054, 0.96, 210],
[0.004008, -0.012334, -0.042045, 0.96, 100],
[0.016397, -0.011913, -0.039054, 0.96, 210],
]
```
Output:
[![A light grey ball with a dark grey five pointed star](https://codegolf.codidact.com/uploads/f3wdgwru6j35ue72x25mazxikfx2)](https://codegolf.codidact.com/uploads/f3wdgwru6j35ue72x25mazxikfx2)
### Combination scene
Input:
```text
[
[0, 0, 1000, 2, 192],
[-2, 2, 100, 3, 98],
[2, 2, 100, 3, 98],
[0.5, -0.5, 500, 0.2, 255],
[-0.349469, -0.1, 20.000279, 0.6, 64],
[-0.349775, -0.100273, 20.000485, 0.6, 85],
[-0.350214, -0.100386, 20.000407, 0.6, 106],
[-0.350527, -0.100273, 20.000091, 0.6, 127],
[-0.350531, -0.1, 19.999721, 0.6, 148],
[-0.350225, -0.099727, 19.999515, 0.6, 169],
[-0.349786, -0.099614, 19.999593, 0.6, 190],
[-0.349473, -0.099727, 19.999909, 0.6, 211],
[0.20032, 0.349824, 15.000456, 0.3, 0],
[0.20032, 0.349517, 14.999924, 0.3, 0],
[0.19968, 0.350483, 15.000076, 0.3, 0],
[0.19968, 0.350176, 14.999544, 0.3, 0],
[0.200578, 0.350073, 14.999958, 0.3, 0],
[0.199817, 0.349519, 15.000277, 0.3, 0],
[0.200183, 0.350481, 14.999723, 0.3, 0],
[0.199422, 0.349927, 15.000042, 0.3, 0],
[0.200235, 0.35042, 15.000332, 0.3, 0],
[0.199765, 0.350078, 15.000529, 0.3, 0],
[0.200235, 0.349922, 14.999471, 0.3, 0],
[0.199765, 0.34958, 14.999668, 0.3, 0],
[0.200429, 0.350419, 15.000005, 0.3, 255],
[0.199571, 0.349795, 15.000366, 0.3, 255],
[0.200429, 0.350205, 14.999634, 0.3, 255],
[0.199571, 0.349581, 14.999995, 0.3, 255],
[0.200524, 0.349747, 15.000146, 0.3, 255],
[0.200196, 0.349509, 15.000283, 0.3, 255],
[0.199804, 0.350491, 14.999717, 0.3, 255],
[0.199476, 0.350253, 14.999854, 0.3, 255],
[0.200138, 0.349578, 14.999597, 0.3, 255],
[0.200138, 0.350138, 15.000567, 0.3, 255],
[0.199862, 0.349862, 14.999433, 0.3, 255],
[0.199862, 0.350422, 15.000403, 0.3, 255],
[0.200488, 0.350136, 15.000321, 0.3, 255],
[0.199957, 0.349751, 15.000544, 0.3, 255],
[0.200488, 0.34979, 14.999721, 0.3, 255],
[0.199957, 0.349404, 14.999944, 0.3, 255],
[0.200043, 0.350596, 15.000056, 0.3, 255],
[0.199512, 0.35021, 15.000279, 0.3, 255],
[0.200043, 0.350249, 14.999456, 0.3, 255],
[0.199512, 0.349864, 14.999679, 0.3, 255],
[-0.5, 0.500442, 10.000442, 0.2, 0],
[-0.5, 0.5, 10, 0.1995, 255],
[-0.499966, 0.49971, 9.999563, 0.199, 0],
[-0.500034, 0.499563, 9.99971, 0.199, 0],
[-0.499934, 0.499432, 9.999144, 0.1985, 255],
[-0.500066, 0.499144, 9.999432, 0.1985, 255],
[0.6, 0.5, 10, 0.25, 210],
[0.600122, 0.4999, 10.000122, 0.25, 210],
[0.600061, 0.49995, 10.000061, 0.25003, 50],
[0.595856, 0.505256, 9.991271, 0.24, 100],
[0.597973, 0.507461, 9.992175, 0.24, 210],
[0.595362, 0.507926, 9.993945, 0.24, 100],
[0.594618, 0.509055, 9.996832, 0.24, 210],
[0.592382, 0.506906, 9.996092, 0.24, 100],
[0.590438, 0.504882, 9.997604, 0.24, 210],
[0.591035, 0.503605, 9.994744, 0.24, 100],
[0.59121, 0.500708, 9.993425, 0.24, 210],
[0.593181, 0.502585, 9.991764, 0.24, 100],
[0.595867, 0.502302, 9.990069, 0.24, 210],
]
```
Output:
[![A beachball, football, eightball, and starball with the sun behind hills in the background](https://codegolf.codidact.com/uploads/usv9npm1t5bjpcaknn6rbzgkqwmi)](https://codegolf.codidact.com/uploads/usv9npm1t5bjpcaknn6rbzgkqwmi)
## Accuracy
The outputs for the test cases were generated using 64 bit floating point numbers. The following is the last test case using 32 bit floating point rather than 64 bit floating point:
[![A beachball, football, eightball, and starball with the sun behind hills in the background](https://codegolf.codidact.com/uploads/7uuxjxq7a9oxc9q0jf2bi8l68qk3)](https://codegolf.codidact.com/uploads/7uuxjxq7a9oxc9q0jf2bi8l68qk3)
The inaccuracies in the 32 bit floating point version are subtle. You may need to click on the image to open it full size to see the occasional inaccurate pixel. For example, around the inside of the holes in the 8 on the eightball. These occasional inaccurate pixels are acceptable in your output (allowing languages which only support 32 bit floating point to compete).
More widespread speckling as in the following image would not be valid, so you may need to be careful if changing the order of calculations during golfing:
[![A scene with significant areas of inaccurate speckles obscuring the intended image](https://codegolf.codidact.com/uploads/1sa7i6z0cv2kgyrmw2tj2ytr1ps1)](https://codegolf.codidact.com/uploads/1sa7i6z0cv2kgyrmw2tj2ytr1ps1)
This exaggerated inaccuracy was achieved by adding 1000 to and then subtracting 1000 from the distance calculation, exceeding the available precision of 32 bit floating point and causing some distances to appear in the wrong order. Note that this same approach causes no inaccuracy with 64 bit floating point, so if your language supports both you may have more leeway for rearranging calculations if you use 64 bit floating point.
## Explanatory animations
The example images were produced from plain grey spheres with no surface pattern or texture. The following animations show how they break down into their component plain spheres.
<details>
<summary>Animations</summary>
TODO
</details>
## Scoring
This is a [code golf challenge](https://codegolf.codidact.com/categories/49/tags/4274). Your score is the number of bytes in your code. Lowest score for each language wins.
> Explanations are optional, but I'm more likely to upvote answers that have one.

Given a list of spheres, output an image.
Produce a greyscale [orthographic](https://en.wikipedia.org/wiki/Parallel_projection#Orthographic_projection) (parallel projection - no perspective) ray caster, that renders only plain grey spheres with no lighting.
## Input
- A list of spheres, each consisting of:
- The coordinates of its centre $(x,y,z)$.
- Its radius $r$.
- Its greyscale value $g$, an integer from $0$ to $255$ (both inclusive).
- Only the greyscale value will be restricted to integer values. The coordinates and radius may have fractional values.
## Output
- An image $1024$ by $1024$ pixels with greyscale values from $0$ to $255$.
- The image covers x and y values from $-1$ to $1$.
- A sphere's greyscale value is applied to every visible part of it. There is no variation over the surface of the sphere, so it appears flat like a disk.
- You may output whatever image format is convenient, including simple text formats such as [PGM](https://en.wikipedia.org/wiki/Netpbm#PGM_example) or a list of pixel greyscale values. If you want to include an example of your output as an image in your answer, the image formats that Codidact supports are PNG, JPEG, and GIF. You could output PGM and then convert to PNG to upload to your answer. If you output just the greyscale values as a list of numbers from $0$ to $255$, I've made a `[ TODO page to convert these to greyscale PNG]`.
- You are free to output an image reflected or rotated by a multiple of 90 degrees. However, the z axis is required to be increasing in the forwards direction (into the image) otherwise some of the test cases will look incorrect.
## Calculating a pixel value
Imagine a ray passing into the image through a pixel, perpendicular to the image plane. If it hits a sphere, it shows the greyscale value of the first sphere it hits. Otherwise, it shows a value of zero (black).
For example, for each pixel:
- The pixel $x,y$ coordinates will each be from $0$ to $1023$. Scale these to the range $-1$ to $1$ (for example, divide by $512$ and subtract $1$).
- If these scaled $x,y$ coordinates are within the radius $r$ of the $x,y$ coordinates of a sphere's centre, the ray intersects that sphere (the $z$ coordinate is not relevant to this test because the image is an orthographic projection - the rays are all parallel to the $z$ axis). The distance $a$ of the ray from the sphere's centre can be calculated as follows:
$$a=\sqrt{(x_{ray}-x_{sphere})^2+(y_{ray}-y_{sphere})^2}$$
```text
CONSIDER A DIAGRAM HERE
```
- For each sphere that the ray intersects, find the $z$ coordinate of the nearest intersection (there will often be $2$ per sphere, $1$ on the front of the sphere and $1$ on the back of the sphere, as the ray passes through - only the nearest is relevant). This can be calculated as follows:
- The distance $d$ to the nearest intersection point can be calculated from the $z$ coordinate of the sphere's centre, the radius $r$, and the distance $a$ from the $x,y$ coordinates of the sphere's centre to the scaled $x,y$ coordinates of the pixel:
$$d=z_{sphere}-\sqrt{r_{sphere}^2-a^2}$$
```text
CONSIDER A DIAGRAM HERE
```
- Of all the spheres that the ray intersects, the sphere with the intersection point with the smallest $z$ coordinate $d$ is the one to display. Set the pixel to the greyscale value of this sphere.
- If the ray did not intersect any spheres, set the pixel to zero (black).
## Example implementations
These are ungolfed code to supplement understanding of the requirements. You are free to golf these or implement your own approach that matches the test cases up to reflection & rotation.
The Python implementation uses 64 bit floating point. The Rust implementation uses 32 bit floating point to show the difference in output. I've tried to optimise the test cases for compatibility with 32 bit floating point, so the differences are restricted to being a pixel out in some places.
<details>
<summary>Python</summary>
The function `grey_values` would be sufficient as an entry to this challenge. The rest of the code saves the output as a PNG file. The function itself has no dependence on the `import png` so it would not be required in an entry.
Instructions for installing the `png` package can be found on its [project page on PyPI](https://pypi.org/project/pypng/).
```python
def grey_values(spheres):
values = []
for y_pixel in range(1024):
for x_pixel in range(1024):
x_ray, y_ray = x_pixel / 512 - 1, y_pixel / 512 - 1
min_distance, hit_grey_value = None, 0
for sphere in spheres:
x_sphere, y_sphere, z_sphere, radius, grey = sphere
a = (
(x_ray - x_sphere) ** 2 + (y_ray - y_sphere) ** 2
) ** 0.5
if a < radius:
distance = z_sphere - (radius ** 2 - a ** 2) ** 0.5
if min_distance is None or distance < min_distance:
min_distance = distance
hit_grey_value = grey
values.append(hit_grey_value)
return values
import png
test_case = [
[0, 0, 1000, 2, 192],
[-2, 2, 100, 3, 98],
[2, 2, 100, 3, 98],
[0.5, -0.5, 500, 0.2, 255],
[-0.349469, -0.1, 20.000279, 0.6, 64],
[-0.349775, -0.100273, 20.000485, 0.6, 85],
[-0.350214, -0.100386, 20.000407, 0.6, 106],
[-0.350527, -0.100273, 20.000091, 0.6, 127],
[-0.350531, -0.1, 19.999721, 0.6, 148],
[-0.350225, -0.099727, 19.999515, 0.6, 169],
[-0.349786, -0.099614, 19.999593, 0.6, 190],
[-0.349473, -0.099727, 19.999909, 0.6, 211],
[0.20032, 0.349824, 15.000456, 0.3, 0],
[0.20032, 0.349517, 14.999924, 0.3, 0],
[0.19968, 0.350483, 15.000076, 0.3, 0],
[0.19968, 0.350176, 14.999544, 0.3, 0],
[0.200578, 0.350073, 14.999958, 0.3, 0],
[0.199817, 0.349519, 15.000277, 0.3, 0],
[0.200183, 0.350481, 14.999723, 0.3, 0],
[0.199422, 0.349927, 15.000042, 0.3, 0],
[0.200235, 0.35042, 15.000332, 0.3, 0],
[0.199765, 0.350078, 15.000529, 0.3, 0],
[0.200235, 0.349922, 14.999471, 0.3, 0],
[0.199765, 0.34958, 14.999668, 0.3, 0],
[0.200429, 0.350419, 15.000005, 0.3, 255],
[0.199571, 0.349795, 15.000366, 0.3, 255],
[0.200429, 0.350205, 14.999634, 0.3, 255],
[0.199571, 0.349581, 14.999995, 0.3, 255],
[0.200524, 0.349747, 15.000146, 0.3, 255],
[0.200196, 0.349509, 15.000283, 0.3, 255],
[0.199804, 0.350491, 14.999717, 0.3, 255],
[0.199476, 0.350253, 14.999854, 0.3, 255],
[0.200138, 0.349578, 14.999597, 0.3, 255],
[0.200138, 0.350138, 15.000567, 0.3, 255],
[0.199862, 0.349862, 14.999433, 0.3, 255],
[0.199862, 0.350422, 15.000403, 0.3, 255],
[0.200488, 0.350136, 15.000321, 0.3, 255],
[0.199957, 0.349751, 15.000544, 0.3, 255],
[0.200488, 0.34979, 14.999721, 0.3, 255],
[0.199957, 0.349404, 14.999944, 0.3, 255],
[0.200043, 0.350596, 15.000056, 0.3, 255],
[0.199512, 0.35021, 15.000279, 0.3, 255],
[0.200043, 0.350249, 14.999456, 0.3, 255],
[0.199512, 0.349864, 14.999679, 0.3, 255],
[-0.5, 0.500442, 10.000442, 0.2, 0],
[-0.5, 0.5, 10, 0.1995, 255],
[-0.499966, 0.49971, 9.999563, 0.199, 0],
[-0.500034, 0.499563, 9.99971, 0.199, 0],
[-0.499934, 0.499432, 9.999144, 0.1985, 255],
[-0.500066, 0.499144, 9.999432, 0.1985, 255],
[0.6, 0.5, 10, 0.25, 210],
[0.600122, 0.4999, 10.000122, 0.25, 210],
[0.600061, 0.49995, 10.000061, 0.25003, 50],
[0.595856, 0.505256, 9.991271, 0.24, 100],
[0.597973, 0.507461, 9.992175, 0.24, 210],
[0.595362, 0.507926, 9.993945, 0.24, 100],
[0.594618, 0.509055, 9.996832, 0.24, 210],
[0.592382, 0.506906, 9.996092, 0.24, 100],
[0.590438, 0.504882, 9.997604, 0.24, 210],
[0.591035, 0.503605, 9.994744, 0.24, 100],
[0.59121, 0.500708, 9.993425, 0.24, 210],
[0.593181, 0.502585, 9.991764, 0.24, 100],
[0.595867, 0.502302, 9.990069, 0.24, 210],
]
values = grey_values(test_case)
rows = [values[row * 1024:row * 1024 + 1024] for row in range(1024)]
with open('combination_scene.png', 'wb') as file:
w = png.Writer(1024, 1024, greyscale=True)
w.write(file, rows)
```
</details>
<details>
<summary>Rust</summary>
The function `grey_values` would be sufficient as an entry to this challenge. The rest of the code saves the output as a PNG file.
The `png` crate can be installed using `cargo add png`. The `png` crate is not required for the `grey_values` function, only for saving as a PNG file.
```rust
fn grey_values(spheres: &Vec<[f32; 5]>) -> Vec<u8> {
let mut values = vec![];
for y_pixel in 0..1024 {
for x_pixel in 0..1024 {
let x_ray = x_pixel as f32 / 512.0 - 1.0;
let y_ray = y_pixel as f32 / 512.0 - 1.0;
let mut min_distance = f32::INFINITY;
let mut hit_grey_value = 0;
for sphere in spheres {
let [
x_sphere, y_sphere, z_sphere, radius, grey
] = *sphere;
let a = (
(x_ray - x_sphere).powf(2.0)
+ (y_ray - y_sphere).powf(2.0)
).sqrt();
if a < radius {
let distance = z_sphere - (
radius.powf(2.0) - a.powf(2.0)
).sqrt();
if distance < min_distance {
min_distance = distance;
hit_grey_value = grey as u8;
}
}
}
values.push(hit_grey_value);
}
}
values
}
fn main() {
let test_case: Vec<[f32; 5]> = vec![
[0.0, 0.0, 1000.0, 2.0, 192.0],
[-2.0, 2.0, 100.0, 3.0, 98.0],
[2.0, 2.0, 100.0, 3.0, 98.0],
[0.5, -0.5, 500.0, 0.2, 255.0],
[-0.349469, -0.1, 20.000279, 0.6, 64.0],
[-0.349775, -0.100273, 20.000485, 0.6, 85.0],
[-0.350214, -0.100386, 20.000407, 0.6, 106.0],
[-0.350527, -0.100273, 20.000091, 0.6, 127.0],
[-0.350531, -0.1, 19.999721, 0.6, 148.0],
[-0.350225, -0.099727, 19.999515, 0.6, 169.0],
[-0.349786, -0.099614, 19.999593, 0.6, 190.0],
[-0.349473, -0.099727, 19.999909, 0.6, 211.0],
[0.20032, 0.349824, 15.000456, 0.3, 0.0],
[0.20032, 0.349517, 14.999924, 0.3, 0.0],
[0.19968, 0.350483, 15.000076, 0.3, 0.0],
[0.19968, 0.350176, 14.999544, 0.3, 0.0],
[0.200578, 0.350073, 14.999958, 0.3, 0.0],
[0.199817, 0.349519, 15.000277, 0.3, 0.0],
[0.200183, 0.350481, 14.999723, 0.3, 0.0],
[0.199422, 0.349927, 15.000042, 0.3, 0.0],
[0.200235, 0.35042, 15.000332, 0.3, 0.0],
[0.199765, 0.350078, 15.000529, 0.3, 0.0],
[0.200235, 0.349922, 14.999471, 0.3, 0.0],
[0.199765, 0.34958, 14.999668, 0.3, 0.0],
[0.200429, 0.350419, 15.000005, 0.3, 255.0],
[0.199571, 0.349795, 15.000366, 0.3, 255.0],
[0.200429, 0.350205, 14.999634, 0.3, 255.0],
[0.199571, 0.349581, 14.999995, 0.3, 255.0],
[0.200524, 0.349747, 15.000146, 0.3, 255.0],
[0.200196, 0.349509, 15.000283, 0.3, 255.0],
[0.199804, 0.350491, 14.999717, 0.3, 255.0],
[0.199476, 0.350253, 14.999854, 0.3, 255.0],
[0.200138, 0.349578, 14.999597, 0.3, 255.0],
[0.200138, 0.350138, 15.000567, 0.3, 255.0],
[0.199862, 0.349862, 14.999433, 0.3, 255.0],
[0.199862, 0.350422, 15.000403, 0.3, 255.0],
[0.200488, 0.350136, 15.000321, 0.3, 255.0],
[0.199957, 0.349751, 15.000544, 0.3, 255.0],
[0.200488, 0.34979, 14.999721, 0.3, 255.0],
[0.199957, 0.349404, 14.999944, 0.3, 255.0],
[0.200043, 0.350596, 15.000056, 0.3, 255.0],
[0.199512, 0.35021, 15.000279, 0.3, 255.0],
[0.200043, 0.350249, 14.999456, 0.3, 255.0],
[0.199512, 0.349864, 14.999679, 0.3, 255.0],
[-0.5, 0.500442, 10.000442, 0.2, 0.0],
[-0.5, 0.5, 10.0, 0.1995, 255.0],
[-0.499966, 0.49971, 9.999563, 0.199, 0.0],
[-0.500034, 0.499563, 9.99971, 0.199, 0.0],
[-0.499934, 0.499432, 9.999144, 0.1985, 255.0],
[-0.500066, 0.499144, 9.999432, 0.1985, 255.0],
[0.6, 0.5, 10.0, 0.25, 210.0],
[0.600122, 0.4999, 10.000122, 0.25, 210.0],
[0.600061, 0.49995, 10.000061, 0.25003, 50.0],
[0.595856, 0.505256, 9.991271, 0.24, 100.0],
[0.597973, 0.507461, 9.992175, 0.24, 210.0],
[0.595362, 0.507926, 9.993945, 0.24, 100.0],
[0.594618, 0.509055, 9.996832, 0.24, 210.0],
[0.592382, 0.506906, 9.996092, 0.24, 100.0],
[0.590438, 0.504882, 9.997604, 0.24, 210.0],
[0.591035, 0.503605, 9.994744, 0.24, 100.0],
[0.59121, 0.500708, 9.993425, 0.24, 210.0],
[0.593181, 0.502585, 9.991764, 0.24, 100.0],
[0.595867, 0.502302, 9.990069, 0.24, 210.0],
];
let values = grey_values(&test_case);
let path = std::path::Path::new("combination_scene.png");
let file = std::fs::File::create(path).unwrap();
let w = &mut std::io::BufWriter::new(file);
let mut encoder = png::Encoder::new(w, 1024, 1024);
encoder.set_color(png::ColorType::Grayscale);
encoder.set_depth(png::BitDepth::Eight);
let mut writer = encoder.write_header().unwrap();
writer.write_image_data(&values).unwrap();
}
```
</details>
## Test cases
The output images may be scaled down to fit this page. Click on any image to open it full size.
### Single sphere
Input:
```text
[
[0, 0, 0, 1, 128],
]
```
Output:
[![Grey circle on a black background](https://codegolf.codidact.com/uploads/jlru163pa2pgm7aaq70b2tff16uf)](https://codegolf.codidact.com/uploads/jlru163pa2pgm7aaq70b2tff16uf)
### Two separated spheres
Input:
```text
[
[0, -0.55, 0, 0.4, 100],
[0, 0.55, 0, 0.4, 200],
]
```
Output:
[![Light and dark grey circles on a black background](https://codegolf.codidact.com/uploads/8yw7agli78y65kiomb1u5yebs7c5)](https://codegolf.codidact.com/uploads/8yw7agli78y65kiomb1u5yebs7c5)
### Ring of spheres
Input:
```text
[
[0.75, 0, 0, 0.25, 64],
[0.53033, 0.375, -0.375, 0.25, 185],
[0, 0.53033, -0.53033, 0.25, 64],
[-0.53033, 0.375, -0.375, 0.25, 185],
[-0.75, 0, 0, 0.25, 64],
[-0.53033, -0.375, 0.375, 0.25, 185],
[0, -0.53033, 0.53033, 0.25, 64],
[0.53033, -0.375, 0.375, 0.25, 185],
]
```
Output:
[![Eight circles in alternating light and dark grey in a ring some overlapping](https://codegolf.codidact.com/uploads/0lsenynig27au0sryf3ruozt1uwh)](https://codegolf.codidact.com/uploads/0lsenynig27au0sryf3ruozt1uwh)
### Beach ball
Input:
```text
[
[0.001, 0, 0, 1, 64],
[0.000707, 0.000707, 0, 1, 85],
[0, 0.001, 0, 1, 106],
[-0.000707, 0.000707, 0, 1, 127],
[-0.001, 0, 0, 1, 148],
[-0.000707, -0.000707, 0, 1, 169],
[0, -0.001, 0, 1, 190],
[0.000707, -0.000707, 0, 1, 211],
]
```
Output:
[![A circle with eight sectors in different shades of grey](https://codegolf.codidact.com/uploads/51a3ni0cfwsqcdw5b49gy6282qno)](https://codegolf.codidact.com/uploads/51a3ni0cfwsqcdw5b49gy6282qno)
### Football
Input:
```text
[
[0, 0.000058, 0.001947, 1, 0],
[0, -0.001715, 0.000923, 1, 0],
[0, 0.001715, -0.000923, 1, 0],
[0, -0.000058, -0.001947, 1, 0],
[0.001657, -0.000512, 0.000887, 1, 0],
[-0.001657, -0.000512, 0.000887, 1, 0],
[0.001657, 0.000512, -0.000887, 1, 0],
[-0.001657, 0.000512, -0.000887, 1, 0],
[0.001024, 0.001435, 0.000829, 1, 0],
[-0.001024, 0.001435, 0.000829, 1, 0],
[0.001024, -0.001435, -0.000829, 1, 0],
[-0.001024, -0.001435, -0.000829, 1, 0],
[0.001868, 0.000618, 0.000357, 1, 255],
[-0.001868, 0.000618, 0.000357, 1, 255],
[0.001868, -0.000618, -0.000357, 1, 255],
[-0.001868, -0.000618, -0.000357, 1, 255],
[0.000714, -0.000934, 0.001618, 1, 255],
[-0.000714, -0.000934, 0.001618, 1, 255],
[0.000714, 0.000934, -0.001618, 1, 255],
[-0.000714, 0.000934, -0.001618, 1, 255],
[0, -0.001975, -0.000316, 1, 255],
[0, 0.001261, 0.001552, 1, 255],
[0, -0.001261, -0.001552, 1, 255],
[0, 0.001975, 0.000316, 1, 255],
[0.001155, 0.000423, 0.001577, 1, 255],
[-0.001155, 0.000423, 0.001577, 1, 255],
[0.001155, -0.001577, 0.000423, 1, 255],
[-0.001155, -0.001577, 0.000423, 1, 255],
[0.001155, 0.001577, -0.000423, 1, 255],
[-0.001155, 0.001577, -0.000423, 1, 255],
[0.001155, -0.000423, -0.001577, 1, 255],
[-0.001155, -0.000423, -0.001577, 1, 255],
]
```
Output:
[![Black pentagons on a white ball](https://codegolf.codidact.com/uploads/bmrbsw65p59imvwu8ssd42uvvx1r)](https://codegolf.codidact.com/uploads/bmrbsw65p59imvwu8ssd42uvvx1r)
### Eight ball
Input:
```text
[
[0, 0, 1000, 2, 128],
[0, 0, 0.003125, 1, 0],
[0, 0, 0, 0.9975, 255],
[0, 0.000547, -0.002573, 0.995, 0],
[0, -0.000547, -0.002573, 0.995, 0],
[0, 0.001071, -0.005037, 0.9925, 255],
[0, -0.001071, -0.005037, 0.9925, 255],
]
```
Output:
[![A ball with the number 8 in a white circle](https://codegolf.codidact.com/uploads/dvl68crpmy62rqt94mwanditt4wc)](https://codegolf.codidact.com/uploads/dvl68crpmy62rqt94mwanditt4wc)
### Star ball
Input:
```text
[
[0, 0, 0, 1, 210],
[0, 0, 0.0008, 1, 210],
[0, 0, 0.0004, 1.00012, 50],
[0.012969, 0, -0.042045, 0.96, 100],
[0.016397, 0.011913, -0.039054, 0.96, 210],
[0.004008, 0.012334, -0.042045, 0.96, 100],
[-0.006263, 0.019276, -0.039054, 0.96, 210],
[-0.010492, 0.007623, -0.042045, 0.96, 100],
[-0.020268, 0, -0.039054, 0.96, 210],
[-0.010492, -0.007623, -0.042045, 0.96, 100],
[-0.006263, -0.019276, -0.039054, 0.96, 210],
[0.004008, -0.012334, -0.042045, 0.96, 100],
[0.016397, -0.011913, -0.039054, 0.96, 210],
]
```
Output:
[![A light grey ball with a dark grey five pointed star](https://codegolf.codidact.com/uploads/f3wdgwru6j35ue72x25mazxikfx2)](https://codegolf.codidact.com/uploads/f3wdgwru6j35ue72x25mazxikfx2)
### Combination scene
Input:
```text
[
[0, 0, 1000, 2, 192],
[-2, 2, 100, 3, 98],
[2, 2, 100, 3, 98],
[0.5, -0.5, 500, 0.2, 255],
[-0.349469, -0.1, 20.000279, 0.6, 64],
[-0.349775, -0.100273, 20.000485, 0.6, 85],
[-0.350214, -0.100386, 20.000407, 0.6, 106],
[-0.350527, -0.100273, 20.000091, 0.6, 127],
[-0.350531, -0.1, 19.999721, 0.6, 148],
[-0.350225, -0.099727, 19.999515, 0.6, 169],
[-0.349786, -0.099614, 19.999593, 0.6, 190],
[-0.349473, -0.099727, 19.999909, 0.6, 211],
[0.20032, 0.349824, 15.000456, 0.3, 0],
[0.20032, 0.349517, 14.999924, 0.3, 0],
[0.19968, 0.350483, 15.000076, 0.3, 0],
[0.19968, 0.350176, 14.999544, 0.3, 0],
[0.200578, 0.350073, 14.999958, 0.3, 0],
[0.199817, 0.349519, 15.000277, 0.3, 0],
[0.200183, 0.350481, 14.999723, 0.3, 0],
[0.199422, 0.349927, 15.000042, 0.3, 0],
[0.200235, 0.35042, 15.000332, 0.3, 0],
[0.199765, 0.350078, 15.000529, 0.3, 0],
[0.200235, 0.349922, 14.999471, 0.3, 0],
[0.199765, 0.34958, 14.999668, 0.3, 0],
[0.200429, 0.350419, 15.000005, 0.3, 255],
[0.199571, 0.349795, 15.000366, 0.3, 255],
[0.200429, 0.350205, 14.999634, 0.3, 255],
[0.199571, 0.349581, 14.999995, 0.3, 255],
[0.200524, 0.349747, 15.000146, 0.3, 255],
[0.200196, 0.349509, 15.000283, 0.3, 255],
[0.199804, 0.350491, 14.999717, 0.3, 255],
[0.199476, 0.350253, 14.999854, 0.3, 255],
[0.200138, 0.349578, 14.999597, 0.3, 255],
[0.200138, 0.350138, 15.000567, 0.3, 255],
[0.199862, 0.349862, 14.999433, 0.3, 255],
[0.199862, 0.350422, 15.000403, 0.3, 255],
[0.200488, 0.350136, 15.000321, 0.3, 255],
[0.199957, 0.349751, 15.000544, 0.3, 255],
[0.200488, 0.34979, 14.999721, 0.3, 255],
[0.199957, 0.349404, 14.999944, 0.3, 255],
[0.200043, 0.350596, 15.000056, 0.3, 255],
[0.199512, 0.35021, 15.000279, 0.3, 255],
[0.200043, 0.350249, 14.999456, 0.3, 255],
[0.199512, 0.349864, 14.999679, 0.3, 255],
[-0.5, 0.500442, 10.000442, 0.2, 0],
[-0.5, 0.5, 10, 0.1995, 255],
[-0.499966, 0.49971, 9.999563, 0.199, 0],
[-0.500034, 0.499563, 9.99971, 0.199, 0],
[-0.499934, 0.499432, 9.999144, 0.1985, 255],
[-0.500066, 0.499144, 9.999432, 0.1985, 255],
[0.6, 0.5, 10, 0.25, 210],
[0.600122, 0.4999, 10.000122, 0.25, 210],
[0.600061, 0.49995, 10.000061, 0.25003, 50],
[0.595856, 0.505256, 9.991271, 0.24, 100],
[0.597973, 0.507461, 9.992175, 0.24, 210],
[0.595362, 0.507926, 9.993945, 0.24, 100],
[0.594618, 0.509055, 9.996832, 0.24, 210],
[0.592382, 0.506906, 9.996092, 0.24, 100],
[0.590438, 0.504882, 9.997604, 0.24, 210],
[0.591035, 0.503605, 9.994744, 0.24, 100],
[0.59121, 0.500708, 9.993425, 0.24, 210],
[0.593181, 0.502585, 9.991764, 0.24, 100],
[0.595867, 0.502302, 9.990069, 0.24, 210],
]
```
Output:
[![A beachball, football, eightball, and starball with the sun behind hills in the background](https://codegolf.codidact.com/uploads/usv9npm1t5bjpcaknn6rbzgkqwmi)](https://codegolf.codidact.com/uploads/usv9npm1t5bjpcaknn6rbzgkqwmi)
## Accuracy
The outputs for the test cases were generated using 64 bit floating point numbers. The following is the last test case using 32 bit floating point rather than 64 bit floating point:
[![A beachball, football, eightball, and starball with the sun behind hills in the background](https://codegolf.codidact.com/uploads/7uuxjxq7a9oxc9q0jf2bi8l68qk3)](https://codegolf.codidact.com/uploads/7uuxjxq7a9oxc9q0jf2bi8l68qk3)
The inaccuracies in the 32 bit floating point version are subtle. You may need to click on the image to open it full size to see the occasional inaccurate pixel. For example, around the inside of the holes in the 8 on the eightball. These occasional inaccurate pixels are acceptable in your output (allowing languages which only support 32 bit floating point to compete).
More widespread speckling as in the following image would not be valid, so you may need to be careful if changing the order of calculations during golfing:
[![A scene with significant areas of inaccurate speckles obscuring the intended image](https://codegolf.codidact.com/uploads/1sa7i6z0cv2kgyrmw2tj2ytr1ps1)](https://codegolf.codidact.com/uploads/1sa7i6z0cv2kgyrmw2tj2ytr1ps1)
This exaggerated inaccuracy was achieved by adding 1000 to and then subtracting 1000 from the distance calculation, exceeding the available precision of 32 bit floating point and causing some distances to appear in the wrong order. Note that this same approach causes no inaccuracy with 64 bit floating point, so if your language supports both you may have more leeway for rearranging calculations if you use 64 bit floating point.
## Explanatory animations
The example images were produced from plain grey spheres with no surface pattern or texture. The following animations show how they break down into their component plain spheres.
<details>
<summary>Animations</summary>
TODO
</details>
## Scoring
This is a [code golf challenge](https://codegolf.codidact.com/categories/49/tags/4274). Your score is the number of bytes in your code. Lowest score for each language wins.
> Explanations are optional, but I'm more likely to upvote answers that have one.

#16: Post edited by

trichoplax‭ · 2024-08-09T18:30:57Z (4 months ago)
Separate accuracy section from test cases section

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Raw

Markdown

Given a list of spheres, output an image.
Produce a greyscale [orthographic](https://en.wikipedia.org/wiki/Parallel_projection#Orthographic_projection) (parallel projection - no perspective) ray caster, that renders only plain grey spheres with no lighting.
## Input
- A list of spheres, each consisting of:
- The coordinates of its centre $(x,y,z)$.
- Its radius $r$.
- Its greyscale value $g$, an integer from $0$ to $255$ (both inclusive).
- Only the greyscale value will be restricted to integer values. The coordinates and radius may have fractional values.
## Output
- An image $1024$ by $1024$ pixels with greyscale values from $0$ to $255$.
- The image covers x and y values from $-1$ to $1$.
- A sphere's greyscale value is applied to every visible part of it. There is no variation over the surface of the sphere, so it appears flat like a disk.
- You may output whatever image format is convenient, including simple text formats such as [PGM](https://en.wikipedia.org/wiki/Netpbm#PGM_example) or a list of pixel greyscale values. If you want to include an example of your output as an image in your answer, the image formats that Codidact supports are PNG, JPEG, and GIF. You could output PGM and then convert to PNG to upload to your answer. If you output just the greyscale values as a list of numbers from $0$ to $255$, I've made a `[ TODO page to convert these to greyscale PNG]`.
- You are free to output an image reflected or rotated by a multiple of 90 degrees. However, the z axis is required to be increasing in the forwards direction (into the image) otherwise some of the test cases will look incorrect.
## Calculating a pixel value
Imagine a ray passing into the image through a pixel, perpendicular to the image plane. If it hits a sphere, it shows the greyscale value of the first sphere it hits. Otherwise, it shows a value of zero (black).
For example, for each pixel:
- The pixel $x,y$ coordinates will each be from $0$ to $1023$. Scale these to the range $-1$ to $1$ (for example, divide by $512$ and subtract $1$).
- If these scaled $x,y$ coordinates are within the radius $r$ of the $x,y$ coordinates of a sphere's centre, the ray intersects that sphere (the $z$ coordinate is not relevant to this test because the image is an orthographic projection - the rays are all parallel to the $z$ axis). The distance $a$ of the ray from the sphere's centre can be calculated as follows:
$$a=\sqrt{(x_{ray}-x_{sphere})^2+(y_{ray}-y_{sphere})^2}$$
```text
CONSIDER A DIAGRAM HERE
```
- For each sphere that the ray intersects, find the $z$ coordinate of the nearest intersection (there will often be $2$ per sphere, $1$ on the front of the sphere and $1$ on the back of the sphere, as the ray passes through - only the nearest is relevant). This can be calculated as follows:
- The distance $d$ to the nearest intersection point can be calculated from the $z$ coordinate of the sphere's centre, the radius $r$, and the distance $a$ from the $x,y$ coordinates of the sphere's centre to the scaled $x,y$ coordinates of the pixel:
$$d=z_{sphere}-\sqrt{r_{sphere}^2-a^2}$$
```text
CONSIDER A DIAGRAM HERE
```
- Of all the spheres that the ray intersects, the sphere with the intersection point with the smallest $z$ coordinate $d$ is the one to display. Set the pixel to the greyscale value of this sphere.
- If the ray did not intersect any spheres, set the pixel to zero (black).
## Example implementations
These are ungolfed code to supplement understanding of the requirements. You are free to golf these or implement your own approach that matches the test cases up to reflection & rotation.
The Python implementation uses 64 bit floating point. The Rust implementation uses 32 bit floating point to show the difference in output. I've tried to optimise the test cases for compatibility with 32 bit floating point, so the differences are restricted to being a pixel out in some places.
### Python
The function `grey_values` would be sufficient as an entry to this challenge. The rest of the code saves the output as a PNG file. The function itself has no dependence on the `import png` so it would not be required in an entry.
Instructions for installing the `png` package can be found on its [project page on PyPI](https://pypi.org/project/pypng/).
```python
def grey_values(spheres):
values = []
for y_pixel in range(1024):
for x_pixel in range(1024):
x_ray, y_ray = x_pixel / 512 - 1, y_pixel / 512 - 1
min_distance, hit_grey_value = None, 0
for sphere in spheres:
x_sphere, y_sphere, z_sphere, radius, grey = sphere
a = (
(x_ray - x_sphere) ** 2 + (y_ray - y_sphere) ** 2
) ** 0.5
if a < radius:
distance = z_sphere - (radius ** 2 - a ** 2) ** 0.5
if min_distance is None or distance < min_distance:
min_distance = distance
hit_grey_value = grey
values.append(hit_grey_value)
return values
import png
test_case = [
[0, 0, 1000, 2, 192],
[-2, 2, 100, 3, 98],
[2, 2, 100, 3, 98],
[0.5, -0.5, 500, 0.2, 255],
[-0.349469, -0.1, 20.000279, 0.6, 64],
[-0.349775, -0.100273, 20.000485, 0.6, 85],
[-0.350214, -0.100386, 20.000407, 0.6, 106],
[-0.350527, -0.100273, 20.000091, 0.6, 127],
[-0.350531, -0.1, 19.999721, 0.6, 148],
[-0.350225, -0.099727, 19.999515, 0.6, 169],
[-0.349786, -0.099614, 19.999593, 0.6, 190],
[-0.349473, -0.099727, 19.999909, 0.6, 211],
[0.20032, 0.349824, 15.000456, 0.3, 0],
[0.20032, 0.349517, 14.999924, 0.3, 0],
[0.19968, 0.350483, 15.000076, 0.3, 0],
[0.19968, 0.350176, 14.999544, 0.3, 0],
[0.200578, 0.350073, 14.999958, 0.3, 0],
[0.199817, 0.349519, 15.000277, 0.3, 0],
[0.200183, 0.350481, 14.999723, 0.3, 0],
[0.199422, 0.349927, 15.000042, 0.3, 0],
[0.200235, 0.35042, 15.000332, 0.3, 0],
[0.199765, 0.350078, 15.000529, 0.3, 0],
[0.200235, 0.349922, 14.999471, 0.3, 0],
[0.199765, 0.34958, 14.999668, 0.3, 0],
[0.200429, 0.350419, 15.000005, 0.3, 255],
[0.199571, 0.349795, 15.000366, 0.3, 255],
[0.200429, 0.350205, 14.999634, 0.3, 255],
[0.199571, 0.349581, 14.999995, 0.3, 255],
[0.200524, 0.349747, 15.000146, 0.3, 255],
[0.200196, 0.349509, 15.000283, 0.3, 255],
[0.199804, 0.350491, 14.999717, 0.3, 255],
[0.199476, 0.350253, 14.999854, 0.3, 255],
[0.200138, 0.349578, 14.999597, 0.3, 255],
[0.200138, 0.350138, 15.000567, 0.3, 255],
[0.199862, 0.349862, 14.999433, 0.3, 255],
[0.199862, 0.350422, 15.000403, 0.3, 255],
[0.200488, 0.350136, 15.000321, 0.3, 255],
[0.199957, 0.349751, 15.000544, 0.3, 255],
[0.200488, 0.34979, 14.999721, 0.3, 255],
[0.199957, 0.349404, 14.999944, 0.3, 255],
[0.200043, 0.350596, 15.000056, 0.3, 255],
[0.199512, 0.35021, 15.000279, 0.3, 255],
[0.200043, 0.350249, 14.999456, 0.3, 255],
[0.199512, 0.349864, 14.999679, 0.3, 255],
[-0.5, 0.500442, 10.000442, 0.2, 0],
[-0.5, 0.5, 10, 0.1995, 255],
[-0.499966, 0.49971, 9.999563, 0.199, 0],
[-0.500034, 0.499563, 9.99971, 0.199, 0],
[-0.499934, 0.499432, 9.999144, 0.1985, 255],
[-0.500066, 0.499144, 9.999432, 0.1985, 255],
[0.6, 0.5, 10, 0.25, 210],
[0.600122, 0.4999, 10.000122, 0.25, 210],
[0.600061, 0.49995, 10.000061, 0.25003, 50],
[0.595856, 0.505256, 9.991271, 0.24, 100],
[0.597973, 0.507461, 9.992175, 0.24, 210],
[0.595362, 0.507926, 9.993945, 0.24, 100],
[0.594618, 0.509055, 9.996832, 0.24, 210],
[0.592382, 0.506906, 9.996092, 0.24, 100],
[0.590438, 0.504882, 9.997604, 0.24, 210],
[0.591035, 0.503605, 9.994744, 0.24, 100],
[0.59121, 0.500708, 9.993425, 0.24, 210],
[0.593181, 0.502585, 9.991764, 0.24, 100],
[0.595867, 0.502302, 9.990069, 0.24, 210],
]
values = grey_values(test_case)
rows = [values[row * 1024:row * 1024 + 1024] for row in range(1024)]
with open('combination_scene.png', 'wb') as file:
w = png.Writer(1024, 1024, greyscale=True)
w.write(file, rows)
```
### Rust
The function `grey_values` would be sufficient as an entry to this challenge. The rest of the code saves the output as a PNG file.
The `png` crate can be installed using `cargo add png`. The `png` crate is not required for the `grey_values` function, only for saving as a PNG file.
```rust
fn grey_values(spheres: &Vec<[f32; 5]>) -> Vec<u8> {
let mut values = vec![];
for y_pixel in 0..1024 {
for x_pixel in 0..1024 {
let x_ray = x_pixel as f32 / 512.0 - 1.0;
let y_ray = y_pixel as f32 / 512.0 - 1.0;
let mut min_distance = f32::INFINITY;
let mut hit_grey_value = 0;
for sphere in spheres {
let [
x_sphere, y_sphere, z_sphere, radius, grey
] = *sphere;
let a = (
(x_ray - x_sphere).powf(2.0)
+ (y_ray - y_sphere).powf(2.0)
).sqrt();
if a < radius {
let distance = z_sphere - (
radius.powf(2.0) - a.powf(2.0)
).sqrt();
if distance < min_distance {
min_distance = distance;
hit_grey_value = grey as u8;
}
}
}
values.push(hit_grey_value);
}
}
values
}
fn main() {
let test_case: Vec<[f32; 5]> = vec![
[0.0, 0.0, 1000.0, 2.0, 192.0],
[-2.0, 2.0, 100.0, 3.0, 98.0],
[2.0, 2.0, 100.0, 3.0, 98.0],
[0.5, -0.5, 500.0, 0.2, 255.0],
[-0.349469, -0.1, 20.000279, 0.6, 64.0],
[-0.349775, -0.100273, 20.000485, 0.6, 85.0],
[-0.350214, -0.100386, 20.000407, 0.6, 106.0],
[-0.350527, -0.100273, 20.000091, 0.6, 127.0],
[-0.350531, -0.1, 19.999721, 0.6, 148.0],
[-0.350225, -0.099727, 19.999515, 0.6, 169.0],
[-0.349786, -0.099614, 19.999593, 0.6, 190.0],
[-0.349473, -0.099727, 19.999909, 0.6, 211.0],
[0.20032, 0.349824, 15.000456, 0.3, 0.0],
[0.20032, 0.349517, 14.999924, 0.3, 0.0],
[0.19968, 0.350483, 15.000076, 0.3, 0.0],
[0.19968, 0.350176, 14.999544, 0.3, 0.0],
[0.200578, 0.350073, 14.999958, 0.3, 0.0],
[0.199817, 0.349519, 15.000277, 0.3, 0.0],
[0.200183, 0.350481, 14.999723, 0.3, 0.0],
[0.199422, 0.349927, 15.000042, 0.3, 0.0],
[0.200235, 0.35042, 15.000332, 0.3, 0.0],
[0.199765, 0.350078, 15.000529, 0.3, 0.0],
[0.200235, 0.349922, 14.999471, 0.3, 0.0],
[0.199765, 0.34958, 14.999668, 0.3, 0.0],
[0.200429, 0.350419, 15.000005, 0.3, 255.0],
[0.199571, 0.349795, 15.000366, 0.3, 255.0],
[0.200429, 0.350205, 14.999634, 0.3, 255.0],
[0.199571, 0.349581, 14.999995, 0.3, 255.0],
[0.200524, 0.349747, 15.000146, 0.3, 255.0],
[0.200196, 0.349509, 15.000283, 0.3, 255.0],
[0.199804, 0.350491, 14.999717, 0.3, 255.0],
[0.199476, 0.350253, 14.999854, 0.3, 255.0],
[0.200138, 0.349578, 14.999597, 0.3, 255.0],
[0.200138, 0.350138, 15.000567, 0.3, 255.0],
[0.199862, 0.349862, 14.999433, 0.3, 255.0],
[0.199862, 0.350422, 15.000403, 0.3, 255.0],
[0.200488, 0.350136, 15.000321, 0.3, 255.0],
[0.199957, 0.349751, 15.000544, 0.3, 255.0],
[0.200488, 0.34979, 14.999721, 0.3, 255.0],
[0.199957, 0.349404, 14.999944, 0.3, 255.0],
[0.200043, 0.350596, 15.000056, 0.3, 255.0],
[0.199512, 0.35021, 15.000279, 0.3, 255.0],
[0.200043, 0.350249, 14.999456, 0.3, 255.0],
[0.199512, 0.349864, 14.999679, 0.3, 255.0],
[-0.5, 0.500442, 10.000442, 0.2, 0.0],
[-0.5, 0.5, 10.0, 0.1995, 255.0],
[-0.499966, 0.49971, 9.999563, 0.199, 0.0],
[-0.500034, 0.499563, 9.99971, 0.199, 0.0],
[-0.499934, 0.499432, 9.999144, 0.1985, 255.0],
[-0.500066, 0.499144, 9.999432, 0.1985, 255.0],
[0.6, 0.5, 10.0, 0.25, 210.0],
[0.600122, 0.4999, 10.000122, 0.25, 210.0],
[0.600061, 0.49995, 10.000061, 0.25003, 50.0],
[0.595856, 0.505256, 9.991271, 0.24, 100.0],
[0.597973, 0.507461, 9.992175, 0.24, 210.0],
[0.595362, 0.507926, 9.993945, 0.24, 100.0],
[0.594618, 0.509055, 9.996832, 0.24, 210.0],
[0.592382, 0.506906, 9.996092, 0.24, 100.0],
[0.590438, 0.504882, 9.997604, 0.24, 210.0],
[0.591035, 0.503605, 9.994744, 0.24, 100.0],
[0.59121, 0.500708, 9.993425, 0.24, 210.0],
[0.593181, 0.502585, 9.991764, 0.24, 100.0],
[0.595867, 0.502302, 9.990069, 0.24, 210.0],
];
let values = grey_values(&test_case);
let path = std::path::Path::new("combination_scene.png");
let file = std::fs::File::create(path).unwrap();
let w = &mut std::io::BufWriter::new(file);
let mut encoder = png::Encoder::new(w, 1024, 1024);
encoder.set_color(png::ColorType::Grayscale);
encoder.set_depth(png::BitDepth::Eight);
let mut writer = encoder.write_header().unwrap();
writer.write_image_data(&values).unwrap();
}
```
## Test cases
The output images may be scaled down to fit this page. Click on any image to open it full size.
### Single sphere
Input:
```text
[
[0, 0, 0, 1, 128],
]
```
Output:
[![Grey circle on a black background](https://codegolf.codidact.com/uploads/jlru163pa2pgm7aaq70b2tff16uf)](https://codegolf.codidact.com/uploads/jlru163pa2pgm7aaq70b2tff16uf)
### Two separated spheres
Input:
```text
[
[0, -0.55, 0, 0.4, 100],
[0, 0.55, 0, 0.4, 200],
]
```
Output:
[![Light and dark grey circles on a black background](https://codegolf.codidact.com/uploads/8yw7agli78y65kiomb1u5yebs7c5)](https://codegolf.codidact.com/uploads/8yw7agli78y65kiomb1u5yebs7c5)
### Ring of spheres
Input:
```text
[
[0.75, 0, 0, 0.25, 64],
[0.53033, 0.375, -0.375, 0.25, 185],
[0, 0.53033, -0.53033, 0.25, 64],
[-0.53033, 0.375, -0.375, 0.25, 185],
[-0.75, 0, 0, 0.25, 64],
[-0.53033, -0.375, 0.375, 0.25, 185],
[0, -0.53033, 0.53033, 0.25, 64],
[0.53033, -0.375, 0.375, 0.25, 185],
]
```
Output:
[![Eight circles in alternating light and dark grey in a ring some overlapping](https://codegolf.codidact.com/uploads/0lsenynig27au0sryf3ruozt1uwh)](https://codegolf.codidact.com/uploads/0lsenynig27au0sryf3ruozt1uwh)
### Beach ball
Input:
```text
[
[0.001, 0, 0, 1, 64],
[0.000707, 0.000707, 0, 1, 85],
[0, 0.001, 0, 1, 106],
[-0.000707, 0.000707, 0, 1, 127],
[-0.001, 0, 0, 1, 148],
[-0.000707, -0.000707, 0, 1, 169],
[0, -0.001, 0, 1, 190],
[0.000707, -0.000707, 0, 1, 211],
]
```
Output:
[![A circle with eight sectors in different shades of grey](https://codegolf.codidact.com/uploads/51a3ni0cfwsqcdw5b49gy6282qno)](https://codegolf.codidact.com/uploads/51a3ni0cfwsqcdw5b49gy6282qno)
### Football
Input:
```text
[
[0, 0.000058, 0.001947, 1, 0],
[0, -0.001715, 0.000923, 1, 0],
[0, 0.001715, -0.000923, 1, 0],
[0, -0.000058, -0.001947, 1, 0],
[0.001657, -0.000512, 0.000887, 1, 0],
[-0.001657, -0.000512, 0.000887, 1, 0],
[0.001657, 0.000512, -0.000887, 1, 0],
[-0.001657, 0.000512, -0.000887, 1, 0],
[0.001024, 0.001435, 0.000829, 1, 0],
[-0.001024, 0.001435, 0.000829, 1, 0],
[0.001024, -0.001435, -0.000829, 1, 0],
[-0.001024, -0.001435, -0.000829, 1, 0],
[0.001868, 0.000618, 0.000357, 1, 255],
[-0.001868, 0.000618, 0.000357, 1, 255],
[0.001868, -0.000618, -0.000357, 1, 255],
[-0.001868, -0.000618, -0.000357, 1, 255],
[0.000714, -0.000934, 0.001618, 1, 255],
[-0.000714, -0.000934, 0.001618, 1, 255],
[0.000714, 0.000934, -0.001618, 1, 255],
[-0.000714, 0.000934, -0.001618, 1, 255],
[0, -0.001975, -0.000316, 1, 255],
[0, 0.001261, 0.001552, 1, 255],
[0, -0.001261, -0.001552, 1, 255],
[0, 0.001975, 0.000316, 1, 255],
[0.001155, 0.000423, 0.001577, 1, 255],
[-0.001155, 0.000423, 0.001577, 1, 255],
[0.001155, -0.001577, 0.000423, 1, 255],
[-0.001155, -0.001577, 0.000423, 1, 255],
[0.001155, 0.001577, -0.000423, 1, 255],
[-0.001155, 0.001577, -0.000423, 1, 255],
[0.001155, -0.000423, -0.001577, 1, 255],
[-0.001155, -0.000423, -0.001577, 1, 255],
]
```
Output:
[![Black pentagons on a white ball](https://codegolf.codidact.com/uploads/bmrbsw65p59imvwu8ssd42uvvx1r)](https://codegolf.codidact.com/uploads/bmrbsw65p59imvwu8ssd42uvvx1r)
### Eight ball
Input:
```text
[
[0, 0, 1000, 2, 128],
[0, 0, 0.003125, 1, 0],
[0, 0, 0, 0.9975, 255],
[0, 0.000547, -0.002573, 0.995, 0],
[0, -0.000547, -0.002573, 0.995, 0],
[0, 0.001071, -0.005037, 0.9925, 255],
[0, -0.001071, -0.005037, 0.9925, 255],
]
```
Output:
[![A ball with the number 8 in a white circle](https://codegolf.codidact.com/uploads/dvl68crpmy62rqt94mwanditt4wc)](https://codegolf.codidact.com/uploads/dvl68crpmy62rqt94mwanditt4wc)
### Star ball
Input:
```text
[
[0, 0, 0, 1, 210],
[0, 0, 0.0008, 1, 210],
[0, 0, 0.0004, 1.00012, 50],
[0.012969, 0, -0.042045, 0.96, 100],
[0.016397, 0.011913, -0.039054, 0.96, 210],
[0.004008, 0.012334, -0.042045, 0.96, 100],
[-0.006263, 0.019276, -0.039054, 0.96, 210],
[-0.010492, 0.007623, -0.042045, 0.96, 100],
[-0.020268, 0, -0.039054, 0.96, 210],
[-0.010492, -0.007623, -0.042045, 0.96, 100],
[-0.006263, -0.019276, -0.039054, 0.96, 210],
[0.004008, -0.012334, -0.042045, 0.96, 100],
[0.016397, -0.011913, -0.039054, 0.96, 210],
]
```
Output:
[![A light grey ball with a dark grey five pointed star](https://codegolf.codidact.com/uploads/f3wdgwru6j35ue72x25mazxikfx2)](https://codegolf.codidact.com/uploads/f3wdgwru6j35ue72x25mazxikfx2)
### Combination scene
Input:
```text
[
[0, 0, 1000, 2, 192],
[-2, 2, 100, 3, 98],
[2, 2, 100, 3, 98],
[0.5, -0.5, 500, 0.2, 255],
[-0.349469, -0.1, 20.000279, 0.6, 64],
[-0.349775, -0.100273, 20.000485, 0.6, 85],
[-0.350214, -0.100386, 20.000407, 0.6, 106],
[-0.350527, -0.100273, 20.000091, 0.6, 127],
[-0.350531, -0.1, 19.999721, 0.6, 148],
[-0.350225, -0.099727, 19.999515, 0.6, 169],
[-0.349786, -0.099614, 19.999593, 0.6, 190],
[-0.349473, -0.099727, 19.999909, 0.6, 211],
[0.20032, 0.349824, 15.000456, 0.3, 0],
[0.20032, 0.349517, 14.999924, 0.3, 0],
[0.19968, 0.350483, 15.000076, 0.3, 0],
[0.19968, 0.350176, 14.999544, 0.3, 0],
[0.200578, 0.350073, 14.999958, 0.3, 0],
[0.199817, 0.349519, 15.000277, 0.3, 0],
[0.200183, 0.350481, 14.999723, 0.3, 0],
[0.199422, 0.349927, 15.000042, 0.3, 0],
[0.200235, 0.35042, 15.000332, 0.3, 0],
[0.199765, 0.350078, 15.000529, 0.3, 0],
[0.200235, 0.349922, 14.999471, 0.3, 0],
[0.199765, 0.34958, 14.999668, 0.3, 0],
[0.200429, 0.350419, 15.000005, 0.3, 255],
[0.199571, 0.349795, 15.000366, 0.3, 255],
[0.200429, 0.350205, 14.999634, 0.3, 255],
[0.199571, 0.349581, 14.999995, 0.3, 255],
[0.200524, 0.349747, 15.000146, 0.3, 255],
[0.200196, 0.349509, 15.000283, 0.3, 255],
[0.199804, 0.350491, 14.999717, 0.3, 255],
[0.199476, 0.350253, 14.999854, 0.3, 255],
[0.200138, 0.349578, 14.999597, 0.3, 255],
[0.200138, 0.350138, 15.000567, 0.3, 255],
[0.199862, 0.349862, 14.999433, 0.3, 255],
[0.199862, 0.350422, 15.000403, 0.3, 255],
[0.200488, 0.350136, 15.000321, 0.3, 255],
[0.199957, 0.349751, 15.000544, 0.3, 255],
[0.200488, 0.34979, 14.999721, 0.3, 255],
[0.199957, 0.349404, 14.999944, 0.3, 255],
[0.200043, 0.350596, 15.000056, 0.3, 255],
[0.199512, 0.35021, 15.000279, 0.3, 255],
[0.200043, 0.350249, 14.999456, 0.3, 255],
[0.199512, 0.349864, 14.999679, 0.3, 255],
[-0.5, 0.500442, 10.000442, 0.2, 0],
[-0.5, 0.5, 10, 0.1995, 255],
[-0.499966, 0.49971, 9.999563, 0.199, 0],
[-0.500034, 0.499563, 9.99971, 0.199, 0],
[-0.499934, 0.499432, 9.999144, 0.1985, 255],
[-0.500066, 0.499144, 9.999432, 0.1985, 255],
[0.6, 0.5, 10, 0.25, 210],
[0.600122, 0.4999, 10.000122, 0.25, 210],
[0.600061, 0.49995, 10.000061, 0.25003, 50],
[0.595856, 0.505256, 9.991271, 0.24, 100],
[0.597973, 0.507461, 9.992175, 0.24, 210],
[0.595362, 0.507926, 9.993945, 0.24, 100],
[0.594618, 0.509055, 9.996832, 0.24, 210],
[0.592382, 0.506906, 9.996092, 0.24, 100],
[0.590438, 0.504882, 9.997604, 0.24, 210],
[0.591035, 0.503605, 9.994744, 0.24, 100],
[0.59121, 0.500708, 9.993425, 0.24, 210],
[0.593181, 0.502585, 9.991764, 0.24, 100],
[0.595867, 0.502302, 9.990069, 0.24, 210],
]
```
Output:
[![A beachball, football, eightball, and starball with the sun behind hills in the background](https://codegolf.codidact.com/uploads/usv9npm1t5bjpcaknn6rbzgkqwmi)](https://codegolf.codidact.com/uploads/usv9npm1t5bjpcaknn6rbzgkqwmi)
~~Output using 32 bit floating point rather than 64 bit floating point:~~
[![A beachball, football, eightball, and starball with the sun behind hills in the background](https://codegolf.codidact.com/uploads/7uuxjxq7a9oxc9q0jf2bi8l68qk3)](https://codegolf.codidact.com/uploads/7uuxjxq7a9oxc9q0jf2bi8l68qk3)
The inaccuracies in the 32 bit floating point version are subtle. You may need to click on the image to open it full size to see the occasional inaccurate pixel. For example, around the inside of the holes in the 8 on the eightball. These occasional inaccurate pixels are acceptable in your output (allowing languages which only support 32 bit floating point to compete).
More widespread speckling as in the following image would not be valid, so you may need to be careful if changing the order of calculations during golfing:
[![A scene with significant areas of inaccurate speckles obscuring the intended image](https://codegolf.codidact.com/uploads/1sa7i6z0cv2kgyrmw2tj2ytr1ps1)](https://codegolf.codidact.com/uploads/1sa7i6z0cv2kgyrmw2tj2ytr1ps1)
This exaggerated inaccuracy was achieved by adding and subtracting 1000 to the distance calculation, exceeding the available precision of 32 bit floating point and causing some distances to appear in the wrong order. Note that this same approach causes no inaccuracy with 64 bit floating point, so if your language supports both you may have more leeway for rearranging calculations if you use 64 bit floating point.
## Explanatory animations
The example images were produced from plain grey spheres with no surface pattern or texture. The following animations show how they break down into their component plain spheres.
<details>
<summary>Animations</summary>
TODO
</details>
## Scoring
This is a [code golf challenge](https://codegolf.codidact.com/categories/49/tags/4274). Your score is the number of bytes in your code. Lowest score for each language wins.
> Explanations are optional, but I'm more likely to upvote answers that have one.

Given a list of spheres, output an image.
Produce a greyscale [orthographic](https://en.wikipedia.org/wiki/Parallel_projection#Orthographic_projection) (parallel projection - no perspective) ray caster, that renders only plain grey spheres with no lighting.
## Input
- A list of spheres, each consisting of:
- The coordinates of its centre $(x,y,z)$.
- Its radius $r$.
- Its greyscale value $g$, an integer from $0$ to $255$ (both inclusive).
- Only the greyscale value will be restricted to integer values. The coordinates and radius may have fractional values.
## Output
- An image $1024$ by $1024$ pixels with greyscale values from $0$ to $255$.
- The image covers x and y values from $-1$ to $1$.
- A sphere's greyscale value is applied to every visible part of it. There is no variation over the surface of the sphere, so it appears flat like a disk.
- You may output whatever image format is convenient, including simple text formats such as [PGM](https://en.wikipedia.org/wiki/Netpbm#PGM_example) or a list of pixel greyscale values. If you want to include an example of your output as an image in your answer, the image formats that Codidact supports are PNG, JPEG, and GIF. You could output PGM and then convert to PNG to upload to your answer. If you output just the greyscale values as a list of numbers from $0$ to $255$, I've made a `[ TODO page to convert these to greyscale PNG]`.
- You are free to output an image reflected or rotated by a multiple of 90 degrees. However, the z axis is required to be increasing in the forwards direction (into the image) otherwise some of the test cases will look incorrect.
## Calculating a pixel value
Imagine a ray passing into the image through a pixel, perpendicular to the image plane. If it hits a sphere, it shows the greyscale value of the first sphere it hits. Otherwise, it shows a value of zero (black).
For example, for each pixel:
- The pixel $x,y$ coordinates will each be from $0$ to $1023$. Scale these to the range $-1$ to $1$ (for example, divide by $512$ and subtract $1$).
- If these scaled $x,y$ coordinates are within the radius $r$ of the $x,y$ coordinates of a sphere's centre, the ray intersects that sphere (the $z$ coordinate is not relevant to this test because the image is an orthographic projection - the rays are all parallel to the $z$ axis). The distance $a$ of the ray from the sphere's centre can be calculated as follows:
$$a=\sqrt{(x_{ray}-x_{sphere})^2+(y_{ray}-y_{sphere})^2}$$
```text
CONSIDER A DIAGRAM HERE
```
- For each sphere that the ray intersects, find the $z$ coordinate of the nearest intersection (there will often be $2$ per sphere, $1$ on the front of the sphere and $1$ on the back of the sphere, as the ray passes through - only the nearest is relevant). This can be calculated as follows:
- The distance $d$ to the nearest intersection point can be calculated from the $z$ coordinate of the sphere's centre, the radius $r$, and the distance $a$ from the $x,y$ coordinates of the sphere's centre to the scaled $x,y$ coordinates of the pixel:
$$d=z_{sphere}-\sqrt{r_{sphere}^2-a^2}$$
```text
CONSIDER A DIAGRAM HERE
```
- Of all the spheres that the ray intersects, the sphere with the intersection point with the smallest $z$ coordinate $d$ is the one to display. Set the pixel to the greyscale value of this sphere.
- If the ray did not intersect any spheres, set the pixel to zero (black).
## Example implementations
These are ungolfed code to supplement understanding of the requirements. You are free to golf these or implement your own approach that matches the test cases up to reflection & rotation.
The Python implementation uses 64 bit floating point. The Rust implementation uses 32 bit floating point to show the difference in output. I've tried to optimise the test cases for compatibility with 32 bit floating point, so the differences are restricted to being a pixel out in some places.
### Python
The function `grey_values` would be sufficient as an entry to this challenge. The rest of the code saves the output as a PNG file. The function itself has no dependence on the `import png` so it would not be required in an entry.
Instructions for installing the `png` package can be found on its [project page on PyPI](https://pypi.org/project/pypng/).
```python
def grey_values(spheres):
values = []
for y_pixel in range(1024):
for x_pixel in range(1024):
x_ray, y_ray = x_pixel / 512 - 1, y_pixel / 512 - 1
min_distance, hit_grey_value = None, 0
for sphere in spheres:
x_sphere, y_sphere, z_sphere, radius, grey = sphere
a = (
(x_ray - x_sphere) ** 2 + (y_ray - y_sphere) ** 2
) ** 0.5
if a < radius:
distance = z_sphere - (radius ** 2 - a ** 2) ** 0.5
if min_distance is None or distance < min_distance:
min_distance = distance
hit_grey_value = grey
values.append(hit_grey_value)
return values
import png
test_case = [
[0, 0, 1000, 2, 192],
[-2, 2, 100, 3, 98],
[2, 2, 100, 3, 98],
[0.5, -0.5, 500, 0.2, 255],
[-0.349469, -0.1, 20.000279, 0.6, 64],
[-0.349775, -0.100273, 20.000485, 0.6, 85],
[-0.350214, -0.100386, 20.000407, 0.6, 106],
[-0.350527, -0.100273, 20.000091, 0.6, 127],
[-0.350531, -0.1, 19.999721, 0.6, 148],
[-0.350225, -0.099727, 19.999515, 0.6, 169],
[-0.349786, -0.099614, 19.999593, 0.6, 190],
[-0.349473, -0.099727, 19.999909, 0.6, 211],
[0.20032, 0.349824, 15.000456, 0.3, 0],
[0.20032, 0.349517, 14.999924, 0.3, 0],
[0.19968, 0.350483, 15.000076, 0.3, 0],
[0.19968, 0.350176, 14.999544, 0.3, 0],
[0.200578, 0.350073, 14.999958, 0.3, 0],
[0.199817, 0.349519, 15.000277, 0.3, 0],
[0.200183, 0.350481, 14.999723, 0.3, 0],
[0.199422, 0.349927, 15.000042, 0.3, 0],
[0.200235, 0.35042, 15.000332, 0.3, 0],
[0.199765, 0.350078, 15.000529, 0.3, 0],
[0.200235, 0.349922, 14.999471, 0.3, 0],
[0.199765, 0.34958, 14.999668, 0.3, 0],
[0.200429, 0.350419, 15.000005, 0.3, 255],
[0.199571, 0.349795, 15.000366, 0.3, 255],
[0.200429, 0.350205, 14.999634, 0.3, 255],
[0.199571, 0.349581, 14.999995, 0.3, 255],
[0.200524, 0.349747, 15.000146, 0.3, 255],
[0.200196, 0.349509, 15.000283, 0.3, 255],
[0.199804, 0.350491, 14.999717, 0.3, 255],
[0.199476, 0.350253, 14.999854, 0.3, 255],
[0.200138, 0.349578, 14.999597, 0.3, 255],
[0.200138, 0.350138, 15.000567, 0.3, 255],
[0.199862, 0.349862, 14.999433, 0.3, 255],
[0.199862, 0.350422, 15.000403, 0.3, 255],
[0.200488, 0.350136, 15.000321, 0.3, 255],
[0.199957, 0.349751, 15.000544, 0.3, 255],
[0.200488, 0.34979, 14.999721, 0.3, 255],
[0.199957, 0.349404, 14.999944, 0.3, 255],
[0.200043, 0.350596, 15.000056, 0.3, 255],
[0.199512, 0.35021, 15.000279, 0.3, 255],
[0.200043, 0.350249, 14.999456, 0.3, 255],
[0.199512, 0.349864, 14.999679, 0.3, 255],
[-0.5, 0.500442, 10.000442, 0.2, 0],
[-0.5, 0.5, 10, 0.1995, 255],
[-0.499966, 0.49971, 9.999563, 0.199, 0],
[-0.500034, 0.499563, 9.99971, 0.199, 0],
[-0.499934, 0.499432, 9.999144, 0.1985, 255],
[-0.500066, 0.499144, 9.999432, 0.1985, 255],
[0.6, 0.5, 10, 0.25, 210],
[0.600122, 0.4999, 10.000122, 0.25, 210],
[0.600061, 0.49995, 10.000061, 0.25003, 50],
[0.595856, 0.505256, 9.991271, 0.24, 100],
[0.597973, 0.507461, 9.992175, 0.24, 210],
[0.595362, 0.507926, 9.993945, 0.24, 100],
[0.594618, 0.509055, 9.996832, 0.24, 210],
[0.592382, 0.506906, 9.996092, 0.24, 100],
[0.590438, 0.504882, 9.997604, 0.24, 210],
[0.591035, 0.503605, 9.994744, 0.24, 100],
[0.59121, 0.500708, 9.993425, 0.24, 210],
[0.593181, 0.502585, 9.991764, 0.24, 100],
[0.595867, 0.502302, 9.990069, 0.24, 210],
]
values = grey_values(test_case)
rows = [values[row * 1024:row * 1024 + 1024] for row in range(1024)]
with open('combination_scene.png', 'wb') as file:
w = png.Writer(1024, 1024, greyscale=True)
w.write(file, rows)
```
### Rust
The function `grey_values` would be sufficient as an entry to this challenge. The rest of the code saves the output as a PNG file.
The `png` crate can be installed using `cargo add png`. The `png` crate is not required for the `grey_values` function, only for saving as a PNG file.
```rust
fn grey_values(spheres: &Vec<[f32; 5]>) -> Vec<u8> {
let mut values = vec![];
for y_pixel in 0..1024 {
for x_pixel in 0..1024 {
let x_ray = x_pixel as f32 / 512.0 - 1.0;
let y_ray = y_pixel as f32 / 512.0 - 1.0;
let mut min_distance = f32::INFINITY;
let mut hit_grey_value = 0;
for sphere in spheres {
let [
x_sphere, y_sphere, z_sphere, radius, grey
] = *sphere;
let a = (
(x_ray - x_sphere).powf(2.0)
+ (y_ray - y_sphere).powf(2.0)
).sqrt();
if a < radius {
let distance = z_sphere - (
radius.powf(2.0) - a.powf(2.0)
).sqrt();
if distance < min_distance {
min_distance = distance;
hit_grey_value = grey as u8;
}
}
}
values.push(hit_grey_value);
}
}
values
}
fn main() {
let test_case: Vec<[f32; 5]> = vec![
[0.0, 0.0, 1000.0, 2.0, 192.0],
[-2.0, 2.0, 100.0, 3.0, 98.0],
[2.0, 2.0, 100.0, 3.0, 98.0],
[0.5, -0.5, 500.0, 0.2, 255.0],
[-0.349469, -0.1, 20.000279, 0.6, 64.0],
[-0.349775, -0.100273, 20.000485, 0.6, 85.0],
[-0.350214, -0.100386, 20.000407, 0.6, 106.0],
[-0.350527, -0.100273, 20.000091, 0.6, 127.0],
[-0.350531, -0.1, 19.999721, 0.6, 148.0],
[-0.350225, -0.099727, 19.999515, 0.6, 169.0],
[-0.349786, -0.099614, 19.999593, 0.6, 190.0],
[-0.349473, -0.099727, 19.999909, 0.6, 211.0],
[0.20032, 0.349824, 15.000456, 0.3, 0.0],
[0.20032, 0.349517, 14.999924, 0.3, 0.0],
[0.19968, 0.350483, 15.000076, 0.3, 0.0],
[0.19968, 0.350176, 14.999544, 0.3, 0.0],
[0.200578, 0.350073, 14.999958, 0.3, 0.0],
[0.199817, 0.349519, 15.000277, 0.3, 0.0],
[0.200183, 0.350481, 14.999723, 0.3, 0.0],
[0.199422, 0.349927, 15.000042, 0.3, 0.0],
[0.200235, 0.35042, 15.000332, 0.3, 0.0],
[0.199765, 0.350078, 15.000529, 0.3, 0.0],
[0.200235, 0.349922, 14.999471, 0.3, 0.0],
[0.199765, 0.34958, 14.999668, 0.3, 0.0],
[0.200429, 0.350419, 15.000005, 0.3, 255.0],
[0.199571, 0.349795, 15.000366, 0.3, 255.0],
[0.200429, 0.350205, 14.999634, 0.3, 255.0],
[0.199571, 0.349581, 14.999995, 0.3, 255.0],
[0.200524, 0.349747, 15.000146, 0.3, 255.0],
[0.200196, 0.349509, 15.000283, 0.3, 255.0],
[0.199804, 0.350491, 14.999717, 0.3, 255.0],
[0.199476, 0.350253, 14.999854, 0.3, 255.0],
[0.200138, 0.349578, 14.999597, 0.3, 255.0],
[0.200138, 0.350138, 15.000567, 0.3, 255.0],
[0.199862, 0.349862, 14.999433, 0.3, 255.0],
[0.199862, 0.350422, 15.000403, 0.3, 255.0],
[0.200488, 0.350136, 15.000321, 0.3, 255.0],
[0.199957, 0.349751, 15.000544, 0.3, 255.0],
[0.200488, 0.34979, 14.999721, 0.3, 255.0],
[0.199957, 0.349404, 14.999944, 0.3, 255.0],
[0.200043, 0.350596, 15.000056, 0.3, 255.0],
[0.199512, 0.35021, 15.000279, 0.3, 255.0],
[0.200043, 0.350249, 14.999456, 0.3, 255.0],
[0.199512, 0.349864, 14.999679, 0.3, 255.0],
[-0.5, 0.500442, 10.000442, 0.2, 0.0],
[-0.5, 0.5, 10.0, 0.1995, 255.0],
[-0.499966, 0.49971, 9.999563, 0.199, 0.0],
[-0.500034, 0.499563, 9.99971, 0.199, 0.0],
[-0.499934, 0.499432, 9.999144, 0.1985, 255.0],
[-0.500066, 0.499144, 9.999432, 0.1985, 255.0],
[0.6, 0.5, 10.0, 0.25, 210.0],
[0.600122, 0.4999, 10.000122, 0.25, 210.0],
[0.600061, 0.49995, 10.000061, 0.25003, 50.0],
[0.595856, 0.505256, 9.991271, 0.24, 100.0],
[0.597973, 0.507461, 9.992175, 0.24, 210.0],
[0.595362, 0.507926, 9.993945, 0.24, 100.0],
[0.594618, 0.509055, 9.996832, 0.24, 210.0],
[0.592382, 0.506906, 9.996092, 0.24, 100.0],
[0.590438, 0.504882, 9.997604, 0.24, 210.0],
[0.591035, 0.503605, 9.994744, 0.24, 100.0],
[0.59121, 0.500708, 9.993425, 0.24, 210.0],
[0.593181, 0.502585, 9.991764, 0.24, 100.0],
[0.595867, 0.502302, 9.990069, 0.24, 210.0],
];
let values = grey_values(&test_case);
let path = std::path::Path::new("combination_scene.png");
let file = std::fs::File::create(path).unwrap();
let w = &mut std::io::BufWriter::new(file);
let mut encoder = png::Encoder::new(w, 1024, 1024);
encoder.set_color(png::ColorType::Grayscale);
encoder.set_depth(png::BitDepth::Eight);
let mut writer = encoder.write_header().unwrap();
writer.write_image_data(&values).unwrap();
}
```
## Test cases
The output images may be scaled down to fit this page. Click on any image to open it full size.
### Single sphere
Input:
```text
[
[0, 0, 0, 1, 128],
]
```
Output:
[![Grey circle on a black background](https://codegolf.codidact.com/uploads/jlru163pa2pgm7aaq70b2tff16uf)](https://codegolf.codidact.com/uploads/jlru163pa2pgm7aaq70b2tff16uf)
### Two separated spheres
Input:
```text
[
[0, -0.55, 0, 0.4, 100],
[0, 0.55, 0, 0.4, 200],
]
```
Output:
[![Light and dark grey circles on a black background](https://codegolf.codidact.com/uploads/8yw7agli78y65kiomb1u5yebs7c5)](https://codegolf.codidact.com/uploads/8yw7agli78y65kiomb1u5yebs7c5)
### Ring of spheres
Input:
```text
[
[0.75, 0, 0, 0.25, 64],
[0.53033, 0.375, -0.375, 0.25, 185],
[0, 0.53033, -0.53033, 0.25, 64],
[-0.53033, 0.375, -0.375, 0.25, 185],
[-0.75, 0, 0, 0.25, 64],
[-0.53033, -0.375, 0.375, 0.25, 185],
[0, -0.53033, 0.53033, 0.25, 64],
[0.53033, -0.375, 0.375, 0.25, 185],
]
```
Output:
[![Eight circles in alternating light and dark grey in a ring some overlapping](https://codegolf.codidact.com/uploads/0lsenynig27au0sryf3ruozt1uwh)](https://codegolf.codidact.com/uploads/0lsenynig27au0sryf3ruozt1uwh)
### Beach ball
Input:
```text
[
[0.001, 0, 0, 1, 64],
[0.000707, 0.000707, 0, 1, 85],
[0, 0.001, 0, 1, 106],
[-0.000707, 0.000707, 0, 1, 127],
[-0.001, 0, 0, 1, 148],
[-0.000707, -0.000707, 0, 1, 169],
[0, -0.001, 0, 1, 190],
[0.000707, -0.000707, 0, 1, 211],
]
```
Output:
[![A circle with eight sectors in different shades of grey](https://codegolf.codidact.com/uploads/51a3ni0cfwsqcdw5b49gy6282qno)](https://codegolf.codidact.com/uploads/51a3ni0cfwsqcdw5b49gy6282qno)
### Football
Input:
```text
[
[0, 0.000058, 0.001947, 1, 0],
[0, -0.001715, 0.000923, 1, 0],
[0, 0.001715, -0.000923, 1, 0],
[0, -0.000058, -0.001947, 1, 0],
[0.001657, -0.000512, 0.000887, 1, 0],
[-0.001657, -0.000512, 0.000887, 1, 0],
[0.001657, 0.000512, -0.000887, 1, 0],
[-0.001657, 0.000512, -0.000887, 1, 0],
[0.001024, 0.001435, 0.000829, 1, 0],
[-0.001024, 0.001435, 0.000829, 1, 0],
[0.001024, -0.001435, -0.000829, 1, 0],
[-0.001024, -0.001435, -0.000829, 1, 0],
[0.001868, 0.000618, 0.000357, 1, 255],
[-0.001868, 0.000618, 0.000357, 1, 255],
[0.001868, -0.000618, -0.000357, 1, 255],
[-0.001868, -0.000618, -0.000357, 1, 255],
[0.000714, -0.000934, 0.001618, 1, 255],
[-0.000714, -0.000934, 0.001618, 1, 255],
[0.000714, 0.000934, -0.001618, 1, 255],
[-0.000714, 0.000934, -0.001618, 1, 255],
[0, -0.001975, -0.000316, 1, 255],
[0, 0.001261, 0.001552, 1, 255],
[0, -0.001261, -0.001552, 1, 255],
[0, 0.001975, 0.000316, 1, 255],
[0.001155, 0.000423, 0.001577, 1, 255],
[-0.001155, 0.000423, 0.001577, 1, 255],
[0.001155, -0.001577, 0.000423, 1, 255],
[-0.001155, -0.001577, 0.000423, 1, 255],
[0.001155, 0.001577, -0.000423, 1, 255],
[-0.001155, 0.001577, -0.000423, 1, 255],
[0.001155, -0.000423, -0.001577, 1, 255],
[-0.001155, -0.000423, -0.001577, 1, 255],
]
```
Output:
[![Black pentagons on a white ball](https://codegolf.codidact.com/uploads/bmrbsw65p59imvwu8ssd42uvvx1r)](https://codegolf.codidact.com/uploads/bmrbsw65p59imvwu8ssd42uvvx1r)
### Eight ball
Input:
```text
[
[0, 0, 1000, 2, 128],
[0, 0, 0.003125, 1, 0],
[0, 0, 0, 0.9975, 255],
[0, 0.000547, -0.002573, 0.995, 0],
[0, -0.000547, -0.002573, 0.995, 0],
[0, 0.001071, -0.005037, 0.9925, 255],
[0, -0.001071, -0.005037, 0.9925, 255],
]
```
Output:
[![A ball with the number 8 in a white circle](https://codegolf.codidact.com/uploads/dvl68crpmy62rqt94mwanditt4wc)](https://codegolf.codidact.com/uploads/dvl68crpmy62rqt94mwanditt4wc)
### Star ball
Input:
```text
[
[0, 0, 0, 1, 210],
[0, 0, 0.0008, 1, 210],
[0, 0, 0.0004, 1.00012, 50],
[0.012969, 0, -0.042045, 0.96, 100],
[0.016397, 0.011913, -0.039054, 0.96, 210],
[0.004008, 0.012334, -0.042045, 0.96, 100],
[-0.006263, 0.019276, -0.039054, 0.96, 210],
[-0.010492, 0.007623, -0.042045, 0.96, 100],
[-0.020268, 0, -0.039054, 0.96, 210],
[-0.010492, -0.007623, -0.042045, 0.96, 100],
[-0.006263, -0.019276, -0.039054, 0.96, 210],
[0.004008, -0.012334, -0.042045, 0.96, 100],
[0.016397, -0.011913, -0.039054, 0.96, 210],
]
```
Output:
[![A light grey ball with a dark grey five pointed star](https://codegolf.codidact.com/uploads/f3wdgwru6j35ue72x25mazxikfx2)](https://codegolf.codidact.com/uploads/f3wdgwru6j35ue72x25mazxikfx2)
### Combination scene
Input:
```text
[
[0, 0, 1000, 2, 192],
[-2, 2, 100, 3, 98],
[2, 2, 100, 3, 98],
[0.5, -0.5, 500, 0.2, 255],
[-0.349469, -0.1, 20.000279, 0.6, 64],
[-0.349775, -0.100273, 20.000485, 0.6, 85],
[-0.350214, -0.100386, 20.000407, 0.6, 106],
[-0.350527, -0.100273, 20.000091, 0.6, 127],
[-0.350531, -0.1, 19.999721, 0.6, 148],
[-0.350225, -0.099727, 19.999515, 0.6, 169],
[-0.349786, -0.099614, 19.999593, 0.6, 190],
[-0.349473, -0.099727, 19.999909, 0.6, 211],
[0.20032, 0.349824, 15.000456, 0.3, 0],
[0.20032, 0.349517, 14.999924, 0.3, 0],
[0.19968, 0.350483, 15.000076, 0.3, 0],
[0.19968, 0.350176, 14.999544, 0.3, 0],
[0.200578, 0.350073, 14.999958, 0.3, 0],
[0.199817, 0.349519, 15.000277, 0.3, 0],
[0.200183, 0.350481, 14.999723, 0.3, 0],
[0.199422, 0.349927, 15.000042, 0.3, 0],
[0.200235, 0.35042, 15.000332, 0.3, 0],
[0.199765, 0.350078, 15.000529, 0.3, 0],
[0.200235, 0.349922, 14.999471, 0.3, 0],
[0.199765, 0.34958, 14.999668, 0.3, 0],
[0.200429, 0.350419, 15.000005, 0.3, 255],
[0.199571, 0.349795, 15.000366, 0.3, 255],
[0.200429, 0.350205, 14.999634, 0.3, 255],
[0.199571, 0.349581, 14.999995, 0.3, 255],
[0.200524, 0.349747, 15.000146, 0.3, 255],
[0.200196, 0.349509, 15.000283, 0.3, 255],
[0.199804, 0.350491, 14.999717, 0.3, 255],
[0.199476, 0.350253, 14.999854, 0.3, 255],
[0.200138, 0.349578, 14.999597, 0.3, 255],
[0.200138, 0.350138, 15.000567, 0.3, 255],
[0.199862, 0.349862, 14.999433, 0.3, 255],
[0.199862, 0.350422, 15.000403, 0.3, 255],
[0.200488, 0.350136, 15.000321, 0.3, 255],
[0.199957, 0.349751, 15.000544, 0.3, 255],
[0.200488, 0.34979, 14.999721, 0.3, 255],
[0.199957, 0.349404, 14.999944, 0.3, 255],
[0.200043, 0.350596, 15.000056, 0.3, 255],
[0.199512, 0.35021, 15.000279, 0.3, 255],
[0.200043, 0.350249, 14.999456, 0.3, 255],
[0.199512, 0.349864, 14.999679, 0.3, 255],
[-0.5, 0.500442, 10.000442, 0.2, 0],
[-0.5, 0.5, 10, 0.1995, 255],
[-0.499966, 0.49971, 9.999563, 0.199, 0],
[-0.500034, 0.499563, 9.99971, 0.199, 0],
[-0.499934, 0.499432, 9.999144, 0.1985, 255],
[-0.500066, 0.499144, 9.999432, 0.1985, 255],
[0.6, 0.5, 10, 0.25, 210],
[0.600122, 0.4999, 10.000122, 0.25, 210],
[0.600061, 0.49995, 10.000061, 0.25003, 50],
[0.595856, 0.505256, 9.991271, 0.24, 100],
[0.597973, 0.507461, 9.992175, 0.24, 210],
[0.595362, 0.507926, 9.993945, 0.24, 100],
[0.594618, 0.509055, 9.996832, 0.24, 210],
[0.592382, 0.506906, 9.996092, 0.24, 100],
[0.590438, 0.504882, 9.997604, 0.24, 210],
[0.591035, 0.503605, 9.994744, 0.24, 100],
[0.59121, 0.500708, 9.993425, 0.24, 210],
[0.593181, 0.502585, 9.991764, 0.24, 100],
[0.595867, 0.502302, 9.990069, 0.24, 210],
]
```
Output:
[![A beachball, football, eightball, and starball with the sun behind hills in the background](https://codegolf.codidact.com/uploads/usv9npm1t5bjpcaknn6rbzgkqwmi)](https://codegolf.codidact.com/uploads/usv9npm1t5bjpcaknn6rbzgkqwmi)
## Accuracy
The outputs for the test cases were generated using 64 bit floating point numbers. The following is the last test case using 32 bit floating point rather than 64 bit floating point:
[![A beachball, football, eightball, and starball with the sun behind hills in the background](https://codegolf.codidact.com/uploads/7uuxjxq7a9oxc9q0jf2bi8l68qk3)](https://codegolf.codidact.com/uploads/7uuxjxq7a9oxc9q0jf2bi8l68qk3)
The inaccuracies in the 32 bit floating point version are subtle. You may need to click on the image to open it full size to see the occasional inaccurate pixel. For example, around the inside of the holes in the 8 on the eightball. These occasional inaccurate pixels are acceptable in your output (allowing languages which only support 32 bit floating point to compete).
More widespread speckling as in the following image would not be valid, so you may need to be careful if changing the order of calculations during golfing:
[![A scene with significant areas of inaccurate speckles obscuring the intended image](https://codegolf.codidact.com/uploads/1sa7i6z0cv2kgyrmw2tj2ytr1ps1)](https://codegolf.codidact.com/uploads/1sa7i6z0cv2kgyrmw2tj2ytr1ps1)
This exaggerated inaccuracy was achieved by adding 1000 to and then subtracting 1000 from the distance calculation, exceeding the available precision of 32 bit floating point and causing some distances to appear in the wrong order. Note that this same approach causes no inaccuracy with 64 bit floating point, so if your language supports both you may have more leeway for rearranging calculations if you use 64 bit floating point.
## Explanatory animations
The example images were produced from plain grey spheres with no surface pattern or texture. The following animations show how they break down into their component plain spheres.
<details>
<summary>Animations</summary>
TODO
</details>
## Scoring
This is a [code golf challenge](https://codegolf.codidact.com/categories/49/tags/4274). Your score is the number of bytes in your code. Lowest score for each language wins.
> Explanations are optional, but I'm more likely to upvote answers that have one.

#15: Post edited by

trichoplax‭ · 2024-07-26T15:01:16Z (5 months ago)
Include example image with too much inaccuracy

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Given a list of spheres, output an image.
Produce a greyscale [orthographic](https://en.wikipedia.org/wiki/Parallel_projection#Orthographic_projection) (parallel projection - no perspective) ray caster, that renders only plain grey spheres with no lighting.
## Input
- A list of spheres, each consisting of:
- The coordinates of its centre $(x,y,z)$.
- Its radius $r$.
- Its greyscale value $g$, an integer from $0$ to $255$ (both inclusive).
- Only the greyscale value will be restricted to integer values. The coordinates and radius may have fractional values.
## Output
- An image $1024$ by $1024$ pixels with greyscale values from $0$ to $255$.
- The image covers x and y values from $-1$ to $1$.
- A sphere's greyscale value is applied to every visible part of it. There is no variation over the surface of the sphere, so it appears flat like a disk.
- You may output whatever image format is convenient, including simple text formats such as [PGM](https://en.wikipedia.org/wiki/Netpbm#PGM_example) or a list of pixel greyscale values. If you want to include an example of your output as an image in your answer, the image formats that Codidact supports are PNG, JPEG, and GIF. You could output PGM and then convert to PNG to upload to your answer. If you output just the greyscale values as a list of numbers from $0$ to $255$, I've made a `[ TODO page to convert these to greyscale PNG]`.
- You are free to output an image reflected or rotated by a multiple of 90 degrees. However, the z axis is required to be increasing in the forwards direction (into the image) otherwise some of the test cases will look incorrect.
## Calculating a pixel value
Imagine a ray passing into the image through a pixel, perpendicular to the image plane. If it hits a sphere, it shows the greyscale value of the first sphere it hits. Otherwise, it shows a value of zero (black).
For example, for each pixel:
- The pixel $x,y$ coordinates will each be from $0$ to $1023$. Scale these to the range $-1$ to $1$ (for example, divide by $512$ and subtract $1$).
- If these scaled $x,y$ coordinates are within the radius $r$ of the $x,y$ coordinates of a sphere's centre, the ray intersects that sphere (the $z$ coordinate is not relevant to this test because the image is an orthographic projection - the rays are all parallel to the $z$ axis). The distance $a$ of the ray from the sphere's centre can be calculated as follows:
$$a=\sqrt{(x_{ray}-x_{sphere})^2+(y_{ray}-y_{sphere})^2}$$
```text
CONSIDER A DIAGRAM HERE
```
- For each sphere that the ray intersects, find the $z$ coordinate of the nearest intersection (there will often be $2$ per sphere, $1$ on the front of the sphere and $1$ on the back of the sphere, as the ray passes through - only the nearest is relevant). This can be calculated as follows:
- The distance $d$ to the nearest intersection point can be calculated from the $z$ coordinate of the sphere's centre, the radius $r$, and the distance $a$ from the $x,y$ coordinates of the sphere's centre to the scaled $x,y$ coordinates of the pixel:
$$d=z_{sphere}-\sqrt{r_{sphere}^2-a^2}$$
```text
CONSIDER A DIAGRAM HERE
```
- Of all the spheres that the ray intersects, the sphere with the intersection point with the smallest $z$ coordinate $d$ is the one to display. Set the pixel to the greyscale value of this sphere.
- If the ray did not intersect any spheres, set the pixel to zero (black).
## Example implementations
These are ungolfed code to supplement understanding of the requirements. You are free to golf these or implement your own approach that matches the test cases up to reflection & rotation.
The Python implementation uses 64 bit floating point. The Rust implementation uses 32 bit floating point to show the difference in output. I've tried to optimise the test cases for compatibility with 32 bit floating point, so the differences are restricted to being a pixel out in some places.
### Python
The function `grey_values` would be sufficient as an entry to this challenge. The rest of the code saves the output as a PNG file. The function itself has no dependence on the `import png` so it would not be required in an entry.
Instructions for installing the `png` package can be found on its [project page on PyPI](https://pypi.org/project/pypng/).
```python
def grey_values(spheres):
values = []
for y_pixel in range(1024):
for x_pixel in range(1024):
x_ray, y_ray = x_pixel / 512 - 1, y_pixel / 512 - 1
min_distance, hit_grey_value = None, 0
for sphere in spheres:
x_sphere, y_sphere, z_sphere, radius, grey = sphere
a = (
(x_ray - x_sphere) ** 2 + (y_ray - y_sphere) ** 2
) ** 0.5
if a < radius:
distance = z_sphere - (radius ** 2 - a ** 2) ** 0.5
if min_distance is None or distance < min_distance:
min_distance = distance
hit_grey_value = grey
values.append(hit_grey_value)
return values
import png
test_case = [
[0, 0, 1000, 2, 192],
[-2, 2, 100, 3, 98],
[2, 2, 100, 3, 98],
[0.5, -0.5, 500, 0.2, 255],
[-0.349469, -0.1, 20.000279, 0.6, 64],
[-0.349775, -0.100273, 20.000485, 0.6, 85],
[-0.350214, -0.100386, 20.000407, 0.6, 106],
[-0.350527, -0.100273, 20.000091, 0.6, 127],
[-0.350531, -0.1, 19.999721, 0.6, 148],
[-0.350225, -0.099727, 19.999515, 0.6, 169],
[-0.349786, -0.099614, 19.999593, 0.6, 190],
[-0.349473, -0.099727, 19.999909, 0.6, 211],
[0.20032, 0.349824, 15.000456, 0.3, 0],
[0.20032, 0.349517, 14.999924, 0.3, 0],
[0.19968, 0.350483, 15.000076, 0.3, 0],
[0.19968, 0.350176, 14.999544, 0.3, 0],
[0.200578, 0.350073, 14.999958, 0.3, 0],
[0.199817, 0.349519, 15.000277, 0.3, 0],
[0.200183, 0.350481, 14.999723, 0.3, 0],
[0.199422, 0.349927, 15.000042, 0.3, 0],
[0.200235, 0.35042, 15.000332, 0.3, 0],
[0.199765, 0.350078, 15.000529, 0.3, 0],
[0.200235, 0.349922, 14.999471, 0.3, 0],
[0.199765, 0.34958, 14.999668, 0.3, 0],
[0.200429, 0.350419, 15.000005, 0.3, 255],
[0.199571, 0.349795, 15.000366, 0.3, 255],
[0.200429, 0.350205, 14.999634, 0.3, 255],
[0.199571, 0.349581, 14.999995, 0.3, 255],
[0.200524, 0.349747, 15.000146, 0.3, 255],
[0.200196, 0.349509, 15.000283, 0.3, 255],
[0.199804, 0.350491, 14.999717, 0.3, 255],
[0.199476, 0.350253, 14.999854, 0.3, 255],
[0.200138, 0.349578, 14.999597, 0.3, 255],
[0.200138, 0.350138, 15.000567, 0.3, 255],
[0.199862, 0.349862, 14.999433, 0.3, 255],
[0.199862, 0.350422, 15.000403, 0.3, 255],
[0.200488, 0.350136, 15.000321, 0.3, 255],
[0.199957, 0.349751, 15.000544, 0.3, 255],
[0.200488, 0.34979, 14.999721, 0.3, 255],
[0.199957, 0.349404, 14.999944, 0.3, 255],
[0.200043, 0.350596, 15.000056, 0.3, 255],
[0.199512, 0.35021, 15.000279, 0.3, 255],
[0.200043, 0.350249, 14.999456, 0.3, 255],
[0.199512, 0.349864, 14.999679, 0.3, 255],
[-0.5, 0.500442, 10.000442, 0.2, 0],
[-0.5, 0.5, 10, 0.1995, 255],
[-0.499966, 0.49971, 9.999563, 0.199, 0],
[-0.500034, 0.499563, 9.99971, 0.199, 0],
[-0.499934, 0.499432, 9.999144, 0.1985, 255],
[-0.500066, 0.499144, 9.999432, 0.1985, 255],
[0.6, 0.5, 10, 0.25, 210],
[0.600122, 0.4999, 10.000122, 0.25, 210],
[0.600061, 0.49995, 10.000061, 0.25003, 50],
[0.595856, 0.505256, 9.991271, 0.24, 100],
[0.597973, 0.507461, 9.992175, 0.24, 210],
[0.595362, 0.507926, 9.993945, 0.24, 100],
[0.594618, 0.509055, 9.996832, 0.24, 210],
[0.592382, 0.506906, 9.996092, 0.24, 100],
[0.590438, 0.504882, 9.997604, 0.24, 210],
[0.591035, 0.503605, 9.994744, 0.24, 100],
[0.59121, 0.500708, 9.993425, 0.24, 210],
[0.593181, 0.502585, 9.991764, 0.24, 100],
[0.595867, 0.502302, 9.990069, 0.24, 210],
]
values = grey_values(test_case)
rows = [values[row * 1024:row * 1024 + 1024] for row in range(1024)]
with open('combination_scene.png', 'wb') as file:
w = png.Writer(1024, 1024, greyscale=True)
w.write(file, rows)
```
### Rust
The function `grey_values` would be sufficient as an entry to this challenge. The rest of the code saves the output as a PNG file.
The `png` crate can be installed using `cargo add png`. The `png` crate is not required for the `grey_values` function, only for saving as a PNG file.
```rust
fn grey_values(spheres: &Vec<[f32; 5]>) -> Vec<u8> {
let mut values = vec![];
for y_pixel in 0..1024 {
for x_pixel in 0..1024 {
let x_ray = x_pixel as f32 / 512.0 - 1.0;
let y_ray = y_pixel as f32 / 512.0 - 1.0;
let mut min_distance = f32::INFINITY;
let mut hit_grey_value = 0;
for sphere in spheres {
let [
x_sphere, y_sphere, z_sphere, radius, grey
] = *sphere;
let a = (
(x_ray - x_sphere).powf(2.0)
+ (y_ray - y_sphere).powf(2.0)
).sqrt();
if a < radius {
let distance = z_sphere - (
radius.powf(2.0) - a.powf(2.0)
).sqrt();
if distance < min_distance {
min_distance = distance;
hit_grey_value = grey as u8;
}
}
}
values.push(hit_grey_value);
}
}
values
}
fn main() {
let test_case: Vec<[f32; 5]> = vec![
[0.0, 0.0, 1000.0, 2.0, 192.0],
[-2.0, 2.0, 100.0, 3.0, 98.0],
[2.0, 2.0, 100.0, 3.0, 98.0],
[0.5, -0.5, 500.0, 0.2, 255.0],
[-0.349469, -0.1, 20.000279, 0.6, 64.0],
[-0.349775, -0.100273, 20.000485, 0.6, 85.0],
[-0.350214, -0.100386, 20.000407, 0.6, 106.0],
[-0.350527, -0.100273, 20.000091, 0.6, 127.0],
[-0.350531, -0.1, 19.999721, 0.6, 148.0],
[-0.350225, -0.099727, 19.999515, 0.6, 169.0],
[-0.349786, -0.099614, 19.999593, 0.6, 190.0],
[-0.349473, -0.099727, 19.999909, 0.6, 211.0],
[0.20032, 0.349824, 15.000456, 0.3, 0.0],
[0.20032, 0.349517, 14.999924, 0.3, 0.0],
[0.19968, 0.350483, 15.000076, 0.3, 0.0],
[0.19968, 0.350176, 14.999544, 0.3, 0.0],
[0.200578, 0.350073, 14.999958, 0.3, 0.0],
[0.199817, 0.349519, 15.000277, 0.3, 0.0],
[0.200183, 0.350481, 14.999723, 0.3, 0.0],
[0.199422, 0.349927, 15.000042, 0.3, 0.0],
[0.200235, 0.35042, 15.000332, 0.3, 0.0],
[0.199765, 0.350078, 15.000529, 0.3, 0.0],
[0.200235, 0.349922, 14.999471, 0.3, 0.0],
[0.199765, 0.34958, 14.999668, 0.3, 0.0],
[0.200429, 0.350419, 15.000005, 0.3, 255.0],
[0.199571, 0.349795, 15.000366, 0.3, 255.0],
[0.200429, 0.350205, 14.999634, 0.3, 255.0],
[0.199571, 0.349581, 14.999995, 0.3, 255.0],
[0.200524, 0.349747, 15.000146, 0.3, 255.0],
[0.200196, 0.349509, 15.000283, 0.3, 255.0],
[0.199804, 0.350491, 14.999717, 0.3, 255.0],
[0.199476, 0.350253, 14.999854, 0.3, 255.0],
[0.200138, 0.349578, 14.999597, 0.3, 255.0],
[0.200138, 0.350138, 15.000567, 0.3, 255.0],
[0.199862, 0.349862, 14.999433, 0.3, 255.0],
[0.199862, 0.350422, 15.000403, 0.3, 255.0],
[0.200488, 0.350136, 15.000321, 0.3, 255.0],
[0.199957, 0.349751, 15.000544, 0.3, 255.0],
[0.200488, 0.34979, 14.999721, 0.3, 255.0],
[0.199957, 0.349404, 14.999944, 0.3, 255.0],
[0.200043, 0.350596, 15.000056, 0.3, 255.0],
[0.199512, 0.35021, 15.000279, 0.3, 255.0],
[0.200043, 0.350249, 14.999456, 0.3, 255.0],
[0.199512, 0.349864, 14.999679, 0.3, 255.0],
[-0.5, 0.500442, 10.000442, 0.2, 0.0],
[-0.5, 0.5, 10.0, 0.1995, 255.0],
[-0.499966, 0.49971, 9.999563, 0.199, 0.0],
[-0.500034, 0.499563, 9.99971, 0.199, 0.0],
[-0.499934, 0.499432, 9.999144, 0.1985, 255.0],
[-0.500066, 0.499144, 9.999432, 0.1985, 255.0],
[0.6, 0.5, 10.0, 0.25, 210.0],
[0.600122, 0.4999, 10.000122, 0.25, 210.0],
[0.600061, 0.49995, 10.000061, 0.25003, 50.0],
[0.595856, 0.505256, 9.991271, 0.24, 100.0],
[0.597973, 0.507461, 9.992175, 0.24, 210.0],
[0.595362, 0.507926, 9.993945, 0.24, 100.0],
[0.594618, 0.509055, 9.996832, 0.24, 210.0],
[0.592382, 0.506906, 9.996092, 0.24, 100.0],
[0.590438, 0.504882, 9.997604, 0.24, 210.0],
[0.591035, 0.503605, 9.994744, 0.24, 100.0],
[0.59121, 0.500708, 9.993425, 0.24, 210.0],
[0.593181, 0.502585, 9.991764, 0.24, 100.0],
[0.595867, 0.502302, 9.990069, 0.24, 210.0],
];
let values = grey_values(&test_case);
let path = std::path::Path::new("combination_scene.png");
let file = std::fs::File::create(path).unwrap();
let w = &mut std::io::BufWriter::new(file);
let mut encoder = png::Encoder::new(w, 1024, 1024);
encoder.set_color(png::ColorType::Grayscale);
encoder.set_depth(png::BitDepth::Eight);
let mut writer = encoder.write_header().unwrap();
writer.write_image_data(&values).unwrap();
}
```
## Test cases
The output images may be scaled down to fit this page. Click on any image to open it full size.
### Single sphere
Input:
```text
[
[0, 0, 0, 1, 128],
]
```
Output:
[![Grey circle on a black background](https://codegolf.codidact.com/uploads/jlru163pa2pgm7aaq70b2tff16uf)](https://codegolf.codidact.com/uploads/jlru163pa2pgm7aaq70b2tff16uf)
### Two separated spheres
Input:
```text
[
[0, -0.55, 0, 0.4, 100],
[0, 0.55, 0, 0.4, 200],
]
```
Output:
[![Light and dark grey circles on a black background](https://codegolf.codidact.com/uploads/8yw7agli78y65kiomb1u5yebs7c5)](https://codegolf.codidact.com/uploads/8yw7agli78y65kiomb1u5yebs7c5)
### Ring of spheres
Input:
```text
[
[0.75, 0, 0, 0.25, 64],
[0.53033, 0.375, -0.375, 0.25, 185],
[0, 0.53033, -0.53033, 0.25, 64],
[-0.53033, 0.375, -0.375, 0.25, 185],
[-0.75, 0, 0, 0.25, 64],
[-0.53033, -0.375, 0.375, 0.25, 185],
[0, -0.53033, 0.53033, 0.25, 64],
[0.53033, -0.375, 0.375, 0.25, 185],
]
```
Output:
[![Eight circles in alternating light and dark grey in a ring some overlapping](https://codegolf.codidact.com/uploads/0lsenynig27au0sryf3ruozt1uwh)](https://codegolf.codidact.com/uploads/0lsenynig27au0sryf3ruozt1uwh)
### Beach ball
Input:
```text
[
[0.001, 0, 0, 1, 64],
[0.000707, 0.000707, 0, 1, 85],
[0, 0.001, 0, 1, 106],
[-0.000707, 0.000707, 0, 1, 127],
[-0.001, 0, 0, 1, 148],
[-0.000707, -0.000707, 0, 1, 169],
[0, -0.001, 0, 1, 190],
[0.000707, -0.000707, 0, 1, 211],
]
```
Output:
[![A circle with eight sectors in different shades of grey](https://codegolf.codidact.com/uploads/51a3ni0cfwsqcdw5b49gy6282qno)](https://codegolf.codidact.com/uploads/51a3ni0cfwsqcdw5b49gy6282qno)
### Football
Input:
```text
[
[0, 0.000058, 0.001947, 1, 0],
[0, -0.001715, 0.000923, 1, 0],
[0, 0.001715, -0.000923, 1, 0],
[0, -0.000058, -0.001947, 1, 0],
[0.001657, -0.000512, 0.000887, 1, 0],
[-0.001657, -0.000512, 0.000887, 1, 0],
[0.001657, 0.000512, -0.000887, 1, 0],
[-0.001657, 0.000512, -0.000887, 1, 0],
[0.001024, 0.001435, 0.000829, 1, 0],
[-0.001024, 0.001435, 0.000829, 1, 0],
[0.001024, -0.001435, -0.000829, 1, 0],
[-0.001024, -0.001435, -0.000829, 1, 0],
[0.001868, 0.000618, 0.000357, 1, 255],
[-0.001868, 0.000618, 0.000357, 1, 255],
[0.001868, -0.000618, -0.000357, 1, 255],
[-0.001868, -0.000618, -0.000357, 1, 255],
[0.000714, -0.000934, 0.001618, 1, 255],
[-0.000714, -0.000934, 0.001618, 1, 255],
[0.000714, 0.000934, -0.001618, 1, 255],
[-0.000714, 0.000934, -0.001618, 1, 255],
[0, -0.001975, -0.000316, 1, 255],
[0, 0.001261, 0.001552, 1, 255],
[0, -0.001261, -0.001552, 1, 255],
[0, 0.001975, 0.000316, 1, 255],
[0.001155, 0.000423, 0.001577, 1, 255],
[-0.001155, 0.000423, 0.001577, 1, 255],
[0.001155, -0.001577, 0.000423, 1, 255],
[-0.001155, -0.001577, 0.000423, 1, 255],
[0.001155, 0.001577, -0.000423, 1, 255],
[-0.001155, 0.001577, -0.000423, 1, 255],
[0.001155, -0.000423, -0.001577, 1, 255],
[-0.001155, -0.000423, -0.001577, 1, 255],
]
```
Output:
[![Black pentagons on a white ball](https://codegolf.codidact.com/uploads/bmrbsw65p59imvwu8ssd42uvvx1r)](https://codegolf.codidact.com/uploads/bmrbsw65p59imvwu8ssd42uvvx1r)
### Eight ball
Input:
```text
[
[0, 0, 1000, 2, 128],
[0, 0, 0.003125, 1, 0],
[0, 0, 0, 0.9975, 255],
[0, 0.000547, -0.002573, 0.995, 0],
[0, -0.000547, -0.002573, 0.995, 0],
[0, 0.001071, -0.005037, 0.9925, 255],
[0, -0.001071, -0.005037, 0.9925, 255],
]
```
Output:
[![A ball with the number 8 in a white circle](https://codegolf.codidact.com/uploads/dvl68crpmy62rqt94mwanditt4wc)](https://codegolf.codidact.com/uploads/dvl68crpmy62rqt94mwanditt4wc)
### Star ball
Input:
```text
[
[0, 0, 0, 1, 210],
[0, 0, 0.0008, 1, 210],
[0, 0, 0.0004, 1.00012, 50],
[0.012969, 0, -0.042045, 0.96, 100],
[0.016397, 0.011913, -0.039054, 0.96, 210],
[0.004008, 0.012334, -0.042045, 0.96, 100],
[-0.006263, 0.019276, -0.039054, 0.96, 210],
[-0.010492, 0.007623, -0.042045, 0.96, 100],
[-0.020268, 0, -0.039054, 0.96, 210],
[-0.010492, -0.007623, -0.042045, 0.96, 100],
[-0.006263, -0.019276, -0.039054, 0.96, 210],
[0.004008, -0.012334, -0.042045, 0.96, 100],
[0.016397, -0.011913, -0.039054, 0.96, 210],
]
```
Output:
[![A light grey ball with a dark grey five pointed star](https://codegolf.codidact.com/uploads/f3wdgwru6j35ue72x25mazxikfx2)](https://codegolf.codidact.com/uploads/f3wdgwru6j35ue72x25mazxikfx2)
### Combination scene
Input:
```text
[
[0, 0, 1000, 2, 192],
[-2, 2, 100, 3, 98],
[2, 2, 100, 3, 98],
[0.5, -0.5, 500, 0.2, 255],
[-0.349469, -0.1, 20.000279, 0.6, 64],
[-0.349775, -0.100273, 20.000485, 0.6, 85],
[-0.350214, -0.100386, 20.000407, 0.6, 106],
[-0.350527, -0.100273, 20.000091, 0.6, 127],
[-0.350531, -0.1, 19.999721, 0.6, 148],
[-0.350225, -0.099727, 19.999515, 0.6, 169],
[-0.349786, -0.099614, 19.999593, 0.6, 190],
[-0.349473, -0.099727, 19.999909, 0.6, 211],
[0.20032, 0.349824, 15.000456, 0.3, 0],
[0.20032, 0.349517, 14.999924, 0.3, 0],
[0.19968, 0.350483, 15.000076, 0.3, 0],
[0.19968, 0.350176, 14.999544, 0.3, 0],
[0.200578, 0.350073, 14.999958, 0.3, 0],
[0.199817, 0.349519, 15.000277, 0.3, 0],
[0.200183, 0.350481, 14.999723, 0.3, 0],
[0.199422, 0.349927, 15.000042, 0.3, 0],
[0.200235, 0.35042, 15.000332, 0.3, 0],
[0.199765, 0.350078, 15.000529, 0.3, 0],
[0.200235, 0.349922, 14.999471, 0.3, 0],
[0.199765, 0.34958, 14.999668, 0.3, 0],
[0.200429, 0.350419, 15.000005, 0.3, 255],
[0.199571, 0.349795, 15.000366, 0.3, 255],
[0.200429, 0.350205, 14.999634, 0.3, 255],
[0.199571, 0.349581, 14.999995, 0.3, 255],
[0.200524, 0.349747, 15.000146, 0.3, 255],
[0.200196, 0.349509, 15.000283, 0.3, 255],
[0.199804, 0.350491, 14.999717, 0.3, 255],
[0.199476, 0.350253, 14.999854, 0.3, 255],
[0.200138, 0.349578, 14.999597, 0.3, 255],
[0.200138, 0.350138, 15.000567, 0.3, 255],
[0.199862, 0.349862, 14.999433, 0.3, 255],
[0.199862, 0.350422, 15.000403, 0.3, 255],
[0.200488, 0.350136, 15.000321, 0.3, 255],
[0.199957, 0.349751, 15.000544, 0.3, 255],
[0.200488, 0.34979, 14.999721, 0.3, 255],
[0.199957, 0.349404, 14.999944, 0.3, 255],
[0.200043, 0.350596, 15.000056, 0.3, 255],
[0.199512, 0.35021, 15.000279, 0.3, 255],
[0.200043, 0.350249, 14.999456, 0.3, 255],
[0.199512, 0.349864, 14.999679, 0.3, 255],
[-0.5, 0.500442, 10.000442, 0.2, 0],
[-0.5, 0.5, 10, 0.1995, 255],
[-0.499966, 0.49971, 9.999563, 0.199, 0],
[-0.500034, 0.499563, 9.99971, 0.199, 0],
[-0.499934, 0.499432, 9.999144, 0.1985, 255],
[-0.500066, 0.499144, 9.999432, 0.1985, 255],
[0.6, 0.5, 10, 0.25, 210],
[0.600122, 0.4999, 10.000122, 0.25, 210],
[0.600061, 0.49995, 10.000061, 0.25003, 50],
[0.595856, 0.505256, 9.991271, 0.24, 100],
[0.597973, 0.507461, 9.992175, 0.24, 210],
[0.595362, 0.507926, 9.993945, 0.24, 100],
[0.594618, 0.509055, 9.996832, 0.24, 210],
[0.592382, 0.506906, 9.996092, 0.24, 100],
[0.590438, 0.504882, 9.997604, 0.24, 210],
[0.591035, 0.503605, 9.994744, 0.24, 100],
[0.59121, 0.500708, 9.993425, 0.24, 210],
[0.593181, 0.502585, 9.991764, 0.24, 100],
[0.595867, 0.502302, 9.990069, 0.24, 210],
]
```
Output:
[![A beachball, football, eightball, and starball with the sun behind hills in the background](https://codegolf.codidact.com/uploads/usv9npm1t5bjpcaknn6rbzgkqwmi)](https://codegolf.codidact.com/uploads/usv9npm1t5bjpcaknn6rbzgkqwmi)
Output using 32 bit floating point rather than 64 bit floating point:
[![A beachball, football, eightball, and starball with the sun behind hills in the background](https://codegolf.codidact.com/uploads/7uuxjxq7a9oxc9q0jf2bi8l68qk3)](https://codegolf.codidact.com/uploads/7uuxjxq7a9oxc9q0jf2bi8l68qk3)
The inaccuracies in the 32 bit floating point version are subtle. You may need to click on the image to open it full size to see the occasional inaccurate pixel. For example, around the inside of the holes in the 8 on the eightball.
## Explanatory animations
The example images were produced from plain grey spheres with no surface pattern or texture. The following animations show how they break down into their component plain spheres.
<details>
<summary>Animations</summary>
TODO
</details>
## Scoring
This is a [code golf challenge](https://codegolf.codidact.com/categories/49/tags/4274). Your score is the number of bytes in your code. Lowest score for each language wins.
> Explanations are optional, but I'm more likely to upvote answers that have one.

Given a list of spheres, output an image.
Produce a greyscale [orthographic](https://en.wikipedia.org/wiki/Parallel_projection#Orthographic_projection) (parallel projection - no perspective) ray caster, that renders only plain grey spheres with no lighting.
## Input
- A list of spheres, each consisting of:
- The coordinates of its centre $(x,y,z)$.
- Its radius $r$.
- Its greyscale value $g$, an integer from $0$ to $255$ (both inclusive).
- Only the greyscale value will be restricted to integer values. The coordinates and radius may have fractional values.
## Output
- An image $1024$ by $1024$ pixels with greyscale values from $0$ to $255$.
- The image covers x and y values from $-1$ to $1$.
- A sphere's greyscale value is applied to every visible part of it. There is no variation over the surface of the sphere, so it appears flat like a disk.
- You may output whatever image format is convenient, including simple text formats such as [PGM](https://en.wikipedia.org/wiki/Netpbm#PGM_example) or a list of pixel greyscale values. If you want to include an example of your output as an image in your answer, the image formats that Codidact supports are PNG, JPEG, and GIF. You could output PGM and then convert to PNG to upload to your answer. If you output just the greyscale values as a list of numbers from $0$ to $255$, I've made a `[ TODO page to convert these to greyscale PNG]`.
- You are free to output an image reflected or rotated by a multiple of 90 degrees. However, the z axis is required to be increasing in the forwards direction (into the image) otherwise some of the test cases will look incorrect.
## Calculating a pixel value
Imagine a ray passing into the image through a pixel, perpendicular to the image plane. If it hits a sphere, it shows the greyscale value of the first sphere it hits. Otherwise, it shows a value of zero (black).
For example, for each pixel:
- The pixel $x,y$ coordinates will each be from $0$ to $1023$. Scale these to the range $-1$ to $1$ (for example, divide by $512$ and subtract $1$).
- If these scaled $x,y$ coordinates are within the radius $r$ of the $x,y$ coordinates of a sphere's centre, the ray intersects that sphere (the $z$ coordinate is not relevant to this test because the image is an orthographic projection - the rays are all parallel to the $z$ axis). The distance $a$ of the ray from the sphere's centre can be calculated as follows:
$$a=\sqrt{(x_{ray}-x_{sphere})^2+(y_{ray}-y_{sphere})^2}$$
```text
CONSIDER A DIAGRAM HERE
```
- For each sphere that the ray intersects, find the $z$ coordinate of the nearest intersection (there will often be $2$ per sphere, $1$ on the front of the sphere and $1$ on the back of the sphere, as the ray passes through - only the nearest is relevant). This can be calculated as follows:
- The distance $d$ to the nearest intersection point can be calculated from the $z$ coordinate of the sphere's centre, the radius $r$, and the distance $a$ from the $x,y$ coordinates of the sphere's centre to the scaled $x,y$ coordinates of the pixel:
$$d=z_{sphere}-\sqrt{r_{sphere}^2-a^2}$$
```text
CONSIDER A DIAGRAM HERE
```
- Of all the spheres that the ray intersects, the sphere with the intersection point with the smallest $z$ coordinate $d$ is the one to display. Set the pixel to the greyscale value of this sphere.
- If the ray did not intersect any spheres, set the pixel to zero (black).
## Example implementations
These are ungolfed code to supplement understanding of the requirements. You are free to golf these or implement your own approach that matches the test cases up to reflection & rotation.
The Python implementation uses 64 bit floating point. The Rust implementation uses 32 bit floating point to show the difference in output. I've tried to optimise the test cases for compatibility with 32 bit floating point, so the differences are restricted to being a pixel out in some places.
### Python
The function `grey_values` would be sufficient as an entry to this challenge. The rest of the code saves the output as a PNG file. The function itself has no dependence on the `import png` so it would not be required in an entry.
Instructions for installing the `png` package can be found on its [project page on PyPI](https://pypi.org/project/pypng/).
```python
def grey_values(spheres):
values = []
for y_pixel in range(1024):
for x_pixel in range(1024):
x_ray, y_ray = x_pixel / 512 - 1, y_pixel / 512 - 1
min_distance, hit_grey_value = None, 0
for sphere in spheres:
x_sphere, y_sphere, z_sphere, radius, grey = sphere
a = (
(x_ray - x_sphere) ** 2 + (y_ray - y_sphere) ** 2
) ** 0.5
if a < radius:
distance = z_sphere - (radius ** 2 - a ** 2) ** 0.5
if min_distance is None or distance < min_distance:
min_distance = distance
hit_grey_value = grey
values.append(hit_grey_value)
return values
import png
test_case = [
[0, 0, 1000, 2, 192],
[-2, 2, 100, 3, 98],
[2, 2, 100, 3, 98],
[0.5, -0.5, 500, 0.2, 255],
[-0.349469, -0.1, 20.000279, 0.6, 64],
[-0.349775, -0.100273, 20.000485, 0.6, 85],
[-0.350214, -0.100386, 20.000407, 0.6, 106],
[-0.350527, -0.100273, 20.000091, 0.6, 127],
[-0.350531, -0.1, 19.999721, 0.6, 148],
[-0.350225, -0.099727, 19.999515, 0.6, 169],
[-0.349786, -0.099614, 19.999593, 0.6, 190],
[-0.349473, -0.099727, 19.999909, 0.6, 211],
[0.20032, 0.349824, 15.000456, 0.3, 0],
[0.20032, 0.349517, 14.999924, 0.3, 0],
[0.19968, 0.350483, 15.000076, 0.3, 0],
[0.19968, 0.350176, 14.999544, 0.3, 0],
[0.200578, 0.350073, 14.999958, 0.3, 0],
[0.199817, 0.349519, 15.000277, 0.3, 0],
[0.200183, 0.350481, 14.999723, 0.3, 0],
[0.199422, 0.349927, 15.000042, 0.3, 0],
[0.200235, 0.35042, 15.000332, 0.3, 0],
[0.199765, 0.350078, 15.000529, 0.3, 0],
[0.200235, 0.349922, 14.999471, 0.3, 0],
[0.199765, 0.34958, 14.999668, 0.3, 0],
[0.200429, 0.350419, 15.000005, 0.3, 255],
[0.199571, 0.349795, 15.000366, 0.3, 255],
[0.200429, 0.350205, 14.999634, 0.3, 255],
[0.199571, 0.349581, 14.999995, 0.3, 255],
[0.200524, 0.349747, 15.000146, 0.3, 255],
[0.200196, 0.349509, 15.000283, 0.3, 255],
[0.199804, 0.350491, 14.999717, 0.3, 255],
[0.199476, 0.350253, 14.999854, 0.3, 255],
[0.200138, 0.349578, 14.999597, 0.3, 255],
[0.200138, 0.350138, 15.000567, 0.3, 255],
[0.199862, 0.349862, 14.999433, 0.3, 255],
[0.199862, 0.350422, 15.000403, 0.3, 255],
[0.200488, 0.350136, 15.000321, 0.3, 255],
[0.199957, 0.349751, 15.000544, 0.3, 255],
[0.200488, 0.34979, 14.999721, 0.3, 255],
[0.199957, 0.349404, 14.999944, 0.3, 255],
[0.200043, 0.350596, 15.000056, 0.3, 255],
[0.199512, 0.35021, 15.000279, 0.3, 255],
[0.200043, 0.350249, 14.999456, 0.3, 255],
[0.199512, 0.349864, 14.999679, 0.3, 255],
[-0.5, 0.500442, 10.000442, 0.2, 0],
[-0.5, 0.5, 10, 0.1995, 255],
[-0.499966, 0.49971, 9.999563, 0.199, 0],
[-0.500034, 0.499563, 9.99971, 0.199, 0],
[-0.499934, 0.499432, 9.999144, 0.1985, 255],
[-0.500066, 0.499144, 9.999432, 0.1985, 255],
[0.6, 0.5, 10, 0.25, 210],
[0.600122, 0.4999, 10.000122, 0.25, 210],
[0.600061, 0.49995, 10.000061, 0.25003, 50],
[0.595856, 0.505256, 9.991271, 0.24, 100],
[0.597973, 0.507461, 9.992175, 0.24, 210],
[0.595362, 0.507926, 9.993945, 0.24, 100],
[0.594618, 0.509055, 9.996832, 0.24, 210],
[0.592382, 0.506906, 9.996092, 0.24, 100],
[0.590438, 0.504882, 9.997604, 0.24, 210],
[0.591035, 0.503605, 9.994744, 0.24, 100],
[0.59121, 0.500708, 9.993425, 0.24, 210],
[0.593181, 0.502585, 9.991764, 0.24, 100],
[0.595867, 0.502302, 9.990069, 0.24, 210],
]
values = grey_values(test_case)
rows = [values[row * 1024:row * 1024 + 1024] for row in range(1024)]
with open('combination_scene.png', 'wb') as file:
w = png.Writer(1024, 1024, greyscale=True)
w.write(file, rows)
```
### Rust
The function `grey_values` would be sufficient as an entry to this challenge. The rest of the code saves the output as a PNG file.
The `png` crate can be installed using `cargo add png`. The `png` crate is not required for the `grey_values` function, only for saving as a PNG file.
```rust
fn grey_values(spheres: &Vec<[f32; 5]>) -> Vec<u8> {
let mut values = vec![];
for y_pixel in 0..1024 {
for x_pixel in 0..1024 {
let x_ray = x_pixel as f32 / 512.0 - 1.0;
let y_ray = y_pixel as f32 / 512.0 - 1.0;
let mut min_distance = f32::INFINITY;
let mut hit_grey_value = 0;
for sphere in spheres {
let [
x_sphere, y_sphere, z_sphere, radius, grey
] = *sphere;
let a = (
(x_ray - x_sphere).powf(2.0)
+ (y_ray - y_sphere).powf(2.0)
).sqrt();
if a < radius {
let distance = z_sphere - (
radius.powf(2.0) - a.powf(2.0)
).sqrt();
if distance < min_distance {
min_distance = distance;
hit_grey_value = grey as u8;
}
}
}
values.push(hit_grey_value);
}
}
values
}
fn main() {
let test_case: Vec<[f32; 5]> = vec![
[0.0, 0.0, 1000.0, 2.0, 192.0],
[-2.0, 2.0, 100.0, 3.0, 98.0],
[2.0, 2.0, 100.0, 3.0, 98.0],
[0.5, -0.5, 500.0, 0.2, 255.0],
[-0.349469, -0.1, 20.000279, 0.6, 64.0],
[-0.349775, -0.100273, 20.000485, 0.6, 85.0],
[-0.350214, -0.100386, 20.000407, 0.6, 106.0],
[-0.350527, -0.100273, 20.000091, 0.6, 127.0],
[-0.350531, -0.1, 19.999721, 0.6, 148.0],
[-0.350225, -0.099727, 19.999515, 0.6, 169.0],
[-0.349786, -0.099614, 19.999593, 0.6, 190.0],
[-0.349473, -0.099727, 19.999909, 0.6, 211.0],
[0.20032, 0.349824, 15.000456, 0.3, 0.0],
[0.20032, 0.349517, 14.999924, 0.3, 0.0],
[0.19968, 0.350483, 15.000076, 0.3, 0.0],
[0.19968, 0.350176, 14.999544, 0.3, 0.0],
[0.200578, 0.350073, 14.999958, 0.3, 0.0],
[0.199817, 0.349519, 15.000277, 0.3, 0.0],
[0.200183, 0.350481, 14.999723, 0.3, 0.0],
[0.199422, 0.349927, 15.000042, 0.3, 0.0],
[0.200235, 0.35042, 15.000332, 0.3, 0.0],
[0.199765, 0.350078, 15.000529, 0.3, 0.0],
[0.200235, 0.349922, 14.999471, 0.3, 0.0],
[0.199765, 0.34958, 14.999668, 0.3, 0.0],
[0.200429, 0.350419, 15.000005, 0.3, 255.0],
[0.199571, 0.349795, 15.000366, 0.3, 255.0],
[0.200429, 0.350205, 14.999634, 0.3, 255.0],
[0.199571, 0.349581, 14.999995, 0.3, 255.0],
[0.200524, 0.349747, 15.000146, 0.3, 255.0],
[0.200196, 0.349509, 15.000283, 0.3, 255.0],
[0.199804, 0.350491, 14.999717, 0.3, 255.0],
[0.199476, 0.350253, 14.999854, 0.3, 255.0],
[0.200138, 0.349578, 14.999597, 0.3, 255.0],
[0.200138, 0.350138, 15.000567, 0.3, 255.0],
[0.199862, 0.349862, 14.999433, 0.3, 255.0],
[0.199862, 0.350422, 15.000403, 0.3, 255.0],
[0.200488, 0.350136, 15.000321, 0.3, 255.0],
[0.199957, 0.349751, 15.000544, 0.3, 255.0],
[0.200488, 0.34979, 14.999721, 0.3, 255.0],
[0.199957, 0.349404, 14.999944, 0.3, 255.0],
[0.200043, 0.350596, 15.000056, 0.3, 255.0],
[0.199512, 0.35021, 15.000279, 0.3, 255.0],
[0.200043, 0.350249, 14.999456, 0.3, 255.0],
[0.199512, 0.349864, 14.999679, 0.3, 255.0],
[-0.5, 0.500442, 10.000442, 0.2, 0.0],
[-0.5, 0.5, 10.0, 0.1995, 255.0],
[-0.499966, 0.49971, 9.999563, 0.199, 0.0],
[-0.500034, 0.499563, 9.99971, 0.199, 0.0],
[-0.499934, 0.499432, 9.999144, 0.1985, 255.0],
[-0.500066, 0.499144, 9.999432, 0.1985, 255.0],
[0.6, 0.5, 10.0, 0.25, 210.0],
[0.600122, 0.4999, 10.000122, 0.25, 210.0],
[0.600061, 0.49995, 10.000061, 0.25003, 50.0],
[0.595856, 0.505256, 9.991271, 0.24, 100.0],
[0.597973, 0.507461, 9.992175, 0.24, 210.0],
[0.595362, 0.507926, 9.993945, 0.24, 100.0],
[0.594618, 0.509055, 9.996832, 0.24, 210.0],
[0.592382, 0.506906, 9.996092, 0.24, 100.0],
[0.590438, 0.504882, 9.997604, 0.24, 210.0],
[0.591035, 0.503605, 9.994744, 0.24, 100.0],
[0.59121, 0.500708, 9.993425, 0.24, 210.0],
[0.593181, 0.502585, 9.991764, 0.24, 100.0],
[0.595867, 0.502302, 9.990069, 0.24, 210.0],
];
let values = grey_values(&test_case);
let path = std::path::Path::new("combination_scene.png");
let file = std::fs::File::create(path).unwrap();
let w = &mut std::io::BufWriter::new(file);
let mut encoder = png::Encoder::new(w, 1024, 1024);
encoder.set_color(png::ColorType::Grayscale);
encoder.set_depth(png::BitDepth::Eight);
let mut writer = encoder.write_header().unwrap();
writer.write_image_data(&values).unwrap();
}
```
## Test cases
The output images may be scaled down to fit this page. Click on any image to open it full size.
### Single sphere
Input:
```text
[
[0, 0, 0, 1, 128],
]
```
Output:
[![Grey circle on a black background](https://codegolf.codidact.com/uploads/jlru163pa2pgm7aaq70b2tff16uf)](https://codegolf.codidact.com/uploads/jlru163pa2pgm7aaq70b2tff16uf)
### Two separated spheres
Input:
```text
[
[0, -0.55, 0, 0.4, 100],
[0, 0.55, 0, 0.4, 200],
]
```
Output:
[![Light and dark grey circles on a black background](https://codegolf.codidact.com/uploads/8yw7agli78y65kiomb1u5yebs7c5)](https://codegolf.codidact.com/uploads/8yw7agli78y65kiomb1u5yebs7c5)
### Ring of spheres
Input:
```text
[
[0.75, 0, 0, 0.25, 64],
[0.53033, 0.375, -0.375, 0.25, 185],
[0, 0.53033, -0.53033, 0.25, 64],
[-0.53033, 0.375, -0.375, 0.25, 185],
[-0.75, 0, 0, 0.25, 64],
[-0.53033, -0.375, 0.375, 0.25, 185],
[0, -0.53033, 0.53033, 0.25, 64],
[0.53033, -0.375, 0.375, 0.25, 185],
]
```
Output:
[![Eight circles in alternating light and dark grey in a ring some overlapping](https://codegolf.codidact.com/uploads/0lsenynig27au0sryf3ruozt1uwh)](https://codegolf.codidact.com/uploads/0lsenynig27au0sryf3ruozt1uwh)
### Beach ball
Input:
```text
[
[0.001, 0, 0, 1, 64],
[0.000707, 0.000707, 0, 1, 85],
[0, 0.001, 0, 1, 106],
[-0.000707, 0.000707, 0, 1, 127],
[-0.001, 0, 0, 1, 148],
[-0.000707, -0.000707, 0, 1, 169],
[0, -0.001, 0, 1, 190],
[0.000707, -0.000707, 0, 1, 211],
]
```
Output:
[![A circle with eight sectors in different shades of grey](https://codegolf.codidact.com/uploads/51a3ni0cfwsqcdw5b49gy6282qno)](https://codegolf.codidact.com/uploads/51a3ni0cfwsqcdw5b49gy6282qno)
### Football
Input:
```text
[
[0, 0.000058, 0.001947, 1, 0],
[0, -0.001715, 0.000923, 1, 0],
[0, 0.001715, -0.000923, 1, 0],
[0, -0.000058, -0.001947, 1, 0],
[0.001657, -0.000512, 0.000887, 1, 0],
[-0.001657, -0.000512, 0.000887, 1, 0],
[0.001657, 0.000512, -0.000887, 1, 0],
[-0.001657, 0.000512, -0.000887, 1, 0],
[0.001024, 0.001435, 0.000829, 1, 0],
[-0.001024, 0.001435, 0.000829, 1, 0],
[0.001024, -0.001435, -0.000829, 1, 0],
[-0.001024, -0.001435, -0.000829, 1, 0],
[0.001868, 0.000618, 0.000357, 1, 255],
[-0.001868, 0.000618, 0.000357, 1, 255],
[0.001868, -0.000618, -0.000357, 1, 255],
[-0.001868, -0.000618, -0.000357, 1, 255],
[0.000714, -0.000934, 0.001618, 1, 255],
[-0.000714, -0.000934, 0.001618, 1, 255],
[0.000714, 0.000934, -0.001618, 1, 255],
[-0.000714, 0.000934, -0.001618, 1, 255],
[0, -0.001975, -0.000316, 1, 255],
[0, 0.001261, 0.001552, 1, 255],
[0, -0.001261, -0.001552, 1, 255],
[0, 0.001975, 0.000316, 1, 255],
[0.001155, 0.000423, 0.001577, 1, 255],
[-0.001155, 0.000423, 0.001577, 1, 255],
[0.001155, -0.001577, 0.000423, 1, 255],
[-0.001155, -0.001577, 0.000423, 1, 255],
[0.001155, 0.001577, -0.000423, 1, 255],
[-0.001155, 0.001577, -0.000423, 1, 255],
[0.001155, -0.000423, -0.001577, 1, 255],
[-0.001155, -0.000423, -0.001577, 1, 255],
]
```
Output:
[![Black pentagons on a white ball](https://codegolf.codidact.com/uploads/bmrbsw65p59imvwu8ssd42uvvx1r)](https://codegolf.codidact.com/uploads/bmrbsw65p59imvwu8ssd42uvvx1r)
### Eight ball
Input:
```text
[
[0, 0, 1000, 2, 128],
[0, 0, 0.003125, 1, 0],
[0, 0, 0, 0.9975, 255],
[0, 0.000547, -0.002573, 0.995, 0],
[0, -0.000547, -0.002573, 0.995, 0],
[0, 0.001071, -0.005037, 0.9925, 255],
[0, -0.001071, -0.005037, 0.9925, 255],
]
```
Output:
[![A ball with the number 8 in a white circle](https://codegolf.codidact.com/uploads/dvl68crpmy62rqt94mwanditt4wc)](https://codegolf.codidact.com/uploads/dvl68crpmy62rqt94mwanditt4wc)
### Star ball
Input:
```text
[
[0, 0, 0, 1, 210],
[0, 0, 0.0008, 1, 210],
[0, 0, 0.0004, 1.00012, 50],
[0.012969, 0, -0.042045, 0.96, 100],
[0.016397, 0.011913, -0.039054, 0.96, 210],
[0.004008, 0.012334, -0.042045, 0.96, 100],
[-0.006263, 0.019276, -0.039054, 0.96, 210],
[-0.010492, 0.007623, -0.042045, 0.96, 100],
[-0.020268, 0, -0.039054, 0.96, 210],
[-0.010492, -0.007623, -0.042045, 0.96, 100],
[-0.006263, -0.019276, -0.039054, 0.96, 210],
[0.004008, -0.012334, -0.042045, 0.96, 100],
[0.016397, -0.011913, -0.039054, 0.96, 210],
]
```
Output:
[![A light grey ball with a dark grey five pointed star](https://codegolf.codidact.com/uploads/f3wdgwru6j35ue72x25mazxikfx2)](https://codegolf.codidact.com/uploads/f3wdgwru6j35ue72x25mazxikfx2)
### Combination scene
Input:
```text
[
[0, 0, 1000, 2, 192],
[-2, 2, 100, 3, 98],
[2, 2, 100, 3, 98],
[0.5, -0.5, 500, 0.2, 255],
[-0.349469, -0.1, 20.000279, 0.6, 64],
[-0.349775, -0.100273, 20.000485, 0.6, 85],
[-0.350214, -0.100386, 20.000407, 0.6, 106],
[-0.350527, -0.100273, 20.000091, 0.6, 127],
[-0.350531, -0.1, 19.999721, 0.6, 148],
[-0.350225, -0.099727, 19.999515, 0.6, 169],
[-0.349786, -0.099614, 19.999593, 0.6, 190],
[-0.349473, -0.099727, 19.999909, 0.6, 211],
[0.20032, 0.349824, 15.000456, 0.3, 0],
[0.20032, 0.349517, 14.999924, 0.3, 0],
[0.19968, 0.350483, 15.000076, 0.3, 0],
[0.19968, 0.350176, 14.999544, 0.3, 0],
[0.200578, 0.350073, 14.999958, 0.3, 0],
[0.199817, 0.349519, 15.000277, 0.3, 0],
[0.200183, 0.350481, 14.999723, 0.3, 0],
[0.199422, 0.349927, 15.000042, 0.3, 0],
[0.200235, 0.35042, 15.000332, 0.3, 0],
[0.199765, 0.350078, 15.000529, 0.3, 0],
[0.200235, 0.349922, 14.999471, 0.3, 0],
[0.199765, 0.34958, 14.999668, 0.3, 0],
[0.200429, 0.350419, 15.000005, 0.3, 255],
[0.199571, 0.349795, 15.000366, 0.3, 255],
[0.200429, 0.350205, 14.999634, 0.3, 255],
[0.199571, 0.349581, 14.999995, 0.3, 255],
[0.200524, 0.349747, 15.000146, 0.3, 255],
[0.200196, 0.349509, 15.000283, 0.3, 255],
[0.199804, 0.350491, 14.999717, 0.3, 255],
[0.199476, 0.350253, 14.999854, 0.3, 255],
[0.200138, 0.349578, 14.999597, 0.3, 255],
[0.200138, 0.350138, 15.000567, 0.3, 255],
[0.199862, 0.349862, 14.999433, 0.3, 255],
[0.199862, 0.350422, 15.000403, 0.3, 255],
[0.200488, 0.350136, 15.000321, 0.3, 255],
[0.199957, 0.349751, 15.000544, 0.3, 255],
[0.200488, 0.34979, 14.999721, 0.3, 255],
[0.199957, 0.349404, 14.999944, 0.3, 255],
[0.200043, 0.350596, 15.000056, 0.3, 255],
[0.199512, 0.35021, 15.000279, 0.3, 255],
[0.200043, 0.350249, 14.999456, 0.3, 255],
[0.199512, 0.349864, 14.999679, 0.3, 255],
[-0.5, 0.500442, 10.000442, 0.2, 0],
[-0.5, 0.5, 10, 0.1995, 255],
[-0.499966, 0.49971, 9.999563, 0.199, 0],
[-0.500034, 0.499563, 9.99971, 0.199, 0],
[-0.499934, 0.499432, 9.999144, 0.1985, 255],
[-0.500066, 0.499144, 9.999432, 0.1985, 255],
[0.6, 0.5, 10, 0.25, 210],
[0.600122, 0.4999, 10.000122, 0.25, 210],
[0.600061, 0.49995, 10.000061, 0.25003, 50],
[0.595856, 0.505256, 9.991271, 0.24, 100],
[0.597973, 0.507461, 9.992175, 0.24, 210],
[0.595362, 0.507926, 9.993945, 0.24, 100],
[0.594618, 0.509055, 9.996832, 0.24, 210],
[0.592382, 0.506906, 9.996092, 0.24, 100],
[0.590438, 0.504882, 9.997604, 0.24, 210],
[0.591035, 0.503605, 9.994744, 0.24, 100],
[0.59121, 0.500708, 9.993425, 0.24, 210],
[0.593181, 0.502585, 9.991764, 0.24, 100],
[0.595867, 0.502302, 9.990069, 0.24, 210],
]
```
Output:
[![A beachball, football, eightball, and starball with the sun behind hills in the background](https://codegolf.codidact.com/uploads/usv9npm1t5bjpcaknn6rbzgkqwmi)](https://codegolf.codidact.com/uploads/usv9npm1t5bjpcaknn6rbzgkqwmi)
Output using 32 bit floating point rather than 64 bit floating point:
[![A beachball, football, eightball, and starball with the sun behind hills in the background](https://codegolf.codidact.com/uploads/7uuxjxq7a9oxc9q0jf2bi8l68qk3)](https://codegolf.codidact.com/uploads/7uuxjxq7a9oxc9q0jf2bi8l68qk3)
The inaccuracies in the 32 bit floating point version are subtle. You may need to click on the image to open it full size to see the occasional inaccurate pixel. For example, around the inside of the holes in the 8 on the eightball. These occasional inaccurate pixels are acceptable in your output (allowing languages which only support 32 bit floating point to compete).
More widespread speckling as in the following image would not be valid, so you may need to be careful if changing the order of calculations during golfing:
[![A scene with significant areas of inaccurate speckles obscuring the intended image](https://codegolf.codidact.com/uploads/1sa7i6z0cv2kgyrmw2tj2ytr1ps1)](https://codegolf.codidact.com/uploads/1sa7i6z0cv2kgyrmw2tj2ytr1ps1)
This exaggerated inaccuracy was achieved by adding and subtracting 1000 to the distance calculation, exceeding the available precision of 32 bit floating point and causing some distances to appear in the wrong order. Note that this same approach causes no inaccuracy with 64 bit floating point, so if your language supports both you may have more leeway for rearranging calculations if you use 64 bit floating point.
## Explanatory animations
The example images were produced from plain grey spheres with no surface pattern or texture. The following animations show how they break down into their component plain spheres.
<details>
<summary>Animations</summary>
TODO
</details>
## Scoring
This is a [code golf challenge](https://codegolf.codidact.com/categories/49/tags/4274). Your score is the number of bytes in your code. Lowest score for each language wins.
> Explanations are optional, but I'm more likely to upvote answers that have one.

#14: Post edited by

trichoplax‭ · 2024-07-26T14:33:50Z (5 months ago)
Add 32 bit floating point comparison output for final test case

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Given a list of spheres, output an image.
Produce a greyscale [orthographic](https://en.wikipedia.org/wiki/Parallel_projection#Orthographic_projection) (parallel projection - no perspective) ray caster, that renders only plain grey spheres with no lighting.
## Input
- A list of spheres, each consisting of:
- The coordinates of its centre $(x,y,z)$.
- Its radius $r$.
- Its greyscale value $g$, an integer from $0$ to $255$ (both inclusive).
- Only the greyscale value will be restricted to integer values. The coordinates and radius may have fractional values.
## Output
- An image $1024$ by $1024$ pixels with greyscale values from $0$ to $255$.
- The image covers x and y values from $-1$ to $1$.
- A sphere's greyscale value is applied to every visible part of it. There is no variation over the surface of the sphere, so it appears flat like a disk.
- You may output whatever image format is convenient, including simple text formats such as [PGM](https://en.wikipedia.org/wiki/Netpbm#PGM_example) or a list of pixel greyscale values. If you want to include an example of your output as an image in your answer, the image formats that Codidact supports are PNG, JPEG, and GIF. You could output PGM and then convert to PNG to upload to your answer. If you output just the greyscale values as a list of numbers from $0$ to $255$, I've made a `[ TODO page to convert these to greyscale PNG]`.
- You are free to output an image reflected or rotated by a multiple of 90 degrees. However, the z axis is required to be increasing in the forwards direction (into the image) otherwise some of the test cases will look incorrect.
## Calculating a pixel value
Imagine a ray passing into the image through a pixel, perpendicular to the image plane. If it hits a sphere, it shows the greyscale value of the first sphere it hits. Otherwise, it shows a value of zero (black).
For example, for each pixel:
- The pixel $x,y$ coordinates will each be from $0$ to $1023$. Scale these to the range $-1$ to $1$ (for example, divide by $512$ and subtract $1$).
- If these scaled $x,y$ coordinates are within the radius $r$ of the $x,y$ coordinates of a sphere's centre, the ray intersects that sphere (the $z$ coordinate is not relevant to this test because the image is an orthographic projection - the rays are all parallel to the $z$ axis). The distance $a$ of the ray from the sphere's centre can be calculated as follows:
$$a=\sqrt{(x_{ray}-x_{sphere})^2+(y_{ray}-y_{sphere})^2}$$
```text
CONSIDER A DIAGRAM HERE
```
- For each sphere that the ray intersects, find the $z$ coordinate of the nearest intersection (there will often be $2$ per sphere, $1$ on the front of the sphere and $1$ on the back of the sphere, as the ray passes through - only the nearest is relevant). This can be calculated as follows:
- The distance $d$ to the nearest intersection point can be calculated from the $z$ coordinate of the sphere's centre, the radius $r$, and the distance $a$ from the $x,y$ coordinates of the sphere's centre to the scaled $x,y$ coordinates of the pixel:
$$d=z_{sphere}-\sqrt{r_{sphere}^2-a^2}$$
```text
CONSIDER A DIAGRAM HERE
```
- Of all the spheres that the ray intersects, the sphere with the intersection point with the smallest $z$ coordinate $d$ is the one to display. Set the pixel to the greyscale value of this sphere.
- If the ray did not intersect any spheres, set the pixel to zero (black).
## Example implementations
These are ungolfed code to supplement understanding of the requirements. You are free to golf these or implement your own approach that matches the test cases up to reflection & rotation.
The Python implementation uses 64 bit floating point. The Rust implementation uses 32 bit floating point to show the difference in output. I've tried to optimise the test cases for compatibility with 32 bit floating point, so the differences are restricted to being a pixel out in some places.
### Python
The function `grey_values` would be sufficient as an entry to this challenge. The rest of the code saves the output as a PNG file. The function itself has no dependence on the `import png` so it would not be required in an entry.
Instructions for installing the `png` package can be found on its [project page on PyPI](https://pypi.org/project/pypng/).
```python
def grey_values(spheres):
values = []
for y_pixel in range(1024):
for x_pixel in range(1024):
x_ray, y_ray = x_pixel / 512 - 1, y_pixel / 512 - 1
min_distance, hit_grey_value = None, 0
for sphere in spheres:
x_sphere, y_sphere, z_sphere, radius, grey = sphere
a = (
(x_ray - x_sphere) ** 2 + (y_ray - y_sphere) ** 2
) ** 0.5
if a < radius:
distance = z_sphere - (radius ** 2 - a ** 2) ** 0.5
if min_distance is None or distance < min_distance:
min_distance = distance
hit_grey_value = grey
values.append(hit_grey_value)
return values
import png
test_case = [
[0, 0, 1000, 2, 192],
[-2, 2, 100, 3, 98],
[2, 2, 100, 3, 98],
[0.5, -0.5, 500, 0.2, 255],
[-0.349469, -0.1, 20.000279, 0.6, 64],
[-0.349775, -0.100273, 20.000485, 0.6, 85],
[-0.350214, -0.100386, 20.000407, 0.6, 106],
[-0.350527, -0.100273, 20.000091, 0.6, 127],
[-0.350531, -0.1, 19.999721, 0.6, 148],
[-0.350225, -0.099727, 19.999515, 0.6, 169],
[-0.349786, -0.099614, 19.999593, 0.6, 190],
[-0.349473, -0.099727, 19.999909, 0.6, 211],
[0.20032, 0.349824, 15.000456, 0.3, 0],
[0.20032, 0.349517, 14.999924, 0.3, 0],
[0.19968, 0.350483, 15.000076, 0.3, 0],
[0.19968, 0.350176, 14.999544, 0.3, 0],
[0.200578, 0.350073, 14.999958, 0.3, 0],
[0.199817, 0.349519, 15.000277, 0.3, 0],
[0.200183, 0.350481, 14.999723, 0.3, 0],
[0.199422, 0.349927, 15.000042, 0.3, 0],
[0.200235, 0.35042, 15.000332, 0.3, 0],
[0.199765, 0.350078, 15.000529, 0.3, 0],
[0.200235, 0.349922, 14.999471, 0.3, 0],
[0.199765, 0.34958, 14.999668, 0.3, 0],
[0.200429, 0.350419, 15.000005, 0.3, 255],
[0.199571, 0.349795, 15.000366, 0.3, 255],
[0.200429, 0.350205, 14.999634, 0.3, 255],
[0.199571, 0.349581, 14.999995, 0.3, 255],
[0.200524, 0.349747, 15.000146, 0.3, 255],
[0.200196, 0.349509, 15.000283, 0.3, 255],
[0.199804, 0.350491, 14.999717, 0.3, 255],
[0.199476, 0.350253, 14.999854, 0.3, 255],
[0.200138, 0.349578, 14.999597, 0.3, 255],
[0.200138, 0.350138, 15.000567, 0.3, 255],
[0.199862, 0.349862, 14.999433, 0.3, 255],
[0.199862, 0.350422, 15.000403, 0.3, 255],
[0.200488, 0.350136, 15.000321, 0.3, 255],
[0.199957, 0.349751, 15.000544, 0.3, 255],
[0.200488, 0.34979, 14.999721, 0.3, 255],
[0.199957, 0.349404, 14.999944, 0.3, 255],
[0.200043, 0.350596, 15.000056, 0.3, 255],
[0.199512, 0.35021, 15.000279, 0.3, 255],
[0.200043, 0.350249, 14.999456, 0.3, 255],
[0.199512, 0.349864, 14.999679, 0.3, 255],
[-0.5, 0.500442, 10.000442, 0.2, 0],
[-0.5, 0.5, 10, 0.1995, 255],
[-0.499966, 0.49971, 9.999563, 0.199, 0],
[-0.500034, 0.499563, 9.99971, 0.199, 0],
[-0.499934, 0.499432, 9.999144, 0.1985, 255],
[-0.500066, 0.499144, 9.999432, 0.1985, 255],
[0.6, 0.5, 10, 0.25, 210],
[0.600122, 0.4999, 10.000122, 0.25, 210],
[0.600061, 0.49995, 10.000061, 0.25003, 50],
[0.595856, 0.505256, 9.991271, 0.24, 100],
[0.597973, 0.507461, 9.992175, 0.24, 210],
[0.595362, 0.507926, 9.993945, 0.24, 100],
[0.594618, 0.509055, 9.996832, 0.24, 210],
[0.592382, 0.506906, 9.996092, 0.24, 100],
[0.590438, 0.504882, 9.997604, 0.24, 210],
[0.591035, 0.503605, 9.994744, 0.24, 100],
[0.59121, 0.500708, 9.993425, 0.24, 210],
[0.593181, 0.502585, 9.991764, 0.24, 100],
[0.595867, 0.502302, 9.990069, 0.24, 210],
]
values = grey_values(test_case)
rows = [values[row * 1024:row * 1024 + 1024] for row in range(1024)]
with open('combination_scene.png', 'wb') as file:
w = png.Writer(1024, 1024, greyscale=True)
w.write(file, rows)
```
### Rust
The function `grey_values` would be sufficient as an entry to this challenge. The rest of the code saves the output as a PNG file.
The `png` crate can be installed using `cargo add png`. The `png` crate is not required for the `grey_values` function, only for saving as a PNG file.
```rust
fn grey_values(spheres: &Vec<[f32; 5]>) -> Vec<u8> {
let mut values = vec![];
for y_pixel in 0..1024 {
for x_pixel in 0..1024 {
let x_ray = x_pixel as f32 / 512.0 - 1.0;
let y_ray = y_pixel as f32 / 512.0 - 1.0;
let mut min_distance = f32::INFINITY;
let mut hit_grey_value = 0;
for sphere in spheres {
let [
x_sphere, y_sphere, z_sphere, radius, grey
] = *sphere;
let a = (
(x_ray - x_sphere).powf(2.0)
+ (y_ray - y_sphere).powf(2.0)
).sqrt();
if a < radius {
let distance = z_sphere - (
radius.powf(2.0) - a.powf(2.0)
).sqrt();
if distance < min_distance {
min_distance = distance;
hit_grey_value = grey as u8;
}
}
}
values.push(hit_grey_value);
}
}
values
}
fn main() {
let test_case: Vec<[f32; 5]> = vec![
[0.0, 0.0, 1000.0, 2.0, 192.0],
[-2.0, 2.0, 100.0, 3.0, 98.0],
[2.0, 2.0, 100.0, 3.0, 98.0],
[0.5, -0.5, 500.0, 0.2, 255.0],
[-0.349469, -0.1, 20.000279, 0.6, 64.0],
[-0.349775, -0.100273, 20.000485, 0.6, 85.0],
[-0.350214, -0.100386, 20.000407, 0.6, 106.0],
[-0.350527, -0.100273, 20.000091, 0.6, 127.0],
[-0.350531, -0.1, 19.999721, 0.6, 148.0],
[-0.350225, -0.099727, 19.999515, 0.6, 169.0],
[-0.349786, -0.099614, 19.999593, 0.6, 190.0],
[-0.349473, -0.099727, 19.999909, 0.6, 211.0],
[0.20032, 0.349824, 15.000456, 0.3, 0.0],
[0.20032, 0.349517, 14.999924, 0.3, 0.0],
[0.19968, 0.350483, 15.000076, 0.3, 0.0],
[0.19968, 0.350176, 14.999544, 0.3, 0.0],
[0.200578, 0.350073, 14.999958, 0.3, 0.0],
[0.199817, 0.349519, 15.000277, 0.3, 0.0],
[0.200183, 0.350481, 14.999723, 0.3, 0.0],
[0.199422, 0.349927, 15.000042, 0.3, 0.0],
[0.200235, 0.35042, 15.000332, 0.3, 0.0],
[0.199765, 0.350078, 15.000529, 0.3, 0.0],
[0.200235, 0.349922, 14.999471, 0.3, 0.0],
[0.199765, 0.34958, 14.999668, 0.3, 0.0],
[0.200429, 0.350419, 15.000005, 0.3, 255.0],
[0.199571, 0.349795, 15.000366, 0.3, 255.0],
[0.200429, 0.350205, 14.999634, 0.3, 255.0],
[0.199571, 0.349581, 14.999995, 0.3, 255.0],
[0.200524, 0.349747, 15.000146, 0.3, 255.0],
[0.200196, 0.349509, 15.000283, 0.3, 255.0],
[0.199804, 0.350491, 14.999717, 0.3, 255.0],
[0.199476, 0.350253, 14.999854, 0.3, 255.0],
[0.200138, 0.349578, 14.999597, 0.3, 255.0],
[0.200138, 0.350138, 15.000567, 0.3, 255.0],
[0.199862, 0.349862, 14.999433, 0.3, 255.0],
[0.199862, 0.350422, 15.000403, 0.3, 255.0],
[0.200488, 0.350136, 15.000321, 0.3, 255.0],
[0.199957, 0.349751, 15.000544, 0.3, 255.0],
[0.200488, 0.34979, 14.999721, 0.3, 255.0],
[0.199957, 0.349404, 14.999944, 0.3, 255.0],
[0.200043, 0.350596, 15.000056, 0.3, 255.0],
[0.199512, 0.35021, 15.000279, 0.3, 255.0],
[0.200043, 0.350249, 14.999456, 0.3, 255.0],
[0.199512, 0.349864, 14.999679, 0.3, 255.0],
[-0.5, 0.500442, 10.000442, 0.2, 0.0],
[-0.5, 0.5, 10.0, 0.1995, 255.0],
[-0.499966, 0.49971, 9.999563, 0.199, 0.0],
[-0.500034, 0.499563, 9.99971, 0.199, 0.0],
[-0.499934, 0.499432, 9.999144, 0.1985, 255.0],
[-0.500066, 0.499144, 9.999432, 0.1985, 255.0],
[0.6, 0.5, 10.0, 0.25, 210.0],
[0.600122, 0.4999, 10.000122, 0.25, 210.0],
[0.600061, 0.49995, 10.000061, 0.25003, 50.0],
[0.595856, 0.505256, 9.991271, 0.24, 100.0],
[0.597973, 0.507461, 9.992175, 0.24, 210.0],
[0.595362, 0.507926, 9.993945, 0.24, 100.0],
[0.594618, 0.509055, 9.996832, 0.24, 210.0],
[0.592382, 0.506906, 9.996092, 0.24, 100.0],
[0.590438, 0.504882, 9.997604, 0.24, 210.0],
[0.591035, 0.503605, 9.994744, 0.24, 100.0],
[0.59121, 0.500708, 9.993425, 0.24, 210.0],
[0.593181, 0.502585, 9.991764, 0.24, 100.0],
[0.595867, 0.502302, 9.990069, 0.24, 210.0],
];
let values = grey_values(&test_case);
let path = std::path::Path::new("combination_scene.png");
let file = std::fs::File::create(path).unwrap();
let w = &mut std::io::BufWriter::new(file);
let mut encoder = png::Encoder::new(w, 1024, 1024);
encoder.set_color(png::ColorType::Grayscale);
encoder.set_depth(png::BitDepth::Eight);
let mut writer = encoder.write_header().unwrap();
writer.write_image_data(&values).unwrap();
}
```
## Test cases
The output images may be scaled down to fit this page. Click on any image to open it full size.
### Single sphere
Input:
```text
[
[0, 0, 0, 1, 128],
]
```
Output:
[![Grey circle on a black background](https://codegolf.codidact.com/uploads/jlru163pa2pgm7aaq70b2tff16uf)](https://codegolf.codidact.com/uploads/jlru163pa2pgm7aaq70b2tff16uf)
### Two separated spheres
Input:
```text
[
[0, -0.55, 0, 0.4, 100],
[0, 0.55, 0, 0.4, 200],
]
```
Output:
[![Light and dark grey circles on a black background](https://codegolf.codidact.com/uploads/8yw7agli78y65kiomb1u5yebs7c5)](https://codegolf.codidact.com/uploads/8yw7agli78y65kiomb1u5yebs7c5)
### Ring of spheres
Input:
```text
[
[0.75, 0, 0, 0.25, 64],
[0.53033, 0.375, -0.375, 0.25, 185],
[0, 0.53033, -0.53033, 0.25, 64],
[-0.53033, 0.375, -0.375, 0.25, 185],
[-0.75, 0, 0, 0.25, 64],
[-0.53033, -0.375, 0.375, 0.25, 185],
[0, -0.53033, 0.53033, 0.25, 64],
[0.53033, -0.375, 0.375, 0.25, 185],
]
```
Output:
[![Eight circles in alternating light and dark grey in a ring some overlapping](https://codegolf.codidact.com/uploads/0lsenynig27au0sryf3ruozt1uwh)](https://codegolf.codidact.com/uploads/0lsenynig27au0sryf3ruozt1uwh)
### Beach ball
Input:
```text
[
[0.001, 0, 0, 1, 64],
[0.000707, 0.000707, 0, 1, 85],
[0, 0.001, 0, 1, 106],
[-0.000707, 0.000707, 0, 1, 127],
[-0.001, 0, 0, 1, 148],
[-0.000707, -0.000707, 0, 1, 169],
[0, -0.001, 0, 1, 190],
[0.000707, -0.000707, 0, 1, 211],
]
```
Output:
[![A circle with eight sectors in different shades of grey](https://codegolf.codidact.com/uploads/51a3ni0cfwsqcdw5b49gy6282qno)](https://codegolf.codidact.com/uploads/51a3ni0cfwsqcdw5b49gy6282qno)
### Football
Input:
```text
[
[0, 0.000058, 0.001947, 1, 0],
[0, -0.001715, 0.000923, 1, 0],
[0, 0.001715, -0.000923, 1, 0],
[0, -0.000058, -0.001947, 1, 0],
[0.001657, -0.000512, 0.000887, 1, 0],
[-0.001657, -0.000512, 0.000887, 1, 0],
[0.001657, 0.000512, -0.000887, 1, 0],
[-0.001657, 0.000512, -0.000887, 1, 0],
[0.001024, 0.001435, 0.000829, 1, 0],
[-0.001024, 0.001435, 0.000829, 1, 0],
[0.001024, -0.001435, -0.000829, 1, 0],
[-0.001024, -0.001435, -0.000829, 1, 0],
[0.001868, 0.000618, 0.000357, 1, 255],
[-0.001868, 0.000618, 0.000357, 1, 255],
[0.001868, -0.000618, -0.000357, 1, 255],
[-0.001868, -0.000618, -0.000357, 1, 255],
[0.000714, -0.000934, 0.001618, 1, 255],
[-0.000714, -0.000934, 0.001618, 1, 255],
[0.000714, 0.000934, -0.001618, 1, 255],
[-0.000714, 0.000934, -0.001618, 1, 255],
[0, -0.001975, -0.000316, 1, 255],
[0, 0.001261, 0.001552, 1, 255],
[0, -0.001261, -0.001552, 1, 255],
[0, 0.001975, 0.000316, 1, 255],
[0.001155, 0.000423, 0.001577, 1, 255],
[-0.001155, 0.000423, 0.001577, 1, 255],
[0.001155, -0.001577, 0.000423, 1, 255],
[-0.001155, -0.001577, 0.000423, 1, 255],
[0.001155, 0.001577, -0.000423, 1, 255],
[-0.001155, 0.001577, -0.000423, 1, 255],
[0.001155, -0.000423, -0.001577, 1, 255],
[-0.001155, -0.000423, -0.001577, 1, 255],
]
```
Output:
[![Black pentagons on a white ball](https://codegolf.codidact.com/uploads/bmrbsw65p59imvwu8ssd42uvvx1r)](https://codegolf.codidact.com/uploads/bmrbsw65p59imvwu8ssd42uvvx1r)
### Eight ball
Input:
```text
[
[0, 0, 1000, 2, 128],
[0, 0, 0.003125, 1, 0],
[0, 0, 0, 0.9975, 255],
[0, 0.000547, -0.002573, 0.995, 0],
[0, -0.000547, -0.002573, 0.995, 0],
[0, 0.001071, -0.005037, 0.9925, 255],
[0, -0.001071, -0.005037, 0.9925, 255],
]
```
Output:
[![A ball with the number 8 in a white circle](https://codegolf.codidact.com/uploads/dvl68crpmy62rqt94mwanditt4wc)](https://codegolf.codidact.com/uploads/dvl68crpmy62rqt94mwanditt4wc)
### Star ball
Input:
```text
[
[0, 0, 0, 1, 210],
[0, 0, 0.0008, 1, 210],
[0, 0, 0.0004, 1.00012, 50],
[0.012969, 0, -0.042045, 0.96, 100],
[0.016397, 0.011913, -0.039054, 0.96, 210],
[0.004008, 0.012334, -0.042045, 0.96, 100],
[-0.006263, 0.019276, -0.039054, 0.96, 210],
[-0.010492, 0.007623, -0.042045, 0.96, 100],
[-0.020268, 0, -0.039054, 0.96, 210],
[-0.010492, -0.007623, -0.042045, 0.96, 100],
[-0.006263, -0.019276, -0.039054, 0.96, 210],
[0.004008, -0.012334, -0.042045, 0.96, 100],
[0.016397, -0.011913, -0.039054, 0.96, 210],
]
```
Output:
[![A light grey ball with a dark grey five pointed star](https://codegolf.codidact.com/uploads/f3wdgwru6j35ue72x25mazxikfx2)](https://codegolf.codidact.com/uploads/f3wdgwru6j35ue72x25mazxikfx2)
### Combination scene
Input:
```text
[
[0, 0, 1000, 2, 192],
[-2, 2, 100, 3, 98],
[2, 2, 100, 3, 98],
[0.5, -0.5, 500, 0.2, 255],
[-0.349469, -0.1, 20.000279, 0.6, 64],
[-0.349775, -0.100273, 20.000485, 0.6, 85],
[-0.350214, -0.100386, 20.000407, 0.6, 106],
[-0.350527, -0.100273, 20.000091, 0.6, 127],
[-0.350531, -0.1, 19.999721, 0.6, 148],
[-0.350225, -0.099727, 19.999515, 0.6, 169],
[-0.349786, -0.099614, 19.999593, 0.6, 190],
[-0.349473, -0.099727, 19.999909, 0.6, 211],
[0.20032, 0.349824, 15.000456, 0.3, 0],
[0.20032, 0.349517, 14.999924, 0.3, 0],
[0.19968, 0.350483, 15.000076, 0.3, 0],
[0.19968, 0.350176, 14.999544, 0.3, 0],
[0.200578, 0.350073, 14.999958, 0.3, 0],
[0.199817, 0.349519, 15.000277, 0.3, 0],
[0.200183, 0.350481, 14.999723, 0.3, 0],
[0.199422, 0.349927, 15.000042, 0.3, 0],
[0.200235, 0.35042, 15.000332, 0.3, 0],
[0.199765, 0.350078, 15.000529, 0.3, 0],
[0.200235, 0.349922, 14.999471, 0.3, 0],
[0.199765, 0.34958, 14.999668, 0.3, 0],
[0.200429, 0.350419, 15.000005, 0.3, 255],
[0.199571, 0.349795, 15.000366, 0.3, 255],
[0.200429, 0.350205, 14.999634, 0.3, 255],
[0.199571, 0.349581, 14.999995, 0.3, 255],
[0.200524, 0.349747, 15.000146, 0.3, 255],
[0.200196, 0.349509, 15.000283, 0.3, 255],
[0.199804, 0.350491, 14.999717, 0.3, 255],
[0.199476, 0.350253, 14.999854, 0.3, 255],
[0.200138, 0.349578, 14.999597, 0.3, 255],
[0.200138, 0.350138, 15.000567, 0.3, 255],
[0.199862, 0.349862, 14.999433, 0.3, 255],
[0.199862, 0.350422, 15.000403, 0.3, 255],
[0.200488, 0.350136, 15.000321, 0.3, 255],
[0.199957, 0.349751, 15.000544, 0.3, 255],
[0.200488, 0.34979, 14.999721, 0.3, 255],
[0.199957, 0.349404, 14.999944, 0.3, 255],
[0.200043, 0.350596, 15.000056, 0.3, 255],
[0.199512, 0.35021, 15.000279, 0.3, 255],
[0.200043, 0.350249, 14.999456, 0.3, 255],
[0.199512, 0.349864, 14.999679, 0.3, 255],
[-0.5, 0.500442, 10.000442, 0.2, 0],
[-0.5, 0.5, 10, 0.1995, 255],
[-0.499966, 0.49971, 9.999563, 0.199, 0],
[-0.500034, 0.499563, 9.99971, 0.199, 0],
[-0.499934, 0.499432, 9.999144, 0.1985, 255],
[-0.500066, 0.499144, 9.999432, 0.1985, 255],
[0.6, 0.5, 10, 0.25, 210],
[0.600122, 0.4999, 10.000122, 0.25, 210],
[0.600061, 0.49995, 10.000061, 0.25003, 50],
[0.595856, 0.505256, 9.991271, 0.24, 100],
[0.597973, 0.507461, 9.992175, 0.24, 210],
[0.595362, 0.507926, 9.993945, 0.24, 100],
[0.594618, 0.509055, 9.996832, 0.24, 210],
[0.592382, 0.506906, 9.996092, 0.24, 100],
[0.590438, 0.504882, 9.997604, 0.24, 210],
[0.591035, 0.503605, 9.994744, 0.24, 100],
[0.59121, 0.500708, 9.993425, 0.24, 210],
[0.593181, 0.502585, 9.991764, 0.24, 100],
[0.595867, 0.502302, 9.990069, 0.24, 210],
]
```
Output:
[![A beachball, football, eightball, and starball with the sun behind hills in the background](https://codegolf.codidact.com/uploads/usv9npm1t5bjpcaknn6rbzgkqwmi)](https://codegolf.codidact.com/uploads/usv9npm1t5bjpcaknn6rbzgkqwmi)
## Explanatory animations
The example images were produced from plain grey spheres with no surface pattern or texture. The following animations show how they break down into their component plain spheres.
<details>
<summary>Animations</summary>
TODO
</details>
## Scoring
This is a [code golf challenge](https://codegolf.codidact.com/categories/49/tags/4274). Your score is the number of bytes in your code. Lowest score for each language wins.
> Explanations are optional, but I'm more likely to upvote answers that have one.

Given a list of spheres, output an image.
Produce a greyscale [orthographic](https://en.wikipedia.org/wiki/Parallel_projection#Orthographic_projection) (parallel projection - no perspective) ray caster, that renders only plain grey spheres with no lighting.
## Input
- A list of spheres, each consisting of:
- The coordinates of its centre $(x,y,z)$.
- Its radius $r$.
- Its greyscale value $g$, an integer from $0$ to $255$ (both inclusive).
- Only the greyscale value will be restricted to integer values. The coordinates and radius may have fractional values.
## Output
- An image $1024$ by $1024$ pixels with greyscale values from $0$ to $255$.
- The image covers x and y values from $-1$ to $1$.
- A sphere's greyscale value is applied to every visible part of it. There is no variation over the surface of the sphere, so it appears flat like a disk.
- You may output whatever image format is convenient, including simple text formats such as [PGM](https://en.wikipedia.org/wiki/Netpbm#PGM_example) or a list of pixel greyscale values. If you want to include an example of your output as an image in your answer, the image formats that Codidact supports are PNG, JPEG, and GIF. You could output PGM and then convert to PNG to upload to your answer. If you output just the greyscale values as a list of numbers from $0$ to $255$, I've made a `[ TODO page to convert these to greyscale PNG]`.
- You are free to output an image reflected or rotated by a multiple of 90 degrees. However, the z axis is required to be increasing in the forwards direction (into the image) otherwise some of the test cases will look incorrect.
## Calculating a pixel value
Imagine a ray passing into the image through a pixel, perpendicular to the image plane. If it hits a sphere, it shows the greyscale value of the first sphere it hits. Otherwise, it shows a value of zero (black).
For example, for each pixel:
- The pixel $x,y$ coordinates will each be from $0$ to $1023$. Scale these to the range $-1$ to $1$ (for example, divide by $512$ and subtract $1$).
- If these scaled $x,y$ coordinates are within the radius $r$ of the $x,y$ coordinates of a sphere's centre, the ray intersects that sphere (the $z$ coordinate is not relevant to this test because the image is an orthographic projection - the rays are all parallel to the $z$ axis). The distance $a$ of the ray from the sphere's centre can be calculated as follows:
$$a=\sqrt{(x_{ray}-x_{sphere})^2+(y_{ray}-y_{sphere})^2}$$
```text
CONSIDER A DIAGRAM HERE
```
- For each sphere that the ray intersects, find the $z$ coordinate of the nearest intersection (there will often be $2$ per sphere, $1$ on the front of the sphere and $1$ on the back of the sphere, as the ray passes through - only the nearest is relevant). This can be calculated as follows:
- The distance $d$ to the nearest intersection point can be calculated from the $z$ coordinate of the sphere's centre, the radius $r$, and the distance $a$ from the $x,y$ coordinates of the sphere's centre to the scaled $x,y$ coordinates of the pixel:
$$d=z_{sphere}-\sqrt{r_{sphere}^2-a^2}$$
```text
CONSIDER A DIAGRAM HERE
```
- Of all the spheres that the ray intersects, the sphere with the intersection point with the smallest $z$ coordinate $d$ is the one to display. Set the pixel to the greyscale value of this sphere.
- If the ray did not intersect any spheres, set the pixel to zero (black).
## Example implementations
These are ungolfed code to supplement understanding of the requirements. You are free to golf these or implement your own approach that matches the test cases up to reflection & rotation.
The Python implementation uses 64 bit floating point. The Rust implementation uses 32 bit floating point to show the difference in output. I've tried to optimise the test cases for compatibility with 32 bit floating point, so the differences are restricted to being a pixel out in some places.
### Python
The function `grey_values` would be sufficient as an entry to this challenge. The rest of the code saves the output as a PNG file. The function itself has no dependence on the `import png` so it would not be required in an entry.
Instructions for installing the `png` package can be found on its [project page on PyPI](https://pypi.org/project/pypng/).
```python
def grey_values(spheres):
values = []
for y_pixel in range(1024):
for x_pixel in range(1024):
x_ray, y_ray = x_pixel / 512 - 1, y_pixel / 512 - 1
min_distance, hit_grey_value = None, 0
for sphere in spheres:
x_sphere, y_sphere, z_sphere, radius, grey = sphere
a = (
(x_ray - x_sphere) ** 2 + (y_ray - y_sphere) ** 2
) ** 0.5
if a < radius:
distance = z_sphere - (radius ** 2 - a ** 2) ** 0.5
if min_distance is None or distance < min_distance:
min_distance = distance
hit_grey_value = grey
values.append(hit_grey_value)
return values
import png
test_case = [
[0, 0, 1000, 2, 192],
[-2, 2, 100, 3, 98],
[2, 2, 100, 3, 98],
[0.5, -0.5, 500, 0.2, 255],
[-0.349469, -0.1, 20.000279, 0.6, 64],
[-0.349775, -0.100273, 20.000485, 0.6, 85],
[-0.350214, -0.100386, 20.000407, 0.6, 106],
[-0.350527, -0.100273, 20.000091, 0.6, 127],
[-0.350531, -0.1, 19.999721, 0.6, 148],
[-0.350225, -0.099727, 19.999515, 0.6, 169],
[-0.349786, -0.099614, 19.999593, 0.6, 190],
[-0.349473, -0.099727, 19.999909, 0.6, 211],
[0.20032, 0.349824, 15.000456, 0.3, 0],
[0.20032, 0.349517, 14.999924, 0.3, 0],
[0.19968, 0.350483, 15.000076, 0.3, 0],
[0.19968, 0.350176, 14.999544, 0.3, 0],
[0.200578, 0.350073, 14.999958, 0.3, 0],
[0.199817, 0.349519, 15.000277, 0.3, 0],
[0.200183, 0.350481, 14.999723, 0.3, 0],
[0.199422, 0.349927, 15.000042, 0.3, 0],
[0.200235, 0.35042, 15.000332, 0.3, 0],
[0.199765, 0.350078, 15.000529, 0.3, 0],
[0.200235, 0.349922, 14.999471, 0.3, 0],
[0.199765, 0.34958, 14.999668, 0.3, 0],
[0.200429, 0.350419, 15.000005, 0.3, 255],
[0.199571, 0.349795, 15.000366, 0.3, 255],
[0.200429, 0.350205, 14.999634, 0.3, 255],
[0.199571, 0.349581, 14.999995, 0.3, 255],
[0.200524, 0.349747, 15.000146, 0.3, 255],
[0.200196, 0.349509, 15.000283, 0.3, 255],
[0.199804, 0.350491, 14.999717, 0.3, 255],
[0.199476, 0.350253, 14.999854, 0.3, 255],
[0.200138, 0.349578, 14.999597, 0.3, 255],
[0.200138, 0.350138, 15.000567, 0.3, 255],
[0.199862, 0.349862, 14.999433, 0.3, 255],
[0.199862, 0.350422, 15.000403, 0.3, 255],
[0.200488, 0.350136, 15.000321, 0.3, 255],
[0.199957, 0.349751, 15.000544, 0.3, 255],
[0.200488, 0.34979, 14.999721, 0.3, 255],
[0.199957, 0.349404, 14.999944, 0.3, 255],
[0.200043, 0.350596, 15.000056, 0.3, 255],
[0.199512, 0.35021, 15.000279, 0.3, 255],
[0.200043, 0.350249, 14.999456, 0.3, 255],
[0.199512, 0.349864, 14.999679, 0.3, 255],
[-0.5, 0.500442, 10.000442, 0.2, 0],
[-0.5, 0.5, 10, 0.1995, 255],
[-0.499966, 0.49971, 9.999563, 0.199, 0],
[-0.500034, 0.499563, 9.99971, 0.199, 0],
[-0.499934, 0.499432, 9.999144, 0.1985, 255],
[-0.500066, 0.499144, 9.999432, 0.1985, 255],
[0.6, 0.5, 10, 0.25, 210],
[0.600122, 0.4999, 10.000122, 0.25, 210],
[0.600061, 0.49995, 10.000061, 0.25003, 50],
[0.595856, 0.505256, 9.991271, 0.24, 100],
[0.597973, 0.507461, 9.992175, 0.24, 210],
[0.595362, 0.507926, 9.993945, 0.24, 100],
[0.594618, 0.509055, 9.996832, 0.24, 210],
[0.592382, 0.506906, 9.996092, 0.24, 100],
[0.590438, 0.504882, 9.997604, 0.24, 210],
[0.591035, 0.503605, 9.994744, 0.24, 100],
[0.59121, 0.500708, 9.993425, 0.24, 210],
[0.593181, 0.502585, 9.991764, 0.24, 100],
[0.595867, 0.502302, 9.990069, 0.24, 210],
]
values = grey_values(test_case)
rows = [values[row * 1024:row * 1024 + 1024] for row in range(1024)]
with open('combination_scene.png', 'wb') as file:
w = png.Writer(1024, 1024, greyscale=True)
w.write(file, rows)
```
### Rust
The function `grey_values` would be sufficient as an entry to this challenge. The rest of the code saves the output as a PNG file.
The `png` crate can be installed using `cargo add png`. The `png` crate is not required for the `grey_values` function, only for saving as a PNG file.
```rust
fn grey_values(spheres: &Vec<[f32; 5]>) -> Vec<u8> {
let mut values = vec![];
for y_pixel in 0..1024 {
for x_pixel in 0..1024 {
let x_ray = x_pixel as f32 / 512.0 - 1.0;
let y_ray = y_pixel as f32 / 512.0 - 1.0;
let mut min_distance = f32::INFINITY;
let mut hit_grey_value = 0;
for sphere in spheres {
let [
x_sphere, y_sphere, z_sphere, radius, grey
] = *sphere;
let a = (
(x_ray - x_sphere).powf(2.0)
+ (y_ray - y_sphere).powf(2.0)
).sqrt();
if a < radius {
let distance = z_sphere - (
radius.powf(2.0) - a.powf(2.0)
).sqrt();
if distance < min_distance {
min_distance = distance;
hit_grey_value = grey as u8;
}
}
}
values.push(hit_grey_value);
}
}
values
}
fn main() {
let test_case: Vec<[f32; 5]> = vec![
[0.0, 0.0, 1000.0, 2.0, 192.0],
[-2.0, 2.0, 100.0, 3.0, 98.0],
[2.0, 2.0, 100.0, 3.0, 98.0],
[0.5, -0.5, 500.0, 0.2, 255.0],
[-0.349469, -0.1, 20.000279, 0.6, 64.0],
[-0.349775, -0.100273, 20.000485, 0.6, 85.0],
[-0.350214, -0.100386, 20.000407, 0.6, 106.0],
[-0.350527, -0.100273, 20.000091, 0.6, 127.0],
[-0.350531, -0.1, 19.999721, 0.6, 148.0],
[-0.350225, -0.099727, 19.999515, 0.6, 169.0],
[-0.349786, -0.099614, 19.999593, 0.6, 190.0],
[-0.349473, -0.099727, 19.999909, 0.6, 211.0],
[0.20032, 0.349824, 15.000456, 0.3, 0.0],
[0.20032, 0.349517, 14.999924, 0.3, 0.0],
[0.19968, 0.350483, 15.000076, 0.3, 0.0],
[0.19968, 0.350176, 14.999544, 0.3, 0.0],
[0.200578, 0.350073, 14.999958, 0.3, 0.0],
[0.199817, 0.349519, 15.000277, 0.3, 0.0],
[0.200183, 0.350481, 14.999723, 0.3, 0.0],
[0.199422, 0.349927, 15.000042, 0.3, 0.0],
[0.200235, 0.35042, 15.000332, 0.3, 0.0],
[0.199765, 0.350078, 15.000529, 0.3, 0.0],
[0.200235, 0.349922, 14.999471, 0.3, 0.0],
[0.199765, 0.34958, 14.999668, 0.3, 0.0],
[0.200429, 0.350419, 15.000005, 0.3, 255.0],
[0.199571, 0.349795, 15.000366, 0.3, 255.0],
[0.200429, 0.350205, 14.999634, 0.3, 255.0],
[0.199571, 0.349581, 14.999995, 0.3, 255.0],
[0.200524, 0.349747, 15.000146, 0.3, 255.0],
[0.200196, 0.349509, 15.000283, 0.3, 255.0],
[0.199804, 0.350491, 14.999717, 0.3, 255.0],
[0.199476, 0.350253, 14.999854, 0.3, 255.0],
[0.200138, 0.349578, 14.999597, 0.3, 255.0],
[0.200138, 0.350138, 15.000567, 0.3, 255.0],
[0.199862, 0.349862, 14.999433, 0.3, 255.0],
[0.199862, 0.350422, 15.000403, 0.3, 255.0],
[0.200488, 0.350136, 15.000321, 0.3, 255.0],
[0.199957, 0.349751, 15.000544, 0.3, 255.0],
[0.200488, 0.34979, 14.999721, 0.3, 255.0],
[0.199957, 0.349404, 14.999944, 0.3, 255.0],
[0.200043, 0.350596, 15.000056, 0.3, 255.0],
[0.199512, 0.35021, 15.000279, 0.3, 255.0],
[0.200043, 0.350249, 14.999456, 0.3, 255.0],
[0.199512, 0.349864, 14.999679, 0.3, 255.0],
[-0.5, 0.500442, 10.000442, 0.2, 0.0],
[-0.5, 0.5, 10.0, 0.1995, 255.0],
[-0.499966, 0.49971, 9.999563, 0.199, 0.0],
[-0.500034, 0.499563, 9.99971, 0.199, 0.0],
[-0.499934, 0.499432, 9.999144, 0.1985, 255.0],
[-0.500066, 0.499144, 9.999432, 0.1985, 255.0],
[0.6, 0.5, 10.0, 0.25, 210.0],
[0.600122, 0.4999, 10.000122, 0.25, 210.0],
[0.600061, 0.49995, 10.000061, 0.25003, 50.0],
[0.595856, 0.505256, 9.991271, 0.24, 100.0],
[0.597973, 0.507461, 9.992175, 0.24, 210.0],
[0.595362, 0.507926, 9.993945, 0.24, 100.0],
[0.594618, 0.509055, 9.996832, 0.24, 210.0],
[0.592382, 0.506906, 9.996092, 0.24, 100.0],
[0.590438, 0.504882, 9.997604, 0.24, 210.0],
[0.591035, 0.503605, 9.994744, 0.24, 100.0],
[0.59121, 0.500708, 9.993425, 0.24, 210.0],
[0.593181, 0.502585, 9.991764, 0.24, 100.0],
[0.595867, 0.502302, 9.990069, 0.24, 210.0],
];
let values = grey_values(&test_case);
let path = std::path::Path::new("combination_scene.png");
let file = std::fs::File::create(path).unwrap();
let w = &mut std::io::BufWriter::new(file);
let mut encoder = png::Encoder::new(w, 1024, 1024);
encoder.set_color(png::ColorType::Grayscale);
encoder.set_depth(png::BitDepth::Eight);
let mut writer = encoder.write_header().unwrap();
writer.write_image_data(&values).unwrap();
}
```
## Test cases
The output images may be scaled down to fit this page. Click on any image to open it full size.
### Single sphere
Input:
```text
[
[0, 0, 0, 1, 128],
]
```
Output:
[![Grey circle on a black background](https://codegolf.codidact.com/uploads/jlru163pa2pgm7aaq70b2tff16uf)](https://codegolf.codidact.com/uploads/jlru163pa2pgm7aaq70b2tff16uf)
### Two separated spheres
Input:
```text
[
[0, -0.55, 0, 0.4, 100],
[0, 0.55, 0, 0.4, 200],
]
```
Output:
[![Light and dark grey circles on a black background](https://codegolf.codidact.com/uploads/8yw7agli78y65kiomb1u5yebs7c5)](https://codegolf.codidact.com/uploads/8yw7agli78y65kiomb1u5yebs7c5)
### Ring of spheres
Input:
```text
[
[0.75, 0, 0, 0.25, 64],
[0.53033, 0.375, -0.375, 0.25, 185],
[0, 0.53033, -0.53033, 0.25, 64],
[-0.53033, 0.375, -0.375, 0.25, 185],
[-0.75, 0, 0, 0.25, 64],
[-0.53033, -0.375, 0.375, 0.25, 185],
[0, -0.53033, 0.53033, 0.25, 64],
[0.53033, -0.375, 0.375, 0.25, 185],
]
```
Output:
[![Eight circles in alternating light and dark grey in a ring some overlapping](https://codegolf.codidact.com/uploads/0lsenynig27au0sryf3ruozt1uwh)](https://codegolf.codidact.com/uploads/0lsenynig27au0sryf3ruozt1uwh)
### Beach ball
Input:
```text
[
[0.001, 0, 0, 1, 64],
[0.000707, 0.000707, 0, 1, 85],
[0, 0.001, 0, 1, 106],
[-0.000707, 0.000707, 0, 1, 127],
[-0.001, 0, 0, 1, 148],
[-0.000707, -0.000707, 0, 1, 169],
[0, -0.001, 0, 1, 190],
[0.000707, -0.000707, 0, 1, 211],
]
```
Output:
[![A circle with eight sectors in different shades of grey](https://codegolf.codidact.com/uploads/51a3ni0cfwsqcdw5b49gy6282qno)](https://codegolf.codidact.com/uploads/51a3ni0cfwsqcdw5b49gy6282qno)
### Football
Input:
```text
[
[0, 0.000058, 0.001947, 1, 0],
[0, -0.001715, 0.000923, 1, 0],
[0, 0.001715, -0.000923, 1, 0],
[0, -0.000058, -0.001947, 1, 0],
[0.001657, -0.000512, 0.000887, 1, 0],
[-0.001657, -0.000512, 0.000887, 1, 0],
[0.001657, 0.000512, -0.000887, 1, 0],
[-0.001657, 0.000512, -0.000887, 1, 0],
[0.001024, 0.001435, 0.000829, 1, 0],
[-0.001024, 0.001435, 0.000829, 1, 0],
[0.001024, -0.001435, -0.000829, 1, 0],
[-0.001024, -0.001435, -0.000829, 1, 0],
[0.001868, 0.000618, 0.000357, 1, 255],
[-0.001868, 0.000618, 0.000357, 1, 255],
[0.001868, -0.000618, -0.000357, 1, 255],
[-0.001868, -0.000618, -0.000357, 1, 255],
[0.000714, -0.000934, 0.001618, 1, 255],
[-0.000714, -0.000934, 0.001618, 1, 255],
[0.000714, 0.000934, -0.001618, 1, 255],
[-0.000714, 0.000934, -0.001618, 1, 255],
[0, -0.001975, -0.000316, 1, 255],
[0, 0.001261, 0.001552, 1, 255],
[0, -0.001261, -0.001552, 1, 255],
[0, 0.001975, 0.000316, 1, 255],
[0.001155, 0.000423, 0.001577, 1, 255],
[-0.001155, 0.000423, 0.001577, 1, 255],
[0.001155, -0.001577, 0.000423, 1, 255],
[-0.001155, -0.001577, 0.000423, 1, 255],
[0.001155, 0.001577, -0.000423, 1, 255],
[-0.001155, 0.001577, -0.000423, 1, 255],
[0.001155, -0.000423, -0.001577, 1, 255],
[-0.001155, -0.000423, -0.001577, 1, 255],
]
```
Output:
[![Black pentagons on a white ball](https://codegolf.codidact.com/uploads/bmrbsw65p59imvwu8ssd42uvvx1r)](https://codegolf.codidact.com/uploads/bmrbsw65p59imvwu8ssd42uvvx1r)
### Eight ball
Input:
```text
[
[0, 0, 1000, 2, 128],
[0, 0, 0.003125, 1, 0],
[0, 0, 0, 0.9975, 255],
[0, 0.000547, -0.002573, 0.995, 0],
[0, -0.000547, -0.002573, 0.995, 0],
[0, 0.001071, -0.005037, 0.9925, 255],
[0, -0.001071, -0.005037, 0.9925, 255],
]
```
Output:
[![A ball with the number 8 in a white circle](https://codegolf.codidact.com/uploads/dvl68crpmy62rqt94mwanditt4wc)](https://codegolf.codidact.com/uploads/dvl68crpmy62rqt94mwanditt4wc)
### Star ball
Input:
```text
[
[0, 0, 0, 1, 210],
[0, 0, 0.0008, 1, 210],
[0, 0, 0.0004, 1.00012, 50],
[0.012969, 0, -0.042045, 0.96, 100],
[0.016397, 0.011913, -0.039054, 0.96, 210],
[0.004008, 0.012334, -0.042045, 0.96, 100],
[-0.006263, 0.019276, -0.039054, 0.96, 210],
[-0.010492, 0.007623, -0.042045, 0.96, 100],
[-0.020268, 0, -0.039054, 0.96, 210],
[-0.010492, -0.007623, -0.042045, 0.96, 100],
[-0.006263, -0.019276, -0.039054, 0.96, 210],
[0.004008, -0.012334, -0.042045, 0.96, 100],
[0.016397, -0.011913, -0.039054, 0.96, 210],
]
```
Output:
[![A light grey ball with a dark grey five pointed star](https://codegolf.codidact.com/uploads/f3wdgwru6j35ue72x25mazxikfx2)](https://codegolf.codidact.com/uploads/f3wdgwru6j35ue72x25mazxikfx2)
### Combination scene
Input:
```text
[
[0, 0, 1000, 2, 192],
[-2, 2, 100, 3, 98],
[2, 2, 100, 3, 98],
[0.5, -0.5, 500, 0.2, 255],
[-0.349469, -0.1, 20.000279, 0.6, 64],
[-0.349775, -0.100273, 20.000485, 0.6, 85],
[-0.350214, -0.100386, 20.000407, 0.6, 106],
[-0.350527, -0.100273, 20.000091, 0.6, 127],
[-0.350531, -0.1, 19.999721, 0.6, 148],
[-0.350225, -0.099727, 19.999515, 0.6, 169],
[-0.349786, -0.099614, 19.999593, 0.6, 190],
[-0.349473, -0.099727, 19.999909, 0.6, 211],
[0.20032, 0.349824, 15.000456, 0.3, 0],
[0.20032, 0.349517, 14.999924, 0.3, 0],
[0.19968, 0.350483, 15.000076, 0.3, 0],
[0.19968, 0.350176, 14.999544, 0.3, 0],
[0.200578, 0.350073, 14.999958, 0.3, 0],
[0.199817, 0.349519, 15.000277, 0.3, 0],
[0.200183, 0.350481, 14.999723, 0.3, 0],
[0.199422, 0.349927, 15.000042, 0.3, 0],
[0.200235, 0.35042, 15.000332, 0.3, 0],
[0.199765, 0.350078, 15.000529, 0.3, 0],
[0.200235, 0.349922, 14.999471, 0.3, 0],
[0.199765, 0.34958, 14.999668, 0.3, 0],
[0.200429, 0.350419, 15.000005, 0.3, 255],
[0.199571, 0.349795, 15.000366, 0.3, 255],
[0.200429, 0.350205, 14.999634, 0.3, 255],
[0.199571, 0.349581, 14.999995, 0.3, 255],
[0.200524, 0.349747, 15.000146, 0.3, 255],
[0.200196, 0.349509, 15.000283, 0.3, 255],
[0.199804, 0.350491, 14.999717, 0.3, 255],
[0.199476, 0.350253, 14.999854, 0.3, 255],
[0.200138, 0.349578, 14.999597, 0.3, 255],
[0.200138, 0.350138, 15.000567, 0.3, 255],
[0.199862, 0.349862, 14.999433, 0.3, 255],
[0.199862, 0.350422, 15.000403, 0.3, 255],
[0.200488, 0.350136, 15.000321, 0.3, 255],
[0.199957, 0.349751, 15.000544, 0.3, 255],
[0.200488, 0.34979, 14.999721, 0.3, 255],
[0.199957, 0.349404, 14.999944, 0.3, 255],
[0.200043, 0.350596, 15.000056, 0.3, 255],
[0.199512, 0.35021, 15.000279, 0.3, 255],
[0.200043, 0.350249, 14.999456, 0.3, 255],
[0.199512, 0.349864, 14.999679, 0.3, 255],
[-0.5, 0.500442, 10.000442, 0.2, 0],
[-0.5, 0.5, 10, 0.1995, 255],
[-0.499966, 0.49971, 9.999563, 0.199, 0],
[-0.500034, 0.499563, 9.99971, 0.199, 0],
[-0.499934, 0.499432, 9.999144, 0.1985, 255],
[-0.500066, 0.499144, 9.999432, 0.1985, 255],
[0.6, 0.5, 10, 0.25, 210],
[0.600122, 0.4999, 10.000122, 0.25, 210],
[0.600061, 0.49995, 10.000061, 0.25003, 50],
[0.595856, 0.505256, 9.991271, 0.24, 100],
[0.597973, 0.507461, 9.992175, 0.24, 210],
[0.595362, 0.507926, 9.993945, 0.24, 100],
[0.594618, 0.509055, 9.996832, 0.24, 210],
[0.592382, 0.506906, 9.996092, 0.24, 100],
[0.590438, 0.504882, 9.997604, 0.24, 210],
[0.591035, 0.503605, 9.994744, 0.24, 100],
[0.59121, 0.500708, 9.993425, 0.24, 210],
[0.593181, 0.502585, 9.991764, 0.24, 100],
[0.595867, 0.502302, 9.990069, 0.24, 210],
]
```
Output:
[![A beachball, football, eightball, and starball with the sun behind hills in the background](https://codegolf.codidact.com/uploads/usv9npm1t5bjpcaknn6rbzgkqwmi)](https://codegolf.codidact.com/uploads/usv9npm1t5bjpcaknn6rbzgkqwmi)
Output using 32 bit floating point rather than 64 bit floating point:
[![A beachball, football, eightball, and starball with the sun behind hills in the background](https://codegolf.codidact.com/uploads/7uuxjxq7a9oxc9q0jf2bi8l68qk3)](https://codegolf.codidact.com/uploads/7uuxjxq7a9oxc9q0jf2bi8l68qk3)
The inaccuracies in the 32 bit floating point version are subtle. You may need to click on the image to open it full size to see the occasional inaccurate pixel. For example, around the inside of the holes in the 8 on the eightball.
## Explanatory animations
The example images were produced from plain grey spheres with no surface pattern or texture. The following animations show how they break down into their component plain spheres.
<details>
<summary>Animations</summary>
TODO
</details>
## Scoring
This is a [code golf challenge](https://codegolf.codidact.com/categories/49/tags/4274). Your score is the number of bytes in your code. Lowest score for each language wins.
> Explanations are optional, but I'm more likely to upvote answers that have one.

#13: Post edited by

trichoplax‭ · 2024-07-26T13:14:40Z (5 months ago)
Clarify wordings

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Raw

Markdown

Given a list of spheres, output an image.
Produce a greyscale [orthographic](https://en.wikipedia.org/wiki/Parallel_projection#Orthographic_projection) (parallel projection - no perspective) ray caster, that renders only plain grey spheres with no lighting.
## Input
- A list of spheres, each consisting of:
- The coordinates of its centre $(x,y,z)$.
- Its radius $r$.
- Its greyscale value $g$, an integer from $0$ to $255$ (both inclusive).
- Only the greyscale value will be restricted to integer values. The coordinates and radius may have fractional values.
## Output
- An image $1024$ by $1024$ pixels with greyscale values from $0$ to $255$.
- The image covers x and y values from $-1$ to $1$.
- A sphere's greyscale value is applied to every visible part of it. There is no variation over the surface of the sphere, so it appears flat like a disk.
- You may output whatever image format is convenient, including simple text formats such as [PGM](https://en.wikipedia.org/wiki/Netpbm#PGM_example) or a list of pixel greyscale values. If you want to include an example of your output as an image in your answer, the image formats that Codidact supports are PNG, JPEG, and GIF. You could output PGM and then convert to PNG to upload to your answer. If you output just the greyscale values as a list of numbers from $0$ to $255$, I've made a `[ TODO page to convert these to greyscale PNG]`.
- You are free to output an image reflected or rotated by a multiple of 90 degrees. However, the z axis is required to be increasing in the forwards direction (into the image) otherwise some of the test cases will look incorrect.
## Calculating a pixel value
Imagine a ray passing into the image through a pixel, perpendicular to the image plane. If it hits a sphere, it shows the greyscale value of the first sphere it hits. Otherwise, it shows a value of zero (black).
For example, for each pixel:
- The pixel $x,y$ coordinates will each be from $0$ to $1023$. Scale these to the range $-1$ to $1$ (for example, divide by $512$ and subtract $1$).
- If these scaled $x,y$ coordinates are within the radius $r$ of the $x,y$ coordinates of a sphere's centre, the ray intersects that sphere (the $z$ coordinate is not relevant to this test because the image is an orthographic projection - the rays are all parallel to the $z$ axis). The distance $a$ of the ray from the sphere's centre can be calculated as follows:
$$a=\sqrt{(x_{ray}-x_{sphere})^2+(y_{ray}-y_{sphere})^2}$$
```text
CONSIDER A DIAGRAM HERE
```
- For each sphere that the ray intersects, find the $z$ coordinate of the nearest intersection (there will often be $2$ per sphere, $1$ on the front of the sphere and $1$ on the back of the sphere, as the ray passes through - only the nearest is relevant). This can be calculated as follows:
- The distance $d$ to the nearest intersection point can be calculated from the $z$ coordinate of the sphere's centre, the radius $r$, and the distance $a$ from the $x,y$ coordinates of the sphere's centre to the scaled $x,y$ coordinates of the pixel:
$$d=z_{sphere}-\sqrt{r_{sphere}^2-a^2}$$
```text
CONSIDER A DIAGRAM HERE
```
- Of all the spheres that the ray intersects, the sphere with the intersection point with the smallest $z$ coordinate $d$ is the one to display. Set the pixel to the greyscale value of this sphere.
- If the ray did not intersect any spheres, set the pixel to zero (black).
## Example implementations
These are ungolfed code to supplement understanding of the requirements. You are free to golf these or implement your own approach that matches the test cases up to reflection & rotation.
The Python implementation uses 64 bit floating point. The Rust implementation uses 32 bit floating point to show the difference in output. I've tried to optimise the test cases for compatibility with 32 bit floating point, so the differences are restricted to being a pixel out in some places.
### Python
The function `grey_values` would be sufficient as an entry to this challenge. The rest of the code saves the outputs as PNG files. The function itself has no dependence on the `import png` so it would not be required in an entry.
Instructions for installing the `png` package can be found on its [project page on PyPI](https://pypi.org/project/pypng/).
```python
def grey_values(spheres):
values = []
for y_pixel in range(1024):
for x_pixel in range(1024):
x_ray, y_ray = x_pixel / 512 - 1, y_pixel / 512 - 1
min_distance, hit_grey_value = None, 0
for sphere in spheres:
x_sphere, y_sphere, z_sphere, radius, grey = sphere
a = (
(x_ray - x_sphere) ** 2 + (y_ray - y_sphere) ** 2
) ** 0.5
if a < radius:
distance = z_sphere - (radius ** 2 - a ** 2) ** 0.5
if min_distance is None or distance < min_distance:
min_distance = distance
hit_grey_value = grey
values.append(hit_grey_value)
return values
import png
test_case = [
[0, 0, 1000, 2, 192],
[-2, 2, 100, 3, 98],
[2, 2, 100, 3, 98],
[0.5, -0.5, 500, 0.2, 255],
[-0.349469, -0.1, 20.000279, 0.6, 64],
[-0.349775, -0.100273, 20.000485, 0.6, 85],
[-0.350214, -0.100386, 20.000407, 0.6, 106],
[-0.350527, -0.100273, 20.000091, 0.6, 127],
[-0.350531, -0.1, 19.999721, 0.6, 148],
[-0.350225, -0.099727, 19.999515, 0.6, 169],
[-0.349786, -0.099614, 19.999593, 0.6, 190],
[-0.349473, -0.099727, 19.999909, 0.6, 211],
[0.20032, 0.349824, 15.000456, 0.3, 0],
[0.20032, 0.349517, 14.999924, 0.3, 0],
[0.19968, 0.350483, 15.000076, 0.3, 0],
[0.19968, 0.350176, 14.999544, 0.3, 0],
[0.200578, 0.350073, 14.999958, 0.3, 0],
[0.199817, 0.349519, 15.000277, 0.3, 0],
[0.200183, 0.350481, 14.999723, 0.3, 0],
[0.199422, 0.349927, 15.000042, 0.3, 0],
[0.200235, 0.35042, 15.000332, 0.3, 0],
[0.199765, 0.350078, 15.000529, 0.3, 0],
[0.200235, 0.349922, 14.999471, 0.3, 0],
[0.199765, 0.34958, 14.999668, 0.3, 0],
[0.200429, 0.350419, 15.000005, 0.3, 255],
[0.199571, 0.349795, 15.000366, 0.3, 255],
[0.200429, 0.350205, 14.999634, 0.3, 255],
[0.199571, 0.349581, 14.999995, 0.3, 255],
[0.200524, 0.349747, 15.000146, 0.3, 255],
[0.200196, 0.349509, 15.000283, 0.3, 255],
[0.199804, 0.350491, 14.999717, 0.3, 255],
[0.199476, 0.350253, 14.999854, 0.3, 255],
[0.200138, 0.349578, 14.999597, 0.3, 255],
[0.200138, 0.350138, 15.000567, 0.3, 255],
[0.199862, 0.349862, 14.999433, 0.3, 255],
[0.199862, 0.350422, 15.000403, 0.3, 255],
[0.200488, 0.350136, 15.000321, 0.3, 255],
[0.199957, 0.349751, 15.000544, 0.3, 255],
[0.200488, 0.34979, 14.999721, 0.3, 255],
[0.199957, 0.349404, 14.999944, 0.3, 255],
[0.200043, 0.350596, 15.000056, 0.3, 255],
[0.199512, 0.35021, 15.000279, 0.3, 255],
[0.200043, 0.350249, 14.999456, 0.3, 255],
[0.199512, 0.349864, 14.999679, 0.3, 255],
[-0.5, 0.500442, 10.000442, 0.2, 0],
[-0.5, 0.5, 10, 0.1995, 255],
[-0.499966, 0.49971, 9.999563, 0.199, 0],
[-0.500034, 0.499563, 9.99971, 0.199, 0],
[-0.499934, 0.499432, 9.999144, 0.1985, 255],
[-0.500066, 0.499144, 9.999432, 0.1985, 255],
[0.6, 0.5, 10, 0.25, 210],
[0.600122, 0.4999, 10.000122, 0.25, 210],
[0.600061, 0.49995, 10.000061, 0.25003, 50],
[0.595856, 0.505256, 9.991271, 0.24, 100],
[0.597973, 0.507461, 9.992175, 0.24, 210],
[0.595362, 0.507926, 9.993945, 0.24, 100],
[0.594618, 0.509055, 9.996832, 0.24, 210],
[0.592382, 0.506906, 9.996092, 0.24, 100],
[0.590438, 0.504882, 9.997604, 0.24, 210],
[0.591035, 0.503605, 9.994744, 0.24, 100],
[0.59121, 0.500708, 9.993425, 0.24, 210],
[0.593181, 0.502585, 9.991764, 0.24, 100],
[0.595867, 0.502302, 9.990069, 0.24, 210],
]
values = grey_values(test_case)
rows = [values[row * 1024:row * 1024 + 1024] for row in range(1024)]
with open('combination_scene.png', 'wb') as file:
w = png.Writer(1024, 1024, greyscale=True)
w.write(file, rows)
```
### Rust
~~The function `grey_values` would be sufficient as an entry to this challenge. The rest of the code saves the outputs as PNG files.~~
~~The `png` crate can be installed using `cargo add png`.~~
```rust
fn grey_values(spheres: &Vec<[f32; 5]>) -> Vec<u8> {
let mut values = vec![];
for y_pixel in 0..1024 {
for x_pixel in 0..1024 {
let x_ray = x_pixel as f32 / 512.0 - 1.0;
let y_ray = y_pixel as f32 / 512.0 - 1.0;
let mut min_distance = f32::INFINITY;
let mut hit_grey_value = 0;
for sphere in spheres {
let [
x_sphere, y_sphere, z_sphere, radius, grey
] = *sphere;
let a = (
(x_ray - x_sphere).powf(2.0)
+ (y_ray - y_sphere).powf(2.0)
).sqrt();
if a < radius {
let distance = z_sphere - (
radius.powf(2.0) - a.powf(2.0)
).sqrt();
if distance < min_distance {
min_distance = distance;
hit_grey_value = grey as u8;
}
}
}
values.push(hit_grey_value);
}
}
values
}
fn main() {
let test_case: Vec<[f32; 5]> = vec![
[0.0, 0.0, 1000.0, 2.0, 192.0],
[-2.0, 2.0, 100.0, 3.0, 98.0],
[2.0, 2.0, 100.0, 3.0, 98.0],
[0.5, -0.5, 500.0, 0.2, 255.0],
[-0.349469, -0.1, 20.000279, 0.6, 64.0],
[-0.349775, -0.100273, 20.000485, 0.6, 85.0],
[-0.350214, -0.100386, 20.000407, 0.6, 106.0],
[-0.350527, -0.100273, 20.000091, 0.6, 127.0],
[-0.350531, -0.1, 19.999721, 0.6, 148.0],
[-0.350225, -0.099727, 19.999515, 0.6, 169.0],
[-0.349786, -0.099614, 19.999593, 0.6, 190.0],
[-0.349473, -0.099727, 19.999909, 0.6, 211.0],
[0.20032, 0.349824, 15.000456, 0.3, 0.0],
[0.20032, 0.349517, 14.999924, 0.3, 0.0],
[0.19968, 0.350483, 15.000076, 0.3, 0.0],
[0.19968, 0.350176, 14.999544, 0.3, 0.0],
[0.200578, 0.350073, 14.999958, 0.3, 0.0],
[0.199817, 0.349519, 15.000277, 0.3, 0.0],
[0.200183, 0.350481, 14.999723, 0.3, 0.0],
[0.199422, 0.349927, 15.000042, 0.3, 0.0],
[0.200235, 0.35042, 15.000332, 0.3, 0.0],
[0.199765, 0.350078, 15.000529, 0.3, 0.0],
[0.200235, 0.349922, 14.999471, 0.3, 0.0],
[0.199765, 0.34958, 14.999668, 0.3, 0.0],
[0.200429, 0.350419, 15.000005, 0.3, 255.0],
[0.199571, 0.349795, 15.000366, 0.3, 255.0],
[0.200429, 0.350205, 14.999634, 0.3, 255.0],
[0.199571, 0.349581, 14.999995, 0.3, 255.0],
[0.200524, 0.349747, 15.000146, 0.3, 255.0],
[0.200196, 0.349509, 15.000283, 0.3, 255.0],
[0.199804, 0.350491, 14.999717, 0.3, 255.0],
[0.199476, 0.350253, 14.999854, 0.3, 255.0],
[0.200138, 0.349578, 14.999597, 0.3, 255.0],
[0.200138, 0.350138, 15.000567, 0.3, 255.0],
[0.199862, 0.349862, 14.999433, 0.3, 255.0],
[0.199862, 0.350422, 15.000403, 0.3, 255.0],
[0.200488, 0.350136, 15.000321, 0.3, 255.0],
[0.199957, 0.349751, 15.000544, 0.3, 255.0],
[0.200488, 0.34979, 14.999721, 0.3, 255.0],
[0.199957, 0.349404, 14.999944, 0.3, 255.0],
[0.200043, 0.350596, 15.000056, 0.3, 255.0],
[0.199512, 0.35021, 15.000279, 0.3, 255.0],
[0.200043, 0.350249, 14.999456, 0.3, 255.0],
[0.199512, 0.349864, 14.999679, 0.3, 255.0],
[-0.5, 0.500442, 10.000442, 0.2, 0.0],
[-0.5, 0.5, 10.0, 0.1995, 255.0],
[-0.499966, 0.49971, 9.999563, 0.199, 0.0],
[-0.500034, 0.499563, 9.99971, 0.199, 0.0],
[-0.499934, 0.499432, 9.999144, 0.1985, 255.0],
[-0.500066, 0.499144, 9.999432, 0.1985, 255.0],
[0.6, 0.5, 10.0, 0.25, 210.0],
[0.600122, 0.4999, 10.000122, 0.25, 210.0],
[0.600061, 0.49995, 10.000061, 0.25003, 50.0],
[0.595856, 0.505256, 9.991271, 0.24, 100.0],
[0.597973, 0.507461, 9.992175, 0.24, 210.0],
[0.595362, 0.507926, 9.993945, 0.24, 100.0],
[0.594618, 0.509055, 9.996832, 0.24, 210.0],
[0.592382, 0.506906, 9.996092, 0.24, 100.0],
[0.590438, 0.504882, 9.997604, 0.24, 210.0],
[0.591035, 0.503605, 9.994744, 0.24, 100.0],
[0.59121, 0.500708, 9.993425, 0.24, 210.0],
[0.593181, 0.502585, 9.991764, 0.24, 100.0],
[0.595867, 0.502302, 9.990069, 0.24, 210.0],
];
let values = grey_values(&test_case);
let path = std::path::Path::new("combination_scene.png");
let file = std::fs::File::create(path).unwrap();
let w = &mut std::io::BufWriter::new(file);
let mut encoder = png::Encoder::new(w, 1024, 1024);
encoder.set_color(png::ColorType::Grayscale);
encoder.set_depth(png::BitDepth::Eight);
let mut writer = encoder.write_header().unwrap();
writer.write_image_data(&values).unwrap();
}
```
## Test cases
~~The output images may be scaled down to fit. Click on any image to open it full size.~~
### Single sphere
Input:
```text
[
[0, 0, 0, 1, 128],
]
```
Output:
[![Grey circle on a black background](https://codegolf.codidact.com/uploads/jlru163pa2pgm7aaq70b2tff16uf)](https://codegolf.codidact.com/uploads/jlru163pa2pgm7aaq70b2tff16uf)
### Two separated spheres
Input:
```text
[
[0, -0.55, 0, 0.4, 100],
[0, 0.55, 0, 0.4, 200],
]
```
Output:
[![Light and dark grey circles on a black background](https://codegolf.codidact.com/uploads/8yw7agli78y65kiomb1u5yebs7c5)](https://codegolf.codidact.com/uploads/8yw7agli78y65kiomb1u5yebs7c5)
### Ring of spheres
Input:
```text
[
[0.75, 0, 0, 0.25, 64],
[0.53033, 0.375, -0.375, 0.25, 185],
[0, 0.53033, -0.53033, 0.25, 64],
[-0.53033, 0.375, -0.375, 0.25, 185],
[-0.75, 0, 0, 0.25, 64],
[-0.53033, -0.375, 0.375, 0.25, 185],
[0, -0.53033, 0.53033, 0.25, 64],
[0.53033, -0.375, 0.375, 0.25, 185],
]
```
Output:
[![Eight circles in alternating light and dark grey in a ring some overlapping](https://codegolf.codidact.com/uploads/0lsenynig27au0sryf3ruozt1uwh)](https://codegolf.codidact.com/uploads/0lsenynig27au0sryf3ruozt1uwh)
### Beach ball
Input:
```text
[
[0.001, 0, 0, 1, 64],
[0.000707, 0.000707, 0, 1, 85],
[0, 0.001, 0, 1, 106],
[-0.000707, 0.000707, 0, 1, 127],
[-0.001, 0, 0, 1, 148],
[-0.000707, -0.000707, 0, 1, 169],
[0, -0.001, 0, 1, 190],
[0.000707, -0.000707, 0, 1, 211],
]
```
Output:
[![A circle with eight sectors in different shades of grey](https://codegolf.codidact.com/uploads/51a3ni0cfwsqcdw5b49gy6282qno)](https://codegolf.codidact.com/uploads/51a3ni0cfwsqcdw5b49gy6282qno)
### Football
Input:
```text
[
[0, 0.000058, 0.001947, 1, 0],
[0, -0.001715, 0.000923, 1, 0],
[0, 0.001715, -0.000923, 1, 0],
[0, -0.000058, -0.001947, 1, 0],
[0.001657, -0.000512, 0.000887, 1, 0],
[-0.001657, -0.000512, 0.000887, 1, 0],
[0.001657, 0.000512, -0.000887, 1, 0],
[-0.001657, 0.000512, -0.000887, 1, 0],
[0.001024, 0.001435, 0.000829, 1, 0],
[-0.001024, 0.001435, 0.000829, 1, 0],
[0.001024, -0.001435, -0.000829, 1, 0],
[-0.001024, -0.001435, -0.000829, 1, 0],
[0.001868, 0.000618, 0.000357, 1, 255],
[-0.001868, 0.000618, 0.000357, 1, 255],
[0.001868, -0.000618, -0.000357, 1, 255],
[-0.001868, -0.000618, -0.000357, 1, 255],
[0.000714, -0.000934, 0.001618, 1, 255],
[-0.000714, -0.000934, 0.001618, 1, 255],
[0.000714, 0.000934, -0.001618, 1, 255],
[-0.000714, 0.000934, -0.001618, 1, 255],
[0, -0.001975, -0.000316, 1, 255],
[0, 0.001261, 0.001552, 1, 255],
[0, -0.001261, -0.001552, 1, 255],
[0, 0.001975, 0.000316, 1, 255],
[0.001155, 0.000423, 0.001577, 1, 255],
[-0.001155, 0.000423, 0.001577, 1, 255],
[0.001155, -0.001577, 0.000423, 1, 255],
[-0.001155, -0.001577, 0.000423, 1, 255],
[0.001155, 0.001577, -0.000423, 1, 255],
[-0.001155, 0.001577, -0.000423, 1, 255],
[0.001155, -0.000423, -0.001577, 1, 255],
[-0.001155, -0.000423, -0.001577, 1, 255],
]
```
Output:
[![Black pentagons on a white ball](https://codegolf.codidact.com/uploads/bmrbsw65p59imvwu8ssd42uvvx1r)](https://codegolf.codidact.com/uploads/bmrbsw65p59imvwu8ssd42uvvx1r)
### Eight ball
Input:
```text
[
[0, 0, 1000, 2, 128],
[0, 0, 0.003125, 1, 0],
[0, 0, 0, 0.9975, 255],
[0, 0.000547, -0.002573, 0.995, 0],
[0, -0.000547, -0.002573, 0.995, 0],
[0, 0.001071, -0.005037, 0.9925, 255],
[0, -0.001071, -0.005037, 0.9925, 255],
]
```
Output:
[![A ball with the number 8 in a white circle](https://codegolf.codidact.com/uploads/dvl68crpmy62rqt94mwanditt4wc)](https://codegolf.codidact.com/uploads/dvl68crpmy62rqt94mwanditt4wc)
### Star ball
Input:
```text
[
[0, 0, 0, 1, 210],
[0, 0, 0.0008, 1, 210],
[0, 0, 0.0004, 1.00012, 50],
[0.012969, 0, -0.042045, 0.96, 100],
[0.016397, 0.011913, -0.039054, 0.96, 210],
[0.004008, 0.012334, -0.042045, 0.96, 100],
[-0.006263, 0.019276, -0.039054, 0.96, 210],
[-0.010492, 0.007623, -0.042045, 0.96, 100],
[-0.020268, 0, -0.039054, 0.96, 210],
[-0.010492, -0.007623, -0.042045, 0.96, 100],
[-0.006263, -0.019276, -0.039054, 0.96, 210],
[0.004008, -0.012334, -0.042045, 0.96, 100],
[0.016397, -0.011913, -0.039054, 0.96, 210],
]
```
Output:
[![A light grey ball with a dark grey five pointed star](https://codegolf.codidact.com/uploads/f3wdgwru6j35ue72x25mazxikfx2)](https://codegolf.codidact.com/uploads/f3wdgwru6j35ue72x25mazxikfx2)
### Combination scene
Input:
```text
[
[0, 0, 1000, 2, 192],
[-2, 2, 100, 3, 98],
[2, 2, 100, 3, 98],
[0.5, -0.5, 500, 0.2, 255],
[-0.349469, -0.1, 20.000279, 0.6, 64],
[-0.349775, -0.100273, 20.000485, 0.6, 85],
[-0.350214, -0.100386, 20.000407, 0.6, 106],
[-0.350527, -0.100273, 20.000091, 0.6, 127],
[-0.350531, -0.1, 19.999721, 0.6, 148],
[-0.350225, -0.099727, 19.999515, 0.6, 169],
[-0.349786, -0.099614, 19.999593, 0.6, 190],
[-0.349473, -0.099727, 19.999909, 0.6, 211],
[0.20032, 0.349824, 15.000456, 0.3, 0],
[0.20032, 0.349517, 14.999924, 0.3, 0],
[0.19968, 0.350483, 15.000076, 0.3, 0],
[0.19968, 0.350176, 14.999544, 0.3, 0],
[0.200578, 0.350073, 14.999958, 0.3, 0],
[0.199817, 0.349519, 15.000277, 0.3, 0],
[0.200183, 0.350481, 14.999723, 0.3, 0],
[0.199422, 0.349927, 15.000042, 0.3, 0],
[0.200235, 0.35042, 15.000332, 0.3, 0],
[0.199765, 0.350078, 15.000529, 0.3, 0],
[0.200235, 0.349922, 14.999471, 0.3, 0],
[0.199765, 0.34958, 14.999668, 0.3, 0],
[0.200429, 0.350419, 15.000005, 0.3, 255],
[0.199571, 0.349795, 15.000366, 0.3, 255],
[0.200429, 0.350205, 14.999634, 0.3, 255],
[0.199571, 0.349581, 14.999995, 0.3, 255],
[0.200524, 0.349747, 15.000146, 0.3, 255],
[0.200196, 0.349509, 15.000283, 0.3, 255],
[0.199804, 0.350491, 14.999717, 0.3, 255],
[0.199476, 0.350253, 14.999854, 0.3, 255],
[0.200138, 0.349578, 14.999597, 0.3, 255],
[0.200138, 0.350138, 15.000567, 0.3, 255],
[0.199862, 0.349862, 14.999433, 0.3, 255],
[0.199862, 0.350422, 15.000403, 0.3, 255],
[0.200488, 0.350136, 15.000321, 0.3, 255],
[0.199957, 0.349751, 15.000544, 0.3, 255],
[0.200488, 0.34979, 14.999721, 0.3, 255],
[0.199957, 0.349404, 14.999944, 0.3, 255],
[0.200043, 0.350596, 15.000056, 0.3, 255],
[0.199512, 0.35021, 15.000279, 0.3, 255],
[0.200043, 0.350249, 14.999456, 0.3, 255],
[0.199512, 0.349864, 14.999679, 0.3, 255],
[-0.5, 0.500442, 10.000442, 0.2, 0],
[-0.5, 0.5, 10, 0.1995, 255],
[-0.499966, 0.49971, 9.999563, 0.199, 0],
[-0.500034, 0.499563, 9.99971, 0.199, 0],
[-0.499934, 0.499432, 9.999144, 0.1985, 255],
[-0.500066, 0.499144, 9.999432, 0.1985, 255],
[0.6, 0.5, 10, 0.25, 210],
[0.600122, 0.4999, 10.000122, 0.25, 210],
[0.600061, 0.49995, 10.000061, 0.25003, 50],
[0.595856, 0.505256, 9.991271, 0.24, 100],
[0.597973, 0.507461, 9.992175, 0.24, 210],
[0.595362, 0.507926, 9.993945, 0.24, 100],
[0.594618, 0.509055, 9.996832, 0.24, 210],
[0.592382, 0.506906, 9.996092, 0.24, 100],
[0.590438, 0.504882, 9.997604, 0.24, 210],
[0.591035, 0.503605, 9.994744, 0.24, 100],
[0.59121, 0.500708, 9.993425, 0.24, 210],
[0.593181, 0.502585, 9.991764, 0.24, 100],
[0.595867, 0.502302, 9.990069, 0.24, 210],
]
```
Output:
[![A beachball, football, eightball, and starball with the sun behind hills in the background](https://codegolf.codidact.com/uploads/usv9npm1t5bjpcaknn6rbzgkqwmi)](https://codegolf.codidact.com/uploads/usv9npm1t5bjpcaknn6rbzgkqwmi)
## Explanatory animations
The example images were produced from plain grey spheres with no surface pattern or texture. The following animations show how they break down into their component plain spheres.
<details>
<summary>Animations</summary>
TODO
</details>
## Scoring
This is a [code golf challenge](https://codegolf.codidact.com/categories/49/tags/4274). Your score is the number of bytes in your code. Lowest score for each language wins.
> Explanations are optional, but I'm more likely to upvote answers that have one.

Given a list of spheres, output an image.
Produce a greyscale [orthographic](https://en.wikipedia.org/wiki/Parallel_projection#Orthographic_projection) (parallel projection - no perspective) ray caster, that renders only plain grey spheres with no lighting.
## Input
- A list of spheres, each consisting of:
- The coordinates of its centre $(x,y,z)$.
- Its radius $r$.
- Its greyscale value $g$, an integer from $0$ to $255$ (both inclusive).
- Only the greyscale value will be restricted to integer values. The coordinates and radius may have fractional values.
## Output
- An image $1024$ by $1024$ pixels with greyscale values from $0$ to $255$.
- The image covers x and y values from $-1$ to $1$.
- A sphere's greyscale value is applied to every visible part of it. There is no variation over the surface of the sphere, so it appears flat like a disk.
- You may output whatever image format is convenient, including simple text formats such as [PGM](https://en.wikipedia.org/wiki/Netpbm#PGM_example) or a list of pixel greyscale values. If you want to include an example of your output as an image in your answer, the image formats that Codidact supports are PNG, JPEG, and GIF. You could output PGM and then convert to PNG to upload to your answer. If you output just the greyscale values as a list of numbers from $0$ to $255$, I've made a `[ TODO page to convert these to greyscale PNG]`.
- You are free to output an image reflected or rotated by a multiple of 90 degrees. However, the z axis is required to be increasing in the forwards direction (into the image) otherwise some of the test cases will look incorrect.
## Calculating a pixel value
Imagine a ray passing into the image through a pixel, perpendicular to the image plane. If it hits a sphere, it shows the greyscale value of the first sphere it hits. Otherwise, it shows a value of zero (black).
For example, for each pixel:
- The pixel $x,y$ coordinates will each be from $0$ to $1023$. Scale these to the range $-1$ to $1$ (for example, divide by $512$ and subtract $1$).
- If these scaled $x,y$ coordinates are within the radius $r$ of the $x,y$ coordinates of a sphere's centre, the ray intersects that sphere (the $z$ coordinate is not relevant to this test because the image is an orthographic projection - the rays are all parallel to the $z$ axis). The distance $a$ of the ray from the sphere's centre can be calculated as follows:
$$a=\sqrt{(x_{ray}-x_{sphere})^2+(y_{ray}-y_{sphere})^2}$$
```text
CONSIDER A DIAGRAM HERE
```
- For each sphere that the ray intersects, find the $z$ coordinate of the nearest intersection (there will often be $2$ per sphere, $1$ on the front of the sphere and $1$ on the back of the sphere, as the ray passes through - only the nearest is relevant). This can be calculated as follows:
- The distance $d$ to the nearest intersection point can be calculated from the $z$ coordinate of the sphere's centre, the radius $r$, and the distance $a$ from the $x,y$ coordinates of the sphere's centre to the scaled $x,y$ coordinates of the pixel:
$$d=z_{sphere}-\sqrt{r_{sphere}^2-a^2}$$
```text
CONSIDER A DIAGRAM HERE
```
- Of all the spheres that the ray intersects, the sphere with the intersection point with the smallest $z$ coordinate $d$ is the one to display. Set the pixel to the greyscale value of this sphere.
- If the ray did not intersect any spheres, set the pixel to zero (black).
## Example implementations
These are ungolfed code to supplement understanding of the requirements. You are free to golf these or implement your own approach that matches the test cases up to reflection & rotation.
The Python implementation uses 64 bit floating point. The Rust implementation uses 32 bit floating point to show the difference in output. I've tried to optimise the test cases for compatibility with 32 bit floating point, so the differences are restricted to being a pixel out in some places.
### Python
The function `grey_values` would be sufficient as an entry to this challenge. The rest of the code saves the output as a PNG file. The function itself has no dependence on the `import png` so it would not be required in an entry.
Instructions for installing the `png` package can be found on its [project page on PyPI](https://pypi.org/project/pypng/).
```python
def grey_values(spheres):
values = []
for y_pixel in range(1024):
for x_pixel in range(1024):
x_ray, y_ray = x_pixel / 512 - 1, y_pixel / 512 - 1
min_distance, hit_grey_value = None, 0
for sphere in spheres:
x_sphere, y_sphere, z_sphere, radius, grey = sphere
a = (
(x_ray - x_sphere) ** 2 + (y_ray - y_sphere) ** 2
) ** 0.5
if a < radius:
distance = z_sphere - (radius ** 2 - a ** 2) ** 0.5
if min_distance is None or distance < min_distance:
min_distance = distance
hit_grey_value = grey
values.append(hit_grey_value)
return values
import png
test_case = [
[0, 0, 1000, 2, 192],
[-2, 2, 100, 3, 98],
[2, 2, 100, 3, 98],
[0.5, -0.5, 500, 0.2, 255],
[-0.349469, -0.1, 20.000279, 0.6, 64],
[-0.349775, -0.100273, 20.000485, 0.6, 85],
[-0.350214, -0.100386, 20.000407, 0.6, 106],
[-0.350527, -0.100273, 20.000091, 0.6, 127],
[-0.350531, -0.1, 19.999721, 0.6, 148],
[-0.350225, -0.099727, 19.999515, 0.6, 169],
[-0.349786, -0.099614, 19.999593, 0.6, 190],
[-0.349473, -0.099727, 19.999909, 0.6, 211],
[0.20032, 0.349824, 15.000456, 0.3, 0],
[0.20032, 0.349517, 14.999924, 0.3, 0],
[0.19968, 0.350483, 15.000076, 0.3, 0],
[0.19968, 0.350176, 14.999544, 0.3, 0],
[0.200578, 0.350073, 14.999958, 0.3, 0],
[0.199817, 0.349519, 15.000277, 0.3, 0],
[0.200183, 0.350481, 14.999723, 0.3, 0],
[0.199422, 0.349927, 15.000042, 0.3, 0],
[0.200235, 0.35042, 15.000332, 0.3, 0],
[0.199765, 0.350078, 15.000529, 0.3, 0],
[0.200235, 0.349922, 14.999471, 0.3, 0],
[0.199765, 0.34958, 14.999668, 0.3, 0],
[0.200429, 0.350419, 15.000005, 0.3, 255],
[0.199571, 0.349795, 15.000366, 0.3, 255],
[0.200429, 0.350205, 14.999634, 0.3, 255],
[0.199571, 0.349581, 14.999995, 0.3, 255],
[0.200524, 0.349747, 15.000146, 0.3, 255],
[0.200196, 0.349509, 15.000283, 0.3, 255],
[0.199804, 0.350491, 14.999717, 0.3, 255],
[0.199476, 0.350253, 14.999854, 0.3, 255],
[0.200138, 0.349578, 14.999597, 0.3, 255],
[0.200138, 0.350138, 15.000567, 0.3, 255],
[0.199862, 0.349862, 14.999433, 0.3, 255],
[0.199862, 0.350422, 15.000403, 0.3, 255],
[0.200488, 0.350136, 15.000321, 0.3, 255],
[0.199957, 0.349751, 15.000544, 0.3, 255],
[0.200488, 0.34979, 14.999721, 0.3, 255],
[0.199957, 0.349404, 14.999944, 0.3, 255],
[0.200043, 0.350596, 15.000056, 0.3, 255],
[0.199512, 0.35021, 15.000279, 0.3, 255],
[0.200043, 0.350249, 14.999456, 0.3, 255],
[0.199512, 0.349864, 14.999679, 0.3, 255],
[-0.5, 0.500442, 10.000442, 0.2, 0],
[-0.5, 0.5, 10, 0.1995, 255],
[-0.499966, 0.49971, 9.999563, 0.199, 0],
[-0.500034, 0.499563, 9.99971, 0.199, 0],
[-0.499934, 0.499432, 9.999144, 0.1985, 255],
[-0.500066, 0.499144, 9.999432, 0.1985, 255],
[0.6, 0.5, 10, 0.25, 210],
[0.600122, 0.4999, 10.000122, 0.25, 210],
[0.600061, 0.49995, 10.000061, 0.25003, 50],
[0.595856, 0.505256, 9.991271, 0.24, 100],
[0.597973, 0.507461, 9.992175, 0.24, 210],
[0.595362, 0.507926, 9.993945, 0.24, 100],
[0.594618, 0.509055, 9.996832, 0.24, 210],
[0.592382, 0.506906, 9.996092, 0.24, 100],
[0.590438, 0.504882, 9.997604, 0.24, 210],
[0.591035, 0.503605, 9.994744, 0.24, 100],
[0.59121, 0.500708, 9.993425, 0.24, 210],
[0.593181, 0.502585, 9.991764, 0.24, 100],
[0.595867, 0.502302, 9.990069, 0.24, 210],
]
values = grey_values(test_case)
rows = [values[row * 1024:row * 1024 + 1024] for row in range(1024)]
with open('combination_scene.png', 'wb') as file:
w = png.Writer(1024, 1024, greyscale=True)
w.write(file, rows)
```
### Rust
The function `grey_values` would be sufficient as an entry to this challenge. The rest of the code saves the output as a PNG file.
The `png` crate can be installed using `cargo add png`. The `png` crate is not required for the `grey_values` function, only for saving as a PNG file.
```rust
fn grey_values(spheres: &Vec<[f32; 5]>) -> Vec<u8> {
let mut values = vec![];
for y_pixel in 0..1024 {
for x_pixel in 0..1024 {
let x_ray = x_pixel as f32 / 512.0 - 1.0;
let y_ray = y_pixel as f32 / 512.0 - 1.0;
let mut min_distance = f32::INFINITY;
let mut hit_grey_value = 0;
for sphere in spheres {
let [
x_sphere, y_sphere, z_sphere, radius, grey
] = *sphere;
let a = (
(x_ray - x_sphere).powf(2.0)
+ (y_ray - y_sphere).powf(2.0)
).sqrt();
if a < radius {
let distance = z_sphere - (
radius.powf(2.0) - a.powf(2.0)
).sqrt();
if distance < min_distance {
min_distance = distance;
hit_grey_value = grey as u8;
}
}
}
values.push(hit_grey_value);
}
}
values
}
fn main() {
let test_case: Vec<[f32; 5]> = vec![
[0.0, 0.0, 1000.0, 2.0, 192.0],
[-2.0, 2.0, 100.0, 3.0, 98.0],
[2.0, 2.0, 100.0, 3.0, 98.0],
[0.5, -0.5, 500.0, 0.2, 255.0],
[-0.349469, -0.1, 20.000279, 0.6, 64.0],
[-0.349775, -0.100273, 20.000485, 0.6, 85.0],
[-0.350214, -0.100386, 20.000407, 0.6, 106.0],
[-0.350527, -0.100273, 20.000091, 0.6, 127.0],
[-0.350531, -0.1, 19.999721, 0.6, 148.0],
[-0.350225, -0.099727, 19.999515, 0.6, 169.0],
[-0.349786, -0.099614, 19.999593, 0.6, 190.0],
[-0.349473, -0.099727, 19.999909, 0.6, 211.0],
[0.20032, 0.349824, 15.000456, 0.3, 0.0],
[0.20032, 0.349517, 14.999924, 0.3, 0.0],
[0.19968, 0.350483, 15.000076, 0.3, 0.0],
[0.19968, 0.350176, 14.999544, 0.3, 0.0],
[0.200578, 0.350073, 14.999958, 0.3, 0.0],
[0.199817, 0.349519, 15.000277, 0.3, 0.0],
[0.200183, 0.350481, 14.999723, 0.3, 0.0],
[0.199422, 0.349927, 15.000042, 0.3, 0.0],
[0.200235, 0.35042, 15.000332, 0.3, 0.0],
[0.199765, 0.350078, 15.000529, 0.3, 0.0],
[0.200235, 0.349922, 14.999471, 0.3, 0.0],
[0.199765, 0.34958, 14.999668, 0.3, 0.0],
[0.200429, 0.350419, 15.000005, 0.3, 255.0],
[0.199571, 0.349795, 15.000366, 0.3, 255.0],
[0.200429, 0.350205, 14.999634, 0.3, 255.0],
[0.199571, 0.349581, 14.999995, 0.3, 255.0],
[0.200524, 0.349747, 15.000146, 0.3, 255.0],
[0.200196, 0.349509, 15.000283, 0.3, 255.0],
[0.199804, 0.350491, 14.999717, 0.3, 255.0],
[0.199476, 0.350253, 14.999854, 0.3, 255.0],
[0.200138, 0.349578, 14.999597, 0.3, 255.0],
[0.200138, 0.350138, 15.000567, 0.3, 255.0],
[0.199862, 0.349862, 14.999433, 0.3, 255.0],
[0.199862, 0.350422, 15.000403, 0.3, 255.0],
[0.200488, 0.350136, 15.000321, 0.3, 255.0],
[0.199957, 0.349751, 15.000544, 0.3, 255.0],
[0.200488, 0.34979, 14.999721, 0.3, 255.0],
[0.199957, 0.349404, 14.999944, 0.3, 255.0],
[0.200043, 0.350596, 15.000056, 0.3, 255.0],
[0.199512, 0.35021, 15.000279, 0.3, 255.0],
[0.200043, 0.350249, 14.999456, 0.3, 255.0],
[0.199512, 0.349864, 14.999679, 0.3, 255.0],
[-0.5, 0.500442, 10.000442, 0.2, 0.0],
[-0.5, 0.5, 10.0, 0.1995, 255.0],
[-0.499966, 0.49971, 9.999563, 0.199, 0.0],
[-0.500034, 0.499563, 9.99971, 0.199, 0.0],
[-0.499934, 0.499432, 9.999144, 0.1985, 255.0],
[-0.500066, 0.499144, 9.999432, 0.1985, 255.0],
[0.6, 0.5, 10.0, 0.25, 210.0],
[0.600122, 0.4999, 10.000122, 0.25, 210.0],
[0.600061, 0.49995, 10.000061, 0.25003, 50.0],
[0.595856, 0.505256, 9.991271, 0.24, 100.0],
[0.597973, 0.507461, 9.992175, 0.24, 210.0],
[0.595362, 0.507926, 9.993945, 0.24, 100.0],
[0.594618, 0.509055, 9.996832, 0.24, 210.0],
[0.592382, 0.506906, 9.996092, 0.24, 100.0],
[0.590438, 0.504882, 9.997604, 0.24, 210.0],
[0.591035, 0.503605, 9.994744, 0.24, 100.0],
[0.59121, 0.500708, 9.993425, 0.24, 210.0],
[0.593181, 0.502585, 9.991764, 0.24, 100.0],
[0.595867, 0.502302, 9.990069, 0.24, 210.0],
];
let values = grey_values(&test_case);
let path = std::path::Path::new("combination_scene.png");
let file = std::fs::File::create(path).unwrap();
let w = &mut std::io::BufWriter::new(file);
let mut encoder = png::Encoder::new(w, 1024, 1024);
encoder.set_color(png::ColorType::Grayscale);
encoder.set_depth(png::BitDepth::Eight);
let mut writer = encoder.write_header().unwrap();
writer.write_image_data(&values).unwrap();
}
```
## Test cases
The output images may be scaled down to fit this page. Click on any image to open it full size.
### Single sphere
Input:
```text
[
[0, 0, 0, 1, 128],
]
```
Output:
[![Grey circle on a black background](https://codegolf.codidact.com/uploads/jlru163pa2pgm7aaq70b2tff16uf)](https://codegolf.codidact.com/uploads/jlru163pa2pgm7aaq70b2tff16uf)
### Two separated spheres
Input:
```text
[
[0, -0.55, 0, 0.4, 100],
[0, 0.55, 0, 0.4, 200],
]
```
Output:
[![Light and dark grey circles on a black background](https://codegolf.codidact.com/uploads/8yw7agli78y65kiomb1u5yebs7c5)](https://codegolf.codidact.com/uploads/8yw7agli78y65kiomb1u5yebs7c5)
### Ring of spheres
Input:
```text
[
[0.75, 0, 0, 0.25, 64],
[0.53033, 0.375, -0.375, 0.25, 185],
[0, 0.53033, -0.53033, 0.25, 64],
[-0.53033, 0.375, -0.375, 0.25, 185],
[-0.75, 0, 0, 0.25, 64],
[-0.53033, -0.375, 0.375, 0.25, 185],
[0, -0.53033, 0.53033, 0.25, 64],
[0.53033, -0.375, 0.375, 0.25, 185],
]
```
Output:
[![Eight circles in alternating light and dark grey in a ring some overlapping](https://codegolf.codidact.com/uploads/0lsenynig27au0sryf3ruozt1uwh)](https://codegolf.codidact.com/uploads/0lsenynig27au0sryf3ruozt1uwh)
### Beach ball
Input:
```text
[
[0.001, 0, 0, 1, 64],
[0.000707, 0.000707, 0, 1, 85],
[0, 0.001, 0, 1, 106],
[-0.000707, 0.000707, 0, 1, 127],
[-0.001, 0, 0, 1, 148],
[-0.000707, -0.000707, 0, 1, 169],
[0, -0.001, 0, 1, 190],
[0.000707, -0.000707, 0, 1, 211],
]
```
Output:
[![A circle with eight sectors in different shades of grey](https://codegolf.codidact.com/uploads/51a3ni0cfwsqcdw5b49gy6282qno)](https://codegolf.codidact.com/uploads/51a3ni0cfwsqcdw5b49gy6282qno)
### Football
Input:
```text
[
[0, 0.000058, 0.001947, 1, 0],
[0, -0.001715, 0.000923, 1, 0],
[0, 0.001715, -0.000923, 1, 0],
[0, -0.000058, -0.001947, 1, 0],
[0.001657, -0.000512, 0.000887, 1, 0],
[-0.001657, -0.000512, 0.000887, 1, 0],
[0.001657, 0.000512, -0.000887, 1, 0],
[-0.001657, 0.000512, -0.000887, 1, 0],
[0.001024, 0.001435, 0.000829, 1, 0],
[-0.001024, 0.001435, 0.000829, 1, 0],
[0.001024, -0.001435, -0.000829, 1, 0],
[-0.001024, -0.001435, -0.000829, 1, 0],
[0.001868, 0.000618, 0.000357, 1, 255],
[-0.001868, 0.000618, 0.000357, 1, 255],
[0.001868, -0.000618, -0.000357, 1, 255],
[-0.001868, -0.000618, -0.000357, 1, 255],
[0.000714, -0.000934, 0.001618, 1, 255],
[-0.000714, -0.000934, 0.001618, 1, 255],
[0.000714, 0.000934, -0.001618, 1, 255],
[-0.000714, 0.000934, -0.001618, 1, 255],
[0, -0.001975, -0.000316, 1, 255],
[0, 0.001261, 0.001552, 1, 255],
[0, -0.001261, -0.001552, 1, 255],
[0, 0.001975, 0.000316, 1, 255],
[0.001155, 0.000423, 0.001577, 1, 255],
[-0.001155, 0.000423, 0.001577, 1, 255],
[0.001155, -0.001577, 0.000423, 1, 255],
[-0.001155, -0.001577, 0.000423, 1, 255],
[0.001155, 0.001577, -0.000423, 1, 255],
[-0.001155, 0.001577, -0.000423, 1, 255],
[0.001155, -0.000423, -0.001577, 1, 255],
[-0.001155, -0.000423, -0.001577, 1, 255],
]
```
Output:
[![Black pentagons on a white ball](https://codegolf.codidact.com/uploads/bmrbsw65p59imvwu8ssd42uvvx1r)](https://codegolf.codidact.com/uploads/bmrbsw65p59imvwu8ssd42uvvx1r)
### Eight ball
Input:
```text
[
[0, 0, 1000, 2, 128],
[0, 0, 0.003125, 1, 0],
[0, 0, 0, 0.9975, 255],
[0, 0.000547, -0.002573, 0.995, 0],
[0, -0.000547, -0.002573, 0.995, 0],
[0, 0.001071, -0.005037, 0.9925, 255],
[0, -0.001071, -0.005037, 0.9925, 255],
]
```
Output:
[![A ball with the number 8 in a white circle](https://codegolf.codidact.com/uploads/dvl68crpmy62rqt94mwanditt4wc)](https://codegolf.codidact.com/uploads/dvl68crpmy62rqt94mwanditt4wc)
### Star ball
Input:
```text
[
[0, 0, 0, 1, 210],
[0, 0, 0.0008, 1, 210],
[0, 0, 0.0004, 1.00012, 50],
[0.012969, 0, -0.042045, 0.96, 100],
[0.016397, 0.011913, -0.039054, 0.96, 210],
[0.004008, 0.012334, -0.042045, 0.96, 100],
[-0.006263, 0.019276, -0.039054, 0.96, 210],
[-0.010492, 0.007623, -0.042045, 0.96, 100],
[-0.020268, 0, -0.039054, 0.96, 210],
[-0.010492, -0.007623, -0.042045, 0.96, 100],
[-0.006263, -0.019276, -0.039054, 0.96, 210],
[0.004008, -0.012334, -0.042045, 0.96, 100],
[0.016397, -0.011913, -0.039054, 0.96, 210],
]
```
Output:
[![A light grey ball with a dark grey five pointed star](https://codegolf.codidact.com/uploads/f3wdgwru6j35ue72x25mazxikfx2)](https://codegolf.codidact.com/uploads/f3wdgwru6j35ue72x25mazxikfx2)
### Combination scene
Input:
```text
[
[0, 0, 1000, 2, 192],
[-2, 2, 100, 3, 98],
[2, 2, 100, 3, 98],
[0.5, -0.5, 500, 0.2, 255],
[-0.349469, -0.1, 20.000279, 0.6, 64],
[-0.349775, -0.100273, 20.000485, 0.6, 85],
[-0.350214, -0.100386, 20.000407, 0.6, 106],
[-0.350527, -0.100273, 20.000091, 0.6, 127],
[-0.350531, -0.1, 19.999721, 0.6, 148],
[-0.350225, -0.099727, 19.999515, 0.6, 169],
[-0.349786, -0.099614, 19.999593, 0.6, 190],
[-0.349473, -0.099727, 19.999909, 0.6, 211],
[0.20032, 0.349824, 15.000456, 0.3, 0],
[0.20032, 0.349517, 14.999924, 0.3, 0],
[0.19968, 0.350483, 15.000076, 0.3, 0],
[0.19968, 0.350176, 14.999544, 0.3, 0],
[0.200578, 0.350073, 14.999958, 0.3, 0],
[0.199817, 0.349519, 15.000277, 0.3, 0],
[0.200183, 0.350481, 14.999723, 0.3, 0],
[0.199422, 0.349927, 15.000042, 0.3, 0],
[0.200235, 0.35042, 15.000332, 0.3, 0],
[0.199765, 0.350078, 15.000529, 0.3, 0],
[0.200235, 0.349922, 14.999471, 0.3, 0],
[0.199765, 0.34958, 14.999668, 0.3, 0],
[0.200429, 0.350419, 15.000005, 0.3, 255],
[0.199571, 0.349795, 15.000366, 0.3, 255],
[0.200429, 0.350205, 14.999634, 0.3, 255],
[0.199571, 0.349581, 14.999995, 0.3, 255],
[0.200524, 0.349747, 15.000146, 0.3, 255],
[0.200196, 0.349509, 15.000283, 0.3, 255],
[0.199804, 0.350491, 14.999717, 0.3, 255],
[0.199476, 0.350253, 14.999854, 0.3, 255],
[0.200138, 0.349578, 14.999597, 0.3, 255],
[0.200138, 0.350138, 15.000567, 0.3, 255],
[0.199862, 0.349862, 14.999433, 0.3, 255],
[0.199862, 0.350422, 15.000403, 0.3, 255],
[0.200488, 0.350136, 15.000321, 0.3, 255],
[0.199957, 0.349751, 15.000544, 0.3, 255],
[0.200488, 0.34979, 14.999721, 0.3, 255],
[0.199957, 0.349404, 14.999944, 0.3, 255],
[0.200043, 0.350596, 15.000056, 0.3, 255],
[0.199512, 0.35021, 15.000279, 0.3, 255],
[0.200043, 0.350249, 14.999456, 0.3, 255],
[0.199512, 0.349864, 14.999679, 0.3, 255],
[-0.5, 0.500442, 10.000442, 0.2, 0],
[-0.5, 0.5, 10, 0.1995, 255],
[-0.499966, 0.49971, 9.999563, 0.199, 0],
[-0.500034, 0.499563, 9.99971, 0.199, 0],
[-0.499934, 0.499432, 9.999144, 0.1985, 255],
[-0.500066, 0.499144, 9.999432, 0.1985, 255],
[0.6, 0.5, 10, 0.25, 210],
[0.600122, 0.4999, 10.000122, 0.25, 210],
[0.600061, 0.49995, 10.000061, 0.25003, 50],
[0.595856, 0.505256, 9.991271, 0.24, 100],
[0.597973, 0.507461, 9.992175, 0.24, 210],
[0.595362, 0.507926, 9.993945, 0.24, 100],
[0.594618, 0.509055, 9.996832, 0.24, 210],
[0.592382, 0.506906, 9.996092, 0.24, 100],
[0.590438, 0.504882, 9.997604, 0.24, 210],
[0.591035, 0.503605, 9.994744, 0.24, 100],
[0.59121, 0.500708, 9.993425, 0.24, 210],
[0.593181, 0.502585, 9.991764, 0.24, 100],
[0.595867, 0.502302, 9.990069, 0.24, 210],
]
```
Output:
[![A beachball, football, eightball, and starball with the sun behind hills in the background](https://codegolf.codidact.com/uploads/usv9npm1t5bjpcaknn6rbzgkqwmi)](https://codegolf.codidact.com/uploads/usv9npm1t5bjpcaknn6rbzgkqwmi)
## Explanatory animations
The example images were produced from plain grey spheres with no surface pattern or texture. The following animations show how they break down into their component plain spheres.
<details>
<summary>Animations</summary>
TODO
</details>
## Scoring
This is a [code golf challenge](https://codegolf.codidact.com/categories/49/tags/4274). Your score is the number of bytes in your code. Lowest score for each language wins.
> Explanations are optional, but I'm more likely to upvote answers that have one.

#12: Post edited by

trichoplax‭ · 2024-07-26T12:43:44Z (5 months ago)
Remove negative zero from test case inputs

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Given a list of spheres, output an image.
Produce a greyscale [orthographic](https://en.wikipedia.org/wiki/Parallel_projection#Orthographic_projection) (parallel projection - no perspective) ray caster, that renders only plain grey spheres with no lighting.
## Input
- A list of spheres, each consisting of:
- The coordinates of its centre $(x,y,z)$.
- Its radius $r$.
- Its greyscale value $g$, an integer from $0$ to $255$ (both inclusive).
- Only the greyscale value will be restricted to integer values. The coordinates and radius may have fractional values.
## Output
- An image $1024$ by $1024$ pixels with greyscale values from $0$ to $255$.
- The image covers x and y values from $-1$ to $1$.
- A sphere's greyscale value is applied to every visible part of it. There is no variation over the surface of the sphere, so it appears flat like a disk.
- You may output whatever image format is convenient, including simple text formats such as [PGM](https://en.wikipedia.org/wiki/Netpbm#PGM_example) or a list of pixel greyscale values. If you want to include an example of your output as an image in your answer, the image formats that Codidact supports are PNG, JPEG, and GIF. You could output PGM and then convert to PNG to upload to your answer. If you output just the greyscale values as a list of numbers from $0$ to $255$, I've made a `[ TODO page to convert these to greyscale PNG]`.
- You are free to output an image reflected or rotated by a multiple of 90 degrees. However, the z axis is required to be increasing in the forwards direction (into the image) otherwise some of the test cases will look incorrect.
## Calculating a pixel value
Imagine a ray passing into the image through a pixel, perpendicular to the image plane. If it hits a sphere, it shows the greyscale value of the first sphere it hits. Otherwise, it shows a value of zero (black).
For example, for each pixel:
- The pixel $x,y$ coordinates will each be from $0$ to $1023$. Scale these to the range $-1$ to $1$ (for example, divide by $512$ and subtract $1$).
- If these scaled $x,y$ coordinates are within the radius $r$ of the $x,y$ coordinates of a sphere's centre, the ray intersects that sphere (the $z$ coordinate is not relevant to this test because the image is an orthographic projection - the rays are all parallel to the $z$ axis). The distance $a$ of the ray from the sphere's centre can be calculated as follows:
$$a=\sqrt{(x_{ray}-x_{sphere})^2+(y_{ray}-y_{sphere})^2}$$
```text
CONSIDER A DIAGRAM HERE
```
- For each sphere that the ray intersects, find the $z$ coordinate of the nearest intersection (there will often be $2$ per sphere, $1$ on the front of the sphere and $1$ on the back of the sphere, as the ray passes through - only the nearest is relevant). This can be calculated as follows:
- The distance $d$ to the nearest intersection point can be calculated from the $z$ coordinate of the sphere's centre, the radius $r$, and the distance $a$ from the $x,y$ coordinates of the sphere's centre to the scaled $x,y$ coordinates of the pixel:
$$d=z_{sphere}-\sqrt{r_{sphere}^2-a^2}$$
```text
CONSIDER A DIAGRAM HERE
```
- Of all the spheres that the ray intersects, the sphere with the intersection point with the smallest $z$ coordinate $d$ is the one to display. Set the pixel to the greyscale value of this sphere.
- If the ray did not intersect any spheres, set the pixel to zero (black).
## Example implementations
These are ungolfed code to supplement understanding of the requirements. You are free to golf these or implement your own approach that matches the test cases up to reflection & rotation.
The Python implementation uses 64 bit floating point. The Rust implementation uses 32 bit floating point to show the difference in output. I've tried to optimise the test cases for compatibility with 32 bit floating point, so the differences are restricted to being a pixel out in some places.
### Python
The function `grey_values` would be sufficient as an entry to this challenge. The rest of the code saves the outputs as PNG files. The function itself has no dependence on the `import png` so it would not be required in an entry.
Instructions for installing the `png` package can be found on its [project page on PyPI](https://pypi.org/project/pypng/).
```python
def grey_values(spheres):
values = []
for y_pixel in range(1024):
for x_pixel in range(1024):
x_ray, y_ray = x_pixel / 512 - 1, y_pixel / 512 - 1
min_distance, hit_grey_value = None, 0
for sphere in spheres:
x_sphere, y_sphere, z_sphere, radius, grey = sphere
a = (
(x_ray - x_sphere) ** 2 + (y_ray - y_sphere) ** 2
) ** 0.5
if a < radius:
distance = z_sphere - (radius ** 2 - a ** 2) ** 0.5
if min_distance is None or distance < min_distance:
min_distance = distance
hit_grey_value = grey
values.append(hit_grey_value)
return values
import png
test_case = [
[0, 0, 1000, 2, 192],
[-2, 2, 100, 3, 98],
[2, 2, 100, 3, 98],
[0.5, -0.5, 500, 0.2, 255],
[-0.349469, -0.1, 20.000279, 0.6, 64],
[-0.349775, -0.100273, 20.000485, 0.6, 85],
[-0.350214, -0.100386, 20.000407, 0.6, 106],
[-0.350527, -0.100273, 20.000091, 0.6, 127],
[-0.350531, -0.1, 19.999721, 0.6, 148],
[-0.350225, -0.099727, 19.999515, 0.6, 169],
[-0.349786, -0.099614, 19.999593, 0.6, 190],
[-0.349473, -0.099727, 19.999909, 0.6, 211],
[0.20032, 0.349824, 15.000456, 0.3, 0],
[0.20032, 0.349517, 14.999924, 0.3, 0],
[0.19968, 0.350483, 15.000076, 0.3, 0],
[0.19968, 0.350176, 14.999544, 0.3, 0],
[0.200578, 0.350073, 14.999958, 0.3, 0],
[0.199817, 0.349519, 15.000277, 0.3, 0],
[0.200183, 0.350481, 14.999723, 0.3, 0],
[0.199422, 0.349927, 15.000042, 0.3, 0],
[0.200235, 0.35042, 15.000332, 0.3, 0],
[0.199765, 0.350078, 15.000529, 0.3, 0],
[0.200235, 0.349922, 14.999471, 0.3, 0],
[0.199765, 0.34958, 14.999668, 0.3, 0],
[0.200429, 0.350419, 15.000005, 0.3, 255],
[0.199571, 0.349795, 15.000366, 0.3, 255],
[0.200429, 0.350205, 14.999634, 0.3, 255],
[0.199571, 0.349581, 14.999995, 0.3, 255],
[0.200524, 0.349747, 15.000146, 0.3, 255],
[0.200196, 0.349509, 15.000283, 0.3, 255],
[0.199804, 0.350491, 14.999717, 0.3, 255],
[0.199476, 0.350253, 14.999854, 0.3, 255],
[0.200138, 0.349578, 14.999597, 0.3, 255],
[0.200138, 0.350138, 15.000567, 0.3, 255],
[0.199862, 0.349862, 14.999433, 0.3, 255],
[0.199862, 0.350422, 15.000403, 0.3, 255],
[0.200488, 0.350136, 15.000321, 0.3, 255],
[0.199957, 0.349751, 15.000544, 0.3, 255],
[0.200488, 0.34979, 14.999721, 0.3, 255],
[0.199957, 0.349404, 14.999944, 0.3, 255],
[0.200043, 0.350596, 15.000056, 0.3, 255],
[0.199512, 0.35021, 15.000279, 0.3, 255],
[0.200043, 0.350249, 14.999456, 0.3, 255],
[0.199512, 0.349864, 14.999679, 0.3, 255],
[-0.5, 0.500442, 10.000442, 0.2, 0],
[-0.5, 0.5, 10, 0.1995, 255],
[-0.499966, 0.49971, 9.999563, 0.199, 0],
[-0.500034, 0.499563, 9.99971, 0.199, 0],
[-0.499934, 0.499432, 9.999144, 0.1985, 255],
[-0.500066, 0.499144, 9.999432, 0.1985, 255],
[0.6, 0.5, 10, 0.25, 210],
[0.600122, 0.4999, 10.000122, 0.25, 210],
[0.600061, 0.49995, 10.000061, 0.25003, 50],
[0.595856, 0.505256, 9.991271, 0.24, 100],
[0.597973, 0.507461, 9.992175, 0.24, 210],
[0.595362, 0.507926, 9.993945, 0.24, 100],
[0.594618, 0.509055, 9.996832, 0.24, 210],
[0.592382, 0.506906, 9.996092, 0.24, 100],
[0.590438, 0.504882, 9.997604, 0.24, 210],
[0.591035, 0.503605, 9.994744, 0.24, 100],
[0.59121, 0.500708, 9.993425, 0.24, 210],
[0.593181, 0.502585, 9.991764, 0.24, 100],
[0.595867, 0.502302, 9.990069, 0.24, 210],
]
values = grey_values(test_case)
rows = [values[row * 1024:row * 1024 + 1024] for row in range(1024)]
with open('combination_scene.png', 'wb') as file:
w = png.Writer(1024, 1024, greyscale=True)
w.write(file, rows)
```
### Rust
The function `grey_values` would be sufficient as an entry to this challenge. The rest of the code saves the outputs as PNG files.
The `png` crate can be installed using `cargo add png`.
```rust
fn grey_values(spheres: &Vec<[f32; 5]>) -> Vec<u8> {
let mut values = vec![];
for y_pixel in 0..1024 {
for x_pixel in 0..1024 {
let x_ray = x_pixel as f32 / 512.0 - 1.0;
let y_ray = y_pixel as f32 / 512.0 - 1.0;
let mut min_distance = f32::INFINITY;
let mut hit_grey_value = 0;
for sphere in spheres {
let [
x_sphere, y_sphere, z_sphere, radius, grey
] = *sphere;
let a = (
(x_ray - x_sphere).powf(2.0)
+ (y_ray - y_sphere).powf(2.0)
).sqrt();
if a < radius {
let distance = z_sphere - (
radius.powf(2.0) - a.powf(2.0)
).sqrt();
if distance < min_distance {
min_distance = distance;
hit_grey_value = grey as u8;
}
}
}
values.push(hit_grey_value);
}
}
values
}
fn main() {
let test_case: Vec<[f32; 5]> = vec![
[0.0, 0.0, 1000.0, 2.0, 192.0],
[-2.0, 2.0, 100.0, 3.0, 98.0],
[2.0, 2.0, 100.0, 3.0, 98.0],
[0.5, -0.5, 500.0, 0.2, 255.0],
[-0.349469, -0.1, 20.000279, 0.6, 64.0],
[-0.349775, -0.100273, 20.000485, 0.6, 85.0],
[-0.350214, -0.100386, 20.000407, 0.6, 106.0],
[-0.350527, -0.100273, 20.000091, 0.6, 127.0],
[-0.350531, -0.1, 19.999721, 0.6, 148.0],
[-0.350225, -0.099727, 19.999515, 0.6, 169.0],
[-0.349786, -0.099614, 19.999593, 0.6, 190.0],
[-0.349473, -0.099727, 19.999909, 0.6, 211.0],
[0.20032, 0.349824, 15.000456, 0.3, 0.0],
[0.20032, 0.349517, 14.999924, 0.3, 0.0],
[0.19968, 0.350483, 15.000076, 0.3, 0.0],
[0.19968, 0.350176, 14.999544, 0.3, 0.0],
[0.200578, 0.350073, 14.999958, 0.3, 0.0],
[0.199817, 0.349519, 15.000277, 0.3, 0.0],
[0.200183, 0.350481, 14.999723, 0.3, 0.0],
[0.199422, 0.349927, 15.000042, 0.3, 0.0],
[0.200235, 0.35042, 15.000332, 0.3, 0.0],
[0.199765, 0.350078, 15.000529, 0.3, 0.0],
[0.200235, 0.349922, 14.999471, 0.3, 0.0],
[0.199765, 0.34958, 14.999668, 0.3, 0.0],
[0.200429, 0.350419, 15.000005, 0.3, 255.0],
[0.199571, 0.349795, 15.000366, 0.3, 255.0],
[0.200429, 0.350205, 14.999634, 0.3, 255.0],
[0.199571, 0.349581, 14.999995, 0.3, 255.0],
[0.200524, 0.349747, 15.000146, 0.3, 255.0],
[0.200196, 0.349509, 15.000283, 0.3, 255.0],
[0.199804, 0.350491, 14.999717, 0.3, 255.0],
[0.199476, 0.350253, 14.999854, 0.3, 255.0],
[0.200138, 0.349578, 14.999597, 0.3, 255.0],
[0.200138, 0.350138, 15.000567, 0.3, 255.0],
[0.199862, 0.349862, 14.999433, 0.3, 255.0],
[0.199862, 0.350422, 15.000403, 0.3, 255.0],
[0.200488, 0.350136, 15.000321, 0.3, 255.0],
[0.199957, 0.349751, 15.000544, 0.3, 255.0],
[0.200488, 0.34979, 14.999721, 0.3, 255.0],
[0.199957, 0.349404, 14.999944, 0.3, 255.0],
[0.200043, 0.350596, 15.000056, 0.3, 255.0],
[0.199512, 0.35021, 15.000279, 0.3, 255.0],
[0.200043, 0.350249, 14.999456, 0.3, 255.0],
[0.199512, 0.349864, 14.999679, 0.3, 255.0],
[-0.5, 0.500442, 10.000442, 0.2, 0.0],
[-0.5, 0.5, 10.0, 0.1995, 255.0],
[-0.499966, 0.49971, 9.999563, 0.199, 0.0],
[-0.500034, 0.499563, 9.99971, 0.199, 0.0],
[-0.499934, 0.499432, 9.999144, 0.1985, 255.0],
[-0.500066, 0.499144, 9.999432, 0.1985, 255.0],
[0.6, 0.5, 10.0, 0.25, 210.0],
[0.600122, 0.4999, 10.000122, 0.25, 210.0],
[0.600061, 0.49995, 10.000061, 0.25003, 50.0],
[0.595856, 0.505256, 9.991271, 0.24, 100.0],
[0.597973, 0.507461, 9.992175, 0.24, 210.0],
[0.595362, 0.507926, 9.993945, 0.24, 100.0],
[0.594618, 0.509055, 9.996832, 0.24, 210.0],
[0.592382, 0.506906, 9.996092, 0.24, 100.0],
[0.590438, 0.504882, 9.997604, 0.24, 210.0],
[0.591035, 0.503605, 9.994744, 0.24, 100.0],
[0.59121, 0.500708, 9.993425, 0.24, 210.0],
[0.593181, 0.502585, 9.991764, 0.24, 100.0],
[0.595867, 0.502302, 9.990069, 0.24, 210.0],
];
let values = grey_values(&test_case);
let path = std::path::Path::new("combination_scene.png");
let file = std::fs::File::create(path).unwrap();
let w = &mut std::io::BufWriter::new(file);
let mut encoder = png::Encoder::new(w, 1024, 1024);
encoder.set_color(png::ColorType::Grayscale);
encoder.set_depth(png::BitDepth::Eight);
let mut writer = encoder.write_header().unwrap();
writer.write_image_data(&values).unwrap();
}
```
## Test cases
The output images may be scaled down to fit. Click on any image to open it full size.
### Single sphere
Input:
```text
[
[0, 0, 0, 1, 128],
]
```
Output:
[![Grey circle on a black background](https://codegolf.codidact.com/uploads/jlru163pa2pgm7aaq70b2tff16uf)](https://codegolf.codidact.com/uploads/jlru163pa2pgm7aaq70b2tff16uf)
### Two separated spheres
Input:
```text
[
[0, -0.55, 0, 0.4, 100],
[0, 0.55, 0, 0.4, 200],
]
```
Output:
[![Light and dark grey circles on a black background](https://codegolf.codidact.com/uploads/8yw7agli78y65kiomb1u5yebs7c5)](https://codegolf.codidact.com/uploads/8yw7agli78y65kiomb1u5yebs7c5)
### Ring of spheres
Input:
```text
[
[0.75, 0, 0, 0.25, 64],
[0.53033, 0.375, -0.375, 0.25, 185],
[0, 0.53033, -0.53033, 0.25, 64],
[-0.53033, 0.375, -0.375, 0.25, 185],
~~[-0.75, 0, -0, 0.25, 64],~~
[-0.53033, -0.375, 0.375, 0.25, 185],
~~[-0, -0.53033, 0.53033, 0.25, 64],~~
[0.53033, -0.375, 0.375, 0.25, 185],
]
```
Output:
[![Eight circles in alternating light and dark grey in a ring some overlapping](https://codegolf.codidact.com/uploads/0lsenynig27au0sryf3ruozt1uwh)](https://codegolf.codidact.com/uploads/0lsenynig27au0sryf3ruozt1uwh)
### Beach ball
Input:
```text
[
[0.001, 0, 0, 1, 64],
~~[0.000707, 0.000707, -0, 1, 85],~~
~~[0, 0.001, -0, 1, 106],~~
~~[-0.000707, 0.000707, -0, 1, 127],~~
~~[-0.001, 0, -0, 1, 148],~~
[-0.000707, -0.000707, 0, 1, 169],
~~[-0, -0.001, 0, 1, 190],~~
[0.000707, -0.000707, 0, 1, 211],
]
```
Output:
[![A circle with eight sectors in different shades of grey](https://codegolf.codidact.com/uploads/51a3ni0cfwsqcdw5b49gy6282qno)](https://codegolf.codidact.com/uploads/51a3ni0cfwsqcdw5b49gy6282qno)
### Football
Input:
```text
[
[0, 0.000058, 0.001947, 1, 0],
[0, -0.001715, 0.000923, 1, 0],
[0, 0.001715, -0.000923, 1, 0],
[0, -0.000058, -0.001947, 1, 0],
[0.001657, -0.000512, 0.000887, 1, 0],
[-0.001657, -0.000512, 0.000887, 1, 0],
[0.001657, 0.000512, -0.000887, 1, 0],
[-0.001657, 0.000512, -0.000887, 1, 0],
[0.001024, 0.001435, 0.000829, 1, 0],
[-0.001024, 0.001435, 0.000829, 1, 0],
[0.001024, -0.001435, -0.000829, 1, 0],
[-0.001024, -0.001435, -0.000829, 1, 0],
[0.001868, 0.000618, 0.000357, 1, 255],
[-0.001868, 0.000618, 0.000357, 1, 255],
[0.001868, -0.000618, -0.000357, 1, 255],
[-0.001868, -0.000618, -0.000357, 1, 255],
[0.000714, -0.000934, 0.001618, 1, 255],
[-0.000714, -0.000934, 0.001618, 1, 255],
[0.000714, 0.000934, -0.001618, 1, 255],
[-0.000714, 0.000934, -0.001618, 1, 255],
[0, -0.001975, -0.000316, 1, 255],
[0, 0.001261, 0.001552, 1, 255],
[0, -0.001261, -0.001552, 1, 255],
[0, 0.001975, 0.000316, 1, 255],
[0.001155, 0.000423, 0.001577, 1, 255],
[-0.001155, 0.000423, 0.001577, 1, 255],
[0.001155, -0.001577, 0.000423, 1, 255],
[-0.001155, -0.001577, 0.000423, 1, 255],
[0.001155, 0.001577, -0.000423, 1, 255],
[-0.001155, 0.001577, -0.000423, 1, 255],
[0.001155, -0.000423, -0.001577, 1, 255],
[-0.001155, -0.000423, -0.001577, 1, 255],
]
```
Output:
[![Black pentagons on a white ball](https://codegolf.codidact.com/uploads/bmrbsw65p59imvwu8ssd42uvvx1r)](https://codegolf.codidact.com/uploads/bmrbsw65p59imvwu8ssd42uvvx1r)
### Eight ball
Input:
```text
[
[0, 0, 1000, 2, 128],
[0, 0, 0.003125, 1, 0],
[0, 0, 0, 0.9975, 255],
[0, 0.000547, -0.002573, 0.995, 0],
~~[-0, -0.000547, -0.002573, 0.995, 0],~~
[0, 0.001071, -0.005037, 0.9925, 255],
~~[-0, -0.001071, -0.005037, 0.9925, 255],~~
]
```
Output:
[![A ball with the number 8 in a white circle](https://codegolf.codidact.com/uploads/dvl68crpmy62rqt94mwanditt4wc)](https://codegolf.codidact.com/uploads/dvl68crpmy62rqt94mwanditt4wc)
### Star ball
Input:
```text
[
[0, 0, 0, 1, 210],
[0, 0, 0.0008, 1, 210],
[0, 0, 0.0004, 1.00012, 50],
~~[0.012969, -0, -0.042045, 0.96, 100],~~
[0.016397, 0.011913, -0.039054, 0.96, 210],
[0.004008, 0.012334, -0.042045, 0.96, 100],
[-0.006263, 0.019276, -0.039054, 0.96, 210],
[-0.010492, 0.007623, -0.042045, 0.96, 100],
[-0.020268, 0, -0.039054, 0.96, 210],
[-0.010492, -0.007623, -0.042045, 0.96, 100],
[-0.006263, -0.019276, -0.039054, 0.96, 210],
[0.004008, -0.012334, -0.042045, 0.96, 100],
[0.016397, -0.011913, -0.039054, 0.96, 210],
]
```
Output:
[![A light grey ball with a dark grey five pointed star](https://codegolf.codidact.com/uploads/f3wdgwru6j35ue72x25mazxikfx2)](https://codegolf.codidact.com/uploads/f3wdgwru6j35ue72x25mazxikfx2)
### Combination scene
Input:
```text
[
[0, 0, 1000, 2, 192],
[-2, 2, 100, 3, 98],
[2, 2, 100, 3, 98],
[0.5, -0.5, 500, 0.2, 255],
[-0.349469, -0.1, 20.000279, 0.6, 64],
[-0.349775, -0.100273, 20.000485, 0.6, 85],
[-0.350214, -0.100386, 20.000407, 0.6, 106],
[-0.350527, -0.100273, 20.000091, 0.6, 127],
[-0.350531, -0.1, 19.999721, 0.6, 148],
[-0.350225, -0.099727, 19.999515, 0.6, 169],
[-0.349786, -0.099614, 19.999593, 0.6, 190],
[-0.349473, -0.099727, 19.999909, 0.6, 211],
[0.20032, 0.349824, 15.000456, 0.3, 0],
[0.20032, 0.349517, 14.999924, 0.3, 0],
[0.19968, 0.350483, 15.000076, 0.3, 0],
[0.19968, 0.350176, 14.999544, 0.3, 0],
[0.200578, 0.350073, 14.999958, 0.3, 0],
[0.199817, 0.349519, 15.000277, 0.3, 0],
[0.200183, 0.350481, 14.999723, 0.3, 0],
[0.199422, 0.349927, 15.000042, 0.3, 0],
[0.200235, 0.35042, 15.000332, 0.3, 0],
[0.199765, 0.350078, 15.000529, 0.3, 0],
[0.200235, 0.349922, 14.999471, 0.3, 0],
[0.199765, 0.34958, 14.999668, 0.3, 0],
[0.200429, 0.350419, 15.000005, 0.3, 255],
[0.199571, 0.349795, 15.000366, 0.3, 255],
[0.200429, 0.350205, 14.999634, 0.3, 255],
[0.199571, 0.349581, 14.999995, 0.3, 255],
[0.200524, 0.349747, 15.000146, 0.3, 255],
[0.200196, 0.349509, 15.000283, 0.3, 255],
[0.199804, 0.350491, 14.999717, 0.3, 255],
[0.199476, 0.350253, 14.999854, 0.3, 255],
[0.200138, 0.349578, 14.999597, 0.3, 255],
[0.200138, 0.350138, 15.000567, 0.3, 255],
[0.199862, 0.349862, 14.999433, 0.3, 255],
[0.199862, 0.350422, 15.000403, 0.3, 255],
[0.200488, 0.350136, 15.000321, 0.3, 255],
[0.199957, 0.349751, 15.000544, 0.3, 255],
[0.200488, 0.34979, 14.999721, 0.3, 255],
[0.199957, 0.349404, 14.999944, 0.3, 255],
[0.200043, 0.350596, 15.000056, 0.3, 255],
[0.199512, 0.35021, 15.000279, 0.3, 255],
[0.200043, 0.350249, 14.999456, 0.3, 255],
[0.199512, 0.349864, 14.999679, 0.3, 255],
[-0.5, 0.500442, 10.000442, 0.2, 0],
[-0.5, 0.5, 10, 0.1995, 255],
[-0.499966, 0.49971, 9.999563, 0.199, 0],
[-0.500034, 0.499563, 9.99971, 0.199, 0],
[-0.499934, 0.499432, 9.999144, 0.1985, 255],
[-0.500066, 0.499144, 9.999432, 0.1985, 255],
[0.6, 0.5, 10, 0.25, 210],
[0.600122, 0.4999, 10.000122, 0.25, 210],
[0.600061, 0.49995, 10.000061, 0.25003, 50],
[0.595856, 0.505256, 9.991271, 0.24, 100],
[0.597973, 0.507461, 9.992175, 0.24, 210],
[0.595362, 0.507926, 9.993945, 0.24, 100],
[0.594618, 0.509055, 9.996832, 0.24, 210],
[0.592382, 0.506906, 9.996092, 0.24, 100],
[0.590438, 0.504882, 9.997604, 0.24, 210],
[0.591035, 0.503605, 9.994744, 0.24, 100],
[0.59121, 0.500708, 9.993425, 0.24, 210],
[0.593181, 0.502585, 9.991764, 0.24, 100],
[0.595867, 0.502302, 9.990069, 0.24, 210],
]
```
Output:
[![A beachball, football, eightball, and starball with the sun behind hills in the background](https://codegolf.codidact.com/uploads/usv9npm1t5bjpcaknn6rbzgkqwmi)](https://codegolf.codidact.com/uploads/usv9npm1t5bjpcaknn6rbzgkqwmi)
## Explanatory animations
The example images were produced from plain grey spheres with no surface pattern or texture. The following animations show how they break down into their component plain spheres.
<details>
<summary>Animations</summary>
TODO
</details>
## Scoring
This is a [code golf challenge](https://codegolf.codidact.com/categories/49/tags/4274). Your score is the number of bytes in your code. Lowest score for each language wins.
> Explanations are optional, but I'm more likely to upvote answers that have one.

Given a list of spheres, output an image.
Produce a greyscale [orthographic](https://en.wikipedia.org/wiki/Parallel_projection#Orthographic_projection) (parallel projection - no perspective) ray caster, that renders only plain grey spheres with no lighting.
## Input
- A list of spheres, each consisting of:
- The coordinates of its centre $(x,y,z)$.
- Its radius $r$.
- Its greyscale value $g$, an integer from $0$ to $255$ (both inclusive).
- Only the greyscale value will be restricted to integer values. The coordinates and radius may have fractional values.
## Output
- An image $1024$ by $1024$ pixels with greyscale values from $0$ to $255$.
- The image covers x and y values from $-1$ to $1$.
- A sphere's greyscale value is applied to every visible part of it. There is no variation over the surface of the sphere, so it appears flat like a disk.
- You may output whatever image format is convenient, including simple text formats such as [PGM](https://en.wikipedia.org/wiki/Netpbm#PGM_example) or a list of pixel greyscale values. If you want to include an example of your output as an image in your answer, the image formats that Codidact supports are PNG, JPEG, and GIF. You could output PGM and then convert to PNG to upload to your answer. If you output just the greyscale values as a list of numbers from $0$ to $255$, I've made a `[ TODO page to convert these to greyscale PNG]`.
- You are free to output an image reflected or rotated by a multiple of 90 degrees. However, the z axis is required to be increasing in the forwards direction (into the image) otherwise some of the test cases will look incorrect.
## Calculating a pixel value
Imagine a ray passing into the image through a pixel, perpendicular to the image plane. If it hits a sphere, it shows the greyscale value of the first sphere it hits. Otherwise, it shows a value of zero (black).
For example, for each pixel:
- The pixel $x,y$ coordinates will each be from $0$ to $1023$. Scale these to the range $-1$ to $1$ (for example, divide by $512$ and subtract $1$).
- If these scaled $x,y$ coordinates are within the radius $r$ of the $x,y$ coordinates of a sphere's centre, the ray intersects that sphere (the $z$ coordinate is not relevant to this test because the image is an orthographic projection - the rays are all parallel to the $z$ axis). The distance $a$ of the ray from the sphere's centre can be calculated as follows:
$$a=\sqrt{(x_{ray}-x_{sphere})^2+(y_{ray}-y_{sphere})^2}$$
```text
CONSIDER A DIAGRAM HERE
```
- For each sphere that the ray intersects, find the $z$ coordinate of the nearest intersection (there will often be $2$ per sphere, $1$ on the front of the sphere and $1$ on the back of the sphere, as the ray passes through - only the nearest is relevant). This can be calculated as follows:
- The distance $d$ to the nearest intersection point can be calculated from the $z$ coordinate of the sphere's centre, the radius $r$, and the distance $a$ from the $x,y$ coordinates of the sphere's centre to the scaled $x,y$ coordinates of the pixel:
$$d=z_{sphere}-\sqrt{r_{sphere}^2-a^2}$$
```text
CONSIDER A DIAGRAM HERE
```
- Of all the spheres that the ray intersects, the sphere with the intersection point with the smallest $z$ coordinate $d$ is the one to display. Set the pixel to the greyscale value of this sphere.
- If the ray did not intersect any spheres, set the pixel to zero (black).
## Example implementations
These are ungolfed code to supplement understanding of the requirements. You are free to golf these or implement your own approach that matches the test cases up to reflection & rotation.
The Python implementation uses 64 bit floating point. The Rust implementation uses 32 bit floating point to show the difference in output. I've tried to optimise the test cases for compatibility with 32 bit floating point, so the differences are restricted to being a pixel out in some places.
### Python
The function `grey_values` would be sufficient as an entry to this challenge. The rest of the code saves the outputs as PNG files. The function itself has no dependence on the `import png` so it would not be required in an entry.
Instructions for installing the `png` package can be found on its [project page on PyPI](https://pypi.org/project/pypng/).
```python
def grey_values(spheres):
values = []
for y_pixel in range(1024):
for x_pixel in range(1024):
x_ray, y_ray = x_pixel / 512 - 1, y_pixel / 512 - 1
min_distance, hit_grey_value = None, 0
for sphere in spheres:
x_sphere, y_sphere, z_sphere, radius, grey = sphere
a = (
(x_ray - x_sphere) ** 2 + (y_ray - y_sphere) ** 2
) ** 0.5
if a < radius:
distance = z_sphere - (radius ** 2 - a ** 2) ** 0.5
if min_distance is None or distance < min_distance:
min_distance = distance
hit_grey_value = grey
values.append(hit_grey_value)
return values
import png
test_case = [
[0, 0, 1000, 2, 192],
[-2, 2, 100, 3, 98],
[2, 2, 100, 3, 98],
[0.5, -0.5, 500, 0.2, 255],
[-0.349469, -0.1, 20.000279, 0.6, 64],
[-0.349775, -0.100273, 20.000485, 0.6, 85],
[-0.350214, -0.100386, 20.000407, 0.6, 106],
[-0.350527, -0.100273, 20.000091, 0.6, 127],
[-0.350531, -0.1, 19.999721, 0.6, 148],
[-0.350225, -0.099727, 19.999515, 0.6, 169],
[-0.349786, -0.099614, 19.999593, 0.6, 190],
[-0.349473, -0.099727, 19.999909, 0.6, 211],
[0.20032, 0.349824, 15.000456, 0.3, 0],
[0.20032, 0.349517, 14.999924, 0.3, 0],
[0.19968, 0.350483, 15.000076, 0.3, 0],
[0.19968, 0.350176, 14.999544, 0.3, 0],
[0.200578, 0.350073, 14.999958, 0.3, 0],
[0.199817, 0.349519, 15.000277, 0.3, 0],
[0.200183, 0.350481, 14.999723, 0.3, 0],
[0.199422, 0.349927, 15.000042, 0.3, 0],
[0.200235, 0.35042, 15.000332, 0.3, 0],
[0.199765, 0.350078, 15.000529, 0.3, 0],
[0.200235, 0.349922, 14.999471, 0.3, 0],
[0.199765, 0.34958, 14.999668, 0.3, 0],
[0.200429, 0.350419, 15.000005, 0.3, 255],
[0.199571, 0.349795, 15.000366, 0.3, 255],
[0.200429, 0.350205, 14.999634, 0.3, 255],
[0.199571, 0.349581, 14.999995, 0.3, 255],
[0.200524, 0.349747, 15.000146, 0.3, 255],
[0.200196, 0.349509, 15.000283, 0.3, 255],
[0.199804, 0.350491, 14.999717, 0.3, 255],
[0.199476, 0.350253, 14.999854, 0.3, 255],
[0.200138, 0.349578, 14.999597, 0.3, 255],
[0.200138, 0.350138, 15.000567, 0.3, 255],
[0.199862, 0.349862, 14.999433, 0.3, 255],
[0.199862, 0.350422, 15.000403, 0.3, 255],
[0.200488, 0.350136, 15.000321, 0.3, 255],
[0.199957, 0.349751, 15.000544, 0.3, 255],
[0.200488, 0.34979, 14.999721, 0.3, 255],
[0.199957, 0.349404, 14.999944, 0.3, 255],
[0.200043, 0.350596, 15.000056, 0.3, 255],
[0.199512, 0.35021, 15.000279, 0.3, 255],
[0.200043, 0.350249, 14.999456, 0.3, 255],
[0.199512, 0.349864, 14.999679, 0.3, 255],
[-0.5, 0.500442, 10.000442, 0.2, 0],
[-0.5, 0.5, 10, 0.1995, 255],
[-0.499966, 0.49971, 9.999563, 0.199, 0],
[-0.500034, 0.499563, 9.99971, 0.199, 0],
[-0.499934, 0.499432, 9.999144, 0.1985, 255],
[-0.500066, 0.499144, 9.999432, 0.1985, 255],
[0.6, 0.5, 10, 0.25, 210],
[0.600122, 0.4999, 10.000122, 0.25, 210],
[0.600061, 0.49995, 10.000061, 0.25003, 50],
[0.595856, 0.505256, 9.991271, 0.24, 100],
[0.597973, 0.507461, 9.992175, 0.24, 210],
[0.595362, 0.507926, 9.993945, 0.24, 100],
[0.594618, 0.509055, 9.996832, 0.24, 210],
[0.592382, 0.506906, 9.996092, 0.24, 100],
[0.590438, 0.504882, 9.997604, 0.24, 210],
[0.591035, 0.503605, 9.994744, 0.24, 100],
[0.59121, 0.500708, 9.993425, 0.24, 210],
[0.593181, 0.502585, 9.991764, 0.24, 100],
[0.595867, 0.502302, 9.990069, 0.24, 210],
]
values = grey_values(test_case)
rows = [values[row * 1024:row * 1024 + 1024] for row in range(1024)]
with open('combination_scene.png', 'wb') as file:
w = png.Writer(1024, 1024, greyscale=True)
w.write(file, rows)
```
### Rust
The function `grey_values` would be sufficient as an entry to this challenge. The rest of the code saves the outputs as PNG files.
The `png` crate can be installed using `cargo add png`.
```rust
fn grey_values(spheres: &Vec<[f32; 5]>) -> Vec<u8> {
let mut values = vec![];
for y_pixel in 0..1024 {
for x_pixel in 0..1024 {
let x_ray = x_pixel as f32 / 512.0 - 1.0;
let y_ray = y_pixel as f32 / 512.0 - 1.0;
let mut min_distance = f32::INFINITY;
let mut hit_grey_value = 0;
for sphere in spheres {
let [
x_sphere, y_sphere, z_sphere, radius, grey
] = *sphere;
let a = (
(x_ray - x_sphere).powf(2.0)
+ (y_ray - y_sphere).powf(2.0)
).sqrt();
if a < radius {
let distance = z_sphere - (
radius.powf(2.0) - a.powf(2.0)
).sqrt();
if distance < min_distance {
min_distance = distance;
hit_grey_value = grey as u8;
}
}
}
values.push(hit_grey_value);
}
}
values
}
fn main() {
let test_case: Vec<[f32; 5]> = vec![
[0.0, 0.0, 1000.0, 2.0, 192.0],
[-2.0, 2.0, 100.0, 3.0, 98.0],
[2.0, 2.0, 100.0, 3.0, 98.0],
[0.5, -0.5, 500.0, 0.2, 255.0],
[-0.349469, -0.1, 20.000279, 0.6, 64.0],
[-0.349775, -0.100273, 20.000485, 0.6, 85.0],
[-0.350214, -0.100386, 20.000407, 0.6, 106.0],
[-0.350527, -0.100273, 20.000091, 0.6, 127.0],
[-0.350531, -0.1, 19.999721, 0.6, 148.0],
[-0.350225, -0.099727, 19.999515, 0.6, 169.0],
[-0.349786, -0.099614, 19.999593, 0.6, 190.0],
[-0.349473, -0.099727, 19.999909, 0.6, 211.0],
[0.20032, 0.349824, 15.000456, 0.3, 0.0],
[0.20032, 0.349517, 14.999924, 0.3, 0.0],
[0.19968, 0.350483, 15.000076, 0.3, 0.0],
[0.19968, 0.350176, 14.999544, 0.3, 0.0],
[0.200578, 0.350073, 14.999958, 0.3, 0.0],
[0.199817, 0.349519, 15.000277, 0.3, 0.0],
[0.200183, 0.350481, 14.999723, 0.3, 0.0],
[0.199422, 0.349927, 15.000042, 0.3, 0.0],
[0.200235, 0.35042, 15.000332, 0.3, 0.0],
[0.199765, 0.350078, 15.000529, 0.3, 0.0],
[0.200235, 0.349922, 14.999471, 0.3, 0.0],
[0.199765, 0.34958, 14.999668, 0.3, 0.0],
[0.200429, 0.350419, 15.000005, 0.3, 255.0],
[0.199571, 0.349795, 15.000366, 0.3, 255.0],
[0.200429, 0.350205, 14.999634, 0.3, 255.0],
[0.199571, 0.349581, 14.999995, 0.3, 255.0],
[0.200524, 0.349747, 15.000146, 0.3, 255.0],
[0.200196, 0.349509, 15.000283, 0.3, 255.0],
[0.199804, 0.350491, 14.999717, 0.3, 255.0],
[0.199476, 0.350253, 14.999854, 0.3, 255.0],
[0.200138, 0.349578, 14.999597, 0.3, 255.0],
[0.200138, 0.350138, 15.000567, 0.3, 255.0],
[0.199862, 0.349862, 14.999433, 0.3, 255.0],
[0.199862, 0.350422, 15.000403, 0.3, 255.0],
[0.200488, 0.350136, 15.000321, 0.3, 255.0],
[0.199957, 0.349751, 15.000544, 0.3, 255.0],
[0.200488, 0.34979, 14.999721, 0.3, 255.0],
[0.199957, 0.349404, 14.999944, 0.3, 255.0],
[0.200043, 0.350596, 15.000056, 0.3, 255.0],
[0.199512, 0.35021, 15.000279, 0.3, 255.0],
[0.200043, 0.350249, 14.999456, 0.3, 255.0],
[0.199512, 0.349864, 14.999679, 0.3, 255.0],
[-0.5, 0.500442, 10.000442, 0.2, 0.0],
[-0.5, 0.5, 10.0, 0.1995, 255.0],
[-0.499966, 0.49971, 9.999563, 0.199, 0.0],
[-0.500034, 0.499563, 9.99971, 0.199, 0.0],
[-0.499934, 0.499432, 9.999144, 0.1985, 255.0],
[-0.500066, 0.499144, 9.999432, 0.1985, 255.0],
[0.6, 0.5, 10.0, 0.25, 210.0],
[0.600122, 0.4999, 10.000122, 0.25, 210.0],
[0.600061, 0.49995, 10.000061, 0.25003, 50.0],
[0.595856, 0.505256, 9.991271, 0.24, 100.0],
[0.597973, 0.507461, 9.992175, 0.24, 210.0],
[0.595362, 0.507926, 9.993945, 0.24, 100.0],
[0.594618, 0.509055, 9.996832, 0.24, 210.0],
[0.592382, 0.506906, 9.996092, 0.24, 100.0],
[0.590438, 0.504882, 9.997604, 0.24, 210.0],
[0.591035, 0.503605, 9.994744, 0.24, 100.0],
[0.59121, 0.500708, 9.993425, 0.24, 210.0],
[0.593181, 0.502585, 9.991764, 0.24, 100.0],
[0.595867, 0.502302, 9.990069, 0.24, 210.0],
];
let values = grey_values(&test_case);
let path = std::path::Path::new("combination_scene.png");
let file = std::fs::File::create(path).unwrap();
let w = &mut std::io::BufWriter::new(file);
let mut encoder = png::Encoder::new(w, 1024, 1024);
encoder.set_color(png::ColorType::Grayscale);
encoder.set_depth(png::BitDepth::Eight);
let mut writer = encoder.write_header().unwrap();
writer.write_image_data(&values).unwrap();
}
```
## Test cases
The output images may be scaled down to fit. Click on any image to open it full size.
### Single sphere
Input:
```text
[
[0, 0, 0, 1, 128],
]
```
Output:
[![Grey circle on a black background](https://codegolf.codidact.com/uploads/jlru163pa2pgm7aaq70b2tff16uf)](https://codegolf.codidact.com/uploads/jlru163pa2pgm7aaq70b2tff16uf)
### Two separated spheres
Input:
```text
[
[0, -0.55, 0, 0.4, 100],
[0, 0.55, 0, 0.4, 200],
]
```
Output:
[![Light and dark grey circles on a black background](https://codegolf.codidact.com/uploads/8yw7agli78y65kiomb1u5yebs7c5)](https://codegolf.codidact.com/uploads/8yw7agli78y65kiomb1u5yebs7c5)
### Ring of spheres
Input:
```text
[
[0.75, 0, 0, 0.25, 64],
[0.53033, 0.375, -0.375, 0.25, 185],
[0, 0.53033, -0.53033, 0.25, 64],
[-0.53033, 0.375, -0.375, 0.25, 185],
[-0.75, 0, 0, 0.25, 64],
[-0.53033, -0.375, 0.375, 0.25, 185],
[0, -0.53033, 0.53033, 0.25, 64],
[0.53033, -0.375, 0.375, 0.25, 185],
]
```
Output:
[![Eight circles in alternating light and dark grey in a ring some overlapping](https://codegolf.codidact.com/uploads/0lsenynig27au0sryf3ruozt1uwh)](https://codegolf.codidact.com/uploads/0lsenynig27au0sryf3ruozt1uwh)
### Beach ball
Input:
```text
[
[0.001, 0, 0, 1, 64],
[0.000707, 0.000707, 0, 1, 85],
[0, 0.001, 0, 1, 106],
[-0.000707, 0.000707, 0, 1, 127],
[-0.001, 0, 0, 1, 148],
[-0.000707, -0.000707, 0, 1, 169],
[0, -0.001, 0, 1, 190],
[0.000707, -0.000707, 0, 1, 211],
]
```
Output:
[![A circle with eight sectors in different shades of grey](https://codegolf.codidact.com/uploads/51a3ni0cfwsqcdw5b49gy6282qno)](https://codegolf.codidact.com/uploads/51a3ni0cfwsqcdw5b49gy6282qno)
### Football
Input:
```text
[
[0, 0.000058, 0.001947, 1, 0],
[0, -0.001715, 0.000923, 1, 0],
[0, 0.001715, -0.000923, 1, 0],
[0, -0.000058, -0.001947, 1, 0],
[0.001657, -0.000512, 0.000887, 1, 0],
[-0.001657, -0.000512, 0.000887, 1, 0],
[0.001657, 0.000512, -0.000887, 1, 0],
[-0.001657, 0.000512, -0.000887, 1, 0],
[0.001024, 0.001435, 0.000829, 1, 0],
[-0.001024, 0.001435, 0.000829, 1, 0],
[0.001024, -0.001435, -0.000829, 1, 0],
[-0.001024, -0.001435, -0.000829, 1, 0],
[0.001868, 0.000618, 0.000357, 1, 255],
[-0.001868, 0.000618, 0.000357, 1, 255],
[0.001868, -0.000618, -0.000357, 1, 255],
[-0.001868, -0.000618, -0.000357, 1, 255],
[0.000714, -0.000934, 0.001618, 1, 255],
[-0.000714, -0.000934, 0.001618, 1, 255],
[0.000714, 0.000934, -0.001618, 1, 255],
[-0.000714, 0.000934, -0.001618, 1, 255],
[0, -0.001975, -0.000316, 1, 255],
[0, 0.001261, 0.001552, 1, 255],
[0, -0.001261, -0.001552, 1, 255],
[0, 0.001975, 0.000316, 1, 255],
[0.001155, 0.000423, 0.001577, 1, 255],
[-0.001155, 0.000423, 0.001577, 1, 255],
[0.001155, -0.001577, 0.000423, 1, 255],
[-0.001155, -0.001577, 0.000423, 1, 255],
[0.001155, 0.001577, -0.000423, 1, 255],
[-0.001155, 0.001577, -0.000423, 1, 255],
[0.001155, -0.000423, -0.001577, 1, 255],
[-0.001155, -0.000423, -0.001577, 1, 255],
]
```
Output:
[![Black pentagons on a white ball](https://codegolf.codidact.com/uploads/bmrbsw65p59imvwu8ssd42uvvx1r)](https://codegolf.codidact.com/uploads/bmrbsw65p59imvwu8ssd42uvvx1r)
### Eight ball
Input:
```text
[
[0, 0, 1000, 2, 128],
[0, 0, 0.003125, 1, 0],
[0, 0, 0, 0.9975, 255],
[0, 0.000547, -0.002573, 0.995, 0],
[0, -0.000547, -0.002573, 0.995, 0],
[0, 0.001071, -0.005037, 0.9925, 255],
[0, -0.001071, -0.005037, 0.9925, 255],
]
```
Output:
[![A ball with the number 8 in a white circle](https://codegolf.codidact.com/uploads/dvl68crpmy62rqt94mwanditt4wc)](https://codegolf.codidact.com/uploads/dvl68crpmy62rqt94mwanditt4wc)
### Star ball
Input:
```text
[
[0, 0, 0, 1, 210],
[0, 0, 0.0008, 1, 210],
[0, 0, 0.0004, 1.00012, 50],
[0.012969, 0, -0.042045, 0.96, 100],
[0.016397, 0.011913, -0.039054, 0.96, 210],
[0.004008, 0.012334, -0.042045, 0.96, 100],
[-0.006263, 0.019276, -0.039054, 0.96, 210],
[-0.010492, 0.007623, -0.042045, 0.96, 100],
[-0.020268, 0, -0.039054, 0.96, 210],
[-0.010492, -0.007623, -0.042045, 0.96, 100],
[-0.006263, -0.019276, -0.039054, 0.96, 210],
[0.004008, -0.012334, -0.042045, 0.96, 100],
[0.016397, -0.011913, -0.039054, 0.96, 210],
]
```
Output:
[![A light grey ball with a dark grey five pointed star](https://codegolf.codidact.com/uploads/f3wdgwru6j35ue72x25mazxikfx2)](https://codegolf.codidact.com/uploads/f3wdgwru6j35ue72x25mazxikfx2)
### Combination scene
Input:
```text
[
[0, 0, 1000, 2, 192],
[-2, 2, 100, 3, 98],
[2, 2, 100, 3, 98],
[0.5, -0.5, 500, 0.2, 255],
[-0.349469, -0.1, 20.000279, 0.6, 64],
[-0.349775, -0.100273, 20.000485, 0.6, 85],
[-0.350214, -0.100386, 20.000407, 0.6, 106],
[-0.350527, -0.100273, 20.000091, 0.6, 127],
[-0.350531, -0.1, 19.999721, 0.6, 148],
[-0.350225, -0.099727, 19.999515, 0.6, 169],
[-0.349786, -0.099614, 19.999593, 0.6, 190],
[-0.349473, -0.099727, 19.999909, 0.6, 211],
[0.20032, 0.349824, 15.000456, 0.3, 0],
[0.20032, 0.349517, 14.999924, 0.3, 0],
[0.19968, 0.350483, 15.000076, 0.3, 0],
[0.19968, 0.350176, 14.999544, 0.3, 0],
[0.200578, 0.350073, 14.999958, 0.3, 0],
[0.199817, 0.349519, 15.000277, 0.3, 0],
[0.200183, 0.350481, 14.999723, 0.3, 0],
[0.199422, 0.349927, 15.000042, 0.3, 0],
[0.200235, 0.35042, 15.000332, 0.3, 0],
[0.199765, 0.350078, 15.000529, 0.3, 0],
[0.200235, 0.349922, 14.999471, 0.3, 0],
[0.199765, 0.34958, 14.999668, 0.3, 0],
[0.200429, 0.350419, 15.000005, 0.3, 255],
[0.199571, 0.349795, 15.000366, 0.3, 255],
[0.200429, 0.350205, 14.999634, 0.3, 255],
[0.199571, 0.349581, 14.999995, 0.3, 255],
[0.200524, 0.349747, 15.000146, 0.3, 255],
[0.200196, 0.349509, 15.000283, 0.3, 255],
[0.199804, 0.350491, 14.999717, 0.3, 255],
[0.199476, 0.350253, 14.999854, 0.3, 255],
[0.200138, 0.349578, 14.999597, 0.3, 255],
[0.200138, 0.350138, 15.000567, 0.3, 255],
[0.199862, 0.349862, 14.999433, 0.3, 255],
[0.199862, 0.350422, 15.000403, 0.3, 255],
[0.200488, 0.350136, 15.000321, 0.3, 255],
[0.199957, 0.349751, 15.000544, 0.3, 255],
[0.200488, 0.34979, 14.999721, 0.3, 255],
[0.199957, 0.349404, 14.999944, 0.3, 255],
[0.200043, 0.350596, 15.000056, 0.3, 255],
[0.199512, 0.35021, 15.000279, 0.3, 255],
[0.200043, 0.350249, 14.999456, 0.3, 255],
[0.199512, 0.349864, 14.999679, 0.3, 255],
[-0.5, 0.500442, 10.000442, 0.2, 0],
[-0.5, 0.5, 10, 0.1995, 255],
[-0.499966, 0.49971, 9.999563, 0.199, 0],
[-0.500034, 0.499563, 9.99971, 0.199, 0],
[-0.499934, 0.499432, 9.999144, 0.1985, 255],
[-0.500066, 0.499144, 9.999432, 0.1985, 255],
[0.6, 0.5, 10, 0.25, 210],
[0.600122, 0.4999, 10.000122, 0.25, 210],
[0.600061, 0.49995, 10.000061, 0.25003, 50],
[0.595856, 0.505256, 9.991271, 0.24, 100],
[0.597973, 0.507461, 9.992175, 0.24, 210],
[0.595362, 0.507926, 9.993945, 0.24, 100],
[0.594618, 0.509055, 9.996832, 0.24, 210],
[0.592382, 0.506906, 9.996092, 0.24, 100],
[0.590438, 0.504882, 9.997604, 0.24, 210],
[0.591035, 0.503605, 9.994744, 0.24, 100],
[0.59121, 0.500708, 9.993425, 0.24, 210],
[0.593181, 0.502585, 9.991764, 0.24, 100],
[0.595867, 0.502302, 9.990069, 0.24, 210],
]
```
Output:
[![A beachball, football, eightball, and starball with the sun behind hills in the background](https://codegolf.codidact.com/uploads/usv9npm1t5bjpcaknn6rbzgkqwmi)](https://codegolf.codidact.com/uploads/usv9npm1t5bjpcaknn6rbzgkqwmi)
## Explanatory animations
The example images were produced from plain grey spheres with no surface pattern or texture. The following animations show how they break down into their component plain spheres.
<details>
<summary>Animations</summary>
TODO
</details>
## Scoring
This is a [code golf challenge](https://codegolf.codidact.com/categories/49/tags/4274). Your score is the number of bytes in your code. Lowest score for each language wins.
> Explanations are optional, but I'm more likely to upvote answers that have one.

#11: Post edited by

trichoplax‭ · 2024-07-26T12:38:58Z (5 months ago)
Add back in animation section now that there is more space

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Raw

Markdown

Given a list of spheres, output an image.
Produce a greyscale [orthographic](https://en.wikipedia.org/wiki/Parallel_projection#Orthographic_projection) (parallel projection - no perspective) ray caster, that renders only plain grey spheres with no lighting.
## Input
- A list of spheres, each consisting of:
- The coordinates of its centre $(x,y,z)$.
- Its radius $r$.
- Its greyscale value $g$, an integer from $0$ to $255$ (both inclusive).
- Only the greyscale value will be restricted to integer values. The coordinates and radius may have fractional values.
## Output
- An image $1024$ by $1024$ pixels with greyscale values from $0$ to $255$.
- The image covers x and y values from $-1$ to $1$.
- A sphere's greyscale value is applied to every visible part of it. There is no variation over the surface of the sphere, so it appears flat like a disk.
- You may output whatever image format is convenient, including simple text formats such as [PGM](https://en.wikipedia.org/wiki/Netpbm#PGM_example) or a list of pixel greyscale values. If you want to include an example of your output as an image in your answer, the image formats that Codidact supports are PNG, JPEG, and GIF. You could output PGM and then convert to PNG to upload to your answer. If you output just the greyscale values as a list of numbers from $0$ to $255$, I've made a `[ TODO page to convert these to greyscale PNG]`.
- You are free to output an image reflected or rotated by a multiple of 90 degrees. However, the z axis is required to be increasing in the forwards direction (into the image) otherwise some of the test cases will look incorrect.
## Calculating a pixel value
Imagine a ray passing into the image through a pixel, perpendicular to the image plane. If it hits a sphere, it shows the greyscale value of the first sphere it hits. Otherwise, it shows a value of zero (black).
For example, for each pixel:
- The pixel $x,y$ coordinates will each be from $0$ to $1023$. Scale these to the range $-1$ to $1$ (for example, divide by $512$ and subtract $1$).
- If these scaled $x,y$ coordinates are within the radius $r$ of the $x,y$ coordinates of a sphere's centre, the ray intersects that sphere (the $z$ coordinate is not relevant to this test because the image is an orthographic projection - the rays are all parallel to the $z$ axis). The distance $a$ of the ray from the sphere's centre can be calculated as follows:
$$a=\sqrt{(x_{ray}-x_{sphere})^2+(y_{ray}-y_{sphere})^2}$$
```text
CONSIDER A DIAGRAM HERE
```
- For each sphere that the ray intersects, find the $z$ coordinate of the nearest intersection (there will often be $2$ per sphere, $1$ on the front of the sphere and $1$ on the back of the sphere, as the ray passes through - only the nearest is relevant). This can be calculated as follows:
- The distance $d$ to the nearest intersection point can be calculated from the $z$ coordinate of the sphere's centre, the radius $r$, and the distance $a$ from the $x,y$ coordinates of the sphere's centre to the scaled $x,y$ coordinates of the pixel:
$$d=z_{sphere}-\sqrt{r_{sphere}^2-a^2}$$
```text
CONSIDER A DIAGRAM HERE
```
- Of all the spheres that the ray intersects, the sphere with the intersection point with the smallest $z$ coordinate $d$ is the one to display. Set the pixel to the greyscale value of this sphere.
- If the ray did not intersect any spheres, set the pixel to zero (black).
## Example implementations
These are ungolfed code to supplement understanding of the requirements. You are free to golf these or implement your own approach that matches the test cases up to reflection & rotation.
The Python implementation uses 64 bit floating point. The Rust implementation uses 32 bit floating point to show the difference in output. I've tried to optimise the test cases for compatibility with 32 bit floating point, so the differences are restricted to being a pixel out in some places.
### Python
The function `grey_values` would be sufficient as an entry to this challenge. The rest of the code saves the outputs as PNG files. The function itself has no dependence on the `import png` so it would not be required in an entry.
Instructions for installing the `png` package can be found on its [project page on PyPI](https://pypi.org/project/pypng/).
```python
def grey_values(spheres):
values = []
for y_pixel in range(1024):
for x_pixel in range(1024):
x_ray, y_ray = x_pixel / 512 - 1, y_pixel / 512 - 1
min_distance, hit_grey_value = None, 0
for sphere in spheres:
x_sphere, y_sphere, z_sphere, radius, grey = sphere
a = (
(x_ray - x_sphere) ** 2 + (y_ray - y_sphere) ** 2
) ** 0.5
if a < radius:
distance = z_sphere - (radius ** 2 - a ** 2) ** 0.5
if min_distance is None or distance < min_distance:
min_distance = distance
hit_grey_value = grey
values.append(hit_grey_value)
return values
import png
test_case = [
[0, 0, 1000, 2, 192],
[-2, 2, 100, 3, 98],
[2, 2, 100, 3, 98],
[0.5, -0.5, 500, 0.2, 255],
[-0.349469, -0.1, 20.000279, 0.6, 64],
[-0.349775, -0.100273, 20.000485, 0.6, 85],
[-0.350214, -0.100386, 20.000407, 0.6, 106],
[-0.350527, -0.100273, 20.000091, 0.6, 127],
[-0.350531, -0.1, 19.999721, 0.6, 148],
[-0.350225, -0.099727, 19.999515, 0.6, 169],
[-0.349786, -0.099614, 19.999593, 0.6, 190],
[-0.349473, -0.099727, 19.999909, 0.6, 211],
[0.20032, 0.349824, 15.000456, 0.3, 0],
[0.20032, 0.349517, 14.999924, 0.3, 0],
[0.19968, 0.350483, 15.000076, 0.3, 0],
[0.19968, 0.350176, 14.999544, 0.3, 0],
[0.200578, 0.350073, 14.999958, 0.3, 0],
[0.199817, 0.349519, 15.000277, 0.3, 0],
[0.200183, 0.350481, 14.999723, 0.3, 0],
[0.199422, 0.349927, 15.000042, 0.3, 0],
[0.200235, 0.35042, 15.000332, 0.3, 0],
[0.199765, 0.350078, 15.000529, 0.3, 0],
[0.200235, 0.349922, 14.999471, 0.3, 0],
[0.199765, 0.34958, 14.999668, 0.3, 0],
[0.200429, 0.350419, 15.000005, 0.3, 255],
[0.199571, 0.349795, 15.000366, 0.3, 255],
[0.200429, 0.350205, 14.999634, 0.3, 255],
[0.199571, 0.349581, 14.999995, 0.3, 255],
[0.200524, 0.349747, 15.000146, 0.3, 255],
[0.200196, 0.349509, 15.000283, 0.3, 255],
[0.199804, 0.350491, 14.999717, 0.3, 255],
[0.199476, 0.350253, 14.999854, 0.3, 255],
[0.200138, 0.349578, 14.999597, 0.3, 255],
[0.200138, 0.350138, 15.000567, 0.3, 255],
[0.199862, 0.349862, 14.999433, 0.3, 255],
[0.199862, 0.350422, 15.000403, 0.3, 255],
[0.200488, 0.350136, 15.000321, 0.3, 255],
[0.199957, 0.349751, 15.000544, 0.3, 255],
[0.200488, 0.34979, 14.999721, 0.3, 255],
[0.199957, 0.349404, 14.999944, 0.3, 255],
[0.200043, 0.350596, 15.000056, 0.3, 255],
[0.199512, 0.35021, 15.000279, 0.3, 255],
[0.200043, 0.350249, 14.999456, 0.3, 255],
[0.199512, 0.349864, 14.999679, 0.3, 255],
[-0.5, 0.500442, 10.000442, 0.2, 0],
[-0.5, 0.5, 10, 0.1995, 255],
[-0.499966, 0.49971, 9.999563, 0.199, 0],
[-0.500034, 0.499563, 9.99971, 0.199, 0],
[-0.499934, 0.499432, 9.999144, 0.1985, 255],
[-0.500066, 0.499144, 9.999432, 0.1985, 255],
[0.6, 0.5, 10, 0.25, 210],
[0.600122, 0.4999, 10.000122, 0.25, 210],
[0.600061, 0.49995, 10.000061, 0.25003, 50],
[0.595856, 0.505256, 9.991271, 0.24, 100],
[0.597973, 0.507461, 9.992175, 0.24, 210],
[0.595362, 0.507926, 9.993945, 0.24, 100],
[0.594618, 0.509055, 9.996832, 0.24, 210],
[0.592382, 0.506906, 9.996092, 0.24, 100],
[0.590438, 0.504882, 9.997604, 0.24, 210],
[0.591035, 0.503605, 9.994744, 0.24, 100],
[0.59121, 0.500708, 9.993425, 0.24, 210],
[0.593181, 0.502585, 9.991764, 0.24, 100],
[0.595867, 0.502302, 9.990069, 0.24, 210],
]
values = grey_values(test_case)
rows = [values[row * 1024:row * 1024 + 1024] for row in range(1024)]
with open('combination_scene.png', 'wb') as file:
w = png.Writer(1024, 1024, greyscale=True)
w.write(file, rows)
```
### Rust
The function `grey_values` would be sufficient as an entry to this challenge. The rest of the code saves the outputs as PNG files.
The `png` crate can be installed using `cargo add png`.
```rust
fn grey_values(spheres: &Vec<[f32; 5]>) -> Vec<u8> {
let mut values = vec![];
for y_pixel in 0..1024 {
for x_pixel in 0..1024 {
let x_ray = x_pixel as f32 / 512.0 - 1.0;
let y_ray = y_pixel as f32 / 512.0 - 1.0;
let mut min_distance = f32::INFINITY;
let mut hit_grey_value = 0;
for sphere in spheres {
let [
x_sphere, y_sphere, z_sphere, radius, grey
] = *sphere;
let a = (
(x_ray - x_sphere).powf(2.0)
+ (y_ray - y_sphere).powf(2.0)
).sqrt();
if a < radius {
let distance = z_sphere - (
radius.powf(2.0) - a.powf(2.0)
).sqrt();
if distance < min_distance {
min_distance = distance;
hit_grey_value = grey as u8;
}
}
}
values.push(hit_grey_value);
}
}
values
}
fn main() {
let test_case: Vec<[f32; 5]> = vec![
[0.0, 0.0, 1000.0, 2.0, 192.0],
[-2.0, 2.0, 100.0, 3.0, 98.0],
[2.0, 2.0, 100.0, 3.0, 98.0],
[0.5, -0.5, 500.0, 0.2, 255.0],
[-0.349469, -0.1, 20.000279, 0.6, 64.0],
[-0.349775, -0.100273, 20.000485, 0.6, 85.0],
[-0.350214, -0.100386, 20.000407, 0.6, 106.0],
[-0.350527, -0.100273, 20.000091, 0.6, 127.0],
[-0.350531, -0.1, 19.999721, 0.6, 148.0],
[-0.350225, -0.099727, 19.999515, 0.6, 169.0],
[-0.349786, -0.099614, 19.999593, 0.6, 190.0],
[-0.349473, -0.099727, 19.999909, 0.6, 211.0],
[0.20032, 0.349824, 15.000456, 0.3, 0.0],
[0.20032, 0.349517, 14.999924, 0.3, 0.0],
[0.19968, 0.350483, 15.000076, 0.3, 0.0],
[0.19968, 0.350176, 14.999544, 0.3, 0.0],
[0.200578, 0.350073, 14.999958, 0.3, 0.0],
[0.199817, 0.349519, 15.000277, 0.3, 0.0],
[0.200183, 0.350481, 14.999723, 0.3, 0.0],
[0.199422, 0.349927, 15.000042, 0.3, 0.0],
[0.200235, 0.35042, 15.000332, 0.3, 0.0],
[0.199765, 0.350078, 15.000529, 0.3, 0.0],
[0.200235, 0.349922, 14.999471, 0.3, 0.0],
[0.199765, 0.34958, 14.999668, 0.3, 0.0],
[0.200429, 0.350419, 15.000005, 0.3, 255.0],
[0.199571, 0.349795, 15.000366, 0.3, 255.0],
[0.200429, 0.350205, 14.999634, 0.3, 255.0],
[0.199571, 0.349581, 14.999995, 0.3, 255.0],
[0.200524, 0.349747, 15.000146, 0.3, 255.0],
[0.200196, 0.349509, 15.000283, 0.3, 255.0],
[0.199804, 0.350491, 14.999717, 0.3, 255.0],
[0.199476, 0.350253, 14.999854, 0.3, 255.0],
[0.200138, 0.349578, 14.999597, 0.3, 255.0],
[0.200138, 0.350138, 15.000567, 0.3, 255.0],
[0.199862, 0.349862, 14.999433, 0.3, 255.0],
[0.199862, 0.350422, 15.000403, 0.3, 255.0],
[0.200488, 0.350136, 15.000321, 0.3, 255.0],
[0.199957, 0.349751, 15.000544, 0.3, 255.0],
[0.200488, 0.34979, 14.999721, 0.3, 255.0],
[0.199957, 0.349404, 14.999944, 0.3, 255.0],
[0.200043, 0.350596, 15.000056, 0.3, 255.0],
[0.199512, 0.35021, 15.000279, 0.3, 255.0],
[0.200043, 0.350249, 14.999456, 0.3, 255.0],
[0.199512, 0.349864, 14.999679, 0.3, 255.0],
[-0.5, 0.500442, 10.000442, 0.2, 0.0],
[-0.5, 0.5, 10.0, 0.1995, 255.0],
[-0.499966, 0.49971, 9.999563, 0.199, 0.0],
[-0.500034, 0.499563, 9.99971, 0.199, 0.0],
[-0.499934, 0.499432, 9.999144, 0.1985, 255.0],
[-0.500066, 0.499144, 9.999432, 0.1985, 255.0],
[0.6, 0.5, 10.0, 0.25, 210.0],
[0.600122, 0.4999, 10.000122, 0.25, 210.0],
[0.600061, 0.49995, 10.000061, 0.25003, 50.0],
[0.595856, 0.505256, 9.991271, 0.24, 100.0],
[0.597973, 0.507461, 9.992175, 0.24, 210.0],
[0.595362, 0.507926, 9.993945, 0.24, 100.0],
[0.594618, 0.509055, 9.996832, 0.24, 210.0],
[0.592382, 0.506906, 9.996092, 0.24, 100.0],
[0.590438, 0.504882, 9.997604, 0.24, 210.0],
[0.591035, 0.503605, 9.994744, 0.24, 100.0],
[0.59121, 0.500708, 9.993425, 0.24, 210.0],
[0.593181, 0.502585, 9.991764, 0.24, 100.0],
[0.595867, 0.502302, 9.990069, 0.24, 210.0],
];
let values = grey_values(&test_case);
let path = std::path::Path::new("combination_scene.png");
let file = std::fs::File::create(path).unwrap();
let w = &mut std::io::BufWriter::new(file);
let mut encoder = png::Encoder::new(w, 1024, 1024);
encoder.set_color(png::ColorType::Grayscale);
encoder.set_depth(png::BitDepth::Eight);
let mut writer = encoder.write_header().unwrap();
writer.write_image_data(&values).unwrap();
}
```
## Test cases
The output images may be scaled down to fit. Click on any image to open it full size.
### Single sphere
Input:
```text
[
[0, 0, 0, 1, 128],
]
```
Output:
[![Grey circle on a black background](https://codegolf.codidact.com/uploads/jlru163pa2pgm7aaq70b2tff16uf)](https://codegolf.codidact.com/uploads/jlru163pa2pgm7aaq70b2tff16uf)
### Two separated spheres
Input:
```text
[
[0, -0.55, 0, 0.4, 100],
[0, 0.55, 0, 0.4, 200],
]
```
Output:
[![Light and dark grey circles on a black background](https://codegolf.codidact.com/uploads/8yw7agli78y65kiomb1u5yebs7c5)](https://codegolf.codidact.com/uploads/8yw7agli78y65kiomb1u5yebs7c5)
### Ring of spheres
Input:
```text
[
[0.75, 0, 0, 0.25, 64],
[0.53033, 0.375, -0.375, 0.25, 185],
[0, 0.53033, -0.53033, 0.25, 64],
[-0.53033, 0.375, -0.375, 0.25, 185],
[-0.75, 0, -0, 0.25, 64],
[-0.53033, -0.375, 0.375, 0.25, 185],
[-0, -0.53033, 0.53033, 0.25, 64],
[0.53033, -0.375, 0.375, 0.25, 185],
]
```
Output:
[![Eight circles in alternating light and dark grey in a ring some overlapping](https://codegolf.codidact.com/uploads/0lsenynig27au0sryf3ruozt1uwh)](https://codegolf.codidact.com/uploads/0lsenynig27au0sryf3ruozt1uwh)
### Beach ball
Input:
```text
[
[0.001, 0, 0, 1, 64],
[0.000707, 0.000707, -0, 1, 85],
[0, 0.001, -0, 1, 106],
[-0.000707, 0.000707, -0, 1, 127],
[-0.001, 0, -0, 1, 148],
[-0.000707, -0.000707, 0, 1, 169],
[-0, -0.001, 0, 1, 190],
[0.000707, -0.000707, 0, 1, 211],
]
```
Output:
[![A circle with eight sectors in different shades of grey](https://codegolf.codidact.com/uploads/51a3ni0cfwsqcdw5b49gy6282qno)](https://codegolf.codidact.com/uploads/51a3ni0cfwsqcdw5b49gy6282qno)
### Football
Input:
```text
[
[0, 0.000058, 0.001947, 1, 0],
[0, -0.001715, 0.000923, 1, 0],
[0, 0.001715, -0.000923, 1, 0],
[0, -0.000058, -0.001947, 1, 0],
[0.001657, -0.000512, 0.000887, 1, 0],
[-0.001657, -0.000512, 0.000887, 1, 0],
[0.001657, 0.000512, -0.000887, 1, 0],
[-0.001657, 0.000512, -0.000887, 1, 0],
[0.001024, 0.001435, 0.000829, 1, 0],
[-0.001024, 0.001435, 0.000829, 1, 0],
[0.001024, -0.001435, -0.000829, 1, 0],
[-0.001024, -0.001435, -0.000829, 1, 0],
[0.001868, 0.000618, 0.000357, 1, 255],
[-0.001868, 0.000618, 0.000357, 1, 255],
[0.001868, -0.000618, -0.000357, 1, 255],
[-0.001868, -0.000618, -0.000357, 1, 255],
[0.000714, -0.000934, 0.001618, 1, 255],
[-0.000714, -0.000934, 0.001618, 1, 255],
[0.000714, 0.000934, -0.001618, 1, 255],
[-0.000714, 0.000934, -0.001618, 1, 255],
[0, -0.001975, -0.000316, 1, 255],
[0, 0.001261, 0.001552, 1, 255],
[0, -0.001261, -0.001552, 1, 255],
[0, 0.001975, 0.000316, 1, 255],
[0.001155, 0.000423, 0.001577, 1, 255],
[-0.001155, 0.000423, 0.001577, 1, 255],
[0.001155, -0.001577, 0.000423, 1, 255],
[-0.001155, -0.001577, 0.000423, 1, 255],
[0.001155, 0.001577, -0.000423, 1, 255],
[-0.001155, 0.001577, -0.000423, 1, 255],
[0.001155, -0.000423, -0.001577, 1, 255],
[-0.001155, -0.000423, -0.001577, 1, 255],
]
```
Output:
[![Black pentagons on a white ball](https://codegolf.codidact.com/uploads/bmrbsw65p59imvwu8ssd42uvvx1r)](https://codegolf.codidact.com/uploads/bmrbsw65p59imvwu8ssd42uvvx1r)
### Eight ball
Input:
```text
[
[0, 0, 1000, 2, 128],
[0, 0, 0.003125, 1, 0],
[0, 0, 0, 0.9975, 255],
[0, 0.000547, -0.002573, 0.995, 0],
[-0, -0.000547, -0.002573, 0.995, 0],
[0, 0.001071, -0.005037, 0.9925, 255],
[-0, -0.001071, -0.005037, 0.9925, 255],
]
```
Output:
[![A ball with the number 8 in a white circle](https://codegolf.codidact.com/uploads/dvl68crpmy62rqt94mwanditt4wc)](https://codegolf.codidact.com/uploads/dvl68crpmy62rqt94mwanditt4wc)
### Star ball
Input:
```text
[
[0, 0, 0, 1, 210],
[0, 0, 0.0008, 1, 210],
[0, 0, 0.0004, 1.00012, 50],
[0.012969, -0, -0.042045, 0.96, 100],
[0.016397, 0.011913, -0.039054, 0.96, 210],
[0.004008, 0.012334, -0.042045, 0.96, 100],
[-0.006263, 0.019276, -0.039054, 0.96, 210],
[-0.010492, 0.007623, -0.042045, 0.96, 100],
[-0.020268, 0, -0.039054, 0.96, 210],
[-0.010492, -0.007623, -0.042045, 0.96, 100],
[-0.006263, -0.019276, -0.039054, 0.96, 210],
[0.004008, -0.012334, -0.042045, 0.96, 100],
[0.016397, -0.011913, -0.039054, 0.96, 210],
]
```
Output:
[![A light grey ball with a dark grey five pointed star](https://codegolf.codidact.com/uploads/f3wdgwru6j35ue72x25mazxikfx2)](https://codegolf.codidact.com/uploads/f3wdgwru6j35ue72x25mazxikfx2)
### Combination scene
Input:
```text
[
[0, 0, 1000, 2, 192],
[-2, 2, 100, 3, 98],
[2, 2, 100, 3, 98],
[0.5, -0.5, 500, 0.2, 255],
[-0.349469, -0.1, 20.000279, 0.6, 64],
[-0.349775, -0.100273, 20.000485, 0.6, 85],
[-0.350214, -0.100386, 20.000407, 0.6, 106],
[-0.350527, -0.100273, 20.000091, 0.6, 127],
[-0.350531, -0.1, 19.999721, 0.6, 148],
[-0.350225, -0.099727, 19.999515, 0.6, 169],
[-0.349786, -0.099614, 19.999593, 0.6, 190],
[-0.349473, -0.099727, 19.999909, 0.6, 211],
[0.20032, 0.349824, 15.000456, 0.3, 0],
[0.20032, 0.349517, 14.999924, 0.3, 0],
[0.19968, 0.350483, 15.000076, 0.3, 0],
[0.19968, 0.350176, 14.999544, 0.3, 0],
[0.200578, 0.350073, 14.999958, 0.3, 0],
[0.199817, 0.349519, 15.000277, 0.3, 0],
[0.200183, 0.350481, 14.999723, 0.3, 0],
[0.199422, 0.349927, 15.000042, 0.3, 0],
[0.200235, 0.35042, 15.000332, 0.3, 0],
[0.199765, 0.350078, 15.000529, 0.3, 0],
[0.200235, 0.349922, 14.999471, 0.3, 0],
[0.199765, 0.34958, 14.999668, 0.3, 0],
[0.200429, 0.350419, 15.000005, 0.3, 255],
[0.199571, 0.349795, 15.000366, 0.3, 255],
[0.200429, 0.350205, 14.999634, 0.3, 255],
[0.199571, 0.349581, 14.999995, 0.3, 255],
[0.200524, 0.349747, 15.000146, 0.3, 255],
[0.200196, 0.349509, 15.000283, 0.3, 255],
[0.199804, 0.350491, 14.999717, 0.3, 255],
[0.199476, 0.350253, 14.999854, 0.3, 255],
[0.200138, 0.349578, 14.999597, 0.3, 255],
[0.200138, 0.350138, 15.000567, 0.3, 255],
[0.199862, 0.349862, 14.999433, 0.3, 255],
[0.199862, 0.350422, 15.000403, 0.3, 255],
[0.200488, 0.350136, 15.000321, 0.3, 255],
[0.199957, 0.349751, 15.000544, 0.3, 255],
[0.200488, 0.34979, 14.999721, 0.3, 255],
[0.199957, 0.349404, 14.999944, 0.3, 255],
[0.200043, 0.350596, 15.000056, 0.3, 255],
[0.199512, 0.35021, 15.000279, 0.3, 255],
[0.200043, 0.350249, 14.999456, 0.3, 255],
[0.199512, 0.349864, 14.999679, 0.3, 255],
[-0.5, 0.500442, 10.000442, 0.2, 0],
[-0.5, 0.5, 10, 0.1995, 255],
[-0.499966, 0.49971, 9.999563, 0.199, 0],
[-0.500034, 0.499563, 9.99971, 0.199, 0],
[-0.499934, 0.499432, 9.999144, 0.1985, 255],
[-0.500066, 0.499144, 9.999432, 0.1985, 255],
[0.6, 0.5, 10, 0.25, 210],
[0.600122, 0.4999, 10.000122, 0.25, 210],
[0.600061, 0.49995, 10.000061, 0.25003, 50],
[0.595856, 0.505256, 9.991271, 0.24, 100],
[0.597973, 0.507461, 9.992175, 0.24, 210],
[0.595362, 0.507926, 9.993945, 0.24, 100],
[0.594618, 0.509055, 9.996832, 0.24, 210],
[0.592382, 0.506906, 9.996092, 0.24, 100],
[0.590438, 0.504882, 9.997604, 0.24, 210],
[0.591035, 0.503605, 9.994744, 0.24, 100],
[0.59121, 0.500708, 9.993425, 0.24, 210],
[0.593181, 0.502585, 9.991764, 0.24, 100],
[0.595867, 0.502302, 9.990069, 0.24, 210],
]
```
Output:
[![A beachball, football, eightball, and starball with the sun behind hills in the background](https://codegolf.codidact.com/uploads/usv9npm1t5bjpcaknn6rbzgkqwmi)](https://codegolf.codidact.com/uploads/usv9npm1t5bjpcaknn6rbzgkqwmi)
## Scoring
This is a [code golf challenge](https://codegolf.codidact.com/categories/49/tags/4274). Your score is the number of bytes in your code. Lowest score for each language wins.
> Explanations are optional, but I'm more likely to upvote answers that have one.

Given a list of spheres, output an image.
Produce a greyscale [orthographic](https://en.wikipedia.org/wiki/Parallel_projection#Orthographic_projection) (parallel projection - no perspective) ray caster, that renders only plain grey spheres with no lighting.
## Input
- A list of spheres, each consisting of:
- The coordinates of its centre $(x,y,z)$.
- Its radius $r$.
- Its greyscale value $g$, an integer from $0$ to $255$ (both inclusive).
- Only the greyscale value will be restricted to integer values. The coordinates and radius may have fractional values.
## Output
- An image $1024$ by $1024$ pixels with greyscale values from $0$ to $255$.
- The image covers x and y values from $-1$ to $1$.
- A sphere's greyscale value is applied to every visible part of it. There is no variation over the surface of the sphere, so it appears flat like a disk.
- You may output whatever image format is convenient, including simple text formats such as [PGM](https://en.wikipedia.org/wiki/Netpbm#PGM_example) or a list of pixel greyscale values. If you want to include an example of your output as an image in your answer, the image formats that Codidact supports are PNG, JPEG, and GIF. You could output PGM and then convert to PNG to upload to your answer. If you output just the greyscale values as a list of numbers from $0$ to $255$, I've made a `[ TODO page to convert these to greyscale PNG]`.
- You are free to output an image reflected or rotated by a multiple of 90 degrees. However, the z axis is required to be increasing in the forwards direction (into the image) otherwise some of the test cases will look incorrect.
## Calculating a pixel value
Imagine a ray passing into the image through a pixel, perpendicular to the image plane. If it hits a sphere, it shows the greyscale value of the first sphere it hits. Otherwise, it shows a value of zero (black).
For example, for each pixel:
- The pixel $x,y$ coordinates will each be from $0$ to $1023$. Scale these to the range $-1$ to $1$ (for example, divide by $512$ and subtract $1$).
- If these scaled $x,y$ coordinates are within the radius $r$ of the $x,y$ coordinates of a sphere's centre, the ray intersects that sphere (the $z$ coordinate is not relevant to this test because the image is an orthographic projection - the rays are all parallel to the $z$ axis). The distance $a$ of the ray from the sphere's centre can be calculated as follows:
$$a=\sqrt{(x_{ray}-x_{sphere})^2+(y_{ray}-y_{sphere})^2}$$
```text
CONSIDER A DIAGRAM HERE
```
- For each sphere that the ray intersects, find the $z$ coordinate of the nearest intersection (there will often be $2$ per sphere, $1$ on the front of the sphere and $1$ on the back of the sphere, as the ray passes through - only the nearest is relevant). This can be calculated as follows:
- The distance $d$ to the nearest intersection point can be calculated from the $z$ coordinate of the sphere's centre, the radius $r$, and the distance $a$ from the $x,y$ coordinates of the sphere's centre to the scaled $x,y$ coordinates of the pixel:
$$d=z_{sphere}-\sqrt{r_{sphere}^2-a^2}$$
```text
CONSIDER A DIAGRAM HERE
```
- Of all the spheres that the ray intersects, the sphere with the intersection point with the smallest $z$ coordinate $d$ is the one to display. Set the pixel to the greyscale value of this sphere.
- If the ray did not intersect any spheres, set the pixel to zero (black).
## Example implementations
These are ungolfed code to supplement understanding of the requirements. You are free to golf these or implement your own approach that matches the test cases up to reflection & rotation.
The Python implementation uses 64 bit floating point. The Rust implementation uses 32 bit floating point to show the difference in output. I've tried to optimise the test cases for compatibility with 32 bit floating point, so the differences are restricted to being a pixel out in some places.
### Python
The function `grey_values` would be sufficient as an entry to this challenge. The rest of the code saves the outputs as PNG files. The function itself has no dependence on the `import png` so it would not be required in an entry.
Instructions for installing the `png` package can be found on its [project page on PyPI](https://pypi.org/project/pypng/).
```python
def grey_values(spheres):
values = []
for y_pixel in range(1024):
for x_pixel in range(1024):
x_ray, y_ray = x_pixel / 512 - 1, y_pixel / 512 - 1
min_distance, hit_grey_value = None, 0
for sphere in spheres:
x_sphere, y_sphere, z_sphere, radius, grey = sphere
a = (
(x_ray - x_sphere) ** 2 + (y_ray - y_sphere) ** 2
) ** 0.5
if a < radius:
distance = z_sphere - (radius ** 2 - a ** 2) ** 0.5
if min_distance is None or distance < min_distance:
min_distance = distance
hit_grey_value = grey
values.append(hit_grey_value)
return values
import png
test_case = [
[0, 0, 1000, 2, 192],
[-2, 2, 100, 3, 98],
[2, 2, 100, 3, 98],
[0.5, -0.5, 500, 0.2, 255],
[-0.349469, -0.1, 20.000279, 0.6, 64],
[-0.349775, -0.100273, 20.000485, 0.6, 85],
[-0.350214, -0.100386, 20.000407, 0.6, 106],
[-0.350527, -0.100273, 20.000091, 0.6, 127],
[-0.350531, -0.1, 19.999721, 0.6, 148],
[-0.350225, -0.099727, 19.999515, 0.6, 169],
[-0.349786, -0.099614, 19.999593, 0.6, 190],
[-0.349473, -0.099727, 19.999909, 0.6, 211],
[0.20032, 0.349824, 15.000456, 0.3, 0],
[0.20032, 0.349517, 14.999924, 0.3, 0],
[0.19968, 0.350483, 15.000076, 0.3, 0],
[0.19968, 0.350176, 14.999544, 0.3, 0],
[0.200578, 0.350073, 14.999958, 0.3, 0],
[0.199817, 0.349519, 15.000277, 0.3, 0],
[0.200183, 0.350481, 14.999723, 0.3, 0],
[0.199422, 0.349927, 15.000042, 0.3, 0],
[0.200235, 0.35042, 15.000332, 0.3, 0],
[0.199765, 0.350078, 15.000529, 0.3, 0],
[0.200235, 0.349922, 14.999471, 0.3, 0],
[0.199765, 0.34958, 14.999668, 0.3, 0],
[0.200429, 0.350419, 15.000005, 0.3, 255],
[0.199571, 0.349795, 15.000366, 0.3, 255],
[0.200429, 0.350205, 14.999634, 0.3, 255],
[0.199571, 0.349581, 14.999995, 0.3, 255],
[0.200524, 0.349747, 15.000146, 0.3, 255],
[0.200196, 0.349509, 15.000283, 0.3, 255],
[0.199804, 0.350491, 14.999717, 0.3, 255],
[0.199476, 0.350253, 14.999854, 0.3, 255],
[0.200138, 0.349578, 14.999597, 0.3, 255],
[0.200138, 0.350138, 15.000567, 0.3, 255],
[0.199862, 0.349862, 14.999433, 0.3, 255],
[0.199862, 0.350422, 15.000403, 0.3, 255],
[0.200488, 0.350136, 15.000321, 0.3, 255],
[0.199957, 0.349751, 15.000544, 0.3, 255],
[0.200488, 0.34979, 14.999721, 0.3, 255],
[0.199957, 0.349404, 14.999944, 0.3, 255],
[0.200043, 0.350596, 15.000056, 0.3, 255],
[0.199512, 0.35021, 15.000279, 0.3, 255],
[0.200043, 0.350249, 14.999456, 0.3, 255],
[0.199512, 0.349864, 14.999679, 0.3, 255],
[-0.5, 0.500442, 10.000442, 0.2, 0],
[-0.5, 0.5, 10, 0.1995, 255],
[-0.499966, 0.49971, 9.999563, 0.199, 0],
[-0.500034, 0.499563, 9.99971, 0.199, 0],
[-0.499934, 0.499432, 9.999144, 0.1985, 255],
[-0.500066, 0.499144, 9.999432, 0.1985, 255],
[0.6, 0.5, 10, 0.25, 210],
[0.600122, 0.4999, 10.000122, 0.25, 210],
[0.600061, 0.49995, 10.000061, 0.25003, 50],
[0.595856, 0.505256, 9.991271, 0.24, 100],
[0.597973, 0.507461, 9.992175, 0.24, 210],
[0.595362, 0.507926, 9.993945, 0.24, 100],
[0.594618, 0.509055, 9.996832, 0.24, 210],
[0.592382, 0.506906, 9.996092, 0.24, 100],
[0.590438, 0.504882, 9.997604, 0.24, 210],
[0.591035, 0.503605, 9.994744, 0.24, 100],
[0.59121, 0.500708, 9.993425, 0.24, 210],
[0.593181, 0.502585, 9.991764, 0.24, 100],
[0.595867, 0.502302, 9.990069, 0.24, 210],
]
values = grey_values(test_case)
rows = [values[row * 1024:row * 1024 + 1024] for row in range(1024)]
with open('combination_scene.png', 'wb') as file:
w = png.Writer(1024, 1024, greyscale=True)
w.write(file, rows)
```
### Rust
The function `grey_values` would be sufficient as an entry to this challenge. The rest of the code saves the outputs as PNG files.
The `png` crate can be installed using `cargo add png`.
```rust
fn grey_values(spheres: &Vec<[f32; 5]>) -> Vec<u8> {
let mut values = vec![];
for y_pixel in 0..1024 {
for x_pixel in 0..1024 {
let x_ray = x_pixel as f32 / 512.0 - 1.0;
let y_ray = y_pixel as f32 / 512.0 - 1.0;
let mut min_distance = f32::INFINITY;
let mut hit_grey_value = 0;
for sphere in spheres {
let [
x_sphere, y_sphere, z_sphere, radius, grey
] = *sphere;
let a = (
(x_ray - x_sphere).powf(2.0)
+ (y_ray - y_sphere).powf(2.0)
).sqrt();
if a < radius {
let distance = z_sphere - (
radius.powf(2.0) - a.powf(2.0)
).sqrt();
if distance < min_distance {
min_distance = distance;
hit_grey_value = grey as u8;
}
}
}
values.push(hit_grey_value);
}
}
values
}
fn main() {
let test_case: Vec<[f32; 5]> = vec![
[0.0, 0.0, 1000.0, 2.0, 192.0],
[-2.0, 2.0, 100.0, 3.0, 98.0],
[2.0, 2.0, 100.0, 3.0, 98.0],
[0.5, -0.5, 500.0, 0.2, 255.0],
[-0.349469, -0.1, 20.000279, 0.6, 64.0],
[-0.349775, -0.100273, 20.000485, 0.6, 85.0],
[-0.350214, -0.100386, 20.000407, 0.6, 106.0],
[-0.350527, -0.100273, 20.000091, 0.6, 127.0],
[-0.350531, -0.1, 19.999721, 0.6, 148.0],
[-0.350225, -0.099727, 19.999515, 0.6, 169.0],
[-0.349786, -0.099614, 19.999593, 0.6, 190.0],
[-0.349473, -0.099727, 19.999909, 0.6, 211.0],
[0.20032, 0.349824, 15.000456, 0.3, 0.0],
[0.20032, 0.349517, 14.999924, 0.3, 0.0],
[0.19968, 0.350483, 15.000076, 0.3, 0.0],
[0.19968, 0.350176, 14.999544, 0.3, 0.0],
[0.200578, 0.350073, 14.999958, 0.3, 0.0],
[0.199817, 0.349519, 15.000277, 0.3, 0.0],
[0.200183, 0.350481, 14.999723, 0.3, 0.0],
[0.199422, 0.349927, 15.000042, 0.3, 0.0],
[0.200235, 0.35042, 15.000332, 0.3, 0.0],
[0.199765, 0.350078, 15.000529, 0.3, 0.0],
[0.200235, 0.349922, 14.999471, 0.3, 0.0],
[0.199765, 0.34958, 14.999668, 0.3, 0.0],
[0.200429, 0.350419, 15.000005, 0.3, 255.0],
[0.199571, 0.349795, 15.000366, 0.3, 255.0],
[0.200429, 0.350205, 14.999634, 0.3, 255.0],
[0.199571, 0.349581, 14.999995, 0.3, 255.0],
[0.200524, 0.349747, 15.000146, 0.3, 255.0],
[0.200196, 0.349509, 15.000283, 0.3, 255.0],
[0.199804, 0.350491, 14.999717, 0.3, 255.0],
[0.199476, 0.350253, 14.999854, 0.3, 255.0],
[0.200138, 0.349578, 14.999597, 0.3, 255.0],
[0.200138, 0.350138, 15.000567, 0.3, 255.0],
[0.199862, 0.349862, 14.999433, 0.3, 255.0],
[0.199862, 0.350422, 15.000403, 0.3, 255.0],
[0.200488, 0.350136, 15.000321, 0.3, 255.0],
[0.199957, 0.349751, 15.000544, 0.3, 255.0],
[0.200488, 0.34979, 14.999721, 0.3, 255.0],
[0.199957, 0.349404, 14.999944, 0.3, 255.0],
[0.200043, 0.350596, 15.000056, 0.3, 255.0],
[0.199512, 0.35021, 15.000279, 0.3, 255.0],
[0.200043, 0.350249, 14.999456, 0.3, 255.0],
[0.199512, 0.349864, 14.999679, 0.3, 255.0],
[-0.5, 0.500442, 10.000442, 0.2, 0.0],
[-0.5, 0.5, 10.0, 0.1995, 255.0],
[-0.499966, 0.49971, 9.999563, 0.199, 0.0],
[-0.500034, 0.499563, 9.99971, 0.199, 0.0],
[-0.499934, 0.499432, 9.999144, 0.1985, 255.0],
[-0.500066, 0.499144, 9.999432, 0.1985, 255.0],
[0.6, 0.5, 10.0, 0.25, 210.0],
[0.600122, 0.4999, 10.000122, 0.25, 210.0],
[0.600061, 0.49995, 10.000061, 0.25003, 50.0],
[0.595856, 0.505256, 9.991271, 0.24, 100.0],
[0.597973, 0.507461, 9.992175, 0.24, 210.0],
[0.595362, 0.507926, 9.993945, 0.24, 100.0],
[0.594618, 0.509055, 9.996832, 0.24, 210.0],
[0.592382, 0.506906, 9.996092, 0.24, 100.0],
[0.590438, 0.504882, 9.997604, 0.24, 210.0],
[0.591035, 0.503605, 9.994744, 0.24, 100.0],
[0.59121, 0.500708, 9.993425, 0.24, 210.0],
[0.593181, 0.502585, 9.991764, 0.24, 100.0],
[0.595867, 0.502302, 9.990069, 0.24, 210.0],
];
let values = grey_values(&test_case);
let path = std::path::Path::new("combination_scene.png");
let file = std::fs::File::create(path).unwrap();
let w = &mut std::io::BufWriter::new(file);
let mut encoder = png::Encoder::new(w, 1024, 1024);
encoder.set_color(png::ColorType::Grayscale);
encoder.set_depth(png::BitDepth::Eight);
let mut writer = encoder.write_header().unwrap();
writer.write_image_data(&values).unwrap();
}
```
## Test cases
The output images may be scaled down to fit. Click on any image to open it full size.
### Single sphere
Input:
```text
[
[0, 0, 0, 1, 128],
]
```
Output:
[![Grey circle on a black background](https://codegolf.codidact.com/uploads/jlru163pa2pgm7aaq70b2tff16uf)](https://codegolf.codidact.com/uploads/jlru163pa2pgm7aaq70b2tff16uf)
### Two separated spheres
Input:
```text
[
[0, -0.55, 0, 0.4, 100],
[0, 0.55, 0, 0.4, 200],
]
```
Output:
[![Light and dark grey circles on a black background](https://codegolf.codidact.com/uploads/8yw7agli78y65kiomb1u5yebs7c5)](https://codegolf.codidact.com/uploads/8yw7agli78y65kiomb1u5yebs7c5)
### Ring of spheres
Input:
```text
[
[0.75, 0, 0, 0.25, 64],
[0.53033, 0.375, -0.375, 0.25, 185],
[0, 0.53033, -0.53033, 0.25, 64],
[-0.53033, 0.375, -0.375, 0.25, 185],
[-0.75, 0, -0, 0.25, 64],
[-0.53033, -0.375, 0.375, 0.25, 185],
[-0, -0.53033, 0.53033, 0.25, 64],
[0.53033, -0.375, 0.375, 0.25, 185],
]
```
Output:
[![Eight circles in alternating light and dark grey in a ring some overlapping](https://codegolf.codidact.com/uploads/0lsenynig27au0sryf3ruozt1uwh)](https://codegolf.codidact.com/uploads/0lsenynig27au0sryf3ruozt1uwh)
### Beach ball
Input:
```text
[
[0.001, 0, 0, 1, 64],
[0.000707, 0.000707, -0, 1, 85],
[0, 0.001, -0, 1, 106],
[-0.000707, 0.000707, -0, 1, 127],
[-0.001, 0, -0, 1, 148],
[-0.000707, -0.000707, 0, 1, 169],
[-0, -0.001, 0, 1, 190],
[0.000707, -0.000707, 0, 1, 211],
]
```
Output:
[![A circle with eight sectors in different shades of grey](https://codegolf.codidact.com/uploads/51a3ni0cfwsqcdw5b49gy6282qno)](https://codegolf.codidact.com/uploads/51a3ni0cfwsqcdw5b49gy6282qno)
### Football
Input:
```text
[
[0, 0.000058, 0.001947, 1, 0],
[0, -0.001715, 0.000923, 1, 0],
[0, 0.001715, -0.000923, 1, 0],
[0, -0.000058, -0.001947, 1, 0],
[0.001657, -0.000512, 0.000887, 1, 0],
[-0.001657, -0.000512, 0.000887, 1, 0],
[0.001657, 0.000512, -0.000887, 1, 0],
[-0.001657, 0.000512, -0.000887, 1, 0],
[0.001024, 0.001435, 0.000829, 1, 0],
[-0.001024, 0.001435, 0.000829, 1, 0],
[0.001024, -0.001435, -0.000829, 1, 0],
[-0.001024, -0.001435, -0.000829, 1, 0],
[0.001868, 0.000618, 0.000357, 1, 255],
[-0.001868, 0.000618, 0.000357, 1, 255],
[0.001868, -0.000618, -0.000357, 1, 255],
[-0.001868, -0.000618, -0.000357, 1, 255],
[0.000714, -0.000934, 0.001618, 1, 255],
[-0.000714, -0.000934, 0.001618, 1, 255],
[0.000714, 0.000934, -0.001618, 1, 255],
[-0.000714, 0.000934, -0.001618, 1, 255],
[0, -0.001975, -0.000316, 1, 255],
[0, 0.001261, 0.001552, 1, 255],
[0, -0.001261, -0.001552, 1, 255],
[0, 0.001975, 0.000316, 1, 255],
[0.001155, 0.000423, 0.001577, 1, 255],
[-0.001155, 0.000423, 0.001577, 1, 255],
[0.001155, -0.001577, 0.000423, 1, 255],
[-0.001155, -0.001577, 0.000423, 1, 255],
[0.001155, 0.001577, -0.000423, 1, 255],
[-0.001155, 0.001577, -0.000423, 1, 255],
[0.001155, -0.000423, -0.001577, 1, 255],
[-0.001155, -0.000423, -0.001577, 1, 255],
]
```
Output:
[![Black pentagons on a white ball](https://codegolf.codidact.com/uploads/bmrbsw65p59imvwu8ssd42uvvx1r)](https://codegolf.codidact.com/uploads/bmrbsw65p59imvwu8ssd42uvvx1r)
### Eight ball
Input:
```text
[
[0, 0, 1000, 2, 128],
[0, 0, 0.003125, 1, 0],
[0, 0, 0, 0.9975, 255],
[0, 0.000547, -0.002573, 0.995, 0],
[-0, -0.000547, -0.002573, 0.995, 0],
[0, 0.001071, -0.005037, 0.9925, 255],
[-0, -0.001071, -0.005037, 0.9925, 255],
]
```
Output:
[![A ball with the number 8 in a white circle](https://codegolf.codidact.com/uploads/dvl68crpmy62rqt94mwanditt4wc)](https://codegolf.codidact.com/uploads/dvl68crpmy62rqt94mwanditt4wc)
### Star ball
Input:
```text
[
[0, 0, 0, 1, 210],
[0, 0, 0.0008, 1, 210],
[0, 0, 0.0004, 1.00012, 50],
[0.012969, -0, -0.042045, 0.96, 100],
[0.016397, 0.011913, -0.039054, 0.96, 210],
[0.004008, 0.012334, -0.042045, 0.96, 100],
[-0.006263, 0.019276, -0.039054, 0.96, 210],
[-0.010492, 0.007623, -0.042045, 0.96, 100],
[-0.020268, 0, -0.039054, 0.96, 210],
[-0.010492, -0.007623, -0.042045, 0.96, 100],
[-0.006263, -0.019276, -0.039054, 0.96, 210],
[0.004008, -0.012334, -0.042045, 0.96, 100],
[0.016397, -0.011913, -0.039054, 0.96, 210],
]
```
Output:
[![A light grey ball with a dark grey five pointed star](https://codegolf.codidact.com/uploads/f3wdgwru6j35ue72x25mazxikfx2)](https://codegolf.codidact.com/uploads/f3wdgwru6j35ue72x25mazxikfx2)
### Combination scene
Input:
```text
[
[0, 0, 1000, 2, 192],
[-2, 2, 100, 3, 98],
[2, 2, 100, 3, 98],
[0.5, -0.5, 500, 0.2, 255],
[-0.349469, -0.1, 20.000279, 0.6, 64],
[-0.349775, -0.100273, 20.000485, 0.6, 85],
[-0.350214, -0.100386, 20.000407, 0.6, 106],
[-0.350527, -0.100273, 20.000091, 0.6, 127],
[-0.350531, -0.1, 19.999721, 0.6, 148],
[-0.350225, -0.099727, 19.999515, 0.6, 169],
[-0.349786, -0.099614, 19.999593, 0.6, 190],
[-0.349473, -0.099727, 19.999909, 0.6, 211],
[0.20032, 0.349824, 15.000456, 0.3, 0],
[0.20032, 0.349517, 14.999924, 0.3, 0],
[0.19968, 0.350483, 15.000076, 0.3, 0],
[0.19968, 0.350176, 14.999544, 0.3, 0],
[0.200578, 0.350073, 14.999958, 0.3, 0],
[0.199817, 0.349519, 15.000277, 0.3, 0],
[0.200183, 0.350481, 14.999723, 0.3, 0],
[0.199422, 0.349927, 15.000042, 0.3, 0],
[0.200235, 0.35042, 15.000332, 0.3, 0],
[0.199765, 0.350078, 15.000529, 0.3, 0],
[0.200235, 0.349922, 14.999471, 0.3, 0],
[0.199765, 0.34958, 14.999668, 0.3, 0],
[0.200429, 0.350419, 15.000005, 0.3, 255],
[0.199571, 0.349795, 15.000366, 0.3, 255],
[0.200429, 0.350205, 14.999634, 0.3, 255],
[0.199571, 0.349581, 14.999995, 0.3, 255],
[0.200524, 0.349747, 15.000146, 0.3, 255],
[0.200196, 0.349509, 15.000283, 0.3, 255],
[0.199804, 0.350491, 14.999717, 0.3, 255],
[0.199476, 0.350253, 14.999854, 0.3, 255],
[0.200138, 0.349578, 14.999597, 0.3, 255],
[0.200138, 0.350138, 15.000567, 0.3, 255],
[0.199862, 0.349862, 14.999433, 0.3, 255],
[0.199862, 0.350422, 15.000403, 0.3, 255],
[0.200488, 0.350136, 15.000321, 0.3, 255],
[0.199957, 0.349751, 15.000544, 0.3, 255],
[0.200488, 0.34979, 14.999721, 0.3, 255],
[0.199957, 0.349404, 14.999944, 0.3, 255],
[0.200043, 0.350596, 15.000056, 0.3, 255],
[0.199512, 0.35021, 15.000279, 0.3, 255],
[0.200043, 0.350249, 14.999456, 0.3, 255],
[0.199512, 0.349864, 14.999679, 0.3, 255],
[-0.5, 0.500442, 10.000442, 0.2, 0],
[-0.5, 0.5, 10, 0.1995, 255],
[-0.499966, 0.49971, 9.999563, 0.199, 0],
[-0.500034, 0.499563, 9.99971, 0.199, 0],
[-0.499934, 0.499432, 9.999144, 0.1985, 255],
[-0.500066, 0.499144, 9.999432, 0.1985, 255],
[0.6, 0.5, 10, 0.25, 210],
[0.600122, 0.4999, 10.000122, 0.25, 210],
[0.600061, 0.49995, 10.000061, 0.25003, 50],
[0.595856, 0.505256, 9.991271, 0.24, 100],
[0.597973, 0.507461, 9.992175, 0.24, 210],
[0.595362, 0.507926, 9.993945, 0.24, 100],
[0.594618, 0.509055, 9.996832, 0.24, 210],
[0.592382, 0.506906, 9.996092, 0.24, 100],
[0.590438, 0.504882, 9.997604, 0.24, 210],
[0.591035, 0.503605, 9.994744, 0.24, 100],
[0.59121, 0.500708, 9.993425, 0.24, 210],
[0.593181, 0.502585, 9.991764, 0.24, 100],
[0.595867, 0.502302, 9.990069, 0.24, 210],
]
```
Output:
[![A beachball, football, eightball, and starball with the sun behind hills in the background](https://codegolf.codidact.com/uploads/usv9npm1t5bjpcaknn6rbzgkqwmi)](https://codegolf.codidact.com/uploads/usv9npm1t5bjpcaknn6rbzgkqwmi)
## Explanatory animations
The example images were produced from plain grey spheres with no surface pattern or texture. The following animations show how they break down into their component plain spheres.
<details>
<summary>Animations</summary>
TODO
</details>
## Scoring
This is a [code golf challenge](https://codegolf.codidact.com/categories/49/tags/4274). Your score is the number of bytes in your code. Lowest score for each language wins.
> Explanations are optional, but I'm more likely to upvote answers that have one.

#10: Post edited by

trichoplax‭ · 2024-07-26T12:34:36Z (5 months ago)
Make test case images links to full size

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Markdown

Given a list of spheres, output an image.
Produce a greyscale [orthographic](https://en.wikipedia.org/wiki/Parallel_projection#Orthographic_projection) (parallel projection - no perspective) ray caster, that renders only plain grey spheres with no lighting.
## Input
- A list of spheres, each consisting of:
- The coordinates of its centre $(x,y,z)$.
- Its radius $r$.
- Its greyscale value $g$, an integer from $0$ to $255$ (both inclusive).
- Only the greyscale value will be restricted to integer values. The coordinates and radius may have fractional values.
## Output
- An image $1024$ by $1024$ pixels with greyscale values from $0$ to $255$.
- The image covers x and y values from $-1$ to $1$.
- A sphere's greyscale value is applied to every visible part of it. There is no variation over the surface of the sphere, so it appears flat like a disk.
- You may output whatever image format is convenient, including simple text formats such as [PGM](https://en.wikipedia.org/wiki/Netpbm#PGM_example) or a list of pixel greyscale values. If you want to include an example of your output as an image in your answer, the image formats that Codidact supports are PNG, JPEG, and GIF. You could output PGM and then convert to PNG to upload to your answer. If you output just the greyscale values as a list of numbers from $0$ to $255$, I've made a `[ TODO page to convert these to greyscale PNG]`.
- You are free to output an image reflected or rotated by a multiple of 90 degrees. However, the z axis is required to be increasing in the forwards direction (into the image) otherwise some of the test cases will look incorrect.
## Calculating a pixel value
Imagine a ray passing into the image through a pixel, perpendicular to the image plane. If it hits a sphere, it shows the greyscale value of the first sphere it hits. Otherwise, it shows a value of zero (black).
For example, for each pixel:
- The pixel $x,y$ coordinates will each be from $0$ to $1023$. Scale these to the range $-1$ to $1$ (for example, divide by $512$ and subtract $1$).
- If these scaled $x,y$ coordinates are within the radius $r$ of the $x,y$ coordinates of a sphere's centre, the ray intersects that sphere (the $z$ coordinate is not relevant to this test because the image is an orthographic projection - the rays are all parallel to the $z$ axis). The distance $a$ of the ray from the sphere's centre can be calculated as follows:
$$a=\sqrt{(x_{ray}-x_{sphere})^2+(y_{ray}-y_{sphere})^2}$$
```text
CONSIDER A DIAGRAM HERE
```
- For each sphere that the ray intersects, find the $z$ coordinate of the nearest intersection (there will often be $2$ per sphere, $1$ on the front of the sphere and $1$ on the back of the sphere, as the ray passes through - only the nearest is relevant). This can be calculated as follows:
- The distance $d$ to the nearest intersection point can be calculated from the $z$ coordinate of the sphere's centre, the radius $r$, and the distance $a$ from the $x,y$ coordinates of the sphere's centre to the scaled $x,y$ coordinates of the pixel:
$$d=z_{sphere}-\sqrt{r_{sphere}^2-a^2}$$
```text
CONSIDER A DIAGRAM HERE
```
- Of all the spheres that the ray intersects, the sphere with the intersection point with the smallest $z$ coordinate $d$ is the one to display. Set the pixel to the greyscale value of this sphere.
- If the ray did not intersect any spheres, set the pixel to zero (black).
## Example implementations
These are ungolfed code to supplement understanding of the requirements. You are free to golf these or implement your own approach that matches the test cases up to reflection & rotation.
The Python implementation uses 64 bit floating point. The Rust implementation uses 32 bit floating point to show the difference in output. I've tried to optimise the test cases for compatibility with 32 bit floating point, so the differences are restricted to being a pixel out in some places.
### Python
The function `grey_values` would be sufficient as an entry to this challenge. The rest of the code saves the outputs as PNG files. The function itself has no dependence on the `import png` so it would not be required in an entry.
Instructions for installing the `png` package can be found on its [project page on PyPI](https://pypi.org/project/pypng/).
```python
def grey_values(spheres):
values = []
for y_pixel in range(1024):
for x_pixel in range(1024):
x_ray, y_ray = x_pixel / 512 - 1, y_pixel / 512 - 1
min_distance, hit_grey_value = None, 0
for sphere in spheres:
x_sphere, y_sphere, z_sphere, radius, grey = sphere
a = (
(x_ray - x_sphere) ** 2 + (y_ray - y_sphere) ** 2
) ** 0.5
if a < radius:
distance = z_sphere - (radius ** 2 - a ** 2) ** 0.5
if min_distance is None or distance < min_distance:
min_distance = distance
hit_grey_value = grey
values.append(hit_grey_value)
return values
import png
test_case = [
[0, 0, 1000, 2, 192],
[-2, 2, 100, 3, 98],
[2, 2, 100, 3, 98],
[0.5, -0.5, 500, 0.2, 255],
[-0.349469, -0.1, 20.000279, 0.6, 64],
[-0.349775, -0.100273, 20.000485, 0.6, 85],
[-0.350214, -0.100386, 20.000407, 0.6, 106],
[-0.350527, -0.100273, 20.000091, 0.6, 127],
[-0.350531, -0.1, 19.999721, 0.6, 148],
[-0.350225, -0.099727, 19.999515, 0.6, 169],
[-0.349786, -0.099614, 19.999593, 0.6, 190],
[-0.349473, -0.099727, 19.999909, 0.6, 211],
[0.20032, 0.349824, 15.000456, 0.3, 0],
[0.20032, 0.349517, 14.999924, 0.3, 0],
[0.19968, 0.350483, 15.000076, 0.3, 0],
[0.19968, 0.350176, 14.999544, 0.3, 0],
[0.200578, 0.350073, 14.999958, 0.3, 0],
[0.199817, 0.349519, 15.000277, 0.3, 0],
[0.200183, 0.350481, 14.999723, 0.3, 0],
[0.199422, 0.349927, 15.000042, 0.3, 0],
[0.200235, 0.35042, 15.000332, 0.3, 0],
[0.199765, 0.350078, 15.000529, 0.3, 0],
[0.200235, 0.349922, 14.999471, 0.3, 0],
[0.199765, 0.34958, 14.999668, 0.3, 0],
[0.200429, 0.350419, 15.000005, 0.3, 255],
[0.199571, 0.349795, 15.000366, 0.3, 255],
[0.200429, 0.350205, 14.999634, 0.3, 255],
[0.199571, 0.349581, 14.999995, 0.3, 255],
[0.200524, 0.349747, 15.000146, 0.3, 255],
[0.200196, 0.349509, 15.000283, 0.3, 255],
[0.199804, 0.350491, 14.999717, 0.3, 255],
[0.199476, 0.350253, 14.999854, 0.3, 255],
[0.200138, 0.349578, 14.999597, 0.3, 255],
[0.200138, 0.350138, 15.000567, 0.3, 255],
[0.199862, 0.349862, 14.999433, 0.3, 255],
[0.199862, 0.350422, 15.000403, 0.3, 255],
[0.200488, 0.350136, 15.000321, 0.3, 255],
[0.199957, 0.349751, 15.000544, 0.3, 255],
[0.200488, 0.34979, 14.999721, 0.3, 255],
[0.199957, 0.349404, 14.999944, 0.3, 255],
[0.200043, 0.350596, 15.000056, 0.3, 255],
[0.199512, 0.35021, 15.000279, 0.3, 255],
[0.200043, 0.350249, 14.999456, 0.3, 255],
[0.199512, 0.349864, 14.999679, 0.3, 255],
[-0.5, 0.500442, 10.000442, 0.2, 0],
[-0.5, 0.5, 10, 0.1995, 255],
[-0.499966, 0.49971, 9.999563, 0.199, 0],
[-0.500034, 0.499563, 9.99971, 0.199, 0],
[-0.499934, 0.499432, 9.999144, 0.1985, 255],
[-0.500066, 0.499144, 9.999432, 0.1985, 255],
[0.6, 0.5, 10, 0.25, 210],
[0.600122, 0.4999, 10.000122, 0.25, 210],
[0.600061, 0.49995, 10.000061, 0.25003, 50],
[0.595856, 0.505256, 9.991271, 0.24, 100],
[0.597973, 0.507461, 9.992175, 0.24, 210],
[0.595362, 0.507926, 9.993945, 0.24, 100],
[0.594618, 0.509055, 9.996832, 0.24, 210],
[0.592382, 0.506906, 9.996092, 0.24, 100],
[0.590438, 0.504882, 9.997604, 0.24, 210],
[0.591035, 0.503605, 9.994744, 0.24, 100],
[0.59121, 0.500708, 9.993425, 0.24, 210],
[0.593181, 0.502585, 9.991764, 0.24, 100],
[0.595867, 0.502302, 9.990069, 0.24, 210],
]
values = grey_values(test_case)
rows = [values[row * 1024:row * 1024 + 1024] for row in range(1024)]
with open('combination_scene.png', 'wb') as file:
w = png.Writer(1024, 1024, greyscale=True)
w.write(file, rows)
```
### Rust
The function `grey_values` would be sufficient as an entry to this challenge. The rest of the code saves the outputs as PNG files.
The `png` crate can be installed using `cargo add png`.
```rust
fn grey_values(spheres: &Vec<[f32; 5]>) -> Vec<u8> {
let mut values = vec![];
for y_pixel in 0..1024 {
for x_pixel in 0..1024 {
let x_ray = x_pixel as f32 / 512.0 - 1.0;
let y_ray = y_pixel as f32 / 512.0 - 1.0;
let mut min_distance = f32::INFINITY;
let mut hit_grey_value = 0;
for sphere in spheres {
let [
x_sphere, y_sphere, z_sphere, radius, grey
] = *sphere;
let a = (
(x_ray - x_sphere).powf(2.0)
+ (y_ray - y_sphere).powf(2.0)
).sqrt();
if a < radius {
let distance = z_sphere - (
radius.powf(2.0) - a.powf(2.0)
).sqrt();
if distance < min_distance {
min_distance = distance;
hit_grey_value = grey as u8;
}
}
}
values.push(hit_grey_value);
}
}
values
}
fn main() {
let test_case: Vec<[f32; 5]> = vec![
[0.0, 0.0, 1000.0, 2.0, 192.0],
[-2.0, 2.0, 100.0, 3.0, 98.0],
[2.0, 2.0, 100.0, 3.0, 98.0],
[0.5, -0.5, 500.0, 0.2, 255.0],
[-0.349469, -0.1, 20.000279, 0.6, 64.0],
[-0.349775, -0.100273, 20.000485, 0.6, 85.0],
[-0.350214, -0.100386, 20.000407, 0.6, 106.0],
[-0.350527, -0.100273, 20.000091, 0.6, 127.0],
[-0.350531, -0.1, 19.999721, 0.6, 148.0],
[-0.350225, -0.099727, 19.999515, 0.6, 169.0],
[-0.349786, -0.099614, 19.999593, 0.6, 190.0],
[-0.349473, -0.099727, 19.999909, 0.6, 211.0],
[0.20032, 0.349824, 15.000456, 0.3, 0.0],
[0.20032, 0.349517, 14.999924, 0.3, 0.0],
[0.19968, 0.350483, 15.000076, 0.3, 0.0],
[0.19968, 0.350176, 14.999544, 0.3, 0.0],
[0.200578, 0.350073, 14.999958, 0.3, 0.0],
[0.199817, 0.349519, 15.000277, 0.3, 0.0],
[0.200183, 0.350481, 14.999723, 0.3, 0.0],
[0.199422, 0.349927, 15.000042, 0.3, 0.0],
[0.200235, 0.35042, 15.000332, 0.3, 0.0],
[0.199765, 0.350078, 15.000529, 0.3, 0.0],
[0.200235, 0.349922, 14.999471, 0.3, 0.0],
[0.199765, 0.34958, 14.999668, 0.3, 0.0],
[0.200429, 0.350419, 15.000005, 0.3, 255.0],
[0.199571, 0.349795, 15.000366, 0.3, 255.0],
[0.200429, 0.350205, 14.999634, 0.3, 255.0],
[0.199571, 0.349581, 14.999995, 0.3, 255.0],
[0.200524, 0.349747, 15.000146, 0.3, 255.0],
[0.200196, 0.349509, 15.000283, 0.3, 255.0],
[0.199804, 0.350491, 14.999717, 0.3, 255.0],
[0.199476, 0.350253, 14.999854, 0.3, 255.0],
[0.200138, 0.349578, 14.999597, 0.3, 255.0],
[0.200138, 0.350138, 15.000567, 0.3, 255.0],
[0.199862, 0.349862, 14.999433, 0.3, 255.0],
[0.199862, 0.350422, 15.000403, 0.3, 255.0],
[0.200488, 0.350136, 15.000321, 0.3, 255.0],
[0.199957, 0.349751, 15.000544, 0.3, 255.0],
[0.200488, 0.34979, 14.999721, 0.3, 255.0],
[0.199957, 0.349404, 14.999944, 0.3, 255.0],
[0.200043, 0.350596, 15.000056, 0.3, 255.0],
[0.199512, 0.35021, 15.000279, 0.3, 255.0],
[0.200043, 0.350249, 14.999456, 0.3, 255.0],
[0.199512, 0.349864, 14.999679, 0.3, 255.0],
[-0.5, 0.500442, 10.000442, 0.2, 0.0],
[-0.5, 0.5, 10.0, 0.1995, 255.0],
[-0.499966, 0.49971, 9.999563, 0.199, 0.0],
[-0.500034, 0.499563, 9.99971, 0.199, 0.0],
[-0.499934, 0.499432, 9.999144, 0.1985, 255.0],
[-0.500066, 0.499144, 9.999432, 0.1985, 255.0],
[0.6, 0.5, 10.0, 0.25, 210.0],
[0.600122, 0.4999, 10.000122, 0.25, 210.0],
[0.600061, 0.49995, 10.000061, 0.25003, 50.0],
[0.595856, 0.505256, 9.991271, 0.24, 100.0],
[0.597973, 0.507461, 9.992175, 0.24, 210.0],
[0.595362, 0.507926, 9.993945, 0.24, 100.0],
[0.594618, 0.509055, 9.996832, 0.24, 210.0],
[0.592382, 0.506906, 9.996092, 0.24, 100.0],
[0.590438, 0.504882, 9.997604, 0.24, 210.0],
[0.591035, 0.503605, 9.994744, 0.24, 100.0],
[0.59121, 0.500708, 9.993425, 0.24, 210.0],
[0.593181, 0.502585, 9.991764, 0.24, 100.0],
[0.595867, 0.502302, 9.990069, 0.24, 210.0],
];
let values = grey_values(&test_case);
let path = std::path::Path::new("combination_scene.png");
let file = std::fs::File::create(path).unwrap();
let w = &mut std::io::BufWriter::new(file);
let mut encoder = png::Encoder::new(w, 1024, 1024);
encoder.set_color(png::ColorType::Grayscale);
encoder.set_depth(png::BitDepth::Eight);
let mut writer = encoder.write_header().unwrap();
writer.write_image_data(&values).unwrap();
}
```
## Test cases
### Single sphere
Input:
```text
[
[0, 0, 0, 1, 128],
]
```
Output:
~~![Grey circle on a black background](https://codegolf.codidact.com/uploads/jlru163pa2pgm7aaq70b2tff16uf)~~
### Two separated spheres
Input:
```text
[
[0, -0.55, 0, 0.4, 100],
[0, 0.55, 0, 0.4, 200],
]
```
Output:
~~![Light and dark grey circles on a black background](https://codegolf.codidact.com/uploads/8yw7agli78y65kiomb1u5yebs7c5)~~
### Ring of spheres
Input:
```text
[
[0.75, 0, 0, 0.25, 64],
[0.53033, 0.375, -0.375, 0.25, 185],
[0, 0.53033, -0.53033, 0.25, 64],
[-0.53033, 0.375, -0.375, 0.25, 185],
[-0.75, 0, -0, 0.25, 64],
[-0.53033, -0.375, 0.375, 0.25, 185],
[-0, -0.53033, 0.53033, 0.25, 64],
[0.53033, -0.375, 0.375, 0.25, 185],
]
```
Output:
~~![Eight circles in alternating light and dark grey in a ring some overlapping](https://codegolf.codidact.com/uploads/0lsenynig27au0sryf3ruozt1uwh)~~
### Beach ball
Input:
```text
[
[0.001, 0, 0, 1, 64],
[0.000707, 0.000707, -0, 1, 85],
[0, 0.001, -0, 1, 106],
[-0.000707, 0.000707, -0, 1, 127],
[-0.001, 0, -0, 1, 148],
[-0.000707, -0.000707, 0, 1, 169],
[-0, -0.001, 0, 1, 190],
[0.000707, -0.000707, 0, 1, 211],
]
```
Output:
~~![A circle with eight sectors in different shades of grey](https://codegolf.codidact.com/uploads/51a3ni0cfwsqcdw5b49gy6282qno)~~
### Football
Input:
```text
[
[0, 0.000058, 0.001947, 1, 0],
[0, -0.001715, 0.000923, 1, 0],
[0, 0.001715, -0.000923, 1, 0],
[0, -0.000058, -0.001947, 1, 0],
[0.001657, -0.000512, 0.000887, 1, 0],
[-0.001657, -0.000512, 0.000887, 1, 0],
[0.001657, 0.000512, -0.000887, 1, 0],
[-0.001657, 0.000512, -0.000887, 1, 0],
[0.001024, 0.001435, 0.000829, 1, 0],
[-0.001024, 0.001435, 0.000829, 1, 0],
[0.001024, -0.001435, -0.000829, 1, 0],
[-0.001024, -0.001435, -0.000829, 1, 0],
[0.001868, 0.000618, 0.000357, 1, 255],
[-0.001868, 0.000618, 0.000357, 1, 255],
[0.001868, -0.000618, -0.000357, 1, 255],
[-0.001868, -0.000618, -0.000357, 1, 255],
[0.000714, -0.000934, 0.001618, 1, 255],
[-0.000714, -0.000934, 0.001618, 1, 255],
[0.000714, 0.000934, -0.001618, 1, 255],
[-0.000714, 0.000934, -0.001618, 1, 255],
[0, -0.001975, -0.000316, 1, 255],
[0, 0.001261, 0.001552, 1, 255],
[0, -0.001261, -0.001552, 1, 255],
[0, 0.001975, 0.000316, 1, 255],
[0.001155, 0.000423, 0.001577, 1, 255],
[-0.001155, 0.000423, 0.001577, 1, 255],
[0.001155, -0.001577, 0.000423, 1, 255],
[-0.001155, -0.001577, 0.000423, 1, 255],
[0.001155, 0.001577, -0.000423, 1, 255],
[-0.001155, 0.001577, -0.000423, 1, 255],
[0.001155, -0.000423, -0.001577, 1, 255],
[-0.001155, -0.000423, -0.001577, 1, 255],
]
```
Output:
~~![Black pentagons on a white ball](https://codegolf.codidact.com/uploads/bmrbsw65p59imvwu8ssd42uvvx1r)~~
### Eight ball
Input:
```text
[
[0, 0, 1000, 2, 128],
[0, 0, 0.003125, 1, 0],
[0, 0, 0, 0.9975, 255],
[0, 0.000547, -0.002573, 0.995, 0],
[-0, -0.000547, -0.002573, 0.995, 0],
[0, 0.001071, -0.005037, 0.9925, 255],
[-0, -0.001071, -0.005037, 0.9925, 255],
]
```
Output:
~~![A ball with the number 8 in a white circle](https://codegolf.codidact.com/uploads/dvl68crpmy62rqt94mwanditt4wc)~~
### Star ball
Input:
```text
[
[0, 0, 0, 1, 210],
[0, 0, 0.0008, 1, 210],
[0, 0, 0.0004, 1.00012, 50],
[0.012969, -0, -0.042045, 0.96, 100],
[0.016397, 0.011913, -0.039054, 0.96, 210],
[0.004008, 0.012334, -0.042045, 0.96, 100],
[-0.006263, 0.019276, -0.039054, 0.96, 210],
[-0.010492, 0.007623, -0.042045, 0.96, 100],
[-0.020268, 0, -0.039054, 0.96, 210],
[-0.010492, -0.007623, -0.042045, 0.96, 100],
[-0.006263, -0.019276, -0.039054, 0.96, 210],
[0.004008, -0.012334, -0.042045, 0.96, 100],
[0.016397, -0.011913, -0.039054, 0.96, 210],
]
```
Output:
~~![A light grey ball with a dark grey five pointed star](https://codegolf.codidact.com/uploads/f3wdgwru6j35ue72x25mazxikfx2)~~
### Combination scene
Input:
```text
[
[0, 0, 1000, 2, 192],
[-2, 2, 100, 3, 98],
[2, 2, 100, 3, 98],
[0.5, -0.5, 500, 0.2, 255],
[-0.349469, -0.1, 20.000279, 0.6, 64],
[-0.349775, -0.100273, 20.000485, 0.6, 85],
[-0.350214, -0.100386, 20.000407, 0.6, 106],
[-0.350527, -0.100273, 20.000091, 0.6, 127],
[-0.350531, -0.1, 19.999721, 0.6, 148],
[-0.350225, -0.099727, 19.999515, 0.6, 169],
[-0.349786, -0.099614, 19.999593, 0.6, 190],
[-0.349473, -0.099727, 19.999909, 0.6, 211],
[0.20032, 0.349824, 15.000456, 0.3, 0],
[0.20032, 0.349517, 14.999924, 0.3, 0],
[0.19968, 0.350483, 15.000076, 0.3, 0],
[0.19968, 0.350176, 14.999544, 0.3, 0],
[0.200578, 0.350073, 14.999958, 0.3, 0],
[0.199817, 0.349519, 15.000277, 0.3, 0],
[0.200183, 0.350481, 14.999723, 0.3, 0],
[0.199422, 0.349927, 15.000042, 0.3, 0],
[0.200235, 0.35042, 15.000332, 0.3, 0],
[0.199765, 0.350078, 15.000529, 0.3, 0],
[0.200235, 0.349922, 14.999471, 0.3, 0],
[0.199765, 0.34958, 14.999668, 0.3, 0],
[0.200429, 0.350419, 15.000005, 0.3, 255],
[0.199571, 0.349795, 15.000366, 0.3, 255],
[0.200429, 0.350205, 14.999634, 0.3, 255],
[0.199571, 0.349581, 14.999995, 0.3, 255],
[0.200524, 0.349747, 15.000146, 0.3, 255],
[0.200196, 0.349509, 15.000283, 0.3, 255],
[0.199804, 0.350491, 14.999717, 0.3, 255],
[0.199476, 0.350253, 14.999854, 0.3, 255],
[0.200138, 0.349578, 14.999597, 0.3, 255],
[0.200138, 0.350138, 15.000567, 0.3, 255],
[0.199862, 0.349862, 14.999433, 0.3, 255],
[0.199862, 0.350422, 15.000403, 0.3, 255],
[0.200488, 0.350136, 15.000321, 0.3, 255],
[0.199957, 0.349751, 15.000544, 0.3, 255],
[0.200488, 0.34979, 14.999721, 0.3, 255],
[0.199957, 0.349404, 14.999944, 0.3, 255],
[0.200043, 0.350596, 15.000056, 0.3, 255],
[0.199512, 0.35021, 15.000279, 0.3, 255],
[0.200043, 0.350249, 14.999456, 0.3, 255],
[0.199512, 0.349864, 14.999679, 0.3, 255],
[-0.5, 0.500442, 10.000442, 0.2, 0],
[-0.5, 0.5, 10, 0.1995, 255],
[-0.499966, 0.49971, 9.999563, 0.199, 0],
[-0.500034, 0.499563, 9.99971, 0.199, 0],
[-0.499934, 0.499432, 9.999144, 0.1985, 255],
[-0.500066, 0.499144, 9.999432, 0.1985, 255],
[0.6, 0.5, 10, 0.25, 210],
[0.600122, 0.4999, 10.000122, 0.25, 210],
[0.600061, 0.49995, 10.000061, 0.25003, 50],
[0.595856, 0.505256, 9.991271, 0.24, 100],
[0.597973, 0.507461, 9.992175, 0.24, 210],
[0.595362, 0.507926, 9.993945, 0.24, 100],
[0.594618, 0.509055, 9.996832, 0.24, 210],
[0.592382, 0.506906, 9.996092, 0.24, 100],
[0.590438, 0.504882, 9.997604, 0.24, 210],
[0.591035, 0.503605, 9.994744, 0.24, 100],
[0.59121, 0.500708, 9.993425, 0.24, 210],
[0.593181, 0.502585, 9.991764, 0.24, 100],
[0.595867, 0.502302, 9.990069, 0.24, 210],
]
```
Output:
~~![A beachball, football, eightball, and starball with the sun behind hills in the background](https://codegolf.codidact.com/uploads/usv9npm1t5bjpcaknn6rbzgkqwmi)~~
## Scoring
This is a [code golf challenge](https://codegolf.codidact.com/categories/49/tags/4274). Your score is the number of bytes in your code. Lowest score for each language wins.
> Explanations are optional, but I'm more likely to upvote answers that have one.

Given a list of spheres, output an image.
Produce a greyscale [orthographic](https://en.wikipedia.org/wiki/Parallel_projection#Orthographic_projection) (parallel projection - no perspective) ray caster, that renders only plain grey spheres with no lighting.
## Input
- A list of spheres, each consisting of:
- The coordinates of its centre $(x,y,z)$.
- Its radius $r$.
- Its greyscale value $g$, an integer from $0$ to $255$ (both inclusive).
- Only the greyscale value will be restricted to integer values. The coordinates and radius may have fractional values.
## Output
- An image $1024$ by $1024$ pixels with greyscale values from $0$ to $255$.
- The image covers x and y values from $-1$ to $1$.
- A sphere's greyscale value is applied to every visible part of it. There is no variation over the surface of the sphere, so it appears flat like a disk.
- You may output whatever image format is convenient, including simple text formats such as [PGM](https://en.wikipedia.org/wiki/Netpbm#PGM_example) or a list of pixel greyscale values. If you want to include an example of your output as an image in your answer, the image formats that Codidact supports are PNG, JPEG, and GIF. You could output PGM and then convert to PNG to upload to your answer. If you output just the greyscale values as a list of numbers from $0$ to $255$, I've made a `[ TODO page to convert these to greyscale PNG]`.
- You are free to output an image reflected or rotated by a multiple of 90 degrees. However, the z axis is required to be increasing in the forwards direction (into the image) otherwise some of the test cases will look incorrect.
## Calculating a pixel value
Imagine a ray passing into the image through a pixel, perpendicular to the image plane. If it hits a sphere, it shows the greyscale value of the first sphere it hits. Otherwise, it shows a value of zero (black).
For example, for each pixel:
- The pixel $x,y$ coordinates will each be from $0$ to $1023$. Scale these to the range $-1$ to $1$ (for example, divide by $512$ and subtract $1$).
- If these scaled $x,y$ coordinates are within the radius $r$ of the $x,y$ coordinates of a sphere's centre, the ray intersects that sphere (the $z$ coordinate is not relevant to this test because the image is an orthographic projection - the rays are all parallel to the $z$ axis). The distance $a$ of the ray from the sphere's centre can be calculated as follows:
$$a=\sqrt{(x_{ray}-x_{sphere})^2+(y_{ray}-y_{sphere})^2}$$
```text
CONSIDER A DIAGRAM HERE
```
- For each sphere that the ray intersects, find the $z$ coordinate of the nearest intersection (there will often be $2$ per sphere, $1$ on the front of the sphere and $1$ on the back of the sphere, as the ray passes through - only the nearest is relevant). This can be calculated as follows:
- The distance $d$ to the nearest intersection point can be calculated from the $z$ coordinate of the sphere's centre, the radius $r$, and the distance $a$ from the $x,y$ coordinates of the sphere's centre to the scaled $x,y$ coordinates of the pixel:
$$d=z_{sphere}-\sqrt{r_{sphere}^2-a^2}$$
```text
CONSIDER A DIAGRAM HERE
```
- Of all the spheres that the ray intersects, the sphere with the intersection point with the smallest $z$ coordinate $d$ is the one to display. Set the pixel to the greyscale value of this sphere.
- If the ray did not intersect any spheres, set the pixel to zero (black).
## Example implementations
These are ungolfed code to supplement understanding of the requirements. You are free to golf these or implement your own approach that matches the test cases up to reflection & rotation.
The Python implementation uses 64 bit floating point. The Rust implementation uses 32 bit floating point to show the difference in output. I've tried to optimise the test cases for compatibility with 32 bit floating point, so the differences are restricted to being a pixel out in some places.
### Python
The function `grey_values` would be sufficient as an entry to this challenge. The rest of the code saves the outputs as PNG files. The function itself has no dependence on the `import png` so it would not be required in an entry.
Instructions for installing the `png` package can be found on its [project page on PyPI](https://pypi.org/project/pypng/).
```python
def grey_values(spheres):
values = []
for y_pixel in range(1024):
for x_pixel in range(1024):
x_ray, y_ray = x_pixel / 512 - 1, y_pixel / 512 - 1
min_distance, hit_grey_value = None, 0
for sphere in spheres:
x_sphere, y_sphere, z_sphere, radius, grey = sphere
a = (
(x_ray - x_sphere) ** 2 + (y_ray - y_sphere) ** 2
) ** 0.5
if a < radius:
distance = z_sphere - (radius ** 2 - a ** 2) ** 0.5
if min_distance is None or distance < min_distance:
min_distance = distance
hit_grey_value = grey
values.append(hit_grey_value)
return values
import png
test_case = [
[0, 0, 1000, 2, 192],
[-2, 2, 100, 3, 98],
[2, 2, 100, 3, 98],
[0.5, -0.5, 500, 0.2, 255],
[-0.349469, -0.1, 20.000279, 0.6, 64],
[-0.349775, -0.100273, 20.000485, 0.6, 85],
[-0.350214, -0.100386, 20.000407, 0.6, 106],
[-0.350527, -0.100273, 20.000091, 0.6, 127],
[-0.350531, -0.1, 19.999721, 0.6, 148],
[-0.350225, -0.099727, 19.999515, 0.6, 169],
[-0.349786, -0.099614, 19.999593, 0.6, 190],
[-0.349473, -0.099727, 19.999909, 0.6, 211],
[0.20032, 0.349824, 15.000456, 0.3, 0],
[0.20032, 0.349517, 14.999924, 0.3, 0],
[0.19968, 0.350483, 15.000076, 0.3, 0],
[0.19968, 0.350176, 14.999544, 0.3, 0],
[0.200578, 0.350073, 14.999958, 0.3, 0],
[0.199817, 0.349519, 15.000277, 0.3, 0],
[0.200183, 0.350481, 14.999723, 0.3, 0],
[0.199422, 0.349927, 15.000042, 0.3, 0],
[0.200235, 0.35042, 15.000332, 0.3, 0],
[0.199765, 0.350078, 15.000529, 0.3, 0],
[0.200235, 0.349922, 14.999471, 0.3, 0],
[0.199765, 0.34958, 14.999668, 0.3, 0],
[0.200429, 0.350419, 15.000005, 0.3, 255],
[0.199571, 0.349795, 15.000366, 0.3, 255],
[0.200429, 0.350205, 14.999634, 0.3, 255],
[0.199571, 0.349581, 14.999995, 0.3, 255],
[0.200524, 0.349747, 15.000146, 0.3, 255],
[0.200196, 0.349509, 15.000283, 0.3, 255],
[0.199804, 0.350491, 14.999717, 0.3, 255],
[0.199476, 0.350253, 14.999854, 0.3, 255],
[0.200138, 0.349578, 14.999597, 0.3, 255],
[0.200138, 0.350138, 15.000567, 0.3, 255],
[0.199862, 0.349862, 14.999433, 0.3, 255],
[0.199862, 0.350422, 15.000403, 0.3, 255],
[0.200488, 0.350136, 15.000321, 0.3, 255],
[0.199957, 0.349751, 15.000544, 0.3, 255],
[0.200488, 0.34979, 14.999721, 0.3, 255],
[0.199957, 0.349404, 14.999944, 0.3, 255],
[0.200043, 0.350596, 15.000056, 0.3, 255],
[0.199512, 0.35021, 15.000279, 0.3, 255],
[0.200043, 0.350249, 14.999456, 0.3, 255],
[0.199512, 0.349864, 14.999679, 0.3, 255],
[-0.5, 0.500442, 10.000442, 0.2, 0],
[-0.5, 0.5, 10, 0.1995, 255],
[-0.499966, 0.49971, 9.999563, 0.199, 0],
[-0.500034, 0.499563, 9.99971, 0.199, 0],
[-0.499934, 0.499432, 9.999144, 0.1985, 255],
[-0.500066, 0.499144, 9.999432, 0.1985, 255],
[0.6, 0.5, 10, 0.25, 210],
[0.600122, 0.4999, 10.000122, 0.25, 210],
[0.600061, 0.49995, 10.000061, 0.25003, 50],
[0.595856, 0.505256, 9.991271, 0.24, 100],
[0.597973, 0.507461, 9.992175, 0.24, 210],
[0.595362, 0.507926, 9.993945, 0.24, 100],
[0.594618, 0.509055, 9.996832, 0.24, 210],
[0.592382, 0.506906, 9.996092, 0.24, 100],
[0.590438, 0.504882, 9.997604, 0.24, 210],
[0.591035, 0.503605, 9.994744, 0.24, 100],
[0.59121, 0.500708, 9.993425, 0.24, 210],
[0.593181, 0.502585, 9.991764, 0.24, 100],
[0.595867, 0.502302, 9.990069, 0.24, 210],
]
values = grey_values(test_case)
rows = [values[row * 1024:row * 1024 + 1024] for row in range(1024)]
with open('combination_scene.png', 'wb') as file:
w = png.Writer(1024, 1024, greyscale=True)
w.write(file, rows)
```
### Rust
The function `grey_values` would be sufficient as an entry to this challenge. The rest of the code saves the outputs as PNG files.
The `png` crate can be installed using `cargo add png`.
```rust
fn grey_values(spheres: &Vec<[f32; 5]>) -> Vec<u8> {
let mut values = vec![];
for y_pixel in 0..1024 {
for x_pixel in 0..1024 {
let x_ray = x_pixel as f32 / 512.0 - 1.0;
let y_ray = y_pixel as f32 / 512.0 - 1.0;
let mut min_distance = f32::INFINITY;
let mut hit_grey_value = 0;
for sphere in spheres {
let [
x_sphere, y_sphere, z_sphere, radius, grey
] = *sphere;
let a = (
(x_ray - x_sphere).powf(2.0)
+ (y_ray - y_sphere).powf(2.0)
).sqrt();
if a < radius {
let distance = z_sphere - (
radius.powf(2.0) - a.powf(2.0)
).sqrt();
if distance < min_distance {
min_distance = distance;
hit_grey_value = grey as u8;
}
}
}
values.push(hit_grey_value);
}
}
values
}
fn main() {
let test_case: Vec<[f32; 5]> = vec![
[0.0, 0.0, 1000.0, 2.0, 192.0],
[-2.0, 2.0, 100.0, 3.0, 98.0],
[2.0, 2.0, 100.0, 3.0, 98.0],
[0.5, -0.5, 500.0, 0.2, 255.0],
[-0.349469, -0.1, 20.000279, 0.6, 64.0],
[-0.349775, -0.100273, 20.000485, 0.6, 85.0],
[-0.350214, -0.100386, 20.000407, 0.6, 106.0],
[-0.350527, -0.100273, 20.000091, 0.6, 127.0],
[-0.350531, -0.1, 19.999721, 0.6, 148.0],
[-0.350225, -0.099727, 19.999515, 0.6, 169.0],
[-0.349786, -0.099614, 19.999593, 0.6, 190.0],
[-0.349473, -0.099727, 19.999909, 0.6, 211.0],
[0.20032, 0.349824, 15.000456, 0.3, 0.0],
[0.20032, 0.349517, 14.999924, 0.3, 0.0],
[0.19968, 0.350483, 15.000076, 0.3, 0.0],
[0.19968, 0.350176, 14.999544, 0.3, 0.0],
[0.200578, 0.350073, 14.999958, 0.3, 0.0],
[0.199817, 0.349519, 15.000277, 0.3, 0.0],
[0.200183, 0.350481, 14.999723, 0.3, 0.0],
[0.199422, 0.349927, 15.000042, 0.3, 0.0],
[0.200235, 0.35042, 15.000332, 0.3, 0.0],
[0.199765, 0.350078, 15.000529, 0.3, 0.0],
[0.200235, 0.349922, 14.999471, 0.3, 0.0],
[0.199765, 0.34958, 14.999668, 0.3, 0.0],
[0.200429, 0.350419, 15.000005, 0.3, 255.0],
[0.199571, 0.349795, 15.000366, 0.3, 255.0],
[0.200429, 0.350205, 14.999634, 0.3, 255.0],
[0.199571, 0.349581, 14.999995, 0.3, 255.0],
[0.200524, 0.349747, 15.000146, 0.3, 255.0],
[0.200196, 0.349509, 15.000283, 0.3, 255.0],
[0.199804, 0.350491, 14.999717, 0.3, 255.0],
[0.199476, 0.350253, 14.999854, 0.3, 255.0],
[0.200138, 0.349578, 14.999597, 0.3, 255.0],
[0.200138, 0.350138, 15.000567, 0.3, 255.0],
[0.199862, 0.349862, 14.999433, 0.3, 255.0],
[0.199862, 0.350422, 15.000403, 0.3, 255.0],
[0.200488, 0.350136, 15.000321, 0.3, 255.0],
[0.199957, 0.349751, 15.000544, 0.3, 255.0],
[0.200488, 0.34979, 14.999721, 0.3, 255.0],
[0.199957, 0.349404, 14.999944, 0.3, 255.0],
[0.200043, 0.350596, 15.000056, 0.3, 255.0],
[0.199512, 0.35021, 15.000279, 0.3, 255.0],
[0.200043, 0.350249, 14.999456, 0.3, 255.0],
[0.199512, 0.349864, 14.999679, 0.3, 255.0],
[-0.5, 0.500442, 10.000442, 0.2, 0.0],
[-0.5, 0.5, 10.0, 0.1995, 255.0],
[-0.499966, 0.49971, 9.999563, 0.199, 0.0],
[-0.500034, 0.499563, 9.99971, 0.199, 0.0],
[-0.499934, 0.499432, 9.999144, 0.1985, 255.0],
[-0.500066, 0.499144, 9.999432, 0.1985, 255.0],
[0.6, 0.5, 10.0, 0.25, 210.0],
[0.600122, 0.4999, 10.000122, 0.25, 210.0],
[0.600061, 0.49995, 10.000061, 0.25003, 50.0],
[0.595856, 0.505256, 9.991271, 0.24, 100.0],
[0.597973, 0.507461, 9.992175, 0.24, 210.0],
[0.595362, 0.507926, 9.993945, 0.24, 100.0],
[0.594618, 0.509055, 9.996832, 0.24, 210.0],
[0.592382, 0.506906, 9.996092, 0.24, 100.0],
[0.590438, 0.504882, 9.997604, 0.24, 210.0],
[0.591035, 0.503605, 9.994744, 0.24, 100.0],
[0.59121, 0.500708, 9.993425, 0.24, 210.0],
[0.593181, 0.502585, 9.991764, 0.24, 100.0],
[0.595867, 0.502302, 9.990069, 0.24, 210.0],
];
let values = grey_values(&test_case);
let path = std::path::Path::new("combination_scene.png");
let file = std::fs::File::create(path).unwrap();
let w = &mut std::io::BufWriter::new(file);
let mut encoder = png::Encoder::new(w, 1024, 1024);
encoder.set_color(png::ColorType::Grayscale);
encoder.set_depth(png::BitDepth::Eight);
let mut writer = encoder.write_header().unwrap();
writer.write_image_data(&values).unwrap();
}
```
## Test cases
The output images may be scaled down to fit. Click on any image to open it full size.
### Single sphere
Input:
```text
[
[0, 0, 0, 1, 128],
]
```
Output:
[![Grey circle on a black background](https://codegolf.codidact.com/uploads/jlru163pa2pgm7aaq70b2tff16uf)](https://codegolf.codidact.com/uploads/jlru163pa2pgm7aaq70b2tff16uf)
### Two separated spheres
Input:
```text
[
[0, -0.55, 0, 0.4, 100],
[0, 0.55, 0, 0.4, 200],
]
```
Output:
[![Light and dark grey circles on a black background](https://codegolf.codidact.com/uploads/8yw7agli78y65kiomb1u5yebs7c5)](https://codegolf.codidact.com/uploads/8yw7agli78y65kiomb1u5yebs7c5)
### Ring of spheres
Input:
```text
[
[0.75, 0, 0, 0.25, 64],
[0.53033, 0.375, -0.375, 0.25, 185],
[0, 0.53033, -0.53033, 0.25, 64],
[-0.53033, 0.375, -0.375, 0.25, 185],
[-0.75, 0, -0, 0.25, 64],
[-0.53033, -0.375, 0.375, 0.25, 185],
[-0, -0.53033, 0.53033, 0.25, 64],
[0.53033, -0.375, 0.375, 0.25, 185],
]
```
Output:
[![Eight circles in alternating light and dark grey in a ring some overlapping](https://codegolf.codidact.com/uploads/0lsenynig27au0sryf3ruozt1uwh)](https://codegolf.codidact.com/uploads/0lsenynig27au0sryf3ruozt1uwh)
### Beach ball
Input:
```text
[
[0.001, 0, 0, 1, 64],
[0.000707, 0.000707, -0, 1, 85],
[0, 0.001, -0, 1, 106],
[-0.000707, 0.000707, -0, 1, 127],
[-0.001, 0, -0, 1, 148],
[-0.000707, -0.000707, 0, 1, 169],
[-0, -0.001, 0, 1, 190],
[0.000707, -0.000707, 0, 1, 211],
]
```
Output:
[![A circle with eight sectors in different shades of grey](https://codegolf.codidact.com/uploads/51a3ni0cfwsqcdw5b49gy6282qno)](https://codegolf.codidact.com/uploads/51a3ni0cfwsqcdw5b49gy6282qno)
### Football
Input:
```text
[
[0, 0.000058, 0.001947, 1, 0],
[0, -0.001715, 0.000923, 1, 0],
[0, 0.001715, -0.000923, 1, 0],
[0, -0.000058, -0.001947, 1, 0],
[0.001657, -0.000512, 0.000887, 1, 0],
[-0.001657, -0.000512, 0.000887, 1, 0],
[0.001657, 0.000512, -0.000887, 1, 0],
[-0.001657, 0.000512, -0.000887, 1, 0],
[0.001024, 0.001435, 0.000829, 1, 0],
[-0.001024, 0.001435, 0.000829, 1, 0],
[0.001024, -0.001435, -0.000829, 1, 0],
[-0.001024, -0.001435, -0.000829, 1, 0],
[0.001868, 0.000618, 0.000357, 1, 255],
[-0.001868, 0.000618, 0.000357, 1, 255],
[0.001868, -0.000618, -0.000357, 1, 255],
[-0.001868, -0.000618, -0.000357, 1, 255],
[0.000714, -0.000934, 0.001618, 1, 255],
[-0.000714, -0.000934, 0.001618, 1, 255],
[0.000714, 0.000934, -0.001618, 1, 255],
[-0.000714, 0.000934, -0.001618, 1, 255],
[0, -0.001975, -0.000316, 1, 255],
[0, 0.001261, 0.001552, 1, 255],
[0, -0.001261, -0.001552, 1, 255],
[0, 0.001975, 0.000316, 1, 255],
[0.001155, 0.000423, 0.001577, 1, 255],
[-0.001155, 0.000423, 0.001577, 1, 255],
[0.001155, -0.001577, 0.000423, 1, 255],
[-0.001155, -0.001577, 0.000423, 1, 255],
[0.001155, 0.001577, -0.000423, 1, 255],
[-0.001155, 0.001577, -0.000423, 1, 255],
[0.001155, -0.000423, -0.001577, 1, 255],
[-0.001155, -0.000423, -0.001577, 1, 255],
]
```
Output:
[![Black pentagons on a white ball](https://codegolf.codidact.com/uploads/bmrbsw65p59imvwu8ssd42uvvx1r)](https://codegolf.codidact.com/uploads/bmrbsw65p59imvwu8ssd42uvvx1r)
### Eight ball
Input:
```text
[
[0, 0, 1000, 2, 128],
[0, 0, 0.003125, 1, 0],
[0, 0, 0, 0.9975, 255],
[0, 0.000547, -0.002573, 0.995, 0],
[-0, -0.000547, -0.002573, 0.995, 0],
[0, 0.001071, -0.005037, 0.9925, 255],
[-0, -0.001071, -0.005037, 0.9925, 255],
]
```
Output:
[![A ball with the number 8 in a white circle](https://codegolf.codidact.com/uploads/dvl68crpmy62rqt94mwanditt4wc)](https://codegolf.codidact.com/uploads/dvl68crpmy62rqt94mwanditt4wc)
### Star ball
Input:
```text
[
[0, 0, 0, 1, 210],
[0, 0, 0.0008, 1, 210],
[0, 0, 0.0004, 1.00012, 50],
[0.012969, -0, -0.042045, 0.96, 100],
[0.016397, 0.011913, -0.039054, 0.96, 210],
[0.004008, 0.012334, -0.042045, 0.96, 100],
[-0.006263, 0.019276, -0.039054, 0.96, 210],
[-0.010492, 0.007623, -0.042045, 0.96, 100],
[-0.020268, 0, -0.039054, 0.96, 210],
[-0.010492, -0.007623, -0.042045, 0.96, 100],
[-0.006263, -0.019276, -0.039054, 0.96, 210],
[0.004008, -0.012334, -0.042045, 0.96, 100],
[0.016397, -0.011913, -0.039054, 0.96, 210],
]
```
Output:
[![A light grey ball with a dark grey five pointed star](https://codegolf.codidact.com/uploads/f3wdgwru6j35ue72x25mazxikfx2)](https://codegolf.codidact.com/uploads/f3wdgwru6j35ue72x25mazxikfx2)
### Combination scene
Input:
```text
[
[0, 0, 1000, 2, 192],
[-2, 2, 100, 3, 98],
[2, 2, 100, 3, 98],
[0.5, -0.5, 500, 0.2, 255],
[-0.349469, -0.1, 20.000279, 0.6, 64],
[-0.349775, -0.100273, 20.000485, 0.6, 85],
[-0.350214, -0.100386, 20.000407, 0.6, 106],
[-0.350527, -0.100273, 20.000091, 0.6, 127],
[-0.350531, -0.1, 19.999721, 0.6, 148],
[-0.350225, -0.099727, 19.999515, 0.6, 169],
[-0.349786, -0.099614, 19.999593, 0.6, 190],
[-0.349473, -0.099727, 19.999909, 0.6, 211],
[0.20032, 0.349824, 15.000456, 0.3, 0],
[0.20032, 0.349517, 14.999924, 0.3, 0],
[0.19968, 0.350483, 15.000076, 0.3, 0],
[0.19968, 0.350176, 14.999544, 0.3, 0],
[0.200578, 0.350073, 14.999958, 0.3, 0],
[0.199817, 0.349519, 15.000277, 0.3, 0],
[0.200183, 0.350481, 14.999723, 0.3, 0],
[0.199422, 0.349927, 15.000042, 0.3, 0],
[0.200235, 0.35042, 15.000332, 0.3, 0],
[0.199765, 0.350078, 15.000529, 0.3, 0],
[0.200235, 0.349922, 14.999471, 0.3, 0],
[0.199765, 0.34958, 14.999668, 0.3, 0],
[0.200429, 0.350419, 15.000005, 0.3, 255],
[0.199571, 0.349795, 15.000366, 0.3, 255],
[0.200429, 0.350205, 14.999634, 0.3, 255],
[0.199571, 0.349581, 14.999995, 0.3, 255],
[0.200524, 0.349747, 15.000146, 0.3, 255],
[0.200196, 0.349509, 15.000283, 0.3, 255],
[0.199804, 0.350491, 14.999717, 0.3, 255],
[0.199476, 0.350253, 14.999854, 0.3, 255],
[0.200138, 0.349578, 14.999597, 0.3, 255],
[0.200138, 0.350138, 15.000567, 0.3, 255],
[0.199862, 0.349862, 14.999433, 0.3, 255],
[0.199862, 0.350422, 15.000403, 0.3, 255],
[0.200488, 0.350136, 15.000321, 0.3, 255],
[0.199957, 0.349751, 15.000544, 0.3, 255],
[0.200488, 0.34979, 14.999721, 0.3, 255],
[0.199957, 0.349404, 14.999944, 0.3, 255],
[0.200043, 0.350596, 15.000056, 0.3, 255],
[0.199512, 0.35021, 15.000279, 0.3, 255],
[0.200043, 0.350249, 14.999456, 0.3, 255],
[0.199512, 0.349864, 14.999679, 0.3, 255],
[-0.5, 0.500442, 10.000442, 0.2, 0],
[-0.5, 0.5, 10, 0.1995, 255],
[-0.499966, 0.49971, 9.999563, 0.199, 0],
[-0.500034, 0.499563, 9.99971, 0.199, 0],
[-0.499934, 0.499432, 9.999144, 0.1985, 255],
[-0.500066, 0.499144, 9.999432, 0.1985, 255],
[0.6, 0.5, 10, 0.25, 210],
[0.600122, 0.4999, 10.000122, 0.25, 210],
[0.600061, 0.49995, 10.000061, 0.25003, 50],
[0.595856, 0.505256, 9.991271, 0.24, 100],
[0.597973, 0.507461, 9.992175, 0.24, 210],
[0.595362, 0.507926, 9.993945, 0.24, 100],
[0.594618, 0.509055, 9.996832, 0.24, 210],
[0.592382, 0.506906, 9.996092, 0.24, 100],
[0.590438, 0.504882, 9.997604, 0.24, 210],
[0.591035, 0.503605, 9.994744, 0.24, 100],
[0.59121, 0.500708, 9.993425, 0.24, 210],
[0.593181, 0.502585, 9.991764, 0.24, 100],
[0.595867, 0.502302, 9.990069, 0.24, 210],
]
```
Output:
[![A beachball, football, eightball, and starball with the sun behind hills in the background](https://codegolf.codidact.com/uploads/usv9npm1t5bjpcaknn6rbzgkqwmi)](https://codegolf.codidact.com/uploads/usv9npm1t5bjpcaknn6rbzgkqwmi)
## Scoring
This is a [code golf challenge](https://codegolf.codidact.com/categories/49/tags/4274). Your score is the number of bytes in your code. Lowest score for each language wins.
> Explanations are optional, but I'm more likely to upvote answers that have one.

#9: Post edited by

trichoplax‭ · 2024-07-26T12:12:11Z (5 months ago)
Add test case outputs

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Markdown

Given a list of spheres, output an image.
Produce a greyscale [orthographic](https://en.wikipedia.org/wiki/Parallel_projection#Orthographic_projection) (parallel projection - no perspective) ray caster, that renders only plain grey spheres with no lighting.
## Input
- A list of spheres, each consisting of:
- The coordinates of its centre $(x,y,z)$.
- Its radius $r$.
- Its greyscale value $g$, an integer from $0$ to $255$ (both inclusive).
- Only the greyscale value will be restricted to integer values. The coordinates and radius may have fractional values.
## Output
- An image $1024$ by $1024$ pixels with greyscale values from $0$ to $255$.
- The image covers x and y values from $-1$ to $1$.
- A sphere's greyscale value is applied to every visible part of it. There is no variation over the surface of the sphere, so it appears flat like a disk.
- You may output whatever image format is convenient, including simple text formats such as [PGM](https://en.wikipedia.org/wiki/Netpbm#PGM_example) or a list of pixel greyscale values. If you want to include an example of your output as an image in your answer, the image formats that Codidact supports are PNG, JPEG, and GIF. You could output PGM and then convert to PNG to upload to your answer. If you output just the greyscale values as a list of numbers from $0$ to $255$, I've made a `[ TODO page to convert these to greyscale PNG]`.
- You are free to output an image reflected or rotated by a multiple of 90 degrees. However, the z axis is required to be increasing in the forwards direction (into the image) otherwise some of the test cases will look incorrect.
## Calculating a pixel value
Imagine a ray passing into the image through a pixel, perpendicular to the image plane. If it hits a sphere, it shows the greyscale value of the first sphere it hits. Otherwise, it shows a value of zero (black).
For example, for each pixel:
- The pixel $x,y$ coordinates will each be from $0$ to $1023$. Scale these to the range $-1$ to $1$ (for example, divide by $512$ and subtract $1$).
- If these scaled $x,y$ coordinates are within the radius $r$ of the $x,y$ coordinates of a sphere's centre, the ray intersects that sphere (the $z$ coordinate is not relevant to this test because the image is an orthographic projection - the rays are all parallel to the $z$ axis). The distance $a$ of the ray from the sphere's centre can be calculated as follows:
$$a=\sqrt{(x_{ray}-x_{sphere})^2+(y_{ray}-y_{sphere})^2}$$
```text
CONSIDER A DIAGRAM HERE
```
- For each sphere that the ray intersects, find the $z$ coordinate of the nearest intersection (there will often be $2$ per sphere, $1$ on the front of the sphere and $1$ on the back of the sphere, as the ray passes through - only the nearest is relevant). This can be calculated as follows:
- The distance $d$ to the nearest intersection point can be calculated from the $z$ coordinate of the sphere's centre, the radius $r$, and the distance $a$ from the $x,y$ coordinates of the sphere's centre to the scaled $x,y$ coordinates of the pixel:
$$d=z_{sphere}-\sqrt{r_{sphere}^2-a^2}$$
```text
CONSIDER A DIAGRAM HERE
```
- Of all the spheres that the ray intersects, the sphere with the intersection point with the smallest $z$ coordinate $d$ is the one to display. Set the pixel to the greyscale value of this sphere.
- If the ray did not intersect any spheres, set the pixel to zero (black).
## Example implementations
These are ungolfed code to supplement understanding of the requirements. You are free to golf these or implement your own approach that matches the test cases up to reflection & rotation.
The Python implementation uses 64 bit floating point. The Rust implementation uses 32 bit floating point to show the difference in output. I've tried to optimise the test cases for compatibility with 32 bit floating point, so the differences are restricted to being a pixel out in some places.
### Python
The function `grey_values` would be sufficient as an entry to this challenge. The rest of the code saves the outputs as PNG files. The function itself has no dependence on the `import png` so it would not be required in an entry.
Instructions for installing the `png` package can be found on its [project page on PyPI](https://pypi.org/project/pypng/).
```python
def grey_values(spheres):
values = []
for y_pixel in range(1024):
for x_pixel in range(1024):
x_ray, y_ray = x_pixel / 512 - 1, y_pixel / 512 - 1
min_distance, hit_grey_value = None, 0
for sphere in spheres:
x_sphere, y_sphere, z_sphere, radius, grey = sphere
a = (
(x_ray - x_sphere) ** 2 + (y_ray - y_sphere) ** 2
) ** 0.5
if a < radius:
distance = z_sphere - (radius ** 2 - a ** 2) ** 0.5
if min_distance is None or distance < min_distance:
min_distance = distance
hit_grey_value = grey
values.append(hit_grey_value)
return values
import png
test_case = [
[0, 0, 1000, 2, 192],
[-2, 2, 100, 3, 98],
[2, 2, 100, 3, 98],
[0.5, -0.5, 500, 0.2, 255],
[-0.349469, -0.1, 20.000279, 0.6, 64],
[-0.349775, -0.100273, 20.000485, 0.6, 85],
[-0.350214, -0.100386, 20.000407, 0.6, 106],
[-0.350527, -0.100273, 20.000091, 0.6, 127],
[-0.350531, -0.1, 19.999721, 0.6, 148],
[-0.350225, -0.099727, 19.999515, 0.6, 169],
[-0.349786, -0.099614, 19.999593, 0.6, 190],
[-0.349473, -0.099727, 19.999909, 0.6, 211],
[0.20032, 0.349824, 15.000456, 0.3, 0],
[0.20032, 0.349517, 14.999924, 0.3, 0],
[0.19968, 0.350483, 15.000076, 0.3, 0],
[0.19968, 0.350176, 14.999544, 0.3, 0],
[0.200578, 0.350073, 14.999958, 0.3, 0],
[0.199817, 0.349519, 15.000277, 0.3, 0],
[0.200183, 0.350481, 14.999723, 0.3, 0],
[0.199422, 0.349927, 15.000042, 0.3, 0],
[0.200235, 0.35042, 15.000332, 0.3, 0],
[0.199765, 0.350078, 15.000529, 0.3, 0],
[0.200235, 0.349922, 14.999471, 0.3, 0],
[0.199765, 0.34958, 14.999668, 0.3, 0],
[0.200429, 0.350419, 15.000005, 0.3, 255],
[0.199571, 0.349795, 15.000366, 0.3, 255],
[0.200429, 0.350205, 14.999634, 0.3, 255],
[0.199571, 0.349581, 14.999995, 0.3, 255],
[0.200524, 0.349747, 15.000146, 0.3, 255],
[0.200196, 0.349509, 15.000283, 0.3, 255],
[0.199804, 0.350491, 14.999717, 0.3, 255],
[0.199476, 0.350253, 14.999854, 0.3, 255],
[0.200138, 0.349578, 14.999597, 0.3, 255],
[0.200138, 0.350138, 15.000567, 0.3, 255],
[0.199862, 0.349862, 14.999433, 0.3, 255],
[0.199862, 0.350422, 15.000403, 0.3, 255],
[0.200488, 0.350136, 15.000321, 0.3, 255],
[0.199957, 0.349751, 15.000544, 0.3, 255],
[0.200488, 0.34979, 14.999721, 0.3, 255],
[0.199957, 0.349404, 14.999944, 0.3, 255],
[0.200043, 0.350596, 15.000056, 0.3, 255],
[0.199512, 0.35021, 15.000279, 0.3, 255],
[0.200043, 0.350249, 14.999456, 0.3, 255],
[0.199512, 0.349864, 14.999679, 0.3, 255],
[-0.5, 0.500442, 10.000442, 0.2, 0],
[-0.5, 0.5, 10, 0.1995, 255],
[-0.499966, 0.49971, 9.999563, 0.199, 0],
[-0.500034, 0.499563, 9.99971, 0.199, 0],
[-0.499934, 0.499432, 9.999144, 0.1985, 255],
[-0.500066, 0.499144, 9.999432, 0.1985, 255],
[0.6, 0.5, 10, 0.25, 210],
[0.600122, 0.4999, 10.000122, 0.25, 210],
[0.600061, 0.49995, 10.000061, 0.25003, 50],
[0.595856, 0.505256, 9.991271, 0.24, 100],
[0.597973, 0.507461, 9.992175, 0.24, 210],
[0.595362, 0.507926, 9.993945, 0.24, 100],
[0.594618, 0.509055, 9.996832, 0.24, 210],
[0.592382, 0.506906, 9.996092, 0.24, 100],
[0.590438, 0.504882, 9.997604, 0.24, 210],
[0.591035, 0.503605, 9.994744, 0.24, 100],
[0.59121, 0.500708, 9.993425, 0.24, 210],
[0.593181, 0.502585, 9.991764, 0.24, 100],
[0.595867, 0.502302, 9.990069, 0.24, 210],
]
values = grey_values(test_case)
rows = [values[row * 1024:row * 1024 + 1024] for row in range(1024)]
with open('combination_scene.png', 'wb') as file:
w = png.Writer(1024, 1024, greyscale=True)
w.write(file, rows)
```
### Rust
The function `grey_values` would be sufficient as an entry to this challenge. The rest of the code saves the outputs as PNG files.
The `png` crate can be installed using `cargo add png`.
```rust
fn grey_values(spheres: &Vec<[f32; 5]>) -> Vec<u8> {
let mut values = vec![];
for y_pixel in 0..1024 {
for x_pixel in 0..1024 {
let x_ray = x_pixel as f32 / 512.0 - 1.0;
let y_ray = y_pixel as f32 / 512.0 - 1.0;
let mut min_distance = f32::INFINITY;
let mut hit_grey_value = 0;
for sphere in spheres {
let [
x_sphere, y_sphere, z_sphere, radius, grey
] = *sphere;
let a = (
(x_ray - x_sphere).powf(2.0)
+ (y_ray - y_sphere).powf(2.0)
).sqrt();
if a < radius {
let distance = z_sphere - (
radius.powf(2.0) - a.powf(2.0)
).sqrt();
if distance < min_distance {
min_distance = distance;
hit_grey_value = grey as u8;
}
}
}
values.push(hit_grey_value);
}
}
values
}
fn main() {
let test_case: Vec<[f32; 5]> = vec![
[0.0, 0.0, 1000.0, 2.0, 192.0],
[-2.0, 2.0, 100.0, 3.0, 98.0],
[2.0, 2.0, 100.0, 3.0, 98.0],
[0.5, -0.5, 500.0, 0.2, 255.0],
[-0.349469, -0.1, 20.000279, 0.6, 64.0],
[-0.349775, -0.100273, 20.000485, 0.6, 85.0],
[-0.350214, -0.100386, 20.000407, 0.6, 106.0],
[-0.350527, -0.100273, 20.000091, 0.6, 127.0],
[-0.350531, -0.1, 19.999721, 0.6, 148.0],
[-0.350225, -0.099727, 19.999515, 0.6, 169.0],
[-0.349786, -0.099614, 19.999593, 0.6, 190.0],
[-0.349473, -0.099727, 19.999909, 0.6, 211.0],
[0.20032, 0.349824, 15.000456, 0.3, 0.0],
[0.20032, 0.349517, 14.999924, 0.3, 0.0],
[0.19968, 0.350483, 15.000076, 0.3, 0.0],
[0.19968, 0.350176, 14.999544, 0.3, 0.0],
[0.200578, 0.350073, 14.999958, 0.3, 0.0],
[0.199817, 0.349519, 15.000277, 0.3, 0.0],
[0.200183, 0.350481, 14.999723, 0.3, 0.0],
[0.199422, 0.349927, 15.000042, 0.3, 0.0],
[0.200235, 0.35042, 15.000332, 0.3, 0.0],
[0.199765, 0.350078, 15.000529, 0.3, 0.0],
[0.200235, 0.349922, 14.999471, 0.3, 0.0],
[0.199765, 0.34958, 14.999668, 0.3, 0.0],
[0.200429, 0.350419, 15.000005, 0.3, 255.0],
[0.199571, 0.349795, 15.000366, 0.3, 255.0],
[0.200429, 0.350205, 14.999634, 0.3, 255.0],
[0.199571, 0.349581, 14.999995, 0.3, 255.0],
[0.200524, 0.349747, 15.000146, 0.3, 255.0],
[0.200196, 0.349509, 15.000283, 0.3, 255.0],
[0.199804, 0.350491, 14.999717, 0.3, 255.0],
[0.199476, 0.350253, 14.999854, 0.3, 255.0],
[0.200138, 0.349578, 14.999597, 0.3, 255.0],
[0.200138, 0.350138, 15.000567, 0.3, 255.0],
[0.199862, 0.349862, 14.999433, 0.3, 255.0],
[0.199862, 0.350422, 15.000403, 0.3, 255.0],
[0.200488, 0.350136, 15.000321, 0.3, 255.0],
[0.199957, 0.349751, 15.000544, 0.3, 255.0],
[0.200488, 0.34979, 14.999721, 0.3, 255.0],
[0.199957, 0.349404, 14.999944, 0.3, 255.0],
[0.200043, 0.350596, 15.000056, 0.3, 255.0],
[0.199512, 0.35021, 15.000279, 0.3, 255.0],
[0.200043, 0.350249, 14.999456, 0.3, 255.0],
[0.199512, 0.349864, 14.999679, 0.3, 255.0],
[-0.5, 0.500442, 10.000442, 0.2, 0.0],
[-0.5, 0.5, 10.0, 0.1995, 255.0],
[-0.499966, 0.49971, 9.999563, 0.199, 0.0],
[-0.500034, 0.499563, 9.99971, 0.199, 0.0],
[-0.499934, 0.499432, 9.999144, 0.1985, 255.0],
[-0.500066, 0.499144, 9.999432, 0.1985, 255.0],
[0.6, 0.5, 10.0, 0.25, 210.0],
[0.600122, 0.4999, 10.000122, 0.25, 210.0],
[0.600061, 0.49995, 10.000061, 0.25003, 50.0],
[0.595856, 0.505256, 9.991271, 0.24, 100.0],
[0.597973, 0.507461, 9.992175, 0.24, 210.0],
[0.595362, 0.507926, 9.993945, 0.24, 100.0],
[0.594618, 0.509055, 9.996832, 0.24, 210.0],
[0.592382, 0.506906, 9.996092, 0.24, 100.0],
[0.590438, 0.504882, 9.997604, 0.24, 210.0],
[0.591035, 0.503605, 9.994744, 0.24, 100.0],
[0.59121, 0.500708, 9.993425, 0.24, 210.0],
[0.593181, 0.502585, 9.991764, 0.24, 100.0],
[0.595867, 0.502302, 9.990069, 0.24, 210.0],
];
let values = grey_values(&test_case);
let path = std::path::Path::new("combination_scene.png");
let file = std::fs::File::create(path).unwrap();
let w = &mut std::io::BufWriter::new(file);
let mut encoder = png::Encoder::new(w, 1024, 1024);
encoder.set_color(png::ColorType::Grayscale);
encoder.set_depth(png::BitDepth::Eight);
let mut writer = encoder.write_header().unwrap();
writer.write_image_data(&values).unwrap();
}
```
## Test cases
~~The images may be scaled down. Clicking or tapping on an image will open it full size.~~
### Single sphere
Input:
```text
[
[0, 0, 0, 1, 128],
]
```
Output:
~~```text~~
~~TODO~~
~~```~~
### Two separated spheres
Input:
```text
[
[0, -0.55, 0, 0.4, 100],
[0, 0.55, 0, 0.4, 200],
]
```
Output:
~~```text~~
~~TODO~~
~~```~~
### Ring of spheres
Input:
```text
[
[0.75, 0, 0, 0.25, 64],
[0.53033, 0.375, -0.375, 0.25, 185],
[0, 0.53033, -0.53033, 0.25, 64],
[-0.53033, 0.375, -0.375, 0.25, 185],
[-0.75, 0, -0, 0.25, 64],
[-0.53033, -0.375, 0.375, 0.25, 185],
[-0, -0.53033, 0.53033, 0.25, 64],
[0.53033, -0.375, 0.375, 0.25, 185],
]
```
Output:
~~```text~~
~~TODO~~
~~```~~
### Beach ball
Input:
```text
[
[0.001, 0, 0, 1, 64],
[0.000707, 0.000707, -0, 1, 85],
[0, 0.001, -0, 1, 106],
[-0.000707, 0.000707, -0, 1, 127],
[-0.001, 0, -0, 1, 148],
[-0.000707, -0.000707, 0, 1, 169],
[-0, -0.001, 0, 1, 190],
[0.000707, -0.000707, 0, 1, 211],
]
```
Output:
~~```text~~
~~TODO~~
~~```~~
### Football
Input:
```text
[
[0, 0.000058, 0.001947, 1, 0],
[0, -0.001715, 0.000923, 1, 0],
[0, 0.001715, -0.000923, 1, 0],
[0, -0.000058, -0.001947, 1, 0],
[0.001657, -0.000512, 0.000887, 1, 0],
[-0.001657, -0.000512, 0.000887, 1, 0],
[0.001657, 0.000512, -0.000887, 1, 0],
[-0.001657, 0.000512, -0.000887, 1, 0],
[0.001024, 0.001435, 0.000829, 1, 0],
[-0.001024, 0.001435, 0.000829, 1, 0],
[0.001024, -0.001435, -0.000829, 1, 0],
[-0.001024, -0.001435, -0.000829, 1, 0],
[0.001868, 0.000618, 0.000357, 1, 255],
[-0.001868, 0.000618, 0.000357, 1, 255],
[0.001868, -0.000618, -0.000357, 1, 255],
[-0.001868, -0.000618, -0.000357, 1, 255],
[0.000714, -0.000934, 0.001618, 1, 255],
[-0.000714, -0.000934, 0.001618, 1, 255],
[0.000714, 0.000934, -0.001618, 1, 255],
[-0.000714, 0.000934, -0.001618, 1, 255],
[0, -0.001975, -0.000316, 1, 255],
[0, 0.001261, 0.001552, 1, 255],
[0, -0.001261, -0.001552, 1, 255],
[0, 0.001975, 0.000316, 1, 255],
[0.001155, 0.000423, 0.001577, 1, 255],
[-0.001155, 0.000423, 0.001577, 1, 255],
[0.001155, -0.001577, 0.000423, 1, 255],
[-0.001155, -0.001577, 0.000423, 1, 255],
[0.001155, 0.001577, -0.000423, 1, 255],
[-0.001155, 0.001577, -0.000423, 1, 255],
[0.001155, -0.000423, -0.001577, 1, 255],
[-0.001155, -0.000423, -0.001577, 1, 255],
]
```
Output:
~~```text~~
~~TODO~~
~~```~~
### Eight ball
Input:
```text
[
[0, 0, 1000, 2, 128],
[0, 0, 0.003125, 1, 0],
[0, 0, 0, 0.9975, 255],
[0, 0.000547, -0.002573, 0.995, 0],
[-0, -0.000547, -0.002573, 0.995, 0],
[0, 0.001071, -0.005037, 0.9925, 255],
[-0, -0.001071, -0.005037, 0.9925, 255],
]
```
Output:
~~```text~~
~~TODO~~
~~```~~
### Star ball
Input:
```text
[
[0, 0, 0, 1, 210],
[0, 0, 0.0008, 1, 210],
[0, 0, 0.0004, 1.00012, 50],
[0.012969, -0, -0.042045, 0.96, 100],
[0.016397, 0.011913, -0.039054, 0.96, 210],
[0.004008, 0.012334, -0.042045, 0.96, 100],
[-0.006263, 0.019276, -0.039054, 0.96, 210],
[-0.010492, 0.007623, -0.042045, 0.96, 100],
[-0.020268, 0, -0.039054, 0.96, 210],
[-0.010492, -0.007623, -0.042045, 0.96, 100],
[-0.006263, -0.019276, -0.039054, 0.96, 210],
[0.004008, -0.012334, -0.042045, 0.96, 100],
[0.016397, -0.011913, -0.039054, 0.96, 210],
]
```
Output:
~~```text~~
~~TODO~~
~~```~~
### Combination scene
Input:
```text
[
[0, 0, 1000, 2, 192],
[-2, 2, 100, 3, 98],
[2, 2, 100, 3, 98],
[0.5, -0.5, 500, 0.2, 255],
[-0.349469, -0.1, 20.000279, 0.6, 64],
[-0.349775, -0.100273, 20.000485, 0.6, 85],
[-0.350214, -0.100386, 20.000407, 0.6, 106],
[-0.350527, -0.100273, 20.000091, 0.6, 127],
[-0.350531, -0.1, 19.999721, 0.6, 148],
[-0.350225, -0.099727, 19.999515, 0.6, 169],
[-0.349786, -0.099614, 19.999593, 0.6, 190],
[-0.349473, -0.099727, 19.999909, 0.6, 211],
[0.20032, 0.349824, 15.000456, 0.3, 0],
[0.20032, 0.349517, 14.999924, 0.3, 0],
[0.19968, 0.350483, 15.000076, 0.3, 0],
[0.19968, 0.350176, 14.999544, 0.3, 0],
[0.200578, 0.350073, 14.999958, 0.3, 0],
[0.199817, 0.349519, 15.000277, 0.3, 0],
[0.200183, 0.350481, 14.999723, 0.3, 0],
[0.199422, 0.349927, 15.000042, 0.3, 0],
[0.200235, 0.35042, 15.000332, 0.3, 0],
[0.199765, 0.350078, 15.000529, 0.3, 0],
[0.200235, 0.349922, 14.999471, 0.3, 0],
[0.199765, 0.34958, 14.999668, 0.3, 0],
[0.200429, 0.350419, 15.000005, 0.3, 255],
[0.199571, 0.349795, 15.000366, 0.3, 255],
[0.200429, 0.350205, 14.999634, 0.3, 255],
[0.199571, 0.349581, 14.999995, 0.3, 255],
[0.200524, 0.349747, 15.000146, 0.3, 255],
[0.200196, 0.349509, 15.000283, 0.3, 255],
[0.199804, 0.350491, 14.999717, 0.3, 255],
[0.199476, 0.350253, 14.999854, 0.3, 255],
[0.200138, 0.349578, 14.999597, 0.3, 255],
[0.200138, 0.350138, 15.000567, 0.3, 255],
[0.199862, 0.349862, 14.999433, 0.3, 255],
[0.199862, 0.350422, 15.000403, 0.3, 255],
[0.200488, 0.350136, 15.000321, 0.3, 255],
[0.199957, 0.349751, 15.000544, 0.3, 255],
[0.200488, 0.34979, 14.999721, 0.3, 255],
[0.199957, 0.349404, 14.999944, 0.3, 255],
[0.200043, 0.350596, 15.000056, 0.3, 255],
[0.199512, 0.35021, 15.000279, 0.3, 255],
[0.200043, 0.350249, 14.999456, 0.3, 255],
[0.199512, 0.349864, 14.999679, 0.3, 255],
[-0.5, 0.500442, 10.000442, 0.2, 0],
[-0.5, 0.5, 10, 0.1995, 255],
[-0.499966, 0.49971, 9.999563, 0.199, 0],
[-0.500034, 0.499563, 9.99971, 0.199, 0],
[-0.499934, 0.499432, 9.999144, 0.1985, 255],
[-0.500066, 0.499144, 9.999432, 0.1985, 255],
[0.6, 0.5, 10, 0.25, 210],
[0.600122, 0.4999, 10.000122, 0.25, 210],
[0.600061, 0.49995, 10.000061, 0.25003, 50],
[0.595856, 0.505256, 9.991271, 0.24, 100],
[0.597973, 0.507461, 9.992175, 0.24, 210],
[0.595362, 0.507926, 9.993945, 0.24, 100],
[0.594618, 0.509055, 9.996832, 0.24, 210],
[0.592382, 0.506906, 9.996092, 0.24, 100],
[0.590438, 0.504882, 9.997604, 0.24, 210],
[0.591035, 0.503605, 9.994744, 0.24, 100],
[0.59121, 0.500708, 9.993425, 0.24, 210],
[0.593181, 0.502585, 9.991764, 0.24, 100],
[0.595867, 0.502302, 9.990069, 0.24, 210],
]
```
Output:
~~```text~~
~~TODO~~
~~```~~
## Scoring
This is a [code golf challenge](https://codegolf.codidact.com/categories/49/tags/4274). Your score is the number of bytes in your code. Lowest score for each language wins.
> Explanations are optional, but I'm more likely to upvote answers that have one.

Given a list of spheres, output an image.
Produce a greyscale [orthographic](https://en.wikipedia.org/wiki/Parallel_projection#Orthographic_projection) (parallel projection - no perspective) ray caster, that renders only plain grey spheres with no lighting.
## Input
- A list of spheres, each consisting of:
- The coordinates of its centre $(x,y,z)$.
- Its radius $r$.
- Its greyscale value $g$, an integer from $0$ to $255$ (both inclusive).
- Only the greyscale value will be restricted to integer values. The coordinates and radius may have fractional values.
## Output
- An image $1024$ by $1024$ pixels with greyscale values from $0$ to $255$.
- The image covers x and y values from $-1$ to $1$.
- A sphere's greyscale value is applied to every visible part of it. There is no variation over the surface of the sphere, so it appears flat like a disk.
- You may output whatever image format is convenient, including simple text formats such as [PGM](https://en.wikipedia.org/wiki/Netpbm#PGM_example) or a list of pixel greyscale values. If you want to include an example of your output as an image in your answer, the image formats that Codidact supports are PNG, JPEG, and GIF. You could output PGM and then convert to PNG to upload to your answer. If you output just the greyscale values as a list of numbers from $0$ to $255$, I've made a `[ TODO page to convert these to greyscale PNG]`.
- You are free to output an image reflected or rotated by a multiple of 90 degrees. However, the z axis is required to be increasing in the forwards direction (into the image) otherwise some of the test cases will look incorrect.
## Calculating a pixel value
Imagine a ray passing into the image through a pixel, perpendicular to the image plane. If it hits a sphere, it shows the greyscale value of the first sphere it hits. Otherwise, it shows a value of zero (black).
For example, for each pixel:
- The pixel $x,y$ coordinates will each be from $0$ to $1023$. Scale these to the range $-1$ to $1$ (for example, divide by $512$ and subtract $1$).
- If these scaled $x,y$ coordinates are within the radius $r$ of the $x,y$ coordinates of a sphere's centre, the ray intersects that sphere (the $z$ coordinate is not relevant to this test because the image is an orthographic projection - the rays are all parallel to the $z$ axis). The distance $a$ of the ray from the sphere's centre can be calculated as follows:
$$a=\sqrt{(x_{ray}-x_{sphere})^2+(y_{ray}-y_{sphere})^2}$$
```text
CONSIDER A DIAGRAM HERE
```
- For each sphere that the ray intersects, find the $z$ coordinate of the nearest intersection (there will often be $2$ per sphere, $1$ on the front of the sphere and $1$ on the back of the sphere, as the ray passes through - only the nearest is relevant). This can be calculated as follows:
- The distance $d$ to the nearest intersection point can be calculated from the $z$ coordinate of the sphere's centre, the radius $r$, and the distance $a$ from the $x,y$ coordinates of the sphere's centre to the scaled $x,y$ coordinates of the pixel:
$$d=z_{sphere}-\sqrt{r_{sphere}^2-a^2}$$
```text
CONSIDER A DIAGRAM HERE
```
- Of all the spheres that the ray intersects, the sphere with the intersection point with the smallest $z$ coordinate $d$ is the one to display. Set the pixel to the greyscale value of this sphere.
- If the ray did not intersect any spheres, set the pixel to zero (black).
## Example implementations
These are ungolfed code to supplement understanding of the requirements. You are free to golf these or implement your own approach that matches the test cases up to reflection & rotation.
The Python implementation uses 64 bit floating point. The Rust implementation uses 32 bit floating point to show the difference in output. I've tried to optimise the test cases for compatibility with 32 bit floating point, so the differences are restricted to being a pixel out in some places.
### Python
The function `grey_values` would be sufficient as an entry to this challenge. The rest of the code saves the outputs as PNG files. The function itself has no dependence on the `import png` so it would not be required in an entry.
Instructions for installing the `png` package can be found on its [project page on PyPI](https://pypi.org/project/pypng/).
```python
def grey_values(spheres):
values = []
for y_pixel in range(1024):
for x_pixel in range(1024):
x_ray, y_ray = x_pixel / 512 - 1, y_pixel / 512 - 1
min_distance, hit_grey_value = None, 0
for sphere in spheres:
x_sphere, y_sphere, z_sphere, radius, grey = sphere
a = (
(x_ray - x_sphere) ** 2 + (y_ray - y_sphere) ** 2
) ** 0.5
if a < radius:
distance = z_sphere - (radius ** 2 - a ** 2) ** 0.5
if min_distance is None or distance < min_distance:
min_distance = distance
hit_grey_value = grey
values.append(hit_grey_value)
return values
import png
test_case = [
[0, 0, 1000, 2, 192],
[-2, 2, 100, 3, 98],
[2, 2, 100, 3, 98],
[0.5, -0.5, 500, 0.2, 255],
[-0.349469, -0.1, 20.000279, 0.6, 64],
[-0.349775, -0.100273, 20.000485, 0.6, 85],
[-0.350214, -0.100386, 20.000407, 0.6, 106],
[-0.350527, -0.100273, 20.000091, 0.6, 127],
[-0.350531, -0.1, 19.999721, 0.6, 148],
[-0.350225, -0.099727, 19.999515, 0.6, 169],
[-0.349786, -0.099614, 19.999593, 0.6, 190],
[-0.349473, -0.099727, 19.999909, 0.6, 211],
[0.20032, 0.349824, 15.000456, 0.3, 0],
[0.20032, 0.349517, 14.999924, 0.3, 0],
[0.19968, 0.350483, 15.000076, 0.3, 0],
[0.19968, 0.350176, 14.999544, 0.3, 0],
[0.200578, 0.350073, 14.999958, 0.3, 0],
[0.199817, 0.349519, 15.000277, 0.3, 0],
[0.200183, 0.350481, 14.999723, 0.3, 0],
[0.199422, 0.349927, 15.000042, 0.3, 0],
[0.200235, 0.35042, 15.000332, 0.3, 0],
[0.199765, 0.350078, 15.000529, 0.3, 0],
[0.200235, 0.349922, 14.999471, 0.3, 0],
[0.199765, 0.34958, 14.999668, 0.3, 0],
[0.200429, 0.350419, 15.000005, 0.3, 255],
[0.199571, 0.349795, 15.000366, 0.3, 255],
[0.200429, 0.350205, 14.999634, 0.3, 255],
[0.199571, 0.349581, 14.999995, 0.3, 255],
[0.200524, 0.349747, 15.000146, 0.3, 255],
[0.200196, 0.349509, 15.000283, 0.3, 255],
[0.199804, 0.350491, 14.999717, 0.3, 255],
[0.199476, 0.350253, 14.999854, 0.3, 255],
[0.200138, 0.349578, 14.999597, 0.3, 255],
[0.200138, 0.350138, 15.000567, 0.3, 255],
[0.199862, 0.349862, 14.999433, 0.3, 255],
[0.199862, 0.350422, 15.000403, 0.3, 255],
[0.200488, 0.350136, 15.000321, 0.3, 255],
[0.199957, 0.349751, 15.000544, 0.3, 255],
[0.200488, 0.34979, 14.999721, 0.3, 255],
[0.199957, 0.349404, 14.999944, 0.3, 255],
[0.200043, 0.350596, 15.000056, 0.3, 255],
[0.199512, 0.35021, 15.000279, 0.3, 255],
[0.200043, 0.350249, 14.999456, 0.3, 255],
[0.199512, 0.349864, 14.999679, 0.3, 255],
[-0.5, 0.500442, 10.000442, 0.2, 0],
[-0.5, 0.5, 10, 0.1995, 255],
[-0.499966, 0.49971, 9.999563, 0.199, 0],
[-0.500034, 0.499563, 9.99971, 0.199, 0],
[-0.499934, 0.499432, 9.999144, 0.1985, 255],
[-0.500066, 0.499144, 9.999432, 0.1985, 255],
[0.6, 0.5, 10, 0.25, 210],
[0.600122, 0.4999, 10.000122, 0.25, 210],
[0.600061, 0.49995, 10.000061, 0.25003, 50],
[0.595856, 0.505256, 9.991271, 0.24, 100],
[0.597973, 0.507461, 9.992175, 0.24, 210],
[0.595362, 0.507926, 9.993945, 0.24, 100],
[0.594618, 0.509055, 9.996832, 0.24, 210],
[0.592382, 0.506906, 9.996092, 0.24, 100],
[0.590438, 0.504882, 9.997604, 0.24, 210],
[0.591035, 0.503605, 9.994744, 0.24, 100],
[0.59121, 0.500708, 9.993425, 0.24, 210],
[0.593181, 0.502585, 9.991764, 0.24, 100],
[0.595867, 0.502302, 9.990069, 0.24, 210],
]
values = grey_values(test_case)
rows = [values[row * 1024:row * 1024 + 1024] for row in range(1024)]
with open('combination_scene.png', 'wb') as file:
w = png.Writer(1024, 1024, greyscale=True)
w.write(file, rows)
```
### Rust
The function `grey_values` would be sufficient as an entry to this challenge. The rest of the code saves the outputs as PNG files.
The `png` crate can be installed using `cargo add png`.
```rust
fn grey_values(spheres: &Vec<[f32; 5]>) -> Vec<u8> {
let mut values = vec![];
for y_pixel in 0..1024 {
for x_pixel in 0..1024 {
let x_ray = x_pixel as f32 / 512.0 - 1.0;
let y_ray = y_pixel as f32 / 512.0 - 1.0;
let mut min_distance = f32::INFINITY;
let mut hit_grey_value = 0;
for sphere in spheres {
let [
x_sphere, y_sphere, z_sphere, radius, grey
] = *sphere;
let a = (
(x_ray - x_sphere).powf(2.0)
+ (y_ray - y_sphere).powf(2.0)
).sqrt();
if a < radius {
let distance = z_sphere - (
radius.powf(2.0) - a.powf(2.0)
).sqrt();
if distance < min_distance {
min_distance = distance;
hit_grey_value = grey as u8;
}
}
}
values.push(hit_grey_value);
}
}
values
}
fn main() {
let test_case: Vec<[f32; 5]> = vec![
[0.0, 0.0, 1000.0, 2.0, 192.0],
[-2.0, 2.0, 100.0, 3.0, 98.0],
[2.0, 2.0, 100.0, 3.0, 98.0],
[0.5, -0.5, 500.0, 0.2, 255.0],
[-0.349469, -0.1, 20.000279, 0.6, 64.0],
[-0.349775, -0.100273, 20.000485, 0.6, 85.0],
[-0.350214, -0.100386, 20.000407, 0.6, 106.0],
[-0.350527, -0.100273, 20.000091, 0.6, 127.0],
[-0.350531, -0.1, 19.999721, 0.6, 148.0],
[-0.350225, -0.099727, 19.999515, 0.6, 169.0],
[-0.349786, -0.099614, 19.999593, 0.6, 190.0],
[-0.349473, -0.099727, 19.999909, 0.6, 211.0],
[0.20032, 0.349824, 15.000456, 0.3, 0.0],
[0.20032, 0.349517, 14.999924, 0.3, 0.0],
[0.19968, 0.350483, 15.000076, 0.3, 0.0],
[0.19968, 0.350176, 14.999544, 0.3, 0.0],
[0.200578, 0.350073, 14.999958, 0.3, 0.0],
[0.199817, 0.349519, 15.000277, 0.3, 0.0],
[0.200183, 0.350481, 14.999723, 0.3, 0.0],
[0.199422, 0.349927, 15.000042, 0.3, 0.0],
[0.200235, 0.35042, 15.000332, 0.3, 0.0],
[0.199765, 0.350078, 15.000529, 0.3, 0.0],
[0.200235, 0.349922, 14.999471, 0.3, 0.0],
[0.199765, 0.34958, 14.999668, 0.3, 0.0],
[0.200429, 0.350419, 15.000005, 0.3, 255.0],
[0.199571, 0.349795, 15.000366, 0.3, 255.0],
[0.200429, 0.350205, 14.999634, 0.3, 255.0],
[0.199571, 0.349581, 14.999995, 0.3, 255.0],
[0.200524, 0.349747, 15.000146, 0.3, 255.0],
[0.200196, 0.349509, 15.000283, 0.3, 255.0],
[0.199804, 0.350491, 14.999717, 0.3, 255.0],
[0.199476, 0.350253, 14.999854, 0.3, 255.0],
[0.200138, 0.349578, 14.999597, 0.3, 255.0],
[0.200138, 0.350138, 15.000567, 0.3, 255.0],
[0.199862, 0.349862, 14.999433, 0.3, 255.0],
[0.199862, 0.350422, 15.000403, 0.3, 255.0],
[0.200488, 0.350136, 15.000321, 0.3, 255.0],
[0.199957, 0.349751, 15.000544, 0.3, 255.0],
[0.200488, 0.34979, 14.999721, 0.3, 255.0],
[0.199957, 0.349404, 14.999944, 0.3, 255.0],
[0.200043, 0.350596, 15.000056, 0.3, 255.0],
[0.199512, 0.35021, 15.000279, 0.3, 255.0],
[0.200043, 0.350249, 14.999456, 0.3, 255.0],
[0.199512, 0.349864, 14.999679, 0.3, 255.0],
[-0.5, 0.500442, 10.000442, 0.2, 0.0],
[-0.5, 0.5, 10.0, 0.1995, 255.0],
[-0.499966, 0.49971, 9.999563, 0.199, 0.0],
[-0.500034, 0.499563, 9.99971, 0.199, 0.0],
[-0.499934, 0.499432, 9.999144, 0.1985, 255.0],
[-0.500066, 0.499144, 9.999432, 0.1985, 255.0],
[0.6, 0.5, 10.0, 0.25, 210.0],
[0.600122, 0.4999, 10.000122, 0.25, 210.0],
[0.600061, 0.49995, 10.000061, 0.25003, 50.0],
[0.595856, 0.505256, 9.991271, 0.24, 100.0],
[0.597973, 0.507461, 9.992175, 0.24, 210.0],
[0.595362, 0.507926, 9.993945, 0.24, 100.0],
[0.594618, 0.509055, 9.996832, 0.24, 210.0],
[0.592382, 0.506906, 9.996092, 0.24, 100.0],
[0.590438, 0.504882, 9.997604, 0.24, 210.0],
[0.591035, 0.503605, 9.994744, 0.24, 100.0],
[0.59121, 0.500708, 9.993425, 0.24, 210.0],
[0.593181, 0.502585, 9.991764, 0.24, 100.0],
[0.595867, 0.502302, 9.990069, 0.24, 210.0],
];
let values = grey_values(&test_case);
let path = std::path::Path::new("combination_scene.png");
let file = std::fs::File::create(path).unwrap();
let w = &mut std::io::BufWriter::new(file);
let mut encoder = png::Encoder::new(w, 1024, 1024);
encoder.set_color(png::ColorType::Grayscale);
encoder.set_depth(png::BitDepth::Eight);
let mut writer = encoder.write_header().unwrap();
writer.write_image_data(&values).unwrap();
}
```
## Test cases
### Single sphere
Input:
```text
[
[0, 0, 0, 1, 128],
]
```
Output:
![Grey circle on a black background](https://codegolf.codidact.com/uploads/jlru163pa2pgm7aaq70b2tff16uf)
### Two separated spheres
Input:
```text
[
[0, -0.55, 0, 0.4, 100],
[0, 0.55, 0, 0.4, 200],
]
```
Output:
![Light and dark grey circles on a black background](https://codegolf.codidact.com/uploads/8yw7agli78y65kiomb1u5yebs7c5)
### Ring of spheres
Input:
```text
[
[0.75, 0, 0, 0.25, 64],
[0.53033, 0.375, -0.375, 0.25, 185],
[0, 0.53033, -0.53033, 0.25, 64],
[-0.53033, 0.375, -0.375, 0.25, 185],
[-0.75, 0, -0, 0.25, 64],
[-0.53033, -0.375, 0.375, 0.25, 185],
[-0, -0.53033, 0.53033, 0.25, 64],
[0.53033, -0.375, 0.375, 0.25, 185],
]
```
Output:
![Eight circles in alternating light and dark grey in a ring some overlapping](https://codegolf.codidact.com/uploads/0lsenynig27au0sryf3ruozt1uwh)
### Beach ball
Input:
```text
[
[0.001, 0, 0, 1, 64],
[0.000707, 0.000707, -0, 1, 85],
[0, 0.001, -0, 1, 106],
[-0.000707, 0.000707, -0, 1, 127],
[-0.001, 0, -0, 1, 148],
[-0.000707, -0.000707, 0, 1, 169],
[-0, -0.001, 0, 1, 190],
[0.000707, -0.000707, 0, 1, 211],
]
```
Output:
![A circle with eight sectors in different shades of grey](https://codegolf.codidact.com/uploads/51a3ni0cfwsqcdw5b49gy6282qno)
### Football
Input:
```text
[
[0, 0.000058, 0.001947, 1, 0],
[0, -0.001715, 0.000923, 1, 0],
[0, 0.001715, -0.000923, 1, 0],
[0, -0.000058, -0.001947, 1, 0],
[0.001657, -0.000512, 0.000887, 1, 0],
[-0.001657, -0.000512, 0.000887, 1, 0],
[0.001657, 0.000512, -0.000887, 1, 0],
[-0.001657, 0.000512, -0.000887, 1, 0],
[0.001024, 0.001435, 0.000829, 1, 0],
[-0.001024, 0.001435, 0.000829, 1, 0],
[0.001024, -0.001435, -0.000829, 1, 0],
[-0.001024, -0.001435, -0.000829, 1, 0],
[0.001868, 0.000618, 0.000357, 1, 255],
[-0.001868, 0.000618, 0.000357, 1, 255],
[0.001868, -0.000618, -0.000357, 1, 255],
[-0.001868, -0.000618, -0.000357, 1, 255],
[0.000714, -0.000934, 0.001618, 1, 255],
[-0.000714, -0.000934, 0.001618, 1, 255],
[0.000714, 0.000934, -0.001618, 1, 255],
[-0.000714, 0.000934, -0.001618, 1, 255],
[0, -0.001975, -0.000316, 1, 255],
[0, 0.001261, 0.001552, 1, 255],
[0, -0.001261, -0.001552, 1, 255],
[0, 0.001975, 0.000316, 1, 255],
[0.001155, 0.000423, 0.001577, 1, 255],
[-0.001155, 0.000423, 0.001577, 1, 255],
[0.001155, -0.001577, 0.000423, 1, 255],
[-0.001155, -0.001577, 0.000423, 1, 255],
[0.001155, 0.001577, -0.000423, 1, 255],
[-0.001155, 0.001577, -0.000423, 1, 255],
[0.001155, -0.000423, -0.001577, 1, 255],
[-0.001155, -0.000423, -0.001577, 1, 255],
]
```
Output:
![Black pentagons on a white ball](https://codegolf.codidact.com/uploads/bmrbsw65p59imvwu8ssd42uvvx1r)
### Eight ball
Input:
```text
[
[0, 0, 1000, 2, 128],
[0, 0, 0.003125, 1, 0],
[0, 0, 0, 0.9975, 255],
[0, 0.000547, -0.002573, 0.995, 0],
[-0, -0.000547, -0.002573, 0.995, 0],
[0, 0.001071, -0.005037, 0.9925, 255],
[-0, -0.001071, -0.005037, 0.9925, 255],
]
```
Output:
![A ball with the number 8 in a white circle](https://codegolf.codidact.com/uploads/dvl68crpmy62rqt94mwanditt4wc)
### Star ball
Input:
```text
[
[0, 0, 0, 1, 210],
[0, 0, 0.0008, 1, 210],
[0, 0, 0.0004, 1.00012, 50],
[0.012969, -0, -0.042045, 0.96, 100],
[0.016397, 0.011913, -0.039054, 0.96, 210],
[0.004008, 0.012334, -0.042045, 0.96, 100],
[-0.006263, 0.019276, -0.039054, 0.96, 210],
[-0.010492, 0.007623, -0.042045, 0.96, 100],
[-0.020268, 0, -0.039054, 0.96, 210],
[-0.010492, -0.007623, -0.042045, 0.96, 100],
[-0.006263, -0.019276, -0.039054, 0.96, 210],
[0.004008, -0.012334, -0.042045, 0.96, 100],
[0.016397, -0.011913, -0.039054, 0.96, 210],
]
```
Output:
![A light grey ball with a dark grey five pointed star](https://codegolf.codidact.com/uploads/f3wdgwru6j35ue72x25mazxikfx2)
### Combination scene
Input:
```text
[
[0, 0, 1000, 2, 192],
[-2, 2, 100, 3, 98],
[2, 2, 100, 3, 98],
[0.5, -0.5, 500, 0.2, 255],
[-0.349469, -0.1, 20.000279, 0.6, 64],
[-0.349775, -0.100273, 20.000485, 0.6, 85],
[-0.350214, -0.100386, 20.000407, 0.6, 106],
[-0.350527, -0.100273, 20.000091, 0.6, 127],
[-0.350531, -0.1, 19.999721, 0.6, 148],
[-0.350225, -0.099727, 19.999515, 0.6, 169],
[-0.349786, -0.099614, 19.999593, 0.6, 190],
[-0.349473, -0.099727, 19.999909, 0.6, 211],
[0.20032, 0.349824, 15.000456, 0.3, 0],
[0.20032, 0.349517, 14.999924, 0.3, 0],
[0.19968, 0.350483, 15.000076, 0.3, 0],
[0.19968, 0.350176, 14.999544, 0.3, 0],
[0.200578, 0.350073, 14.999958, 0.3, 0],
[0.199817, 0.349519, 15.000277, 0.3, 0],
[0.200183, 0.350481, 14.999723, 0.3, 0],
[0.199422, 0.349927, 15.000042, 0.3, 0],
[0.200235, 0.35042, 15.000332, 0.3, 0],
[0.199765, 0.350078, 15.000529, 0.3, 0],
[0.200235, 0.349922, 14.999471, 0.3, 0],
[0.199765, 0.34958, 14.999668, 0.3, 0],
[0.200429, 0.350419, 15.000005, 0.3, 255],
[0.199571, 0.349795, 15.000366, 0.3, 255],
[0.200429, 0.350205, 14.999634, 0.3, 255],
[0.199571, 0.349581, 14.999995, 0.3, 255],
[0.200524, 0.349747, 15.000146, 0.3, 255],
[0.200196, 0.349509, 15.000283, 0.3, 255],
[0.199804, 0.350491, 14.999717, 0.3, 255],
[0.199476, 0.350253, 14.999854, 0.3, 255],
[0.200138, 0.349578, 14.999597, 0.3, 255],
[0.200138, 0.350138, 15.000567, 0.3, 255],
[0.199862, 0.349862, 14.999433, 0.3, 255],
[0.199862, 0.350422, 15.000403, 0.3, 255],
[0.200488, 0.350136, 15.000321, 0.3, 255],
[0.199957, 0.349751, 15.000544, 0.3, 255],
[0.200488, 0.34979, 14.999721, 0.3, 255],
[0.199957, 0.349404, 14.999944, 0.3, 255],
[0.200043, 0.350596, 15.000056, 0.3, 255],
[0.199512, 0.35021, 15.000279, 0.3, 255],
[0.200043, 0.350249, 14.999456, 0.3, 255],
[0.199512, 0.349864, 14.999679, 0.3, 255],
[-0.5, 0.500442, 10.000442, 0.2, 0],
[-0.5, 0.5, 10, 0.1995, 255],
[-0.499966, 0.49971, 9.999563, 0.199, 0],
[-0.500034, 0.499563, 9.99971, 0.199, 0],
[-0.499934, 0.499432, 9.999144, 0.1985, 255],
[-0.500066, 0.499144, 9.999432, 0.1985, 255],
[0.6, 0.5, 10, 0.25, 210],
[0.600122, 0.4999, 10.000122, 0.25, 210],
[0.600061, 0.49995, 10.000061, 0.25003, 50],
[0.595856, 0.505256, 9.991271, 0.24, 100],
[0.597973, 0.507461, 9.992175, 0.24, 210],
[0.595362, 0.507926, 9.993945, 0.24, 100],
[0.594618, 0.509055, 9.996832, 0.24, 210],
[0.592382, 0.506906, 9.996092, 0.24, 100],
[0.590438, 0.504882, 9.997604, 0.24, 210],
[0.591035, 0.503605, 9.994744, 0.24, 100],
[0.59121, 0.500708, 9.993425, 0.24, 210],
[0.593181, 0.502585, 9.991764, 0.24, 100],
[0.595867, 0.502302, 9.990069, 0.24, 210],
]
```
Output:
![A beachball, football, eightball, and starball with the sun behind hills in the background](https://codegolf.codidact.com/uploads/usv9npm1t5bjpcaknn6rbzgkqwmi)
## Scoring
This is a [code golf challenge](https://codegolf.codidact.com/categories/49/tags/4274). Your score is the number of bytes in your code. Lowest score for each language wins.
> Explanations are optional, but I'm more likely to upvote answers that have one.

#8: Post edited by

trichoplax‭ · 2024-07-26T11:48:56Z (5 months ago)
Reduce example implementations to one test case

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Given a list of spheres, output an image.
Produce a greyscale [orthographic](https://en.wikipedia.org/wiki/Parallel_projection#Orthographic_projection) (parallel projection - no perspective) ray caster, that renders only plain grey spheres with no lighting.
## Input
- A list of spheres, each consisting of:
- The coordinates of its centre $(x,y,z)$.
- Its radius $r$.
- Its greyscale value $g$, an integer from $0$ to $255$ (both inclusive).
- Only the greyscale value will be restricted to integer values. The coordinates and radius may have fractional values.
## Output
- An image $1024$ by $1024$ pixels with greyscale values from $0$ to $255$.
- The image covers x and y values from $-1$ to $1$.
- A sphere's greyscale value is applied to every visible part of it. There is no variation over the surface of the sphere, so it appears flat like a disk.
- You may output whatever image format is convenient, including simple text formats such as [PGM](https://en.wikipedia.org/wiki/Netpbm#PGM_example) or a list of pixel greyscale values. If you want to include an example of your output as an image in your answer, the image formats that Codidact supports are PNG, JPEG, and GIF. You could output PGM and then convert to PNG to upload to your answer. If you output just the greyscale values as a list of numbers from $0$ to $255$, I've made a `[ TODO page to convert these to greyscale PNG]`.
- You are free to output an image reflected or rotated by a multiple of 90 degrees. However, the z axis is required to be increasing in the forwards direction (into the image) otherwise some of the test cases will look incorrect.
## Calculating a pixel value
Imagine a ray passing into the image through a pixel, perpendicular to the image plane. If it hits a sphere, it shows the greyscale value of the first sphere it hits. Otherwise, it shows a value of zero (black).
For example, for each pixel:
- The pixel $x,y$ coordinates will each be from $0$ to $1023$. Scale these to the range $-1$ to $1$ (for example, divide by $512$ and subtract $1$).
- If these scaled $x,y$ coordinates are within the radius $r$ of the $x,y$ coordinates of a sphere's centre, the ray intersects that sphere (the $z$ coordinate is not relevant to this test because the image is an orthographic projection - the rays are all parallel to the $z$ axis). The distance $a$ of the ray from the sphere's centre can be calculated as follows:
$$a=\sqrt{(x_{ray}-x_{sphere})^2+(y_{ray}-y_{sphere})^2}$$
```text
CONSIDER A DIAGRAM HERE
```
- For each sphere that the ray intersects, find the $z$ coordinate of the nearest intersection (there will often be $2$ per sphere, $1$ on the front of the sphere and $1$ on the back of the sphere, as the ray passes through - only the nearest is relevant). This can be calculated as follows:
- The distance $d$ to the nearest intersection point can be calculated from the $z$ coordinate of the sphere's centre, the radius $r$, and the distance $a$ from the $x,y$ coordinates of the sphere's centre to the scaled $x,y$ coordinates of the pixel:
$$d=z_{sphere}-\sqrt{r_{sphere}^2-a^2}$$
```text
CONSIDER A DIAGRAM HERE
```
- Of all the spheres that the ray intersects, the sphere with the intersection point with the smallest $z$ coordinate $d$ is the one to display. Set the pixel to the greyscale value of this sphere.
- If the ray did not intersect any spheres, set the pixel to zero (black).
## Example implementations
These are ungolfed code to supplement understanding of the requirements. You are free to golf these or implement your own approach that matches the test cases up to reflection & rotation.
The Python implementation uses 64 bit floating point. The Rust implementation uses 32 bit floating point to show the difference in output. I've tried to optimise the test cases for compatibility with 32 bit floating point, so the differences are restricted to being a pixel out in some places.
### Python
The function `grey_values` would be sufficient as an entry to this challenge. The rest of the code saves the outputs as PNG files. The function itself has no dependence on the `import png` so it would not be required in an entry.
```python
def grey_values(spheres):
values = []
for y_pixel in range(1024):
for x_pixel in range(1024):
x_ray, y_ray = x_pixel / 512 - 1, y_pixel / 512 - 1
min_distance, hit_grey_value = None, 0
for sphere in spheres:
x_sphere, y_sphere, z_sphere, radius, grey = sphere
a = (
(x_ray - x_sphere) ** 2 + (y_ray - y_sphere) ** 2
) ** 0.5
if a < radius:
distance = z_sphere - (radius ** 2 - a ** 2) ** 0.5
if min_distance is None or distance < min_distance:
min_distance = distance
hit_grey_value = grey
values.append(hit_grey_value)
return values
import png
~~test_cases = [~~
[
~~[0, 0, 0, 1, 128],~~
],
[
~~[0, -0.55, 0, 0.4, 100],~~
~~[0, 0.55, 0, 0.4, 200],~~
],
[
~~[0.75, 0, 0, 0.25, 64],~~
~~[0.53033, 0.375, -0.375, 0.25, 185],~~
~~[0, 0.53033, -0.53033, 0.25, 64],~~
~~[-0.53033, 0.375, -0.375, 0.25, 185],~~
~~[-0.75, 0, -0, 0.25, 64],~~
~~[-0.53033, -0.375, 0.375, 0.25, 185],~~
~~[-0, -0.53033, 0.53033, 0.25, 64],~~
~~[0.53033, -0.375, 0.375, 0.25, 185],~~
],
[
~~[0.001, 0, 0, 1, 64],~~
~~[0.000707, 0.000707, -0, 1, 85],~~
[0, 0.001, -0, 1, 106],
~~[-0.000707, 0.000707, -0, 1, 127],~~
~~[-0.001, 0, -0, 1, 148],~~
~~[-0.000707, -0.000707, 0, 1, 169],~~
[-0, -0.001, 0, 1, 190],
~~[0.000707, -0.000707, 0, 1, 211],~~
],
[
~~[0, 0.000058, 0.001947, 1, 0],~~
~~[0, -0.001715, 0.000923, 1, 0],~~
~~[0, 0.001715, -0.000923, 1, 0],~~
~~[0, -0.000058, -0.001947, 1, 0],~~
~~[0.001657, -0.000512, 0.000887, 1, 0],~~
~~[-0.001657, -0.000512, 0.000887, 1, 0],~~
~~[0.001657, 0.000512, -0.000887, 1, 0],~~
~~[-0.001657, 0.000512, -0.000887, 1, 0],~~
~~[0.001024, 0.001435, 0.000829, 1, 0],~~
[-0.001024, 0.001435, 0.000829, 1, 0],
~~[0.001024, -0.001435, -0.000829, 1, 0],~~
~~[-0.001024, -0.001435, -0.000829, 1, 0],~~
[0.001868, 0.000618, 0.000357, 1, 255],
~~[-0.001868, 0.000618, 0.000357, 1, 255],~~
[0.001868, -0.000618, -0.000357, 1, 255],
~~[-0.001868, -0.000618, -0.000357, 1, 255],~~
[0.000714, -0.000934, 0.001618, 1, 255],
~~[-0.000714, -0.000934, 0.001618, 1, 255],~~
~~[0.000714, 0.000934, -0.001618, 1, 255],~~
[-0.000714, 0.000934, -0.001618, 1, 255],
[0, -0.001975, -0.000316, 1, 255],
~~[0, 0.001261, 0.001552, 1, 255],~~
~~[0, -0.001261, -0.001552, 1, 255],~~
~~[0, 0.001975, 0.000316, 1, 255],~~
~~[0.001155, 0.000423, 0.001577, 1, 255],~~
~~[-0.001155, 0.000423, 0.001577, 1, 255],~~
~~[0.001155, -0.001577, 0.000423, 1, 255],~~
~~[-0.001155, -0.001577, 0.000423, 1, 255],~~
~~[0.001155, 0.001577, -0.000423, 1, 255],~~
~~[-0.001155, 0.001577, -0.000423, 1, 255],~~
~~[0.001155, -0.000423, -0.001577, 1, 255],~~
~~[-0.001155, -0.000423, -0.001577, 1, 255],~~
],
[
~~[0, 0, 1000, 2, 128],~~
~~[0, 0, 0.003125, 1, 0],~~
~~[0, 0, 0, 0.9975, 255],~~
~~[0, 0.000547, -0.002573, 0.995, 0],~~
~~[-0, -0.000547, -0.002573, 0.995, 0],~~
~~[0, 0.001071, -0.005037, 0.9925, 255],~~
~~[-0, -0.001071, -0.005037, 0.9925, 255],~~
],
[
~~[0, 0, 0, 1, 210],~~
~~[0, 0, 0.0008, 1, 210],~~
~~[0, 0, 0.0004, 1.00012, 50],~~
~~[0.012969, -0, -0.042045, 0.96, 100],~~
~~[0.016397, 0.011913, -0.039054, 0.96, 210],~~
~~[0.004008, 0.012334, -0.042045, 0.96, 100],~~
~~[-0.006263, 0.019276, -0.039054, 0.96, 210],~~
[-0.010492, 0.007623, -0.042045, 0.96, 100],
~~[-0.020268, 0, -0.039054, 0.96, 210],~~
~~[-0.010492, -0.007623, -0.042045, 0.96, 100],~~
~~[-0.006263, -0.019276, -0.039054, 0.96, 210],~~
~~[0.004008, -0.012334, -0.042045, 0.96, 100],~~
~~[0.016397, -0.011913, -0.039054, 0.96, 210],~~
],
[
~~[0, 0, 1000, 2, 192],~~
~~[-2, 2, 100, 3, 98],~~
~~[2, 2, 100, 3, 98],~~
~~[0.5, -0.5, 500, 0.2, 255],~~
~~[-0.349469, -0.1, 20.000279, 0.6, 64],~~
~~[-0.349775, -0.100273, 20.000485, 0.6, 85],~~
~~[-0.350214, -0.100386, 20.000407, 0.6, 106],~~
~~[-0.350527, -0.100273, 20.000091, 0.6, 127],~~
~~[-0.350531, -0.1, 19.999721, 0.6, 148],~~
~~[-0.350225, -0.099727, 19.999515, 0.6, 169],~~
~~[-0.349786, -0.099614, 19.999593, 0.6, 190],~~
~~[-0.349473, -0.099727, 19.999909, 0.6, 211],~~
~~[0.20032, 0.349824, 15.000456, 0.3, 0],~~
~~[0.20032, 0.349517, 14.999924, 0.3, 0],~~
~~[0.19968, 0.350483, 15.000076, 0.3, 0],~~
~~[0.19968, 0.350176, 14.999544, 0.3, 0],~~
~~[0.200578, 0.350073, 14.999958, 0.3, 0],~~
~~[0.199817, 0.349519, 15.000277, 0.3, 0],~~
~~[0.200183, 0.350481, 14.999723, 0.3, 0],~~
~~[0.199422, 0.349927, 15.000042, 0.3, 0],~~
~~[0.200235, 0.35042, 15.000332, 0.3, 0],~~
~~[0.199765, 0.350078, 15.000529, 0.3, 0],~~
~~[0.200235, 0.349922, 14.999471, 0.3, 0],~~
~~[0.199765, 0.34958, 14.999668, 0.3, 0],~~
~~[0.200429, 0.350419, 15.000005, 0.3, 255],~~
~~[0.199571, 0.349795, 15.000366, 0.3, 255],~~
~~[0.200429, 0.350205, 14.999634, 0.3, 255],~~
~~[0.199571, 0.349581, 14.999995, 0.3, 255],~~
~~[0.200524, 0.349747, 15.000146, 0.3, 255],~~
~~[0.200196, 0.349509, 15.000283, 0.3, 255],~~
~~[0.199804, 0.350491, 14.999717, 0.3, 255],~~
~~[0.199476, 0.350253, 14.999854, 0.3, 255],~~
~~[0.200138, 0.349578, 14.999597, 0.3, 255],~~
~~[0.200138, 0.350138, 15.000567, 0.3, 255],~~
~~[0.199862, 0.349862, 14.999433, 0.3, 255],~~
~~[0.199862, 0.350422, 15.000403, 0.3, 255],~~
~~[0.200488, 0.350136, 15.000321, 0.3, 255],~~
~~[0.199957, 0.349751, 15.000544, 0.3, 255],~~
~~[0.200488, 0.34979, 14.999721, 0.3, 255],~~
~~[0.199957, 0.349404, 14.999944, 0.3, 255],~~
~~[0.200043, 0.350596, 15.000056, 0.3, 255],~~
~~[0.199512, 0.35021, 15.000279, 0.3, 255],~~
~~[0.200043, 0.350249, 14.999456, 0.3, 255],~~
~~[0.199512, 0.349864, 14.999679, 0.3, 255],~~
~~[-0.5, 0.500442, 10.000442, 0.2, 0],~~
~~[-0.5, 0.5, 10, 0.1995, 255],~~
~~[-0.499966, 0.49971, 9.999563, 0.199, 0],~~
~~[-0.500034, 0.499563, 9.99971, 0.199, 0],~~
~~[-0.499934, 0.499432, 9.999144, 0.1985, 255],~~
~~[-0.500066, 0.499144, 9.999432, 0.1985, 255],~~
~~[0.6, 0.5, 10, 0.25, 210],~~
~~[0.600122, 0.4999, 10.000122, 0.25, 210],~~
~~[0.600061, 0.49995, 10.000061, 0.25003, 50],~~
~~[0.595856, 0.505256, 9.991271, 0.24, 100],~~
~~[0.597973, 0.507461, 9.992175, 0.24, 210],~~
~~[0.595362, 0.507926, 9.993945, 0.24, 100],~~
~~[0.594618, 0.509055, 9.996832, 0.24, 210],~~
~~[0.592382, 0.506906, 9.996092, 0.24, 100],~~
~~[0.590438, 0.504882, 9.997604, 0.24, 210],~~
~~[0.591035, 0.503605, 9.994744, 0.24, 100],~~
~~[0.59121, 0.500708, 9.993425, 0.24, 210],~~
~~[0.593181, 0.502585, 9.991764, 0.24, 100],~~
~~[0.595867, 0.502302, 9.990069, 0.24, 210],~~
],
]
~~scene_names = (~~
~~"single_sphere",~~
~~"two_spheres",~~
~~"ring_of_spheres",~~
~~"beachball",~~
~~"football",~~
~~"eightball",~~
~~"starball",~~
~~"combination_scene",~~
)
~~for i in range(8):~~
~~print(f'Generating {scene_names[i]}.')~~
~~values = grey_values(test_cases[i])~~
~~rows = [values[row * 1024:row * 1024 + 1024] for row in range(1024)]~~
~~with open(f'{scene_names[i]}.png', 'wb') as file:~~
~~w = png.Writer(1024, 1024, greyscale=True)~~
~~w.write(file, rows)~~
```
### Rust
The function `grey_values` would be sufficient as an entry to this challenge. The rest of the code saves the outputs as PNG files.
```rust
fn grey_values(spheres: &Vec<[f32; 5]>) -> Vec<u8> {
let mut values = vec![];
for y_pixel in 0..1024 {
for x_pixel in 0..1024 {
let x_ray = x_pixel as f32 / 512.0 - 1.0;
let y_ray = y_pixel as f32 / 512.0 - 1.0;
let mut min_distance = f32::INFINITY;
let mut hit_grey_value = 0;
for sphere in spheres {
let [
x_sphere, y_sphere, z_sphere, radius, grey
] = *sphere;
let a = (
(x_ray - x_sphere).powf(2.0)
+ (y_ray - y_sphere).powf(2.0)
).sqrt();
if a < radius {
let distance = z_sphere - (
radius.powf(2.0) - a.powf(2.0)
).sqrt();
if distance < min_distance {
min_distance = distance;
hit_grey_value = grey as u8;
}
}
}
values.push(hit_grey_value);
}
}
values
}
fn main() {
~~let test_cases: Vec<Vec<[f32; 5]>> = vec![~~
~~vec![~~
~~[0.0, 0.0, 0.0, 1.0, 128.0]~~
],
~~vec![~~
~~[0.0, -0.55, 0.0, 0.4, 100.0],~~
~~[0.0, 0.55, 0.0, 0.4, 200.0]~~
],
~~vec![~~
~~[0.75, 0.0, 0.0, 0.25, 64.0],~~
~~[0.53033, 0.375, -0.375, 0.25, 185.0],~~
~~[0.0, 0.53033, -0.53033, 0.25, 64.0],~~
~~[-0.53033, 0.375, -0.375, 0.25, 185.0],~~
~~[-0.75, 0.0, 0.0, 0.25, 64.0],~~
~~[-0.53033, -0.375, 0.375, 0.25, 185.0],~~
~~[0.0, -0.53033, 0.53033, 0.25, 64.0],~~
~~[0.53033, -0.375, 0.375, 0.25, 185.0],~~
],
~~vec![~~
~~[0.001, 0.0, 0.0, 1.0, 64.0],~~
~~[0.000707, 0.000707, 0.0, 1.0, 85.0],~~
~~[0.0, 0.001, 0.0, 1.0, 106.0],~~
~~[-0.000707, 0.000707, 0.0, 1.0, 127.0],~~
~~[-0.001, 0.0, 0.0, 1.0, 148.0],~~
~~[-0.000707, -0.000707, 0.0, 1.0, 169.0],~~
~~[0.0, -0.001, 0.0, 1.0, 190.0],~~
~~[0.000707, -0.000707, 0.0, 1.0, 211.0],~~
],
~~vec![~~
~~[0.0, 0.000058, 0.001947, 1.0, 0.0],~~
~~[0.0, -0.001715, 0.000923, 1.0, 0.0],~~
~~[0.0, 0.001715, -0.000923, 1.0, 0.0],~~
~~[0.0, -0.000058, -0.001947, 1.0, 0.0],~~
~~[0.001657, -0.000512, 0.000887, 1.0, 0.0],~~
~~[-0.001657, -0.000512, 0.000887, 1.0, 0.0],~~
~~[0.001657, 0.000512, -0.000887, 1.0, 0.0],~~
~~[-0.001657, 0.000512, -0.000887, 1.0, 0.0],~~
~~[0.001024, 0.001435, 0.000829, 1.0, 0.0],~~
~~[-0.001024, 0.001435, 0.000829, 1.0, 0.0],~~
~~[0.001024, -0.001435, -0.000829, 1.0, 0.0],~~
~~[-0.001024, -0.001435, -0.000829, 1.0, 0.0],~~
~~[0.001868, 0.000618, 0.000357, 1.0, 255.0],~~
~~[-0.001868, 0.000618, 0.000357, 1.0, 255.0],~~
~~[0.001868, -0.000618, -0.000357, 1.0, 255.0],~~
~~[-0.001868, -0.000618, -0.000357, 1.0, 255.0],~~
~~[0.000714, -0.000934, 0.001618, 1.0, 255.0],~~
~~[-0.000714, -0.000934, 0.001618, 1.0, 255.0],~~
~~[0.000714, 0.000934, -0.001618, 1.0, 255.0],~~
~~[-0.000714, 0.000934, -0.001618, 1.0, 255.0],~~
~~[0.0, -0.001975, -0.000316, 1.0, 255.0],~~
~~[0.0, 0.001261, 0.001552, 1.0, 255.0],~~
~~[0.0, -0.001261, -0.001552, 1.0, 255.0],~~
~~[0.0, 0.001975, 0.000316, 1.0, 255.0],~~
~~[0.001155, 0.000423, 0.001577, 1.0, 255.0],~~
~~[-0.001155, 0.000423, 0.001577, 1.0, 255.0],~~
~~[0.001155, -0.001577, 0.000423, 1.0, 255.0],~~
~~[-0.001155, -0.001577, 0.000423, 1.0, 255.0],~~
~~[0.001155, 0.001577, -0.000423, 1.0, 255.0],~~
~~[-0.001155, 0.001577, -0.000423, 1.0, 255.0],~~
~~[0.001155, -0.000423, -0.001577, 1.0, 255.0],~~
~~[-0.001155, -0.000423, -0.001577, 1.0, 255.0],~~
],
~~vec![~~
~~[0.0, 0.0, 1000.0, 2.0, 128.0],~~
~~[0.0, 0.0, 0.003125, 1.0, 0.0],~~
[0.0, 0.0, 0.0, 0.9975, 255.0],
~~[0.0, 0.000547, -0.002573, 0.995, 0.0],~~
[0.0, -0.000547, -0.002573, 0.995, 0.0],
[0.0, 0.001071, -0.005037, 0.9925, 255.0],
~~[0.0, -0.001071, -0.005037, 0.9925, 255.0],~~
],
~~vec![~~
~~[0.0, 0.0, 0.0, 1.0, 210.0],~~
~~[0.0, 0.0, 0.0008, 1.0, 210.0],~~
~~[0.0, 0.0, 0.0004, 1.00012, 50.0],~~
~~[0.012969, 0.0, -0.042045, 0.96, 100.0],~~
~~[0.016397, 0.011913, -0.039054, 0.96, 210.0],~~
~~[0.004008, 0.012334, -0.042045, 0.96, 100.0],~~
~~[-0.006263, 0.019276, -0.039054, 0.96, 210.0],~~
[-0.010492, 0.007623, -0.042045, 0.96, 100.0],
~~[-0.020268, 0.0, -0.039054, 0.96, 210.0],~~
~~[-0.010492, -0.007623, -0.042045, 0.96, 100.0],~~
~~[-0.006263, -0.019276, -0.039054, 0.96, 210.0],~~
~~[0.004008, -0.012334, -0.042045, 0.96, 100.0],~~
~~[0.016397, -0.011913, -0.039054, 0.96, 210.0],~~
],
~~vec![~~
~~[0.0, 0.0, 1000.0, 2.0, 192.0],~~
~~[-2.0, 2.0, 100.0, 3.0, 98.0],~~
~~[2.0, 2.0, 100.0, 3.0, 98.0],~~
~~[0.5, -0.5, 500.0, 0.2, 255.0],~~
~~[-0.349469, -0.1, 20.000279, 0.6, 64.0],~~
~~[-0.349775, -0.100273, 20.000485, 0.6, 85.0],~~
~~[-0.350214, -0.100386, 20.000407, 0.6, 106.0],~~
~~[-0.350527, -0.100273, 20.000091, 0.6, 127.0],~~
~~[-0.350531, -0.1, 19.999721, 0.6, 148.0],~~
~~[-0.350225, -0.099727, 19.999515, 0.6, 169.0],~~
~~[-0.349786, -0.099614, 19.999593, 0.6, 190.0],~~
~~[-0.349473, -0.099727, 19.999909, 0.6, 211.0],~~
~~[0.20032, 0.349824, 15.000456, 0.3, 0.0],~~
~~[0.20032, 0.349517, 14.999924, 0.3, 0.0],~~
~~[0.19968, 0.350483, 15.000076, 0.3, 0.0],~~
~~[0.19968, 0.350176, 14.999544, 0.3, 0.0],~~
~~[0.200578, 0.350073, 14.999958, 0.3, 0.0],~~
~~[0.199817, 0.349519, 15.000277, 0.3, 0.0],~~
~~[0.200183, 0.350481, 14.999723, 0.3, 0.0],~~
~~[0.199422, 0.349927, 15.000042, 0.3, 0.0],~~
~~[0.200235, 0.35042, 15.000332, 0.3, 0.0],~~
~~[0.199765, 0.350078, 15.000529, 0.3, 0.0],~~
~~[0.200235, 0.349922, 14.999471, 0.3, 0.0],~~
~~[0.199765, 0.34958, 14.999668, 0.3, 0.0],~~
~~[0.200429, 0.350419, 15.000005, 0.3, 255.0],~~
~~[0.199571, 0.349795, 15.000366, 0.3, 255.0],~~
~~[0.200429, 0.350205, 14.999634, 0.3, 255.0],~~
~~[0.199571, 0.349581, 14.999995, 0.3, 255.0],~~
~~[0.200524, 0.349747, 15.000146, 0.3, 255.0],~~
~~[0.200196, 0.349509, 15.000283, 0.3, 255.0],~~
~~[0.199804, 0.350491, 14.999717, 0.3, 255.0],~~
~~[0.199476, 0.350253, 14.999854, 0.3, 255.0],~~
~~[0.200138, 0.349578, 14.999597, 0.3, 255.0],~~
~~[0.200138, 0.350138, 15.000567, 0.3, 255.0],~~
~~[0.199862, 0.349862, 14.999433, 0.3, 255.0],~~
~~[0.199862, 0.350422, 15.000403, 0.3, 255.0],~~
~~[0.200488, 0.350136, 15.000321, 0.3, 255.0],~~
~~[0.199957, 0.349751, 15.000544, 0.3, 255.0],~~
~~[0.200488, 0.34979, 14.999721, 0.3, 255.0],~~
~~[0.199957, 0.349404, 14.999944, 0.3, 255.0],~~
~~[0.200043, 0.350596, 15.000056, 0.3, 255.0],~~
~~[0.199512, 0.35021, 15.000279, 0.3, 255.0],~~
~~[0.200043, 0.350249, 14.999456, 0.3, 255.0],~~
~~[0.199512, 0.349864, 14.999679, 0.3, 255.0],~~
~~[-0.5, 0.500442, 10.000442, 0.2, 0.0],~~
~~[-0.5, 0.5, 10.0, 0.1995, 255.0],~~
~~[-0.499966, 0.49971, 9.999563, 0.199, 0.0],~~
~~[-0.500034, 0.499563, 9.99971, 0.199, 0.0],~~
~~[-0.499934, 0.499432, 9.999144, 0.1985, 255.0],~~
~~[-0.500066, 0.499144, 9.999432, 0.1985, 255.0],~~
~~[0.6, 0.5, 10.0, 0.25, 210.0],~~
~~[0.600122, 0.4999, 10.000122, 0.25, 210.0],~~
~~[0.600061, 0.49995, 10.000061, 0.25003, 50.0],~~
~~[0.595856, 0.505256, 9.991271, 0.24, 100.0],~~
~~[0.597973, 0.507461, 9.992175, 0.24, 210.0],~~
~~[0.595362, 0.507926, 9.993945, 0.24, 100.0],~~
~~[0.594618, 0.509055, 9.996832, 0.24, 210.0],~~
~~[0.592382, 0.506906, 9.996092, 0.24, 100.0],~~
~~[0.590438, 0.504882, 9.997604, 0.24, 210.0],~~
~~[0.591035, 0.503605, 9.994744, 0.24, 100.0],~~
~~[0.59121, 0.500708, 9.993425, 0.24, 210.0],~~
~~[0.593181, 0.502585, 9.991764, 0.24, 100.0],~~
~~[0.595867, 0.502302, 9.990069, 0.24, 210.0],~~
],
];
let scene_names = [
~~"single_sphere",~~
~~"two_spheres",~~
~~"ring_of_spheres",~~
~~"beachball",~~
~~"football",~~
~~"eightball",~~
~~"starball",~~
~~"combination_scene",~~
];
~~for i in 0..8 {~~
~~eprintln!("Generating {}", scene_names[i]);~~
let values = grey_values(&test_cases[i]);
let filename = format!("{}.png", scene_names[i]);
~~let path = std::path::Path::new(&filename);~~
~~let file = std::fs::File::create(path).unwrap();~~
~~let w = &mut std::io::BufWriter::new(file);~~
~~let mut encoder = png::Encoder::new(w, 1024, 1024);~~
~~encoder.set_color(png::ColorType::Grayscale);~~
~~encoder.set_depth(png::BitDepth::Eight);~~
~~let mut writer = encoder.write_header().unwrap();~~
~~writer.write_image_data(&values).unwrap();~~
}
}
```
## Test cases
The images may be scaled down. Clicking or tapping on an image will open it full size.
### Single sphere
Input:
```text
[
[0, 0, 0, 1, 128],
]
```
Output:
```text
TODO
```
### Two separated spheres
Input:
```text
[
[0, -0.55, 0, 0.4, 100],
[0, 0.55, 0, 0.4, 200],
]
```
Output:
```text
TODO
```
### Ring of spheres
Input:
```text
[
[0.75, 0, 0, 0.25, 64],
[0.53033, 0.375, -0.375, 0.25, 185],
[0, 0.53033, -0.53033, 0.25, 64],
[-0.53033, 0.375, -0.375, 0.25, 185],
[-0.75, 0, -0, 0.25, 64],
[-0.53033, -0.375, 0.375, 0.25, 185],
[-0, -0.53033, 0.53033, 0.25, 64],
[0.53033, -0.375, 0.375, 0.25, 185],
]
```
Output:
```text
TODO
```
### Beach ball
Input:
```text
[
[0.001, 0, 0, 1, 64],
[0.000707, 0.000707, -0, 1, 85],
[0, 0.001, -0, 1, 106],
[-0.000707, 0.000707, -0, 1, 127],
[-0.001, 0, -0, 1, 148],
[-0.000707, -0.000707, 0, 1, 169],
[-0, -0.001, 0, 1, 190],
[0.000707, -0.000707, 0, 1, 211],
]
```
Output:
```text
TODO
```
### Football
Input:
```text
[
[0, 0.000058, 0.001947, 1, 0],
[0, -0.001715, 0.000923, 1, 0],
[0, 0.001715, -0.000923, 1, 0],
[0, -0.000058, -0.001947, 1, 0],
[0.001657, -0.000512, 0.000887, 1, 0],
[-0.001657, -0.000512, 0.000887, 1, 0],
[0.001657, 0.000512, -0.000887, 1, 0],
[-0.001657, 0.000512, -0.000887, 1, 0],
[0.001024, 0.001435, 0.000829, 1, 0],
[-0.001024, 0.001435, 0.000829, 1, 0],
[0.001024, -0.001435, -0.000829, 1, 0],
[-0.001024, -0.001435, -0.000829, 1, 0],
[0.001868, 0.000618, 0.000357, 1, 255],
[-0.001868, 0.000618, 0.000357, 1, 255],
[0.001868, -0.000618, -0.000357, 1, 255],
[-0.001868, -0.000618, -0.000357, 1, 255],
[0.000714, -0.000934, 0.001618, 1, 255],
[-0.000714, -0.000934, 0.001618, 1, 255],
[0.000714, 0.000934, -0.001618, 1, 255],
[-0.000714, 0.000934, -0.001618, 1, 255],
[0, -0.001975, -0.000316, 1, 255],
[0, 0.001261, 0.001552, 1, 255],
[0, -0.001261, -0.001552, 1, 255],
[0, 0.001975, 0.000316, 1, 255],
[0.001155, 0.000423, 0.001577, 1, 255],
[-0.001155, 0.000423, 0.001577, 1, 255],
[0.001155, -0.001577, 0.000423, 1, 255],
[-0.001155, -0.001577, 0.000423, 1, 255],
[0.001155, 0.001577, -0.000423, 1, 255],
[-0.001155, 0.001577, -0.000423, 1, 255],
[0.001155, -0.000423, -0.001577, 1, 255],
[-0.001155, -0.000423, -0.001577, 1, 255],
]
```
Output:
```text
TODO
```
### Eight ball
Input:
```text
[
[0, 0, 1000, 2, 128],
[0, 0, 0.003125, 1, 0],
[0, 0, 0, 0.9975, 255],
[0, 0.000547, -0.002573, 0.995, 0],
[-0, -0.000547, -0.002573, 0.995, 0],
[0, 0.001071, -0.005037, 0.9925, 255],
[-0, -0.001071, -0.005037, 0.9925, 255],
]
```
Output:
```text
TODO
```
### Star ball
Input:
```text
[
[0, 0, 0, 1, 210],
[0, 0, 0.0008, 1, 210],
[0, 0, 0.0004, 1.00012, 50],
[0.012969, -0, -0.042045, 0.96, 100],
[0.016397, 0.011913, -0.039054, 0.96, 210],
[0.004008, 0.012334, -0.042045, 0.96, 100],
[-0.006263, 0.019276, -0.039054, 0.96, 210],
[-0.010492, 0.007623, -0.042045, 0.96, 100],
[-0.020268, 0, -0.039054, 0.96, 210],
[-0.010492, -0.007623, -0.042045, 0.96, 100],
[-0.006263, -0.019276, -0.039054, 0.96, 210],
[0.004008, -0.012334, -0.042045, 0.96, 100],
[0.016397, -0.011913, -0.039054, 0.96, 210],
]
```
Output:
```text
TODO
```
### Combination scene
Input:
```text
[
[0, 0, 1000, 2, 192],
[-2, 2, 100, 3, 98],
[2, 2, 100, 3, 98],
[0.5, -0.5, 500, 0.2, 255],
[-0.349469, -0.1, 20.000279, 0.6, 64],
[-0.349775, -0.100273, 20.000485, 0.6, 85],
[-0.350214, -0.100386, 20.000407, 0.6, 106],
[-0.350527, -0.100273, 20.000091, 0.6, 127],
[-0.350531, -0.1, 19.999721, 0.6, 148],
[-0.350225, -0.099727, 19.999515, 0.6, 169],
[-0.349786, -0.099614, 19.999593, 0.6, 190],
[-0.349473, -0.099727, 19.999909, 0.6, 211],
[0.20032, 0.349824, 15.000456, 0.3, 0],
[0.20032, 0.349517, 14.999924, 0.3, 0],
[0.19968, 0.350483, 15.000076, 0.3, 0],
[0.19968, 0.350176, 14.999544, 0.3, 0],
[0.200578, 0.350073, 14.999958, 0.3, 0],
[0.199817, 0.349519, 15.000277, 0.3, 0],
[0.200183, 0.350481, 14.999723, 0.3, 0],
[0.199422, 0.349927, 15.000042, 0.3, 0],
[0.200235, 0.35042, 15.000332, 0.3, 0],
[0.199765, 0.350078, 15.000529, 0.3, 0],
[0.200235, 0.349922, 14.999471, 0.3, 0],
[0.199765, 0.34958, 14.999668, 0.3, 0],
[0.200429, 0.350419, 15.000005, 0.3, 255],
[0.199571, 0.349795, 15.000366, 0.3, 255],
[0.200429, 0.350205, 14.999634, 0.3, 255],
[0.199571, 0.349581, 14.999995, 0.3, 255],
[0.200524, 0.349747, 15.000146, 0.3, 255],
[0.200196, 0.349509, 15.000283, 0.3, 255],
[0.199804, 0.350491, 14.999717, 0.3, 255],
[0.199476, 0.350253, 14.999854, 0.3, 255],
[0.200138, 0.349578, 14.999597, 0.3, 255],
[0.200138, 0.350138, 15.000567, 0.3, 255],
[0.199862, 0.349862, 14.999433, 0.3, 255],
[0.199862, 0.350422, 15.000403, 0.3, 255],
[0.200488, 0.350136, 15.000321, 0.3, 255],
[0.199957, 0.349751, 15.000544, 0.3, 255],
[0.200488, 0.34979, 14.999721, 0.3, 255],
[0.199957, 0.349404, 14.999944, 0.3, 255],
[0.200043, 0.350596, 15.000056, 0.3, 255],
[0.199512, 0.35021, 15.000279, 0.3, 255],
[0.200043, 0.350249, 14.999456, 0.3, 255],
[0.199512, 0.349864, 14.999679, 0.3, 255],
[-0.5, 0.500442, 10.000442, 0.2, 0],
[-0.5, 0.5, 10, 0.1995, 255],
[-0.499966, 0.49971, 9.999563, 0.199, 0],
[-0.500034, 0.499563, 9.99971, 0.199, 0],
[-0.499934, 0.499432, 9.999144, 0.1985, 255],
[-0.500066, 0.499144, 9.999432, 0.1985, 255],
[0.6, 0.5, 10, 0.25, 210],
[0.600122, 0.4999, 10.000122, 0.25, 210],
[0.600061, 0.49995, 10.000061, 0.25003, 50],
[0.595856, 0.505256, 9.991271, 0.24, 100],
[0.597973, 0.507461, 9.992175, 0.24, 210],
[0.595362, 0.507926, 9.993945, 0.24, 100],
[0.594618, 0.509055, 9.996832, 0.24, 210],
[0.592382, 0.506906, 9.996092, 0.24, 100],
[0.590438, 0.504882, 9.997604, 0.24, 210],
[0.591035, 0.503605, 9.994744, 0.24, 100],
[0.59121, 0.500708, 9.993425, 0.24, 210],
[0.593181, 0.502585, 9.991764, 0.24, 100],
[0.595867, 0.502302, 9.990069, 0.24, 210],
]
```
Output:
```text
TODO
```
## Scoring
This is a [code golf challenge](https://codegolf.codidact.com/categories/49/tags/4274). Your score is the number of bytes in your code. Lowest score for each language wins.
> Explanations are optional, but I'm more likely to upvote answers that have one.

Given a list of spheres, output an image.
Produce a greyscale [orthographic](https://en.wikipedia.org/wiki/Parallel_projection#Orthographic_projection) (parallel projection - no perspective) ray caster, that renders only plain grey spheres with no lighting.
## Input
- A list of spheres, each consisting of:
- The coordinates of its centre $(x,y,z)$.
- Its radius $r$.
- Its greyscale value $g$, an integer from $0$ to $255$ (both inclusive).
- Only the greyscale value will be restricted to integer values. The coordinates and radius may have fractional values.
## Output
- An image $1024$ by $1024$ pixels with greyscale values from $0$ to $255$.
- The image covers x and y values from $-1$ to $1$.
- A sphere's greyscale value is applied to every visible part of it. There is no variation over the surface of the sphere, so it appears flat like a disk.
- You may output whatever image format is convenient, including simple text formats such as [PGM](https://en.wikipedia.org/wiki/Netpbm#PGM_example) or a list of pixel greyscale values. If you want to include an example of your output as an image in your answer, the image formats that Codidact supports are PNG, JPEG, and GIF. You could output PGM and then convert to PNG to upload to your answer. If you output just the greyscale values as a list of numbers from $0$ to $255$, I've made a `[ TODO page to convert these to greyscale PNG]`.
- You are free to output an image reflected or rotated by a multiple of 90 degrees. However, the z axis is required to be increasing in the forwards direction (into the image) otherwise some of the test cases will look incorrect.
## Calculating a pixel value
Imagine a ray passing into the image through a pixel, perpendicular to the image plane. If it hits a sphere, it shows the greyscale value of the first sphere it hits. Otherwise, it shows a value of zero (black).
For example, for each pixel:
- The pixel $x,y$ coordinates will each be from $0$ to $1023$. Scale these to the range $-1$ to $1$ (for example, divide by $512$ and subtract $1$).
- If these scaled $x,y$ coordinates are within the radius $r$ of the $x,y$ coordinates of a sphere's centre, the ray intersects that sphere (the $z$ coordinate is not relevant to this test because the image is an orthographic projection - the rays are all parallel to the $z$ axis). The distance $a$ of the ray from the sphere's centre can be calculated as follows:
$$a=\sqrt{(x_{ray}-x_{sphere})^2+(y_{ray}-y_{sphere})^2}$$
```text
CONSIDER A DIAGRAM HERE
```
- For each sphere that the ray intersects, find the $z$ coordinate of the nearest intersection (there will often be $2$ per sphere, $1$ on the front of the sphere and $1$ on the back of the sphere, as the ray passes through - only the nearest is relevant). This can be calculated as follows:
- The distance $d$ to the nearest intersection point can be calculated from the $z$ coordinate of the sphere's centre, the radius $r$, and the distance $a$ from the $x,y$ coordinates of the sphere's centre to the scaled $x,y$ coordinates of the pixel:
$$d=z_{sphere}-\sqrt{r_{sphere}^2-a^2}$$
```text
CONSIDER A DIAGRAM HERE
```
- Of all the spheres that the ray intersects, the sphere with the intersection point with the smallest $z$ coordinate $d$ is the one to display. Set the pixel to the greyscale value of this sphere.
- If the ray did not intersect any spheres, set the pixel to zero (black).
## Example implementations
These are ungolfed code to supplement understanding of the requirements. You are free to golf these or implement your own approach that matches the test cases up to reflection & rotation.
The Python implementation uses 64 bit floating point. The Rust implementation uses 32 bit floating point to show the difference in output. I've tried to optimise the test cases for compatibility with 32 bit floating point, so the differences are restricted to being a pixel out in some places.
### Python
The function `grey_values` would be sufficient as an entry to this challenge. The rest of the code saves the outputs as PNG files. The function itself has no dependence on the `import png` so it would not be required in an entry.
Instructions for installing the `png` package can be found on its [project page on PyPI](https://pypi.org/project/pypng/).
```python
def grey_values(spheres):
values = []
for y_pixel in range(1024):
for x_pixel in range(1024):
x_ray, y_ray = x_pixel / 512 - 1, y_pixel / 512 - 1
min_distance, hit_grey_value = None, 0
for sphere in spheres:
x_sphere, y_sphere, z_sphere, radius, grey = sphere
a = (
(x_ray - x_sphere) ** 2 + (y_ray - y_sphere) ** 2
) ** 0.5
if a < radius:
distance = z_sphere - (radius ** 2 - a ** 2) ** 0.5
if min_distance is None or distance < min_distance:
min_distance = distance
hit_grey_value = grey
values.append(hit_grey_value)
return values
import png
test_case = [
[0, 0, 1000, 2, 192],
[-2, 2, 100, 3, 98],
[2, 2, 100, 3, 98],
[0.5, -0.5, 500, 0.2, 255],
[-0.349469, -0.1, 20.000279, 0.6, 64],
[-0.349775, -0.100273, 20.000485, 0.6, 85],
[-0.350214, -0.100386, 20.000407, 0.6, 106],
[-0.350527, -0.100273, 20.000091, 0.6, 127],
[-0.350531, -0.1, 19.999721, 0.6, 148],
[-0.350225, -0.099727, 19.999515, 0.6, 169],
[-0.349786, -0.099614, 19.999593, 0.6, 190],
[-0.349473, -0.099727, 19.999909, 0.6, 211],
[0.20032, 0.349824, 15.000456, 0.3, 0],
[0.20032, 0.349517, 14.999924, 0.3, 0],
[0.19968, 0.350483, 15.000076, 0.3, 0],
[0.19968, 0.350176, 14.999544, 0.3, 0],
[0.200578, 0.350073, 14.999958, 0.3, 0],
[0.199817, 0.349519, 15.000277, 0.3, 0],
[0.200183, 0.350481, 14.999723, 0.3, 0],
[0.199422, 0.349927, 15.000042, 0.3, 0],
[0.200235, 0.35042, 15.000332, 0.3, 0],
[0.199765, 0.350078, 15.000529, 0.3, 0],
[0.200235, 0.349922, 14.999471, 0.3, 0],
[0.199765, 0.34958, 14.999668, 0.3, 0],
[0.200429, 0.350419, 15.000005, 0.3, 255],
[0.199571, 0.349795, 15.000366, 0.3, 255],
[0.200429, 0.350205, 14.999634, 0.3, 255],
[0.199571, 0.349581, 14.999995, 0.3, 255],
[0.200524, 0.349747, 15.000146, 0.3, 255],
[0.200196, 0.349509, 15.000283, 0.3, 255],
[0.199804, 0.350491, 14.999717, 0.3, 255],
[0.199476, 0.350253, 14.999854, 0.3, 255],
[0.200138, 0.349578, 14.999597, 0.3, 255],
[0.200138, 0.350138, 15.000567, 0.3, 255],
[0.199862, 0.349862, 14.999433, 0.3, 255],
[0.199862, 0.350422, 15.000403, 0.3, 255],
[0.200488, 0.350136, 15.000321, 0.3, 255],
[0.199957, 0.349751, 15.000544, 0.3, 255],
[0.200488, 0.34979, 14.999721, 0.3, 255],
[0.199957, 0.349404, 14.999944, 0.3, 255],
[0.200043, 0.350596, 15.000056, 0.3, 255],
[0.199512, 0.35021, 15.000279, 0.3, 255],
[0.200043, 0.350249, 14.999456, 0.3, 255],
[0.199512, 0.349864, 14.999679, 0.3, 255],
[-0.5, 0.500442, 10.000442, 0.2, 0],
[-0.5, 0.5, 10, 0.1995, 255],
[-0.499966, 0.49971, 9.999563, 0.199, 0],
[-0.500034, 0.499563, 9.99971, 0.199, 0],
[-0.499934, 0.499432, 9.999144, 0.1985, 255],
[-0.500066, 0.499144, 9.999432, 0.1985, 255],
[0.6, 0.5, 10, 0.25, 210],
[0.600122, 0.4999, 10.000122, 0.25, 210],
[0.600061, 0.49995, 10.000061, 0.25003, 50],
[0.595856, 0.505256, 9.991271, 0.24, 100],
[0.597973, 0.507461, 9.992175, 0.24, 210],
[0.595362, 0.507926, 9.993945, 0.24, 100],
[0.594618, 0.509055, 9.996832, 0.24, 210],
[0.592382, 0.506906, 9.996092, 0.24, 100],
[0.590438, 0.504882, 9.997604, 0.24, 210],
[0.591035, 0.503605, 9.994744, 0.24, 100],
[0.59121, 0.500708, 9.993425, 0.24, 210],
[0.593181, 0.502585, 9.991764, 0.24, 100],
[0.595867, 0.502302, 9.990069, 0.24, 210],
]
values = grey_values(test_case)
rows = [values[row * 1024:row * 1024 + 1024] for row in range(1024)]
with open('combination_scene.png', 'wb') as file:
w = png.Writer(1024, 1024, greyscale=True)
w.write(file, rows)
```
### Rust
The function `grey_values` would be sufficient as an entry to this challenge. The rest of the code saves the outputs as PNG files.
The `png` crate can be installed using `cargo add png`.
```rust
fn grey_values(spheres: &Vec<[f32; 5]>) -> Vec<u8> {
let mut values = vec![];
for y_pixel in 0..1024 {
for x_pixel in 0..1024 {
let x_ray = x_pixel as f32 / 512.0 - 1.0;
let y_ray = y_pixel as f32 / 512.0 - 1.0;
let mut min_distance = f32::INFINITY;
let mut hit_grey_value = 0;
for sphere in spheres {
let [
x_sphere, y_sphere, z_sphere, radius, grey
] = *sphere;
let a = (
(x_ray - x_sphere).powf(2.0)
+ (y_ray - y_sphere).powf(2.0)
).sqrt();
if a < radius {
let distance = z_sphere - (
radius.powf(2.0) - a.powf(2.0)
).sqrt();
if distance < min_distance {
min_distance = distance;
hit_grey_value = grey as u8;
}
}
}
values.push(hit_grey_value);
}
}
values
}
fn main() {
let test_case: Vec<[f32; 5]> = vec![
[0.0, 0.0, 1000.0, 2.0, 192.0],
[-2.0, 2.0, 100.0, 3.0, 98.0],
[2.0, 2.0, 100.0, 3.0, 98.0],
[0.5, -0.5, 500.0, 0.2, 255.0],
[-0.349469, -0.1, 20.000279, 0.6, 64.0],
[-0.349775, -0.100273, 20.000485, 0.6, 85.0],
[-0.350214, -0.100386, 20.000407, 0.6, 106.0],
[-0.350527, -0.100273, 20.000091, 0.6, 127.0],
[-0.350531, -0.1, 19.999721, 0.6, 148.0],
[-0.350225, -0.099727, 19.999515, 0.6, 169.0],
[-0.349786, -0.099614, 19.999593, 0.6, 190.0],
[-0.349473, -0.099727, 19.999909, 0.6, 211.0],
[0.20032, 0.349824, 15.000456, 0.3, 0.0],
[0.20032, 0.349517, 14.999924, 0.3, 0.0],
[0.19968, 0.350483, 15.000076, 0.3, 0.0],
[0.19968, 0.350176, 14.999544, 0.3, 0.0],
[0.200578, 0.350073, 14.999958, 0.3, 0.0],
[0.199817, 0.349519, 15.000277, 0.3, 0.0],
[0.200183, 0.350481, 14.999723, 0.3, 0.0],
[0.199422, 0.349927, 15.000042, 0.3, 0.0],
[0.200235, 0.35042, 15.000332, 0.3, 0.0],
[0.199765, 0.350078, 15.000529, 0.3, 0.0],
[0.200235, 0.349922, 14.999471, 0.3, 0.0],
[0.199765, 0.34958, 14.999668, 0.3, 0.0],
[0.200429, 0.350419, 15.000005, 0.3, 255.0],
[0.199571, 0.349795, 15.000366, 0.3, 255.0],
[0.200429, 0.350205, 14.999634, 0.3, 255.0],
[0.199571, 0.349581, 14.999995, 0.3, 255.0],
[0.200524, 0.349747, 15.000146, 0.3, 255.0],
[0.200196, 0.349509, 15.000283, 0.3, 255.0],
[0.199804, 0.350491, 14.999717, 0.3, 255.0],
[0.199476, 0.350253, 14.999854, 0.3, 255.0],
[0.200138, 0.349578, 14.999597, 0.3, 255.0],
[0.200138, 0.350138, 15.000567, 0.3, 255.0],
[0.199862, 0.349862, 14.999433, 0.3, 255.0],
[0.199862, 0.350422, 15.000403, 0.3, 255.0],
[0.200488, 0.350136, 15.000321, 0.3, 255.0],
[0.199957, 0.349751, 15.000544, 0.3, 255.0],
[0.200488, 0.34979, 14.999721, 0.3, 255.0],
[0.199957, 0.349404, 14.999944, 0.3, 255.0],
[0.200043, 0.350596, 15.000056, 0.3, 255.0],
[0.199512, 0.35021, 15.000279, 0.3, 255.0],
[0.200043, 0.350249, 14.999456, 0.3, 255.0],
[0.199512, 0.349864, 14.999679, 0.3, 255.0],
[-0.5, 0.500442, 10.000442, 0.2, 0.0],
[-0.5, 0.5, 10.0, 0.1995, 255.0],
[-0.499966, 0.49971, 9.999563, 0.199, 0.0],
[-0.500034, 0.499563, 9.99971, 0.199, 0.0],
[-0.499934, 0.499432, 9.999144, 0.1985, 255.0],
[-0.500066, 0.499144, 9.999432, 0.1985, 255.0],
[0.6, 0.5, 10.0, 0.25, 210.0],
[0.600122, 0.4999, 10.000122, 0.25, 210.0],
[0.600061, 0.49995, 10.000061, 0.25003, 50.0],
[0.595856, 0.505256, 9.991271, 0.24, 100.0],
[0.597973, 0.507461, 9.992175, 0.24, 210.0],
[0.595362, 0.507926, 9.993945, 0.24, 100.0],
[0.594618, 0.509055, 9.996832, 0.24, 210.0],
[0.592382, 0.506906, 9.996092, 0.24, 100.0],
[0.590438, 0.504882, 9.997604, 0.24, 210.0],
[0.591035, 0.503605, 9.994744, 0.24, 100.0],
[0.59121, 0.500708, 9.993425, 0.24, 210.0],
[0.593181, 0.502585, 9.991764, 0.24, 100.0],
[0.595867, 0.502302, 9.990069, 0.24, 210.0],
];
let values = grey_values(&test_case);
let path = std::path::Path::new("combination_scene.png");
let file = std::fs::File::create(path).unwrap();
let w = &mut std::io::BufWriter::new(file);
let mut encoder = png::Encoder::new(w, 1024, 1024);
encoder.set_color(png::ColorType::Grayscale);
encoder.set_depth(png::BitDepth::Eight);
let mut writer = encoder.write_header().unwrap();
writer.write_image_data(&values).unwrap();
}
```
## Test cases
The images may be scaled down. Clicking or tapping on an image will open it full size.
### Single sphere
Input:
```text
[
[0, 0, 0, 1, 128],
]
```
Output:
```text
TODO
```
### Two separated spheres
Input:
```text
[
[0, -0.55, 0, 0.4, 100],
[0, 0.55, 0, 0.4, 200],
]
```
Output:
```text
TODO
```
### Ring of spheres
Input:
```text
[
[0.75, 0, 0, 0.25, 64],
[0.53033, 0.375, -0.375, 0.25, 185],
[0, 0.53033, -0.53033, 0.25, 64],
[-0.53033, 0.375, -0.375, 0.25, 185],
[-0.75, 0, -0, 0.25, 64],
[-0.53033, -0.375, 0.375, 0.25, 185],
[-0, -0.53033, 0.53033, 0.25, 64],
[0.53033, -0.375, 0.375, 0.25, 185],
]
```
Output:
```text
TODO
```
### Beach ball
Input:
```text
[
[0.001, 0, 0, 1, 64],
[0.000707, 0.000707, -0, 1, 85],
[0, 0.001, -0, 1, 106],
[-0.000707, 0.000707, -0, 1, 127],
[-0.001, 0, -0, 1, 148],
[-0.000707, -0.000707, 0, 1, 169],
[-0, -0.001, 0, 1, 190],
[0.000707, -0.000707, 0, 1, 211],
]
```
Output:
```text
TODO
```
### Football
Input:
```text
[
[0, 0.000058, 0.001947, 1, 0],
[0, -0.001715, 0.000923, 1, 0],
[0, 0.001715, -0.000923, 1, 0],
[0, -0.000058, -0.001947, 1, 0],
[0.001657, -0.000512, 0.000887, 1, 0],
[-0.001657, -0.000512, 0.000887, 1, 0],
[0.001657, 0.000512, -0.000887, 1, 0],
[-0.001657, 0.000512, -0.000887, 1, 0],
[0.001024, 0.001435, 0.000829, 1, 0],
[-0.001024, 0.001435, 0.000829, 1, 0],
[0.001024, -0.001435, -0.000829, 1, 0],
[-0.001024, -0.001435, -0.000829, 1, 0],
[0.001868, 0.000618, 0.000357, 1, 255],
[-0.001868, 0.000618, 0.000357, 1, 255],
[0.001868, -0.000618, -0.000357, 1, 255],
[-0.001868, -0.000618, -0.000357, 1, 255],
[0.000714, -0.000934, 0.001618, 1, 255],
[-0.000714, -0.000934, 0.001618, 1, 255],
[0.000714, 0.000934, -0.001618, 1, 255],
[-0.000714, 0.000934, -0.001618, 1, 255],
[0, -0.001975, -0.000316, 1, 255],
[0, 0.001261, 0.001552, 1, 255],
[0, -0.001261, -0.001552, 1, 255],
[0, 0.001975, 0.000316, 1, 255],
[0.001155, 0.000423, 0.001577, 1, 255],
[-0.001155, 0.000423, 0.001577, 1, 255],
[0.001155, -0.001577, 0.000423, 1, 255],
[-0.001155, -0.001577, 0.000423, 1, 255],
[0.001155, 0.001577, -0.000423, 1, 255],
[-0.001155, 0.001577, -0.000423, 1, 255],
[0.001155, -0.000423, -0.001577, 1, 255],
[-0.001155, -0.000423, -0.001577, 1, 255],
]
```
Output:
```text
TODO
```
### Eight ball
Input:
```text
[
[0, 0, 1000, 2, 128],
[0, 0, 0.003125, 1, 0],
[0, 0, 0, 0.9975, 255],
[0, 0.000547, -0.002573, 0.995, 0],
[-0, -0.000547, -0.002573, 0.995, 0],
[0, 0.001071, -0.005037, 0.9925, 255],
[-0, -0.001071, -0.005037, 0.9925, 255],
]
```
Output:
```text
TODO
```
### Star ball
Input:
```text
[
[0, 0, 0, 1, 210],
[0, 0, 0.0008, 1, 210],
[0, 0, 0.0004, 1.00012, 50],
[0.012969, -0, -0.042045, 0.96, 100],
[0.016397, 0.011913, -0.039054, 0.96, 210],
[0.004008, 0.012334, -0.042045, 0.96, 100],
[-0.006263, 0.019276, -0.039054, 0.96, 210],
[-0.010492, 0.007623, -0.042045, 0.96, 100],
[-0.020268, 0, -0.039054, 0.96, 210],
[-0.010492, -0.007623, -0.042045, 0.96, 100],
[-0.006263, -0.019276, -0.039054, 0.96, 210],
[0.004008, -0.012334, -0.042045, 0.96, 100],
[0.016397, -0.011913, -0.039054, 0.96, 210],
]
```
Output:
```text
TODO
```
### Combination scene
Input:
```text
[
[0, 0, 1000, 2, 192],
[-2, 2, 100, 3, 98],
[2, 2, 100, 3, 98],
[0.5, -0.5, 500, 0.2, 255],
[-0.349469, -0.1, 20.000279, 0.6, 64],
[-0.349775, -0.100273, 20.000485, 0.6, 85],
[-0.350214, -0.100386, 20.000407, 0.6, 106],
[-0.350527, -0.100273, 20.000091, 0.6, 127],
[-0.350531, -0.1, 19.999721, 0.6, 148],
[-0.350225, -0.099727, 19.999515, 0.6, 169],
[-0.349786, -0.099614, 19.999593, 0.6, 190],
[-0.349473, -0.099727, 19.999909, 0.6, 211],
[0.20032, 0.349824, 15.000456, 0.3, 0],
[0.20032, 0.349517, 14.999924, 0.3, 0],
[0.19968, 0.350483, 15.000076, 0.3, 0],
[0.19968, 0.350176, 14.999544, 0.3, 0],
[0.200578, 0.350073, 14.999958, 0.3, 0],
[0.199817, 0.349519, 15.000277, 0.3, 0],
[0.200183, 0.350481, 14.999723, 0.3, 0],
[0.199422, 0.349927, 15.000042, 0.3, 0],
[0.200235, 0.35042, 15.000332, 0.3, 0],
[0.199765, 0.350078, 15.000529, 0.3, 0],
[0.200235, 0.349922, 14.999471, 0.3, 0],
[0.199765, 0.34958, 14.999668, 0.3, 0],
[0.200429, 0.350419, 15.000005, 0.3, 255],
[0.199571, 0.349795, 15.000366, 0.3, 255],
[0.200429, 0.350205, 14.999634, 0.3, 255],
[0.199571, 0.349581, 14.999995, 0.3, 255],
[0.200524, 0.349747, 15.000146, 0.3, 255],
[0.200196, 0.349509, 15.000283, 0.3, 255],
[0.199804, 0.350491, 14.999717, 0.3, 255],
[0.199476, 0.350253, 14.999854, 0.3, 255],
[0.200138, 0.349578, 14.999597, 0.3, 255],
[0.200138, 0.350138, 15.000567, 0.3, 255],
[0.199862, 0.349862, 14.999433, 0.3, 255],
[0.199862, 0.350422, 15.000403, 0.3, 255],
[0.200488, 0.350136, 15.000321, 0.3, 255],
[0.199957, 0.349751, 15.000544, 0.3, 255],
[0.200488, 0.34979, 14.999721, 0.3, 255],
[0.199957, 0.349404, 14.999944, 0.3, 255],
[0.200043, 0.350596, 15.000056, 0.3, 255],
[0.199512, 0.35021, 15.000279, 0.3, 255],
[0.200043, 0.350249, 14.999456, 0.3, 255],
[0.199512, 0.349864, 14.999679, 0.3, 255],
[-0.5, 0.500442, 10.000442, 0.2, 0],
[-0.5, 0.5, 10, 0.1995, 255],
[-0.499966, 0.49971, 9.999563, 0.199, 0],
[-0.500034, 0.499563, 9.99971, 0.199, 0],
[-0.499934, 0.499432, 9.999144, 0.1985, 255],
[-0.500066, 0.499144, 9.999432, 0.1985, 255],
[0.6, 0.5, 10, 0.25, 210],
[0.600122, 0.4999, 10.000122, 0.25, 210],
[0.600061, 0.49995, 10.000061, 0.25003, 50],
[0.595856, 0.505256, 9.991271, 0.24, 100],
[0.597973, 0.507461, 9.992175, 0.24, 210],
[0.595362, 0.507926, 9.993945, 0.24, 100],
[0.594618, 0.509055, 9.996832, 0.24, 210],
[0.592382, 0.506906, 9.996092, 0.24, 100],
[0.590438, 0.504882, 9.997604, 0.24, 210],
[0.591035, 0.503605, 9.994744, 0.24, 100],
[0.59121, 0.500708, 9.993425, 0.24, 210],
[0.593181, 0.502585, 9.991764, 0.24, 100],
[0.595867, 0.502302, 9.990069, 0.24, 210],
]
```
Output:
```text
TODO
```
## Scoring
This is a [code golf challenge](https://codegolf.codidact.com/categories/49/tags/4274). Your score is the number of bytes in your code. Lowest score for each language wins.
> Explanations are optional, but I'm more likely to upvote answers that have one.

#7: Post edited by

trichoplax‭ · 2024-07-26T11:28:19Z (5 months ago)
Add test cases

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Raw

Markdown

~~Given a list of spheres, output an image of them.~~
The challenge is to produce a greyscale [orthographic](https://en.wikipedia.org/wiki/Parallel_projection#Orthographic_projection) (parallel projection - no perspective) ray caster, that renders only plain grey spheres with no lighting, no shading, no shadows, no surface patterns or textures, no reflections, and no transparency.
## Input
- A list of spheres, each consisting of:
~~- The coordinates of its centre $(x, y, z)$.~~
- Its radius $r$.
- Its greyscale value $g$, an integer from $0$ to $255$ (both inclusive).
~~- The input may include up to $100$ spheres.~~
- Only the greyscale value will be restricted to integer values. The coordinates and radius may have fractional values.
~~- Here "integer" refers to the values the greyscale value make take. You are not required to use an integer variable type.[]()~~
## Output
~~- A single greyscale image:~~
~~- $1024$ by $1024$ pixels.~~
- Each pixel has a single greyscale value from $0$ to $255$.
- Horizontally, the image covers x values from $-1$ to $1$.
- Vertically, the image covers y values from $-1$ to $1$.
- The image is orthographic, so there is no perspective. Changing the $z$ coordinate of a sphere does not change its size in the image, instead only affecting whether it is in front of or behind other spheres.
- A sphere's greyscale value is applied to every visible part of it. There is no variation over the surface of the sphere, so it appears flat like a disk.
- The point of the challenge is producing the pixel values, not manipulating image file formats, so you may output whatever image format is convenient, including simple text formats such as [PGM] or even just a list of pixel greyscale values. If you want to include an example of your output as an image in your answer, the image formats that Codidact supports are PNG, JPEG, and GIF. You could output PGM and then convert to PNG to upload to your answer. If you output just the greyscale values as a list of numbers from $0$ to $255$, I've made a `[ TODO page to convert these to greyscale PNG]`.
- The test cases, which match the reference implementations, happen to have the x axis increasing from left to right, and the y axis increasing from top to bottom. You are free to instead interpret these as increasing from right to left for x or from bottom to top for y, resulting in a reflected image. You are also free to output an image rotated by a multiple of 90 degrees, allowing for swapping the x and y axes. However, the z axis is required to be increasing in the forwards direction (into the image) otherwise some of the test cases will look incorrect.
## Calculating a pixel value
Imagine a ray passing into the image through a pixel, perpendicular to the image plane. If it hits a sphere, it shows the greyscale value of the first sphere it hits. Otherwise, it shows a value of zero (black). One possible way of calculating this is as follows:
~~For each pixel:~~
~~- The pixel $x, y$ coordinates will each be from $0$ to $1023$. Scale these to the range $-1$ to $1$ (for example, divide by $512$ and subtract $1$).~~
- If these scaled $x, y$ coordinates are within the radius $r$ of the $x, y$ coordinates of a sphere's centre, the ray intersects that sphere (the $z$ coordinate is not relevant to this test because the image is an orthographic projection - the rays are all parallel to the $z$ axis). The distance $a$ of the ray from the sphere's centre can be calculated as follows:
$$a=\sqrt{(x_{ray}-x_{sphere})^2+(y_{ray}-y_{sphere})^2}$$
```text
CONSIDER A DIAGRAM HERE
```
- For each sphere that the ray intersects, find the $z$ coordinate of the nearest intersection (there will often be $2$ per sphere, $1$ on the front of the sphere and $1$ on the back of the sphere, as the ray passes through - only the nearest is relevant). This can be calculated as follows:
- The distance $d$ to the nearest intersection point can be calculated from the $z$ coordinate of the sphere's centre, the radius $r$, and the distance $a$ from the $x, y$ coordinates of the sphere's centre to the scaled $x, y$ coordinates of the pixel:
$$d=z_{sphere}-\sqrt{r_{sphere}^2-a^2}$$
```text
CONSIDER A DIAGRAM HERE
```
- Of all the spheres that the ray intersects, the sphere with the intersection point with the smallest $z$ coordinate $d$ is the one to display. Set the pixel to the greyscale value of this sphere.
- If the ray did not intersect any spheres, set the pixel to zero (black).
## Example implementations
These are ungolfed implementations designed to supplement understanding of the requirements. You are free to golf these or to implement your own approach that matches the test cases up to reflection and rotation.
The Python implementation uses 64 bit floating point giving accurate images. The Rust implementation deliberately uses 32 bit floating point to show the difference in output. I've tried to optimise the test cases for compatibility with 32 bit floating point, so the differences are restricted to being a pixel out in some places.
### Python
The function `grey_values` would be sufficient as an entry to this challenge. The rest of the code saves the outputs as PNG files. The function itself has no dependence on the `import png` so it would not be required in an entry.
```python
def grey_values(spheres):
values = []
for y_pixel in range(1024):
for x_pixel in range(1024):
x_ray, y_ray = x_pixel / 512 - 1, y_pixel / 512 - 1
min_distance, hit_grey_value = None, 0
for sphere in spheres:
x_sphere, y_sphere, z_sphere, radius, grey = sphere
a = (
(x_ray - x_sphere) ** 2 + (y_ray - y_sphere) ** 2
) ** 0.5
if a < radius:
distance = z_sphere - (radius ** 2 - a ** 2) ** 0.5
if min_distance is None or distance < min_distance:
min_distance = distance
hit_grey_value = grey
values.append(hit_grey_value)
return values
import png
test_cases = [
[
[0, 0, 0, 1, 128],
],
[
[0, -0.55, 0, 0.4, 100],
[0, 0.55, 0, 0.4, 200],
],
[
[0.75, 0, 0, 0.25, 64],
[0.53033, 0.375, -0.375, 0.25, 185],
[0, 0.53033, -0.53033, 0.25, 64],
[-0.53033, 0.375, -0.375, 0.25, 185],
[-0.75, 0, -0, 0.25, 64],
[-0.53033, -0.375, 0.375, 0.25, 185],
[-0, -0.53033, 0.53033, 0.25, 64],
[0.53033, -0.375, 0.375, 0.25, 185],
],
[
[0.001, 0, 0, 1, 64],
[0.000707, 0.000707, -0, 1, 85],
[0, 0.001, -0, 1, 106],
[-0.000707, 0.000707, -0, 1, 127],
[-0.001, 0, -0, 1, 148],
[-0.000707, -0.000707, 0, 1, 169],
[-0, -0.001, 0, 1, 190],
[0.000707, -0.000707, 0, 1, 211],
],
[
[0, 0.000058, 0.001947, 1, 0],
[0, -0.001715, 0.000923, 1, 0],
[0, 0.001715, -0.000923, 1, 0],
[0, -0.000058, -0.001947, 1, 0],
[0.001657, -0.000512, 0.000887, 1, 0],
[-0.001657, -0.000512, 0.000887, 1, 0],
[0.001657, 0.000512, -0.000887, 1, 0],
[-0.001657, 0.000512, -0.000887, 1, 0],
[0.001024, 0.001435, 0.000829, 1, 0],
[-0.001024, 0.001435, 0.000829, 1, 0],
[0.001024, -0.001435, -0.000829, 1, 0],
[-0.001024, -0.001435, -0.000829, 1, 0],
[0.001868, 0.000618, 0.000357, 1, 255],
[-0.001868, 0.000618, 0.000357, 1, 255],
[0.001868, -0.000618, -0.000357, 1, 255],
[-0.001868, -0.000618, -0.000357, 1, 255],
[0.000714, -0.000934, 0.001618, 1, 255],
[-0.000714, -0.000934, 0.001618, 1, 255],
[0.000714, 0.000934, -0.001618, 1, 255],
[-0.000714, 0.000934, -0.001618, 1, 255],
[0, -0.001975, -0.000316, 1, 255],
[0, 0.001261, 0.001552, 1, 255],
[0, -0.001261, -0.001552, 1, 255],
[0, 0.001975, 0.000316, 1, 255],
[0.001155, 0.000423, 0.001577, 1, 255],
[-0.001155, 0.000423, 0.001577, 1, 255],
[0.001155, -0.001577, 0.000423, 1, 255],
[-0.001155, -0.001577, 0.000423, 1, 255],
[0.001155, 0.001577, -0.000423, 1, 255],
[-0.001155, 0.001577, -0.000423, 1, 255],
[0.001155, -0.000423, -0.001577, 1, 255],
[-0.001155, -0.000423, -0.001577, 1, 255],
],
[
[0, 0, 1000, 2, 128],
[0, 0, 0.003125, 1, 0],
[0, 0, 0, 0.9975, 255],
[0, 0.000547, -0.002573, 0.995, 0],
[-0, -0.000547, -0.002573, 0.995, 0],
[0, 0.001071, -0.005037, 0.9925, 255],
[-0, -0.001071, -0.005037, 0.9925, 255],
],
[
[0, 0, 0, 1, 210],
[0, 0, 0.0008, 1, 210],
[0, 0, 0.0004, 1.00012, 50],
[0.012969, -0, -0.042045, 0.96, 100],
[0.016397, 0.011913, -0.039054, 0.96, 210],
[0.004008, 0.012334, -0.042045, 0.96, 100],
[-0.006263, 0.019276, -0.039054, 0.96, 210],
[-0.010492, 0.007623, -0.042045, 0.96, 100],
[-0.020268, 0, -0.039054, 0.96, 210],
[-0.010492, -0.007623, -0.042045, 0.96, 100],
[-0.006263, -0.019276, -0.039054, 0.96, 210],
[0.004008, -0.012334, -0.042045, 0.96, 100],
[0.016397, -0.011913, -0.039054, 0.96, 210],
],
[
[0, 0, 1000, 2, 192],
[-2, 2, 100, 3, 98],
[2, 2, 100, 3, 98],
[0.5, -0.5, 500, 0.2, 255],
[-0.349469, -0.1, 20.000279, 0.6, 64],
[-0.349775, -0.100273, 20.000485, 0.6, 85],
[-0.350214, -0.100386, 20.000407, 0.6, 106],
[-0.350527, -0.100273, 20.000091, 0.6, 127],
[-0.350531, -0.1, 19.999721, 0.6, 148],
[-0.350225, -0.099727, 19.999515, 0.6, 169],
[-0.349786, -0.099614, 19.999593, 0.6, 190],
[-0.349473, -0.099727, 19.999909, 0.6, 211],
[0.20032, 0.349824, 15.000456, 0.3, 0],
[0.20032, 0.349517, 14.999924, 0.3, 0],
[0.19968, 0.350483, 15.000076, 0.3, 0],
[0.19968, 0.350176, 14.999544, 0.3, 0],
[0.200578, 0.350073, 14.999958, 0.3, 0],
[0.199817, 0.349519, 15.000277, 0.3, 0],
[0.200183, 0.350481, 14.999723, 0.3, 0],
[0.199422, 0.349927, 15.000042, 0.3, 0],
[0.200235, 0.35042, 15.000332, 0.3, 0],
[0.199765, 0.350078, 15.000529, 0.3, 0],
[0.200235, 0.349922, 14.999471, 0.3, 0],
[0.199765, 0.34958, 14.999668, 0.3, 0],
[0.200429, 0.350419, 15.000005, 0.3, 255],
[0.199571, 0.349795, 15.000366, 0.3, 255],
[0.200429, 0.350205, 14.999634, 0.3, 255],
[0.199571, 0.349581, 14.999995, 0.3, 255],
[0.200524, 0.349747, 15.000146, 0.3, 255],
[0.200196, 0.349509, 15.000283, 0.3, 255],
[0.199804, 0.350491, 14.999717, 0.3, 255],
[0.199476, 0.350253, 14.999854, 0.3, 255],
[0.200138, 0.349578, 14.999597, 0.3, 255],
[0.200138, 0.350138, 15.000567, 0.3, 255],
[0.199862, 0.349862, 14.999433, 0.3, 255],
[0.199862, 0.350422, 15.000403, 0.3, 255],
[0.200488, 0.350136, 15.000321, 0.3, 255],
[0.199957, 0.349751, 15.000544, 0.3, 255],
[0.200488, 0.34979, 14.999721, 0.3, 255],
[0.199957, 0.349404, 14.999944, 0.3, 255],
[0.200043, 0.350596, 15.000056, 0.3, 255],
[0.199512, 0.35021, 15.000279, 0.3, 255],
[0.200043, 0.350249, 14.999456, 0.3, 255],
[0.199512, 0.349864, 14.999679, 0.3, 255],
[-0.5, 0.500442, 10.000442, 0.2, 0],
[-0.5, 0.5, 10, 0.1995, 255],
[-0.499966, 0.49971, 9.999563, 0.199, 0],
[-0.500034, 0.499563, 9.99971, 0.199, 0],
[-0.499934, 0.499432, 9.999144, 0.1985, 255],
[-0.500066, 0.499144, 9.999432, 0.1985, 255],
[0.6, 0.5, 10, 0.25, 210],
[0.600122, 0.4999, 10.000122, 0.25, 210],
[0.600061, 0.49995, 10.000061, 0.25003, 50],
[0.595856, 0.505256, 9.991271, 0.24, 100],
[0.597973, 0.507461, 9.992175, 0.24, 210],
[0.595362, 0.507926, 9.993945, 0.24, 100],
[0.594618, 0.509055, 9.996832, 0.24, 210],
[0.592382, 0.506906, 9.996092, 0.24, 100],
[0.590438, 0.504882, 9.997604, 0.24, 210],
[0.591035, 0.503605, 9.994744, 0.24, 100],
[0.59121, 0.500708, 9.993425, 0.24, 210],
[0.593181, 0.502585, 9.991764, 0.24, 100],
[0.595867, 0.502302, 9.990069, 0.24, 210],
],
]
scene_names = (
"single_sphere",
"two_spheres",
"ring_of_spheres",
"beachball",
"football",
"eightball",
"starball",
"combination_scene",
)
for i in range(8):
print(f'Generating {scene_names[i]}.')
values = grey_values(test_cases[i])
rows = [values[row * 1024:row * 1024 + 1024] for row in range(1024)]
with open(f'{scene_names[i]}.png', 'wb') as file:
w = png.Writer(1024, 1024, greyscale=True)
w.write(file, rows)
```
### Rust
The function `grey_values` would be sufficient as an entry to this challenge. The rest of the code saves the outputs as PNG files.
```rust
fn grey_values(spheres: &Vec<[f32; 5]>) -> Vec<u8> {
let mut values = vec![];
for y_pixel in 0..1024 {
for x_pixel in 0..1024 {
let x_ray = x_pixel as f32 / 512.0 - 1.0;
let y_ray = y_pixel as f32 / 512.0 - 1.0;
let mut min_distance = f32::INFINITY;
let mut hit_grey_value = 0;
for sphere in spheres {
let [
x_sphere, y_sphere, z_sphere, radius, grey
] = *sphere;
let a = (
(x_ray - x_sphere).powf(2.0)
+ (y_ray - y_sphere).powf(2.0)
).sqrt();
if a < radius {
let distance = z_sphere - (
radius.powf(2.0) - a.powf(2.0)
).sqrt();
if distance < min_distance {
min_distance = distance;
hit_grey_value = grey as u8;
}
}
}
values.push(hit_grey_value);
}
}
values
}
fn main() {
let test_cases: Vec<Vec<[f32; 5]>> = vec![
vec![
[0.0, 0.0, 0.0, 1.0, 128.0]
],
vec![
[0.0, -0.55, 0.0, 0.4, 100.0],
[0.0, 0.55, 0.0, 0.4, 200.0]
],
vec![
[0.75, 0.0, 0.0, 0.25, 64.0],
[0.53033, 0.375, -0.375, 0.25, 185.0],
[0.0, 0.53033, -0.53033, 0.25, 64.0],
[-0.53033, 0.375, -0.375, 0.25, 185.0],
[-0.75, 0.0, 0.0, 0.25, 64.0],
[-0.53033, -0.375, 0.375, 0.25, 185.0],
[0.0, -0.53033, 0.53033, 0.25, 64.0],
[0.53033, -0.375, 0.375, 0.25, 185.0],
],
vec![
[0.001, 0.0, 0.0, 1.0, 64.0],
[0.000707, 0.000707, 0.0, 1.0, 85.0],
[0.0, 0.001, 0.0, 1.0, 106.0],
[-0.000707, 0.000707, 0.0, 1.0, 127.0],
[-0.001, 0.0, 0.0, 1.0, 148.0],
[-0.000707, -0.000707, 0.0, 1.0, 169.0],
[0.0, -0.001, 0.0, 1.0, 190.0],
[0.000707, -0.000707, 0.0, 1.0, 211.0],
],
vec![
[0.0, 0.000058, 0.001947, 1.0, 0.0],
[0.0, -0.001715, 0.000923, 1.0, 0.0],
[0.0, 0.001715, -0.000923, 1.0, 0.0],
[0.0, -0.000058, -0.001947, 1.0, 0.0],
[0.001657, -0.000512, 0.000887, 1.0, 0.0],
[-0.001657, -0.000512, 0.000887, 1.0, 0.0],
[0.001657, 0.000512, -0.000887, 1.0, 0.0],
[-0.001657, 0.000512, -0.000887, 1.0, 0.0],
[0.001024, 0.001435, 0.000829, 1.0, 0.0],
[-0.001024, 0.001435, 0.000829, 1.0, 0.0],
[0.001024, -0.001435, -0.000829, 1.0, 0.0],
[-0.001024, -0.001435, -0.000829, 1.0, 0.0],
[0.001868, 0.000618, 0.000357, 1.0, 255.0],
[-0.001868, 0.000618, 0.000357, 1.0, 255.0],
[0.001868, -0.000618, -0.000357, 1.0, 255.0],
[-0.001868, -0.000618, -0.000357, 1.0, 255.0],
[0.000714, -0.000934, 0.001618, 1.0, 255.0],
[-0.000714, -0.000934, 0.001618, 1.0, 255.0],
[0.000714, 0.000934, -0.001618, 1.0, 255.0],
[-0.000714, 0.000934, -0.001618, 1.0, 255.0],
[0.0, -0.001975, -0.000316, 1.0, 255.0],
[0.0, 0.001261, 0.001552, 1.0, 255.0],
[0.0, -0.001261, -0.001552, 1.0, 255.0],
[0.0, 0.001975, 0.000316, 1.0, 255.0],
[0.001155, 0.000423, 0.001577, 1.0, 255.0],
[-0.001155, 0.000423, 0.001577, 1.0, 255.0],
[0.001155, -0.001577, 0.000423, 1.0, 255.0],
[-0.001155, -0.001577, 0.000423, 1.0, 255.0],
[0.001155, 0.001577, -0.000423, 1.0, 255.0],
[-0.001155, 0.001577, -0.000423, 1.0, 255.0],
[0.001155, -0.000423, -0.001577, 1.0, 255.0],
[-0.001155, -0.000423, -0.001577, 1.0, 255.0],
],
vec![
[0.0, 0.0, 1000.0, 2.0, 128.0],
[0.0, 0.0, 0.003125, 1.0, 0.0],
[0.0, 0.0, 0.0, 0.9975, 255.0],
[0.0, 0.000547, -0.002573, 0.995, 0.0],
[0.0, -0.000547, -0.002573, 0.995, 0.0],
[0.0, 0.001071, -0.005037, 0.9925, 255.0],
[0.0, -0.001071, -0.005037, 0.9925, 255.0],
],
vec![
[0.0, 0.0, 0.0, 1.0, 210.0],
[0.0, 0.0, 0.0008, 1.0, 210.0],
[0.0, 0.0, 0.0004, 1.00012, 50.0],
[0.012969, 0.0, -0.042045, 0.96, 100.0],
[0.016397, 0.011913, -0.039054, 0.96, 210.0],
[0.004008, 0.012334, -0.042045, 0.96, 100.0],
[-0.006263, 0.019276, -0.039054, 0.96, 210.0],
[-0.010492, 0.007623, -0.042045, 0.96, 100.0],
[-0.020268, 0.0, -0.039054, 0.96, 210.0],
[-0.010492, -0.007623, -0.042045, 0.96, 100.0],
[-0.006263, -0.019276, -0.039054, 0.96, 210.0],
[0.004008, -0.012334, -0.042045, 0.96, 100.0],
[0.016397, -0.011913, -0.039054, 0.96, 210.0],
],
vec![
[0.0, 0.0, 1000.0, 2.0, 192.0],
[-2.0, 2.0, 100.0, 3.0, 98.0],
[2.0, 2.0, 100.0, 3.0, 98.0],
[0.5, -0.5, 500.0, 0.2, 255.0],
[-0.349469, -0.1, 20.000279, 0.6, 64.0],
[-0.349775, -0.100273, 20.000485, 0.6, 85.0],
[-0.350214, -0.100386, 20.000407, 0.6, 106.0],
[-0.350527, -0.100273, 20.000091, 0.6, 127.0],
[-0.350531, -0.1, 19.999721, 0.6, 148.0],
[-0.350225, -0.099727, 19.999515, 0.6, 169.0],
[-0.349786, -0.099614, 19.999593, 0.6, 190.0],
[-0.349473, -0.099727, 19.999909, 0.6, 211.0],
[0.20032, 0.349824, 15.000456, 0.3, 0.0],
[0.20032, 0.349517, 14.999924, 0.3, 0.0],
[0.19968, 0.350483, 15.000076, 0.3, 0.0],
[0.19968, 0.350176, 14.999544, 0.3, 0.0],
[0.200578, 0.350073, 14.999958, 0.3, 0.0],
[0.199817, 0.349519, 15.000277, 0.3, 0.0],
[0.200183, 0.350481, 14.999723, 0.3, 0.0],
[0.199422, 0.349927, 15.000042, 0.3, 0.0],
[0.200235, 0.35042, 15.000332, 0.3, 0.0],
[0.199765, 0.350078, 15.000529, 0.3, 0.0],
[0.200235, 0.349922, 14.999471, 0.3, 0.0],
[0.199765, 0.34958, 14.999668, 0.3, 0.0],
[0.200429, 0.350419, 15.000005, 0.3, 255.0],
[0.199571, 0.349795, 15.000366, 0.3, 255.0],
[0.200429, 0.350205, 14.999634, 0.3, 255.0],
[0.199571, 0.349581, 14.999995, 0.3, 255.0],
[0.200524, 0.349747, 15.000146, 0.3, 255.0],
[0.200196, 0.349509, 15.000283, 0.3, 255.0],
[0.199804, 0.350491, 14.999717, 0.3, 255.0],
[0.199476, 0.350253, 14.999854, 0.3, 255.0],
[0.200138, 0.349578, 14.999597, 0.3, 255.0],
[0.200138, 0.350138, 15.000567, 0.3, 255.0],
[0.199862, 0.349862, 14.999433, 0.3, 255.0],
[0.199862, 0.350422, 15.000403, 0.3, 255.0],
[0.200488, 0.350136, 15.000321, 0.3, 255.0],
[0.199957, 0.349751, 15.000544, 0.3, 255.0],
[0.200488, 0.34979, 14.999721, 0.3, 255.0],
[0.199957, 0.349404, 14.999944, 0.3, 255.0],
[0.200043, 0.350596, 15.000056, 0.3, 255.0],
[0.199512, 0.35021, 15.000279, 0.3, 255.0],
[0.200043, 0.350249, 14.999456, 0.3, 255.0],
[0.199512, 0.349864, 14.999679, 0.3, 255.0],
[-0.5, 0.500442, 10.000442, 0.2, 0.0],
[-0.5, 0.5, 10.0, 0.1995, 255.0],
[-0.499966, 0.49971, 9.999563, 0.199, 0.0],
[-0.500034, 0.499563, 9.99971, 0.199, 0.0],
[-0.499934, 0.499432, 9.999144, 0.1985, 255.0],
[-0.500066, 0.499144, 9.999432, 0.1985, 255.0],
[0.6, 0.5, 10.0, 0.25, 210.0],
[0.600122, 0.4999, 10.000122, 0.25, 210.0],
[0.600061, 0.49995, 10.000061, 0.25003, 50.0],
[0.595856, 0.505256, 9.991271, 0.24, 100.0],
[0.597973, 0.507461, 9.992175, 0.24, 210.0],
[0.595362, 0.507926, 9.993945, 0.24, 100.0],
[0.594618, 0.509055, 9.996832, 0.24, 210.0],
[0.592382, 0.506906, 9.996092, 0.24, 100.0],
[0.590438, 0.504882, 9.997604, 0.24, 210.0],
[0.591035, 0.503605, 9.994744, 0.24, 100.0],
[0.59121, 0.500708, 9.993425, 0.24, 210.0],
[0.593181, 0.502585, 9.991764, 0.24, 100.0],
[0.595867, 0.502302, 9.990069, 0.24, 210.0],
],
];
let scene_names = [
"single_sphere",
"two_spheres",
"ring_of_spheres",
"beachball",
"football",
"eightball",
"starball",
"combination_scene",
];
for i in 0..8 {
eprintln!("Generating {}", scene_names[i]);
let values = grey_values(&test_cases[i]);
let filename = format!("{}.png", scene_names[i]);
let path = std::path::Path::new(&filename);
let file = std::fs::File::create(path).unwrap();
let w = &mut std::io::BufWriter::new(file);
let mut encoder = png::Encoder::new(w, 1024, 1024);
encoder.set_color(png::ColorType::Grayscale);
encoder.set_depth(png::BitDepth::Eight);
let mut writer = encoder.write_header().unwrap();
writer.write_image_data(&values).unwrap();
}
}
```
## Test cases
The images may be scaled down. Clicking or tapping on an image will open it full size.
### Single sphere
Input:
```text
~~TODO~~
```
Output:
```text
TODO
```
### Two separated spheres
Input:
```text
~~TODO~~
```
Output:
```text
TODO
```
~~### Eight spheres in a ring~~
Input:
```text
~~TODO~~
```
Output:
```text
TODO
```
### Beach ball
Input:
```text
~~TODO~~
```
Output:
```text
TODO
```
### Football
Input:
```text
~~TODO~~
```
Output:
```text
TODO
```
### Eight ball
Input:
```text
~~TODO~~
```
Output:
```text
TODO
```
### Star ball
Input:
```text
~~TODO~~
```
Output:
```text
TODO
```
### Combination scene
Input:
```text
~~TODO~~
```
Output:
```text
TODO
```
~~## Explanatory animations~~
~~The example images were produced from plain grey spheres with no surface pattern or texture. The following animations show how they break down into their component plain spheres.~~
This section is not necessary to write valid code for this challenge. If your code accepts the parameters of plain grey spheres and renders them correctly, all of the test cases will look right without needing to understand how they were constructed. This section is mainly out of interest, and to make clear that your code *only* needs to render plain grey spheres.
~~```text~~
~~CONSIDER HOSTING THESE OFF SITE AS THEY ARE NOT REQUIRED FOR THE CHALLENGE~~
~~AND EVEN THE SCALED DOWN VERSIONS ARE UNNECESSARY DATA USE FOR ANYONE~~
~~CHOOSING NOT TO VIEW THEM.~~
~~```~~
~~<details>~~
~~<summary>Animations</summary>~~
~~<details>~~
~~<summary>Beach ball</summary>~~
~~This animation is reduced to 256 by 256 pixels. Clicking or tapping on the animation will open it at full size (______MB).~~
~~```text~~
~~TODO~~
~~```~~
~~</details>~~
~~<details>~~
~~<summary>Football</summary>~~
~~This animation is reduced to 256 by 256 pixels. Clicking or tapping on the animation will open it at full size (______MB).~~
~~```text~~
~~TODO~~
~~```~~
~~</details>~~
~~<details>~~
~~<summary>Eight ball</summary>~~
~~This animation is reduced to 256 by 256 pixels. Clicking or tapping on the animation will open it at full size (______MB).~~
~~```text~~
~~TODO~~
~~```~~
~~</details>~~
~~<details>~~
~~<summary>Star ball</summary>~~
~~This animation is reduced to 256 by 256 pixels. Clicking or tapping on the animation will open it at full size (______MB).~~
~~```text~~
~~TODO~~
~~```~~
~~</details>~~
~~</details>~~
## Scoring
~~This is a [code golf challenge]. Your score is the number of bytes in your code. Lowest score for each language wins.~~
~~> Explanations are optional, but I'm more likely to upvote answers that have one.~~
~~[code golf challenge]: https://codegolf.codidact.com/categories/49/tags/4274 "The code-golf tag"~~
~~[PGM]: https://en.wikipedia.org/wiki/Netpbm#PGM_example~~

Given a list of spheres, output an image.
Produce a greyscale [orthographic](https://en.wikipedia.org/wiki/Parallel_projection#Orthographic_projection) (parallel projection - no perspective) ray caster, that renders only plain grey spheres with no lighting.
## Input
- A list of spheres, each consisting of:
- The coordinates of its centre $(x,y,z)$.
- Its radius $r$.
- Its greyscale value $g$, an integer from $0$ to $255$ (both inclusive).
- Only the greyscale value will be restricted to integer values. The coordinates and radius may have fractional values.
## Output
- An image $1024$ by $1024$ pixels with greyscale values from $0$ to $255$.
- The image covers x and y values from $-1$ to $1$.
- A sphere's greyscale value is applied to every visible part of it. There is no variation over the surface of the sphere, so it appears flat like a disk.
- You may output whatever image format is convenient, including simple text formats such as [PGM](https://en.wikipedia.org/wiki/Netpbm#PGM_example) or a list of pixel greyscale values. If you want to include an example of your output as an image in your answer, the image formats that Codidact supports are PNG, JPEG, and GIF. You could output PGM and then convert to PNG to upload to your answer. If you output just the greyscale values as a list of numbers from $0$ to $255$, I've made a `[ TODO page to convert these to greyscale PNG]`.
- You are free to output an image reflected or rotated by a multiple of 90 degrees. However, the z axis is required to be increasing in the forwards direction (into the image) otherwise some of the test cases will look incorrect.
## Calculating a pixel value
Imagine a ray passing into the image through a pixel, perpendicular to the image plane. If it hits a sphere, it shows the greyscale value of the first sphere it hits. Otherwise, it shows a value of zero (black).
For example, for each pixel:
- The pixel $x,y$ coordinates will each be from $0$ to $1023$. Scale these to the range $-1$ to $1$ (for example, divide by $512$ and subtract $1$).
- If these scaled $x,y$ coordinates are within the radius $r$ of the $x,y$ coordinates of a sphere's centre, the ray intersects that sphere (the $z$ coordinate is not relevant to this test because the image is an orthographic projection - the rays are all parallel to the $z$ axis). The distance $a$ of the ray from the sphere's centre can be calculated as follows:
$$a=\sqrt{(x_{ray}-x_{sphere})^2+(y_{ray}-y_{sphere})^2}$$
```text
CONSIDER A DIAGRAM HERE
```
- For each sphere that the ray intersects, find the $z$ coordinate of the nearest intersection (there will often be $2$ per sphere, $1$ on the front of the sphere and $1$ on the back of the sphere, as the ray passes through - only the nearest is relevant). This can be calculated as follows:
- The distance $d$ to the nearest intersection point can be calculated from the $z$ coordinate of the sphere's centre, the radius $r$, and the distance $a$ from the $x,y$ coordinates of the sphere's centre to the scaled $x,y$ coordinates of the pixel:
$$d=z_{sphere}-\sqrt{r_{sphere}^2-a^2}$$
```text
CONSIDER A DIAGRAM HERE
```
- Of all the spheres that the ray intersects, the sphere with the intersection point with the smallest $z$ coordinate $d$ is the one to display. Set the pixel to the greyscale value of this sphere.
- If the ray did not intersect any spheres, set the pixel to zero (black).
## Example implementations
These are ungolfed code to supplement understanding of the requirements. You are free to golf these or implement your own approach that matches the test cases up to reflection & rotation.
The Python implementation uses 64 bit floating point. The Rust implementation uses 32 bit floating point to show the difference in output. I've tried to optimise the test cases for compatibility with 32 bit floating point, so the differences are restricted to being a pixel out in some places.
### Python
The function `grey_values` would be sufficient as an entry to this challenge. The rest of the code saves the outputs as PNG files. The function itself has no dependence on the `import png` so it would not be required in an entry.
```python
def grey_values(spheres):
values = []
for y_pixel in range(1024):
for x_pixel in range(1024):
x_ray, y_ray = x_pixel / 512 - 1, y_pixel / 512 - 1
min_distance, hit_grey_value = None, 0
for sphere in spheres:
x_sphere, y_sphere, z_sphere, radius, grey = sphere
a = (
(x_ray - x_sphere) ** 2 + (y_ray - y_sphere) ** 2
) ** 0.5
if a < radius:
distance = z_sphere - (radius ** 2 - a ** 2) ** 0.5
if min_distance is None or distance < min_distance:
min_distance = distance
hit_grey_value = grey
values.append(hit_grey_value)
return values
import png
test_cases = [
[
[0, 0, 0, 1, 128],
],
[
[0, -0.55, 0, 0.4, 100],
[0, 0.55, 0, 0.4, 200],
],
[
[0.75, 0, 0, 0.25, 64],
[0.53033, 0.375, -0.375, 0.25, 185],
[0, 0.53033, -0.53033, 0.25, 64],
[-0.53033, 0.375, -0.375, 0.25, 185],
[-0.75, 0, -0, 0.25, 64],
[-0.53033, -0.375, 0.375, 0.25, 185],
[-0, -0.53033, 0.53033, 0.25, 64],
[0.53033, -0.375, 0.375, 0.25, 185],
],
[
[0.001, 0, 0, 1, 64],
[0.000707, 0.000707, -0, 1, 85],
[0, 0.001, -0, 1, 106],
[-0.000707, 0.000707, -0, 1, 127],
[-0.001, 0, -0, 1, 148],
[-0.000707, -0.000707, 0, 1, 169],
[-0, -0.001, 0, 1, 190],
[0.000707, -0.000707, 0, 1, 211],
],
[
[0, 0.000058, 0.001947, 1, 0],
[0, -0.001715, 0.000923, 1, 0],
[0, 0.001715, -0.000923, 1, 0],
[0, -0.000058, -0.001947, 1, 0],
[0.001657, -0.000512, 0.000887, 1, 0],
[-0.001657, -0.000512, 0.000887, 1, 0],
[0.001657, 0.000512, -0.000887, 1, 0],
[-0.001657, 0.000512, -0.000887, 1, 0],
[0.001024, 0.001435, 0.000829, 1, 0],
[-0.001024, 0.001435, 0.000829, 1, 0],
[0.001024, -0.001435, -0.000829, 1, 0],
[-0.001024, -0.001435, -0.000829, 1, 0],
[0.001868, 0.000618, 0.000357, 1, 255],
[-0.001868, 0.000618, 0.000357, 1, 255],
[0.001868, -0.000618, -0.000357, 1, 255],
[-0.001868, -0.000618, -0.000357, 1, 255],
[0.000714, -0.000934, 0.001618, 1, 255],
[-0.000714, -0.000934, 0.001618, 1, 255],
[0.000714, 0.000934, -0.001618, 1, 255],
[-0.000714, 0.000934, -0.001618, 1, 255],
[0, -0.001975, -0.000316, 1, 255],
[0, 0.001261, 0.001552, 1, 255],
[0, -0.001261, -0.001552, 1, 255],
[0, 0.001975, 0.000316, 1, 255],
[0.001155, 0.000423, 0.001577, 1, 255],
[-0.001155, 0.000423, 0.001577, 1, 255],
[0.001155, -0.001577, 0.000423, 1, 255],
[-0.001155, -0.001577, 0.000423, 1, 255],
[0.001155, 0.001577, -0.000423, 1, 255],
[-0.001155, 0.001577, -0.000423, 1, 255],
[0.001155, -0.000423, -0.001577, 1, 255],
[-0.001155, -0.000423, -0.001577, 1, 255],
],
[
[0, 0, 1000, 2, 128],
[0, 0, 0.003125, 1, 0],
[0, 0, 0, 0.9975, 255],
[0, 0.000547, -0.002573, 0.995, 0],
[-0, -0.000547, -0.002573, 0.995, 0],
[0, 0.001071, -0.005037, 0.9925, 255],
[-0, -0.001071, -0.005037, 0.9925, 255],
],
[
[0, 0, 0, 1, 210],
[0, 0, 0.0008, 1, 210],
[0, 0, 0.0004, 1.00012, 50],
[0.012969, -0, -0.042045, 0.96, 100],
[0.016397, 0.011913, -0.039054, 0.96, 210],
[0.004008, 0.012334, -0.042045, 0.96, 100],
[-0.006263, 0.019276, -0.039054, 0.96, 210],
[-0.010492, 0.007623, -0.042045, 0.96, 100],
[-0.020268, 0, -0.039054, 0.96, 210],
[-0.010492, -0.007623, -0.042045, 0.96, 100],
[-0.006263, -0.019276, -0.039054, 0.96, 210],
[0.004008, -0.012334, -0.042045, 0.96, 100],
[0.016397, -0.011913, -0.039054, 0.96, 210],
],
[
[0, 0, 1000, 2, 192],
[-2, 2, 100, 3, 98],
[2, 2, 100, 3, 98],
[0.5, -0.5, 500, 0.2, 255],
[-0.349469, -0.1, 20.000279, 0.6, 64],
[-0.349775, -0.100273, 20.000485, 0.6, 85],
[-0.350214, -0.100386, 20.000407, 0.6, 106],
[-0.350527, -0.100273, 20.000091, 0.6, 127],
[-0.350531, -0.1, 19.999721, 0.6, 148],
[-0.350225, -0.099727, 19.999515, 0.6, 169],
[-0.349786, -0.099614, 19.999593, 0.6, 190],
[-0.349473, -0.099727, 19.999909, 0.6, 211],
[0.20032, 0.349824, 15.000456, 0.3, 0],
[0.20032, 0.349517, 14.999924, 0.3, 0],
[0.19968, 0.350483, 15.000076, 0.3, 0],
[0.19968, 0.350176, 14.999544, 0.3, 0],
[0.200578, 0.350073, 14.999958, 0.3, 0],
[0.199817, 0.349519, 15.000277, 0.3, 0],
[0.200183, 0.350481, 14.999723, 0.3, 0],
[0.199422, 0.349927, 15.000042, 0.3, 0],
[0.200235, 0.35042, 15.000332, 0.3, 0],
[0.199765, 0.350078, 15.000529, 0.3, 0],
[0.200235, 0.349922, 14.999471, 0.3, 0],
[0.199765, 0.34958, 14.999668, 0.3, 0],
[0.200429, 0.350419, 15.000005, 0.3, 255],
[0.199571, 0.349795, 15.000366, 0.3, 255],
[0.200429, 0.350205, 14.999634, 0.3, 255],
[0.199571, 0.349581, 14.999995, 0.3, 255],
[0.200524, 0.349747, 15.000146, 0.3, 255],
[0.200196, 0.349509, 15.000283, 0.3, 255],
[0.199804, 0.350491, 14.999717, 0.3, 255],
[0.199476, 0.350253, 14.999854, 0.3, 255],
[0.200138, 0.349578, 14.999597, 0.3, 255],
[0.200138, 0.350138, 15.000567, 0.3, 255],
[0.199862, 0.349862, 14.999433, 0.3, 255],
[0.199862, 0.350422, 15.000403, 0.3, 255],
[0.200488, 0.350136, 15.000321, 0.3, 255],
[0.199957, 0.349751, 15.000544, 0.3, 255],
[0.200488, 0.34979, 14.999721, 0.3, 255],
[0.199957, 0.349404, 14.999944, 0.3, 255],
[0.200043, 0.350596, 15.000056, 0.3, 255],
[0.199512, 0.35021, 15.000279, 0.3, 255],
[0.200043, 0.350249, 14.999456, 0.3, 255],
[0.199512, 0.349864, 14.999679, 0.3, 255],
[-0.5, 0.500442, 10.000442, 0.2, 0],
[-0.5, 0.5, 10, 0.1995, 255],
[-0.499966, 0.49971, 9.999563, 0.199, 0],
[-0.500034, 0.499563, 9.99971, 0.199, 0],
[-0.499934, 0.499432, 9.999144, 0.1985, 255],
[-0.500066, 0.499144, 9.999432, 0.1985, 255],
[0.6, 0.5, 10, 0.25, 210],
[0.600122, 0.4999, 10.000122, 0.25, 210],
[0.600061, 0.49995, 10.000061, 0.25003, 50],
[0.595856, 0.505256, 9.991271, 0.24, 100],
[0.597973, 0.507461, 9.992175, 0.24, 210],
[0.595362, 0.507926, 9.993945, 0.24, 100],
[0.594618, 0.509055, 9.996832, 0.24, 210],
[0.592382, 0.506906, 9.996092, 0.24, 100],
[0.590438, 0.504882, 9.997604, 0.24, 210],
[0.591035, 0.503605, 9.994744, 0.24, 100],
[0.59121, 0.500708, 9.993425, 0.24, 210],
[0.593181, 0.502585, 9.991764, 0.24, 100],
[0.595867, 0.502302, 9.990069, 0.24, 210],
],
]
scene_names = (
"single_sphere",
"two_spheres",
"ring_of_spheres",
"beachball",
"football",
"eightball",
"starball",
"combination_scene",
)
for i in range(8):
print(f'Generating {scene_names[i]}.')
values = grey_values(test_cases[i])
rows = [values[row * 1024:row * 1024 + 1024] for row in range(1024)]
with open(f'{scene_names[i]}.png', 'wb') as file:
w = png.Writer(1024, 1024, greyscale=True)
w.write(file, rows)
```
### Rust
The function `grey_values` would be sufficient as an entry to this challenge. The rest of the code saves the outputs as PNG files.
```rust
fn grey_values(spheres: &Vec<[f32; 5]>) -> Vec<u8> {
let mut values = vec![];
for y_pixel in 0..1024 {
for x_pixel in 0..1024 {
let x_ray = x_pixel as f32 / 512.0 - 1.0;
let y_ray = y_pixel as f32 / 512.0 - 1.0;
let mut min_distance = f32::INFINITY;
let mut hit_grey_value = 0;
for sphere in spheres {
let [
x_sphere, y_sphere, z_sphere, radius, grey
] = *sphere;
let a = (
(x_ray - x_sphere).powf(2.0)
+ (y_ray - y_sphere).powf(2.0)
).sqrt();
if a < radius {
let distance = z_sphere - (
radius.powf(2.0) - a.powf(2.0)
).sqrt();
if distance < min_distance {
min_distance = distance;
hit_grey_value = grey as u8;
}
}
}
values.push(hit_grey_value);
}
}
values
}
fn main() {
let test_cases: Vec<Vec<[f32; 5]>> = vec![
vec![
[0.0, 0.0, 0.0, 1.0, 128.0]
],
vec![
[0.0, -0.55, 0.0, 0.4, 100.0],
[0.0, 0.55, 0.0, 0.4, 200.0]
],
vec![
[0.75, 0.0, 0.0, 0.25, 64.0],
[0.53033, 0.375, -0.375, 0.25, 185.0],
[0.0, 0.53033, -0.53033, 0.25, 64.0],
[-0.53033, 0.375, -0.375, 0.25, 185.0],
[-0.75, 0.0, 0.0, 0.25, 64.0],
[-0.53033, -0.375, 0.375, 0.25, 185.0],
[0.0, -0.53033, 0.53033, 0.25, 64.0],
[0.53033, -0.375, 0.375, 0.25, 185.0],
],
vec![
[0.001, 0.0, 0.0, 1.0, 64.0],
[0.000707, 0.000707, 0.0, 1.0, 85.0],
[0.0, 0.001, 0.0, 1.0, 106.0],
[-0.000707, 0.000707, 0.0, 1.0, 127.0],
[-0.001, 0.0, 0.0, 1.0, 148.0],
[-0.000707, -0.000707, 0.0, 1.0, 169.0],
[0.0, -0.001, 0.0, 1.0, 190.0],
[0.000707, -0.000707, 0.0, 1.0, 211.0],
],
vec![
[0.0, 0.000058, 0.001947, 1.0, 0.0],
[0.0, -0.001715, 0.000923, 1.0, 0.0],
[0.0, 0.001715, -0.000923, 1.0, 0.0],
[0.0, -0.000058, -0.001947, 1.0, 0.0],
[0.001657, -0.000512, 0.000887, 1.0, 0.0],
[-0.001657, -0.000512, 0.000887, 1.0, 0.0],
[0.001657, 0.000512, -0.000887, 1.0, 0.0],
[-0.001657, 0.000512, -0.000887, 1.0, 0.0],
[0.001024, 0.001435, 0.000829, 1.0, 0.0],
[-0.001024, 0.001435, 0.000829, 1.0, 0.0],
[0.001024, -0.001435, -0.000829, 1.0, 0.0],
[-0.001024, -0.001435, -0.000829, 1.0, 0.0],
[0.001868, 0.000618, 0.000357, 1.0, 255.0],
[-0.001868, 0.000618, 0.000357, 1.0, 255.0],
[0.001868, -0.000618, -0.000357, 1.0, 255.0],
[-0.001868, -0.000618, -0.000357, 1.0, 255.0],
[0.000714, -0.000934, 0.001618, 1.0, 255.0],
[-0.000714, -0.000934, 0.001618, 1.0, 255.0],
[0.000714, 0.000934, -0.001618, 1.0, 255.0],
[-0.000714, 0.000934, -0.001618, 1.0, 255.0],
[0.0, -0.001975, -0.000316, 1.0, 255.0],
[0.0, 0.001261, 0.001552, 1.0, 255.0],
[0.0, -0.001261, -0.001552, 1.0, 255.0],
[0.0, 0.001975, 0.000316, 1.0, 255.0],
[0.001155, 0.000423, 0.001577, 1.0, 255.0],
[-0.001155, 0.000423, 0.001577, 1.0, 255.0],
[0.001155, -0.001577, 0.000423, 1.0, 255.0],
[-0.001155, -0.001577, 0.000423, 1.0, 255.0],
[0.001155, 0.001577, -0.000423, 1.0, 255.0],
[-0.001155, 0.001577, -0.000423, 1.0, 255.0],
[0.001155, -0.000423, -0.001577, 1.0, 255.0],
[-0.001155, -0.000423, -0.001577, 1.0, 255.0],
],
vec![
[0.0, 0.0, 1000.0, 2.0, 128.0],
[0.0, 0.0, 0.003125, 1.0, 0.0],
[0.0, 0.0, 0.0, 0.9975, 255.0],
[0.0, 0.000547, -0.002573, 0.995, 0.0],
[0.0, -0.000547, -0.002573, 0.995, 0.0],
[0.0, 0.001071, -0.005037, 0.9925, 255.0],
[0.0, -0.001071, -0.005037, 0.9925, 255.0],
],
vec![
[0.0, 0.0, 0.0, 1.0, 210.0],
[0.0, 0.0, 0.0008, 1.0, 210.0],
[0.0, 0.0, 0.0004, 1.00012, 50.0],
[0.012969, 0.0, -0.042045, 0.96, 100.0],
[0.016397, 0.011913, -0.039054, 0.96, 210.0],
[0.004008, 0.012334, -0.042045, 0.96, 100.0],
[-0.006263, 0.019276, -0.039054, 0.96, 210.0],
[-0.010492, 0.007623, -0.042045, 0.96, 100.0],
[-0.020268, 0.0, -0.039054, 0.96, 210.0],
[-0.010492, -0.007623, -0.042045, 0.96, 100.0],
[-0.006263, -0.019276, -0.039054, 0.96, 210.0],
[0.004008, -0.012334, -0.042045, 0.96, 100.0],
[0.016397, -0.011913, -0.039054, 0.96, 210.0],
],
vec![
[0.0, 0.0, 1000.0, 2.0, 192.0],
[-2.0, 2.0, 100.0, 3.0, 98.0],
[2.0, 2.0, 100.0, 3.0, 98.0],
[0.5, -0.5, 500.0, 0.2, 255.0],
[-0.349469, -0.1, 20.000279, 0.6, 64.0],
[-0.349775, -0.100273, 20.000485, 0.6, 85.0],
[-0.350214, -0.100386, 20.000407, 0.6, 106.0],
[-0.350527, -0.100273, 20.000091, 0.6, 127.0],
[-0.350531, -0.1, 19.999721, 0.6, 148.0],
[-0.350225, -0.099727, 19.999515, 0.6, 169.0],
[-0.349786, -0.099614, 19.999593, 0.6, 190.0],
[-0.349473, -0.099727, 19.999909, 0.6, 211.0],
[0.20032, 0.349824, 15.000456, 0.3, 0.0],
[0.20032, 0.349517, 14.999924, 0.3, 0.0],
[0.19968, 0.350483, 15.000076, 0.3, 0.0],
[0.19968, 0.350176, 14.999544, 0.3, 0.0],
[0.200578, 0.350073, 14.999958, 0.3, 0.0],
[0.199817, 0.349519, 15.000277, 0.3, 0.0],
[0.200183, 0.350481, 14.999723, 0.3, 0.0],
[0.199422, 0.349927, 15.000042, 0.3, 0.0],
[0.200235, 0.35042, 15.000332, 0.3, 0.0],
[0.199765, 0.350078, 15.000529, 0.3, 0.0],
[0.200235, 0.349922, 14.999471, 0.3, 0.0],
[0.199765, 0.34958, 14.999668, 0.3, 0.0],
[0.200429, 0.350419, 15.000005, 0.3, 255.0],
[0.199571, 0.349795, 15.000366, 0.3, 255.0],
[0.200429, 0.350205, 14.999634, 0.3, 255.0],
[0.199571, 0.349581, 14.999995, 0.3, 255.0],
[0.200524, 0.349747, 15.000146, 0.3, 255.0],
[0.200196, 0.349509, 15.000283, 0.3, 255.0],
[0.199804, 0.350491, 14.999717, 0.3, 255.0],
[0.199476, 0.350253, 14.999854, 0.3, 255.0],
[0.200138, 0.349578, 14.999597, 0.3, 255.0],
[0.200138, 0.350138, 15.000567, 0.3, 255.0],
[0.199862, 0.349862, 14.999433, 0.3, 255.0],
[0.199862, 0.350422, 15.000403, 0.3, 255.0],
[0.200488, 0.350136, 15.000321, 0.3, 255.0],
[0.199957, 0.349751, 15.000544, 0.3, 255.0],
[0.200488, 0.34979, 14.999721, 0.3, 255.0],
[0.199957, 0.349404, 14.999944, 0.3, 255.0],
[0.200043, 0.350596, 15.000056, 0.3, 255.0],
[0.199512, 0.35021, 15.000279, 0.3, 255.0],
[0.200043, 0.350249, 14.999456, 0.3, 255.0],
[0.199512, 0.349864, 14.999679, 0.3, 255.0],
[-0.5, 0.500442, 10.000442, 0.2, 0.0],
[-0.5, 0.5, 10.0, 0.1995, 255.0],
[-0.499966, 0.49971, 9.999563, 0.199, 0.0],
[-0.500034, 0.499563, 9.99971, 0.199, 0.0],
[-0.499934, 0.499432, 9.999144, 0.1985, 255.0],
[-0.500066, 0.499144, 9.999432, 0.1985, 255.0],
[0.6, 0.5, 10.0, 0.25, 210.0],
[0.600122, 0.4999, 10.000122, 0.25, 210.0],
[0.600061, 0.49995, 10.000061, 0.25003, 50.0],
[0.595856, 0.505256, 9.991271, 0.24, 100.0],
[0.597973, 0.507461, 9.992175, 0.24, 210.0],
[0.595362, 0.507926, 9.993945, 0.24, 100.0],
[0.594618, 0.509055, 9.996832, 0.24, 210.0],
[0.592382, 0.506906, 9.996092, 0.24, 100.0],
[0.590438, 0.504882, 9.997604, 0.24, 210.0],
[0.591035, 0.503605, 9.994744, 0.24, 100.0],
[0.59121, 0.500708, 9.993425, 0.24, 210.0],
[0.593181, 0.502585, 9.991764, 0.24, 100.0],
[0.595867, 0.502302, 9.990069, 0.24, 210.0],
],
];
let scene_names = [
"single_sphere",
"two_spheres",
"ring_of_spheres",
"beachball",
"football",
"eightball",
"starball",
"combination_scene",
];
for i in 0..8 {
eprintln!("Generating {}", scene_names[i]);
let values = grey_values(&test_cases[i]);
let filename = format!("{}.png", scene_names[i]);
let path = std::path::Path::new(&filename);
let file = std::fs::File::create(path).unwrap();
let w = &mut std::io::BufWriter::new(file);
let mut encoder = png::Encoder::new(w, 1024, 1024);
encoder.set_color(png::ColorType::Grayscale);
encoder.set_depth(png::BitDepth::Eight);
let mut writer = encoder.write_header().unwrap();
writer.write_image_data(&values).unwrap();
}
}
```
## Test cases
The images may be scaled down. Clicking or tapping on an image will open it full size.
### Single sphere
Input:
```text
[
[0, 0, 0, 1, 128],
]
```
Output:
```text
TODO
```
### Two separated spheres
Input:
```text
[
[0, -0.55, 0, 0.4, 100],
[0, 0.55, 0, 0.4, 200],
]
```
Output:
```text
TODO
```
### Ring of spheres
Input:
```text
[
[0.75, 0, 0, 0.25, 64],
[0.53033, 0.375, -0.375, 0.25, 185],
[0, 0.53033, -0.53033, 0.25, 64],
[-0.53033, 0.375, -0.375, 0.25, 185],
[-0.75, 0, -0, 0.25, 64],
[-0.53033, -0.375, 0.375, 0.25, 185],
[-0, -0.53033, 0.53033, 0.25, 64],
[0.53033, -0.375, 0.375, 0.25, 185],
]
```
Output:
```text
TODO
```
### Beach ball
Input:
```text
[
[0.001, 0, 0, 1, 64],
[0.000707, 0.000707, -0, 1, 85],
[0, 0.001, -0, 1, 106],
[-0.000707, 0.000707, -0, 1, 127],
[-0.001, 0, -0, 1, 148],
[-0.000707, -0.000707, 0, 1, 169],
[-0, -0.001, 0, 1, 190],
[0.000707, -0.000707, 0, 1, 211],
]
```
Output:
```text
TODO
```
### Football
Input:
```text
[
[0, 0.000058, 0.001947, 1, 0],
[0, -0.001715, 0.000923, 1, 0],
[0, 0.001715, -0.000923, 1, 0],
[0, -0.000058, -0.001947, 1, 0],
[0.001657, -0.000512, 0.000887, 1, 0],
[-0.001657, -0.000512, 0.000887, 1, 0],
[0.001657, 0.000512, -0.000887, 1, 0],
[-0.001657, 0.000512, -0.000887, 1, 0],
[0.001024, 0.001435, 0.000829, 1, 0],
[-0.001024, 0.001435, 0.000829, 1, 0],
[0.001024, -0.001435, -0.000829, 1, 0],
[-0.001024, -0.001435, -0.000829, 1, 0],
[0.001868, 0.000618, 0.000357, 1, 255],
[-0.001868, 0.000618, 0.000357, 1, 255],
[0.001868, -0.000618, -0.000357, 1, 255],
[-0.001868, -0.000618, -0.000357, 1, 255],
[0.000714, -0.000934, 0.001618, 1, 255],
[-0.000714, -0.000934, 0.001618, 1, 255],
[0.000714, 0.000934, -0.001618, 1, 255],
[-0.000714, 0.000934, -0.001618, 1, 255],
[0, -0.001975, -0.000316, 1, 255],
[0, 0.001261, 0.001552, 1, 255],
[0, -0.001261, -0.001552, 1, 255],
[0, 0.001975, 0.000316, 1, 255],
[0.001155, 0.000423, 0.001577, 1, 255],
[-0.001155, 0.000423, 0.001577, 1, 255],
[0.001155, -0.001577, 0.000423, 1, 255],
[-0.001155, -0.001577, 0.000423, 1, 255],
[0.001155, 0.001577, -0.000423, 1, 255],
[-0.001155, 0.001577, -0.000423, 1, 255],
[0.001155, -0.000423, -0.001577, 1, 255],
[-0.001155, -0.000423, -0.001577, 1, 255],
]
```
Output:
```text
TODO
```
### Eight ball
Input:
```text
[
[0, 0, 1000, 2, 128],
[0, 0, 0.003125, 1, 0],
[0, 0, 0, 0.9975, 255],
[0, 0.000547, -0.002573, 0.995, 0],
[-0, -0.000547, -0.002573, 0.995, 0],
[0, 0.001071, -0.005037, 0.9925, 255],
[-0, -0.001071, -0.005037, 0.9925, 255],
]
```
Output:
```text
TODO
```
### Star ball
Input:
```text
[
[0, 0, 0, 1, 210],
[0, 0, 0.0008, 1, 210],
[0, 0, 0.0004, 1.00012, 50],
[0.012969, -0, -0.042045, 0.96, 100],
[0.016397, 0.011913, -0.039054, 0.96, 210],
[0.004008, 0.012334, -0.042045, 0.96, 100],
[-0.006263, 0.019276, -0.039054, 0.96, 210],
[-0.010492, 0.007623, -0.042045, 0.96, 100],
[-0.020268, 0, -0.039054, 0.96, 210],
[-0.010492, -0.007623, -0.042045, 0.96, 100],
[-0.006263, -0.019276, -0.039054, 0.96, 210],
[0.004008, -0.012334, -0.042045, 0.96, 100],
[0.016397, -0.011913, -0.039054, 0.96, 210],
]
```
Output:
```text
TODO
```
### Combination scene
Input:
```text
[
[0, 0, 1000, 2, 192],
[-2, 2, 100, 3, 98],
[2, 2, 100, 3, 98],
[0.5, -0.5, 500, 0.2, 255],
[-0.349469, -0.1, 20.000279, 0.6, 64],
[-0.349775, -0.100273, 20.000485, 0.6, 85],
[-0.350214, -0.100386, 20.000407, 0.6, 106],
[-0.350527, -0.100273, 20.000091, 0.6, 127],
[-0.350531, -0.1, 19.999721, 0.6, 148],
[-0.350225, -0.099727, 19.999515, 0.6, 169],
[-0.349786, -0.099614, 19.999593, 0.6, 190],
[-0.349473, -0.099727, 19.999909, 0.6, 211],
[0.20032, 0.349824, 15.000456, 0.3, 0],
[0.20032, 0.349517, 14.999924, 0.3, 0],
[0.19968, 0.350483, 15.000076, 0.3, 0],
[0.19968, 0.350176, 14.999544, 0.3, 0],
[0.200578, 0.350073, 14.999958, 0.3, 0],
[0.199817, 0.349519, 15.000277, 0.3, 0],
[0.200183, 0.350481, 14.999723, 0.3, 0],
[0.199422, 0.349927, 15.000042, 0.3, 0],
[0.200235, 0.35042, 15.000332, 0.3, 0],
[0.199765, 0.350078, 15.000529, 0.3, 0],
[0.200235, 0.349922, 14.999471, 0.3, 0],
[0.199765, 0.34958, 14.999668, 0.3, 0],
[0.200429, 0.350419, 15.000005, 0.3, 255],
[0.199571, 0.349795, 15.000366, 0.3, 255],
[0.200429, 0.350205, 14.999634, 0.3, 255],
[0.199571, 0.349581, 14.999995, 0.3, 255],
[0.200524, 0.349747, 15.000146, 0.3, 255],
[0.200196, 0.349509, 15.000283, 0.3, 255],
[0.199804, 0.350491, 14.999717, 0.3, 255],
[0.199476, 0.350253, 14.999854, 0.3, 255],
[0.200138, 0.349578, 14.999597, 0.3, 255],
[0.200138, 0.350138, 15.000567, 0.3, 255],
[0.199862, 0.349862, 14.999433, 0.3, 255],
[0.199862, 0.350422, 15.000403, 0.3, 255],
[0.200488, 0.350136, 15.000321, 0.3, 255],
[0.199957, 0.349751, 15.000544, 0.3, 255],
[0.200488, 0.34979, 14.999721, 0.3, 255],
[0.199957, 0.349404, 14.999944, 0.3, 255],
[0.200043, 0.350596, 15.000056, 0.3, 255],
[0.199512, 0.35021, 15.000279, 0.3, 255],
[0.200043, 0.350249, 14.999456, 0.3, 255],
[0.199512, 0.349864, 14.999679, 0.3, 255],
[-0.5, 0.500442, 10.000442, 0.2, 0],
[-0.5, 0.5, 10, 0.1995, 255],
[-0.499966, 0.49971, 9.999563, 0.199, 0],
[-0.500034, 0.499563, 9.99971, 0.199, 0],
[-0.499934, 0.499432, 9.999144, 0.1985, 255],
[-0.500066, 0.499144, 9.999432, 0.1985, 255],
[0.6, 0.5, 10, 0.25, 210],
[0.600122, 0.4999, 10.000122, 0.25, 210],
[0.600061, 0.49995, 10.000061, 0.25003, 50],
[0.595856, 0.505256, 9.991271, 0.24, 100],
[0.597973, 0.507461, 9.992175, 0.24, 210],
[0.595362, 0.507926, 9.993945, 0.24, 100],
[0.594618, 0.509055, 9.996832, 0.24, 210],
[0.592382, 0.506906, 9.996092, 0.24, 100],
[0.590438, 0.504882, 9.997604, 0.24, 210],
[0.591035, 0.503605, 9.994744, 0.24, 100],
[0.59121, 0.500708, 9.993425, 0.24, 210],
[0.593181, 0.502585, 9.991764, 0.24, 100],
[0.595867, 0.502302, 9.990069, 0.24, 210],
]
```
Output:
```text
TODO
```
## Scoring
This is a [code golf challenge](https://codegolf.codidact.com/categories/49/tags/4274). Your score is the number of bytes in your code. Lowest score for each language wins.
> Explanations are optional, but I'm more likely to upvote answers that have one.

#6: Post edited by

trichoplax‭ · 2024-07-26T11:00:26Z (5 months ago)
Use consistent variable names between Python and Rust

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Given a list of spheres, output an image of them.
The challenge is to produce a greyscale [orthographic](https://en.wikipedia.org/wiki/Parallel_projection#Orthographic_projection) (parallel projection - no perspective) ray caster, that renders only plain grey spheres with no lighting, no shading, no shadows, no surface patterns or textures, no reflections, and no transparency.
## Input
- A list of spheres, each consisting of:
- The coordinates of its centre $(x, y, z)$.
- Its radius $r$.
- Its greyscale value $g$, an integer from $0$ to $255$ (both inclusive).
- The input may include up to $100$ spheres.
- Only the greyscale value will be restricted to integer values. The coordinates and radius may have fractional values.
- Here "integer" refers to the values the greyscale value make take. You are not required to use an integer variable type.[]()
## Output
- A single greyscale image:
- $1024$ by $1024$ pixels.
- Each pixel has a single greyscale value from $0$ to $255$.
- Horizontally, the image covers x values from $-1$ to $1$.
- Vertically, the image covers y values from $-1$ to $1$.
- The image is orthographic, so there is no perspective. Changing the $z$ coordinate of a sphere does not change its size in the image, instead only affecting whether it is in front of or behind other spheres.
- A sphere's greyscale value is applied to every visible part of it. There is no variation over the surface of the sphere, so it appears flat like a disk.
- The point of the challenge is producing the pixel values, not manipulating image file formats, so you may output whatever image format is convenient, including simple text formats such as [PGM] or even just a list of pixel greyscale values. If you want to include an example of your output as an image in your answer, the image formats that Codidact supports are PNG, JPEG, and GIF. You could output PGM and then convert to PNG to upload to your answer. If you output just the greyscale values as a list of numbers from $0$ to $255$, I've made a `[ TODO page to convert these to greyscale PNG]`.
- The test cases, which match the reference implementations, happen to have the x axis increasing from left to right, and the y axis increasing from top to bottom. You are free to instead interpret these as increasing from right to left for x or from bottom to top for y, resulting in a reflected image. You are also free to output an image rotated by a multiple of 90 degrees, allowing for swapping the x and y axes. However, the z axis is required to be increasing in the forwards direction (into the image) otherwise some of the test cases will look incorrect.
## Calculating a pixel value
Imagine a ray passing into the image through a pixel, perpendicular to the image plane. If it hits a sphere, it shows the greyscale value of the first sphere it hits. Otherwise, it shows a value of zero (black). One possible way of calculating this is as follows:
For each pixel:
- The pixel $x, y$ coordinates will each be from $0$ to $1023$. Scale these to the range $-1$ to $1$ (for example, divide by $512$ and subtract $1$).
- If these scaled $x, y$ coordinates are within the radius $r$ of the $x, y$ coordinates of a sphere's centre, the ray intersects that sphere (the $z$ coordinate is not relevant to this test because the image is an orthographic projection - the rays are all parallel to the $z$ axis). The distance $a$ of the ray from the sphere's centre can be calculated as follows:
$$a=\sqrt{(x_{ray}-x_{sphere})^2+(y_{ray}-y_{sphere})^2}$$
```text
CONSIDER A DIAGRAM HERE
```
- For each sphere that the ray intersects, find the $z$ coordinate of the nearest intersection (there will often be $2$ per sphere, $1$ on the front of the sphere and $1$ on the back of the sphere, as the ray passes through - only the nearest is relevant). This can be calculated as follows:
- The distance $d$ to the nearest intersection point can be calculated from the $z$ coordinate of the sphere's centre, the radius $r$, and the distance $a$ from the $x, y$ coordinates of the sphere's centre to the scaled $x, y$ coordinates of the pixel:
$$d=z_{sphere}-\sqrt{r_{sphere}^2-a^2}$$
```text
CONSIDER A DIAGRAM HERE
```
- Of all the spheres that the ray intersects, the sphere with the intersection point with the smallest $z$ coordinate $d$ is the one to display. Set the pixel to the greyscale value of this sphere.
- If the ray did not intersect any spheres, set the pixel to zero (black).
## Example implementations
These are ungolfed implementations designed to supplement understanding of the requirements. You are free to golf these or to implement your own approach that matches the test cases up to reflection and rotation.
The Python implementation uses 64 bit floating point giving accurate images. The Rust implementation deliberately uses 32 bit floating point to show the difference in output. I've tried to optimise the test cases for compatibility with 32 bit floating point, so the differences are restricted to being a pixel out in some places.
### Python
The function `grey_values` would be sufficient as an entry to this challenge. The rest of the code saves the outputs as PNG files. The function itself has no dependence on the `import png` so it would not be required in an entry.
```python
def grey_values(spheres):
values = []
for y_pixel in range(1024):
for x_pixel in range(1024):
x_ray, y_ray = x_pixel / 512 - 1, y_pixel / 512 - 1
min_distance, hit_grey_value = None, 0
for sphere in spheres:
(
~~x_sphere, y_sphere, z_sphere, radius, grey_value~~
~~) = sphere~~
a = (
(x_ray - x_sphere) ** 2 + (y_ray - y_sphere) ** 2
) ** 0.5
if a < radius:
distance = z_sphere - (radius ** 2 - a ** 2) ** 0.5
if min_distance is None or distance < min_distance:
min_distance = distance
~~hit_grey_value = grey_value~~
values.append(hit_grey_value)
return values
import png
test_cases = [
[
[0, 0, 0, 1, 128],
],
[
[0, -0.55, 0, 0.4, 100],
[0, 0.55, 0, 0.4, 200],
],
[
[0.75, 0, 0, 0.25, 64],
[0.53033, 0.375, -0.375, 0.25, 185],
[0, 0.53033, -0.53033, 0.25, 64],
[-0.53033, 0.375, -0.375, 0.25, 185],
[-0.75, 0, -0, 0.25, 64],
[-0.53033, -0.375, 0.375, 0.25, 185],
[-0, -0.53033, 0.53033, 0.25, 64],
[0.53033, -0.375, 0.375, 0.25, 185],
],
[
[0.001, 0, 0, 1, 64],
[0.000707, 0.000707, -0, 1, 85],
[0, 0.001, -0, 1, 106],
[-0.000707, 0.000707, -0, 1, 127],
[-0.001, 0, -0, 1, 148],
[-0.000707, -0.000707, 0, 1, 169],
[-0, -0.001, 0, 1, 190],
[0.000707, -0.000707, 0, 1, 211],
],
[
[0, 0.000058, 0.001947, 1, 0],
[0, -0.001715, 0.000923, 1, 0],
[0, 0.001715, -0.000923, 1, 0],
[0, -0.000058, -0.001947, 1, 0],
[0.001657, -0.000512, 0.000887, 1, 0],
[-0.001657, -0.000512, 0.000887, 1, 0],
[0.001657, 0.000512, -0.000887, 1, 0],
[-0.001657, 0.000512, -0.000887, 1, 0],
[0.001024, 0.001435, 0.000829, 1, 0],
[-0.001024, 0.001435, 0.000829, 1, 0],
[0.001024, -0.001435, -0.000829, 1, 0],
[-0.001024, -0.001435, -0.000829, 1, 0],
[0.001868, 0.000618, 0.000357, 1, 255],
[-0.001868, 0.000618, 0.000357, 1, 255],
[0.001868, -0.000618, -0.000357, 1, 255],
[-0.001868, -0.000618, -0.000357, 1, 255],
[0.000714, -0.000934, 0.001618, 1, 255],
[-0.000714, -0.000934, 0.001618, 1, 255],
[0.000714, 0.000934, -0.001618, 1, 255],
[-0.000714, 0.000934, -0.001618, 1, 255],
[0, -0.001975, -0.000316, 1, 255],
[0, 0.001261, 0.001552, 1, 255],
[0, -0.001261, -0.001552, 1, 255],
[0, 0.001975, 0.000316, 1, 255],
[0.001155, 0.000423, 0.001577, 1, 255],
[-0.001155, 0.000423, 0.001577, 1, 255],
[0.001155, -0.001577, 0.000423, 1, 255],
[-0.001155, -0.001577, 0.000423, 1, 255],
[0.001155, 0.001577, -0.000423, 1, 255],
[-0.001155, 0.001577, -0.000423, 1, 255],
[0.001155, -0.000423, -0.001577, 1, 255],
[-0.001155, -0.000423, -0.001577, 1, 255],
],
[
[0, 0, 1000, 2, 128],
[0, 0, 0.003125, 1, 0],
[0, 0, 0, 0.9975, 255],
[0, 0.000547, -0.002573, 0.995, 0],
[-0, -0.000547, -0.002573, 0.995, 0],
[0, 0.001071, -0.005037, 0.9925, 255],
[-0, -0.001071, -0.005037, 0.9925, 255],
],
[
[0, 0, 0, 1, 210],
[0, 0, 0.0008, 1, 210],
[0, 0, 0.0004, 1.00012, 50],
[0.012969, -0, -0.042045, 0.96, 100],
[0.016397, 0.011913, -0.039054, 0.96, 210],
[0.004008, 0.012334, -0.042045, 0.96, 100],
[-0.006263, 0.019276, -0.039054, 0.96, 210],
[-0.010492, 0.007623, -0.042045, 0.96, 100],
[-0.020268, 0, -0.039054, 0.96, 210],
[-0.010492, -0.007623, -0.042045, 0.96, 100],
[-0.006263, -0.019276, -0.039054, 0.96, 210],
[0.004008, -0.012334, -0.042045, 0.96, 100],
[0.016397, -0.011913, -0.039054, 0.96, 210],
],
[
[0, 0, 1000, 2, 192],
[-2, 2, 100, 3, 98],
[2, 2, 100, 3, 98],
[0.5, -0.5, 500, 0.2, 255],
[-0.349469, -0.1, 20.000279, 0.6, 64],
[-0.349775, -0.100273, 20.000485, 0.6, 85],
[-0.350214, -0.100386, 20.000407, 0.6, 106],
[-0.350527, -0.100273, 20.000091, 0.6, 127],
[-0.350531, -0.1, 19.999721, 0.6, 148],
[-0.350225, -0.099727, 19.999515, 0.6, 169],
[-0.349786, -0.099614, 19.999593, 0.6, 190],
[-0.349473, -0.099727, 19.999909, 0.6, 211],
[0.20032, 0.349824, 15.000456, 0.3, 0],
[0.20032, 0.349517, 14.999924, 0.3, 0],
[0.19968, 0.350483, 15.000076, 0.3, 0],
[0.19968, 0.350176, 14.999544, 0.3, 0],
[0.200578, 0.350073, 14.999958, 0.3, 0],
[0.199817, 0.349519, 15.000277, 0.3, 0],
[0.200183, 0.350481, 14.999723, 0.3, 0],
[0.199422, 0.349927, 15.000042, 0.3, 0],
[0.200235, 0.35042, 15.000332, 0.3, 0],
[0.199765, 0.350078, 15.000529, 0.3, 0],
[0.200235, 0.349922, 14.999471, 0.3, 0],
[0.199765, 0.34958, 14.999668, 0.3, 0],
[0.200429, 0.350419, 15.000005, 0.3, 255],
[0.199571, 0.349795, 15.000366, 0.3, 255],
[0.200429, 0.350205, 14.999634, 0.3, 255],
[0.199571, 0.349581, 14.999995, 0.3, 255],
[0.200524, 0.349747, 15.000146, 0.3, 255],
[0.200196, 0.349509, 15.000283, 0.3, 255],
[0.199804, 0.350491, 14.999717, 0.3, 255],
[0.199476, 0.350253, 14.999854, 0.3, 255],
[0.200138, 0.349578, 14.999597, 0.3, 255],
[0.200138, 0.350138, 15.000567, 0.3, 255],
[0.199862, 0.349862, 14.999433, 0.3, 255],
[0.199862, 0.350422, 15.000403, 0.3, 255],
[0.200488, 0.350136, 15.000321, 0.3, 255],
[0.199957, 0.349751, 15.000544, 0.3, 255],
[0.200488, 0.34979, 14.999721, 0.3, 255],
[0.199957, 0.349404, 14.999944, 0.3, 255],
[0.200043, 0.350596, 15.000056, 0.3, 255],
[0.199512, 0.35021, 15.000279, 0.3, 255],
[0.200043, 0.350249, 14.999456, 0.3, 255],
[0.199512, 0.349864, 14.999679, 0.3, 255],
[-0.5, 0.500442, 10.000442, 0.2, 0],
[-0.5, 0.5, 10, 0.1995, 255],
[-0.499966, 0.49971, 9.999563, 0.199, 0],
[-0.500034, 0.499563, 9.99971, 0.199, 0],
[-0.499934, 0.499432, 9.999144, 0.1985, 255],
[-0.500066, 0.499144, 9.999432, 0.1985, 255],
[0.6, 0.5, 10, 0.25, 210],
[0.600122, 0.4999, 10.000122, 0.25, 210],
[0.600061, 0.49995, 10.000061, 0.25003, 50],
[0.595856, 0.505256, 9.991271, 0.24, 100],
[0.597973, 0.507461, 9.992175, 0.24, 210],
[0.595362, 0.507926, 9.993945, 0.24, 100],
[0.594618, 0.509055, 9.996832, 0.24, 210],
[0.592382, 0.506906, 9.996092, 0.24, 100],
[0.590438, 0.504882, 9.997604, 0.24, 210],
[0.591035, 0.503605, 9.994744, 0.24, 100],
[0.59121, 0.500708, 9.993425, 0.24, 210],
[0.593181, 0.502585, 9.991764, 0.24, 100],
[0.595867, 0.502302, 9.990069, 0.24, 210],
],
]
scene_names = (
"single_sphere",
"two_spheres",
"ring_of_spheres",
"beachball",
"football",
"eightball",
"starball",
"combination_scene",
)
for i in range(8):
print(f'Generating {scene_names[i]}.')
values = grey_values(test_cases[i])
rows = [values[row * 1024:row * 1024 + 1024] for row in range(1024)]
with open(f'{scene_names[i]}.png', 'wb') as file:
w = png.Writer(1024, 1024, greyscale=True)
w.write(file, rows)
```
### Rust
The function `grey_values` would be sufficient as an entry to this challenge. The rest of the code saves the outputs as PNG files.
```rust
fn grey_values(spheres: &Vec<[f32; 5]>) -> Vec<u8> {
let mut values = vec![];
for y_pixel in 0..1024 {
for x_pixel in 0..1024 {
let x_ray = x_pixel as f32 / 512.0 - 1.0;
let y_ray = y_pixel as f32 / 512.0 - 1.0;
let mut min_distance = f32::INFINITY;
let mut hit_grey_value = 0;
for sphere in spheres {
let [
x_sphere, y_sphere, z_sphere, radius, grey
] = *sphere;
let a = (
(x_ray - x_sphere).powf(2.0)
+ (y_ray - y_sphere).powf(2.0)
).sqrt();
if a < radius {
let distance = z_sphere - (
radius.powf(2.0) - a.powf(2.0)
).sqrt();
if distance < min_distance {
min_distance = distance;
hit_grey_value = grey as u8;
}
}
}
values.push(hit_grey_value);
}
}
values
}
fn main() {
let test_cases: Vec<Vec<[f32; 5]>> = vec![
vec![
[0.0, 0.0, 0.0, 1.0, 128.0]
],
vec![
[0.0, -0.55, 0.0, 0.4, 100.0],
[0.0, 0.55, 0.0, 0.4, 200.0]
],
vec![
[0.75, 0.0, 0.0, 0.25, 64.0],
[0.53033, 0.375, -0.375, 0.25, 185.0],
[0.0, 0.53033, -0.53033, 0.25, 64.0],
[-0.53033, 0.375, -0.375, 0.25, 185.0],
[-0.75, 0.0, 0.0, 0.25, 64.0],
[-0.53033, -0.375, 0.375, 0.25, 185.0],
[0.0, -0.53033, 0.53033, 0.25, 64.0],
[0.53033, -0.375, 0.375, 0.25, 185.0],
],
vec![
[0.001, 0.0, 0.0, 1.0, 64.0],
[0.000707, 0.000707, 0.0, 1.0, 85.0],
[0.0, 0.001, 0.0, 1.0, 106.0],
[-0.000707, 0.000707, 0.0, 1.0, 127.0],
[-0.001, 0.0, 0.0, 1.0, 148.0],
[-0.000707, -0.000707, 0.0, 1.0, 169.0],
[0.0, -0.001, 0.0, 1.0, 190.0],
[0.000707, -0.000707, 0.0, 1.0, 211.0],
],
vec![
[0.0, 0.000058, 0.001947, 1.0, 0.0],
[0.0, -0.001715, 0.000923, 1.0, 0.0],
[0.0, 0.001715, -0.000923, 1.0, 0.0],
[0.0, -0.000058, -0.001947, 1.0, 0.0],
[0.001657, -0.000512, 0.000887, 1.0, 0.0],
[-0.001657, -0.000512, 0.000887, 1.0, 0.0],
[0.001657, 0.000512, -0.000887, 1.0, 0.0],
[-0.001657, 0.000512, -0.000887, 1.0, 0.0],
[0.001024, 0.001435, 0.000829, 1.0, 0.0],
[-0.001024, 0.001435, 0.000829, 1.0, 0.0],
[0.001024, -0.001435, -0.000829, 1.0, 0.0],
[-0.001024, -0.001435, -0.000829, 1.0, 0.0],
[0.001868, 0.000618, 0.000357, 1.0, 255.0],
[-0.001868, 0.000618, 0.000357, 1.0, 255.0],
[0.001868, -0.000618, -0.000357, 1.0, 255.0],
[-0.001868, -0.000618, -0.000357, 1.0, 255.0],
[0.000714, -0.000934, 0.001618, 1.0, 255.0],
[-0.000714, -0.000934, 0.001618, 1.0, 255.0],
[0.000714, 0.000934, -0.001618, 1.0, 255.0],
[-0.000714, 0.000934, -0.001618, 1.0, 255.0],
[0.0, -0.001975, -0.000316, 1.0, 255.0],
[0.0, 0.001261, 0.001552, 1.0, 255.0],
[0.0, -0.001261, -0.001552, 1.0, 255.0],
[0.0, 0.001975, 0.000316, 1.0, 255.0],
[0.001155, 0.000423, 0.001577, 1.0, 255.0],
[-0.001155, 0.000423, 0.001577, 1.0, 255.0],
[0.001155, -0.001577, 0.000423, 1.0, 255.0],
[-0.001155, -0.001577, 0.000423, 1.0, 255.0],
[0.001155, 0.001577, -0.000423, 1.0, 255.0],
[-0.001155, 0.001577, -0.000423, 1.0, 255.0],
[0.001155, -0.000423, -0.001577, 1.0, 255.0],
[-0.001155, -0.000423, -0.001577, 1.0, 255.0],
],
vec![
[0.0, 0.0, 1000.0, 2.0, 128.0],
[0.0, 0.0, 0.003125, 1.0, 0.0],
[0.0, 0.0, 0.0, 0.9975, 255.0],
[0.0, 0.000547, -0.002573, 0.995, 0.0],
[0.0, -0.000547, -0.002573, 0.995, 0.0],
[0.0, 0.001071, -0.005037, 0.9925, 255.0],
[0.0, -0.001071, -0.005037, 0.9925, 255.0],
],
vec![
[0.0, 0.0, 0.0, 1.0, 210.0],
[0.0, 0.0, 0.0008, 1.0, 210.0],
[0.0, 0.0, 0.0004, 1.00012, 50.0],
[0.012969, 0.0, -0.042045, 0.96, 100.0],
[0.016397, 0.011913, -0.039054, 0.96, 210.0],
[0.004008, 0.012334, -0.042045, 0.96, 100.0],
[-0.006263, 0.019276, -0.039054, 0.96, 210.0],
[-0.010492, 0.007623, -0.042045, 0.96, 100.0],
[-0.020268, 0.0, -0.039054, 0.96, 210.0],
[-0.010492, -0.007623, -0.042045, 0.96, 100.0],
[-0.006263, -0.019276, -0.039054, 0.96, 210.0],
[0.004008, -0.012334, -0.042045, 0.96, 100.0],
[0.016397, -0.011913, -0.039054, 0.96, 210.0],
],
vec![
[0.0, 0.0, 1000.0, 2.0, 192.0],
[-2.0, 2.0, 100.0, 3.0, 98.0],
[2.0, 2.0, 100.0, 3.0, 98.0],
[0.5, -0.5, 500.0, 0.2, 255.0],
[-0.349469, -0.1, 20.000279, 0.6, 64.0],
[-0.349775, -0.100273, 20.000485, 0.6, 85.0],
[-0.350214, -0.100386, 20.000407, 0.6, 106.0],
[-0.350527, -0.100273, 20.000091, 0.6, 127.0],
[-0.350531, -0.1, 19.999721, 0.6, 148.0],
[-0.350225, -0.099727, 19.999515, 0.6, 169.0],
[-0.349786, -0.099614, 19.999593, 0.6, 190.0],
[-0.349473, -0.099727, 19.999909, 0.6, 211.0],
[0.20032, 0.349824, 15.000456, 0.3, 0.0],
[0.20032, 0.349517, 14.999924, 0.3, 0.0],
[0.19968, 0.350483, 15.000076, 0.3, 0.0],
[0.19968, 0.350176, 14.999544, 0.3, 0.0],
[0.200578, 0.350073, 14.999958, 0.3, 0.0],
[0.199817, 0.349519, 15.000277, 0.3, 0.0],
[0.200183, 0.350481, 14.999723, 0.3, 0.0],
[0.199422, 0.349927, 15.000042, 0.3, 0.0],
[0.200235, 0.35042, 15.000332, 0.3, 0.0],
[0.199765, 0.350078, 15.000529, 0.3, 0.0],
[0.200235, 0.349922, 14.999471, 0.3, 0.0],
[0.199765, 0.34958, 14.999668, 0.3, 0.0],
[0.200429, 0.350419, 15.000005, 0.3, 255.0],
[0.199571, 0.349795, 15.000366, 0.3, 255.0],
[0.200429, 0.350205, 14.999634, 0.3, 255.0],
[0.199571, 0.349581, 14.999995, 0.3, 255.0],
[0.200524, 0.349747, 15.000146, 0.3, 255.0],
[0.200196, 0.349509, 15.000283, 0.3, 255.0],
[0.199804, 0.350491, 14.999717, 0.3, 255.0],
[0.199476, 0.350253, 14.999854, 0.3, 255.0],
[0.200138, 0.349578, 14.999597, 0.3, 255.0],
[0.200138, 0.350138, 15.000567, 0.3, 255.0],
[0.199862, 0.349862, 14.999433, 0.3, 255.0],
[0.199862, 0.350422, 15.000403, 0.3, 255.0],
[0.200488, 0.350136, 15.000321, 0.3, 255.0],
[0.199957, 0.349751, 15.000544, 0.3, 255.0],
[0.200488, 0.34979, 14.999721, 0.3, 255.0],
[0.199957, 0.349404, 14.999944, 0.3, 255.0],
[0.200043, 0.350596, 15.000056, 0.3, 255.0],
[0.199512, 0.35021, 15.000279, 0.3, 255.0],
[0.200043, 0.350249, 14.999456, 0.3, 255.0],
[0.199512, 0.349864, 14.999679, 0.3, 255.0],
[-0.5, 0.500442, 10.000442, 0.2, 0.0],
[-0.5, 0.5, 10.0, 0.1995, 255.0],
[-0.499966, 0.49971, 9.999563, 0.199, 0.0],
[-0.500034, 0.499563, 9.99971, 0.199, 0.0],
[-0.499934, 0.499432, 9.999144, 0.1985, 255.0],
[-0.500066, 0.499144, 9.999432, 0.1985, 255.0],
[0.6, 0.5, 10.0, 0.25, 210.0],
[0.600122, 0.4999, 10.000122, 0.25, 210.0],
[0.600061, 0.49995, 10.000061, 0.25003, 50.0],
[0.595856, 0.505256, 9.991271, 0.24, 100.0],
[0.597973, 0.507461, 9.992175, 0.24, 210.0],
[0.595362, 0.507926, 9.993945, 0.24, 100.0],
[0.594618, 0.509055, 9.996832, 0.24, 210.0],
[0.592382, 0.506906, 9.996092, 0.24, 100.0],
[0.590438, 0.504882, 9.997604, 0.24, 210.0],
[0.591035, 0.503605, 9.994744, 0.24, 100.0],
[0.59121, 0.500708, 9.993425, 0.24, 210.0],
[0.593181, 0.502585, 9.991764, 0.24, 100.0],
[0.595867, 0.502302, 9.990069, 0.24, 210.0],
],
];
let scene_names = [
"single_sphere",
"two_spheres",
"ring_of_spheres",
"beachball",
"football",
"eightball",
"starball",
"combination_scene",
];
for i in 0..8 {
eprintln!("Generating {}", scene_names[i]);
let values = grey_values(&test_cases[i]);
let filename = format!("{}.png", scene_names[i]);
let path = std::path::Path::new(&filename);
let file = std::fs::File::create(path).unwrap();
let w = &mut std::io::BufWriter::new(file);
let mut encoder = png::Encoder::new(w, 1024, 1024);
encoder.set_color(png::ColorType::Grayscale);
encoder.set_depth(png::BitDepth::Eight);
let mut writer = encoder.write_header().unwrap();
writer.write_image_data(&values).unwrap();
}
}
```
## Test cases
The images may be scaled down. Clicking or tapping on an image will open it full size.
### Single sphere
Input:
```text
TODO
```
Output:
```text
TODO
```
### Two separated spheres
Input:
```text
TODO
```
Output:
```text
TODO
```
### Eight spheres in a ring
Input:
```text
TODO
```
Output:
```text
TODO
```
### Beach ball
Input:
```text
TODO
```
Output:
```text
TODO
```
### Football
Input:
```text
TODO
```
Output:
```text
TODO
```
### Eight ball
Input:
```text
TODO
```
Output:
```text
TODO
```
### Star ball
Input:
```text
TODO
```
Output:
```text
TODO
```
### Combination scene
Input:
```text
TODO
```
Output:
```text
TODO
```
## Explanatory animations
The example images were produced from plain grey spheres with no surface pattern or texture. The following animations show how they break down into their component plain spheres.
This section is not necessary to write valid code for this challenge. If your code accepts the parameters of plain grey spheres and renders them correctly, all of the test cases will look right without needing to understand how they were constructed. This section is mainly out of interest, and to make clear that your code *only* needs to render plain grey spheres.
```text
CONSIDER HOSTING THESE OFF SITE AS THEY ARE NOT REQUIRED FOR THE CHALLENGE
AND EVEN THE SCALED DOWN VERSIONS ARE UNNECESSARY DATA USE FOR ANYONE
CHOOSING NOT TO VIEW THEM.
```
<details>
<summary>Animations</summary>
<details>
<summary>Beach ball</summary>
This animation is reduced to 256 by 256 pixels. Clicking or tapping on the animation will open it at full size (______MB).
```text
TODO
```
</details>
<details>
<summary>Football</summary>
This animation is reduced to 256 by 256 pixels. Clicking or tapping on the animation will open it at full size (______MB).
```text
TODO
```
</details>
<details>
<summary>Eight ball</summary>
This animation is reduced to 256 by 256 pixels. Clicking or tapping on the animation will open it at full size (______MB).
```text
TODO
```
</details>
<details>
<summary>Star ball</summary>
This animation is reduced to 256 by 256 pixels. Clicking or tapping on the animation will open it at full size (______MB).
```text
TODO
```
</details>
</details>
## Scoring
This is a [code golf challenge]. Your score is the number of bytes in your code. Lowest score for each language wins.
> Explanations are optional, but I'm more likely to upvote answers that have one.
[code golf challenge]: https://codegolf.codidact.com/categories/49/tags/4274 "The code-golf tag"
[PGM]: https://en.wikipedia.org/wiki/Netpbm#PGM_example

Given a list of spheres, output an image of them.
The challenge is to produce a greyscale [orthographic](https://en.wikipedia.org/wiki/Parallel_projection#Orthographic_projection) (parallel projection - no perspective) ray caster, that renders only plain grey spheres with no lighting, no shading, no shadows, no surface patterns or textures, no reflections, and no transparency.
## Input
- A list of spheres, each consisting of:
- The coordinates of its centre $(x, y, z)$.
- Its radius $r$.
- Its greyscale value $g$, an integer from $0$ to $255$ (both inclusive).
- The input may include up to $100$ spheres.
- Only the greyscale value will be restricted to integer values. The coordinates and radius may have fractional values.
- Here "integer" refers to the values the greyscale value make take. You are not required to use an integer variable type.[]()
## Output
- A single greyscale image:
- $1024$ by $1024$ pixels.
- Each pixel has a single greyscale value from $0$ to $255$.
- Horizontally, the image covers x values from $-1$ to $1$.
- Vertically, the image covers y values from $-1$ to $1$.
- The image is orthographic, so there is no perspective. Changing the $z$ coordinate of a sphere does not change its size in the image, instead only affecting whether it is in front of or behind other spheres.
- A sphere's greyscale value is applied to every visible part of it. There is no variation over the surface of the sphere, so it appears flat like a disk.
- The point of the challenge is producing the pixel values, not manipulating image file formats, so you may output whatever image format is convenient, including simple text formats such as [PGM] or even just a list of pixel greyscale values. If you want to include an example of your output as an image in your answer, the image formats that Codidact supports are PNG, JPEG, and GIF. You could output PGM and then convert to PNG to upload to your answer. If you output just the greyscale values as a list of numbers from $0$ to $255$, I've made a `[ TODO page to convert these to greyscale PNG]`.
- The test cases, which match the reference implementations, happen to have the x axis increasing from left to right, and the y axis increasing from top to bottom. You are free to instead interpret these as increasing from right to left for x or from bottom to top for y, resulting in a reflected image. You are also free to output an image rotated by a multiple of 90 degrees, allowing for swapping the x and y axes. However, the z axis is required to be increasing in the forwards direction (into the image) otherwise some of the test cases will look incorrect.
## Calculating a pixel value
Imagine a ray passing into the image through a pixel, perpendicular to the image plane. If it hits a sphere, it shows the greyscale value of the first sphere it hits. Otherwise, it shows a value of zero (black). One possible way of calculating this is as follows:
For each pixel:
- The pixel $x, y$ coordinates will each be from $0$ to $1023$. Scale these to the range $-1$ to $1$ (for example, divide by $512$ and subtract $1$).
- If these scaled $x, y$ coordinates are within the radius $r$ of the $x, y$ coordinates of a sphere's centre, the ray intersects that sphere (the $z$ coordinate is not relevant to this test because the image is an orthographic projection - the rays are all parallel to the $z$ axis). The distance $a$ of the ray from the sphere's centre can be calculated as follows:
$$a=\sqrt{(x_{ray}-x_{sphere})^2+(y_{ray}-y_{sphere})^2}$$
```text
CONSIDER A DIAGRAM HERE
```
- For each sphere that the ray intersects, find the $z$ coordinate of the nearest intersection (there will often be $2$ per sphere, $1$ on the front of the sphere and $1$ on the back of the sphere, as the ray passes through - only the nearest is relevant). This can be calculated as follows:
- The distance $d$ to the nearest intersection point can be calculated from the $z$ coordinate of the sphere's centre, the radius $r$, and the distance $a$ from the $x, y$ coordinates of the sphere's centre to the scaled $x, y$ coordinates of the pixel:
$$d=z_{sphere}-\sqrt{r_{sphere}^2-a^2}$$
```text
CONSIDER A DIAGRAM HERE
```
- Of all the spheres that the ray intersects, the sphere with the intersection point with the smallest $z$ coordinate $d$ is the one to display. Set the pixel to the greyscale value of this sphere.
- If the ray did not intersect any spheres, set the pixel to zero (black).
## Example implementations
These are ungolfed implementations designed to supplement understanding of the requirements. You are free to golf these or to implement your own approach that matches the test cases up to reflection and rotation.
The Python implementation uses 64 bit floating point giving accurate images. The Rust implementation deliberately uses 32 bit floating point to show the difference in output. I've tried to optimise the test cases for compatibility with 32 bit floating point, so the differences are restricted to being a pixel out in some places.
### Python
The function `grey_values` would be sufficient as an entry to this challenge. The rest of the code saves the outputs as PNG files. The function itself has no dependence on the `import png` so it would not be required in an entry.
```python
def grey_values(spheres):
values = []
for y_pixel in range(1024):
for x_pixel in range(1024):
x_ray, y_ray = x_pixel / 512 - 1, y_pixel / 512 - 1
min_distance, hit_grey_value = None, 0
for sphere in spheres:
x_sphere, y_sphere, z_sphere, radius, grey = sphere
a = (
(x_ray - x_sphere) ** 2 + (y_ray - y_sphere) ** 2
) ** 0.5
if a < radius:
distance = z_sphere - (radius ** 2 - a ** 2) ** 0.5
if min_distance is None or distance < min_distance:
min_distance = distance
hit_grey_value = grey
values.append(hit_grey_value)
return values
import png
test_cases = [
[
[0, 0, 0, 1, 128],
],
[
[0, -0.55, 0, 0.4, 100],
[0, 0.55, 0, 0.4, 200],
],
[
[0.75, 0, 0, 0.25, 64],
[0.53033, 0.375, -0.375, 0.25, 185],
[0, 0.53033, -0.53033, 0.25, 64],
[-0.53033, 0.375, -0.375, 0.25, 185],
[-0.75, 0, -0, 0.25, 64],
[-0.53033, -0.375, 0.375, 0.25, 185],
[-0, -0.53033, 0.53033, 0.25, 64],
[0.53033, -0.375, 0.375, 0.25, 185],
],
[
[0.001, 0, 0, 1, 64],
[0.000707, 0.000707, -0, 1, 85],
[0, 0.001, -0, 1, 106],
[-0.000707, 0.000707, -0, 1, 127],
[-0.001, 0, -0, 1, 148],
[-0.000707, -0.000707, 0, 1, 169],
[-0, -0.001, 0, 1, 190],
[0.000707, -0.000707, 0, 1, 211],
],
[
[0, 0.000058, 0.001947, 1, 0],
[0, -0.001715, 0.000923, 1, 0],
[0, 0.001715, -0.000923, 1, 0],
[0, -0.000058, -0.001947, 1, 0],
[0.001657, -0.000512, 0.000887, 1, 0],
[-0.001657, -0.000512, 0.000887, 1, 0],
[0.001657, 0.000512, -0.000887, 1, 0],
[-0.001657, 0.000512, -0.000887, 1, 0],
[0.001024, 0.001435, 0.000829, 1, 0],
[-0.001024, 0.001435, 0.000829, 1, 0],
[0.001024, -0.001435, -0.000829, 1, 0],
[-0.001024, -0.001435, -0.000829, 1, 0],
[0.001868, 0.000618, 0.000357, 1, 255],
[-0.001868, 0.000618, 0.000357, 1, 255],
[0.001868, -0.000618, -0.000357, 1, 255],
[-0.001868, -0.000618, -0.000357, 1, 255],
[0.000714, -0.000934, 0.001618, 1, 255],
[-0.000714, -0.000934, 0.001618, 1, 255],
[0.000714, 0.000934, -0.001618, 1, 255],
[-0.000714, 0.000934, -0.001618, 1, 255],
[0, -0.001975, -0.000316, 1, 255],
[0, 0.001261, 0.001552, 1, 255],
[0, -0.001261, -0.001552, 1, 255],
[0, 0.001975, 0.000316, 1, 255],
[0.001155, 0.000423, 0.001577, 1, 255],
[-0.001155, 0.000423, 0.001577, 1, 255],
[0.001155, -0.001577, 0.000423, 1, 255],
[-0.001155, -0.001577, 0.000423, 1, 255],
[0.001155, 0.001577, -0.000423, 1, 255],
[-0.001155, 0.001577, -0.000423, 1, 255],
[0.001155, -0.000423, -0.001577, 1, 255],
[-0.001155, -0.000423, -0.001577, 1, 255],
],
[
[0, 0, 1000, 2, 128],
[0, 0, 0.003125, 1, 0],
[0, 0, 0, 0.9975, 255],
[0, 0.000547, -0.002573, 0.995, 0],
[-0, -0.000547, -0.002573, 0.995, 0],
[0, 0.001071, -0.005037, 0.9925, 255],
[-0, -0.001071, -0.005037, 0.9925, 255],
],
[
[0, 0, 0, 1, 210],
[0, 0, 0.0008, 1, 210],
[0, 0, 0.0004, 1.00012, 50],
[0.012969, -0, -0.042045, 0.96, 100],
[0.016397, 0.011913, -0.039054, 0.96, 210],
[0.004008, 0.012334, -0.042045, 0.96, 100],
[-0.006263, 0.019276, -0.039054, 0.96, 210],
[-0.010492, 0.007623, -0.042045, 0.96, 100],
[-0.020268, 0, -0.039054, 0.96, 210],
[-0.010492, -0.007623, -0.042045, 0.96, 100],
[-0.006263, -0.019276, -0.039054, 0.96, 210],
[0.004008, -0.012334, -0.042045, 0.96, 100],
[0.016397, -0.011913, -0.039054, 0.96, 210],
],
[
[0, 0, 1000, 2, 192],
[-2, 2, 100, 3, 98],
[2, 2, 100, 3, 98],
[0.5, -0.5, 500, 0.2, 255],
[-0.349469, -0.1, 20.000279, 0.6, 64],
[-0.349775, -0.100273, 20.000485, 0.6, 85],
[-0.350214, -0.100386, 20.000407, 0.6, 106],
[-0.350527, -0.100273, 20.000091, 0.6, 127],
[-0.350531, -0.1, 19.999721, 0.6, 148],
[-0.350225, -0.099727, 19.999515, 0.6, 169],
[-0.349786, -0.099614, 19.999593, 0.6, 190],
[-0.349473, -0.099727, 19.999909, 0.6, 211],
[0.20032, 0.349824, 15.000456, 0.3, 0],
[0.20032, 0.349517, 14.999924, 0.3, 0],
[0.19968, 0.350483, 15.000076, 0.3, 0],
[0.19968, 0.350176, 14.999544, 0.3, 0],
[0.200578, 0.350073, 14.999958, 0.3, 0],
[0.199817, 0.349519, 15.000277, 0.3, 0],
[0.200183, 0.350481, 14.999723, 0.3, 0],
[0.199422, 0.349927, 15.000042, 0.3, 0],
[0.200235, 0.35042, 15.000332, 0.3, 0],
[0.199765, 0.350078, 15.000529, 0.3, 0],
[0.200235, 0.349922, 14.999471, 0.3, 0],
[0.199765, 0.34958, 14.999668, 0.3, 0],
[0.200429, 0.350419, 15.000005, 0.3, 255],
[0.199571, 0.349795, 15.000366, 0.3, 255],
[0.200429, 0.350205, 14.999634, 0.3, 255],
[0.199571, 0.349581, 14.999995, 0.3, 255],
[0.200524, 0.349747, 15.000146, 0.3, 255],
[0.200196, 0.349509, 15.000283, 0.3, 255],
[0.199804, 0.350491, 14.999717, 0.3, 255],
[0.199476, 0.350253, 14.999854, 0.3, 255],
[0.200138, 0.349578, 14.999597, 0.3, 255],
[0.200138, 0.350138, 15.000567, 0.3, 255],
[0.199862, 0.349862, 14.999433, 0.3, 255],
[0.199862, 0.350422, 15.000403, 0.3, 255],
[0.200488, 0.350136, 15.000321, 0.3, 255],
[0.199957, 0.349751, 15.000544, 0.3, 255],
[0.200488, 0.34979, 14.999721, 0.3, 255],
[0.199957, 0.349404, 14.999944, 0.3, 255],
[0.200043, 0.350596, 15.000056, 0.3, 255],
[0.199512, 0.35021, 15.000279, 0.3, 255],
[0.200043, 0.350249, 14.999456, 0.3, 255],
[0.199512, 0.349864, 14.999679, 0.3, 255],
[-0.5, 0.500442, 10.000442, 0.2, 0],
[-0.5, 0.5, 10, 0.1995, 255],
[-0.499966, 0.49971, 9.999563, 0.199, 0],
[-0.500034, 0.499563, 9.99971, 0.199, 0],
[-0.499934, 0.499432, 9.999144, 0.1985, 255],
[-0.500066, 0.499144, 9.999432, 0.1985, 255],
[0.6, 0.5, 10, 0.25, 210],
[0.600122, 0.4999, 10.000122, 0.25, 210],
[0.600061, 0.49995, 10.000061, 0.25003, 50],
[0.595856, 0.505256, 9.991271, 0.24, 100],
[0.597973, 0.507461, 9.992175, 0.24, 210],
[0.595362, 0.507926, 9.993945, 0.24, 100],
[0.594618, 0.509055, 9.996832, 0.24, 210],
[0.592382, 0.506906, 9.996092, 0.24, 100],
[0.590438, 0.504882, 9.997604, 0.24, 210],
[0.591035, 0.503605, 9.994744, 0.24, 100],
[0.59121, 0.500708, 9.993425, 0.24, 210],
[0.593181, 0.502585, 9.991764, 0.24, 100],
[0.595867, 0.502302, 9.990069, 0.24, 210],
],
]
scene_names = (
"single_sphere",
"two_spheres",
"ring_of_spheres",
"beachball",
"football",
"eightball",
"starball",
"combination_scene",
)
for i in range(8):
print(f'Generating {scene_names[i]}.')
values = grey_values(test_cases[i])
rows = [values[row * 1024:row * 1024 + 1024] for row in range(1024)]
with open(f'{scene_names[i]}.png', 'wb') as file:
w = png.Writer(1024, 1024, greyscale=True)
w.write(file, rows)
```
### Rust
The function `grey_values` would be sufficient as an entry to this challenge. The rest of the code saves the outputs as PNG files.
```rust
fn grey_values(spheres: &Vec<[f32; 5]>) -> Vec<u8> {
let mut values = vec![];
for y_pixel in 0..1024 {
for x_pixel in 0..1024 {
let x_ray = x_pixel as f32 / 512.0 - 1.0;
let y_ray = y_pixel as f32 / 512.0 - 1.0;
let mut min_distance = f32::INFINITY;
let mut hit_grey_value = 0;
for sphere in spheres {
let [
x_sphere, y_sphere, z_sphere, radius, grey
] = *sphere;
let a = (
(x_ray - x_sphere).powf(2.0)
+ (y_ray - y_sphere).powf(2.0)
).sqrt();
if a < radius {
let distance = z_sphere - (
radius.powf(2.0) - a.powf(2.0)
).sqrt();
if distance < min_distance {
min_distance = distance;
hit_grey_value = grey as u8;
}
}
}
values.push(hit_grey_value);
}
}
values
}
fn main() {
let test_cases: Vec<Vec<[f32; 5]>> = vec![
vec![
[0.0, 0.0, 0.0, 1.0, 128.0]
],
vec![
[0.0, -0.55, 0.0, 0.4, 100.0],
[0.0, 0.55, 0.0, 0.4, 200.0]
],
vec![
[0.75, 0.0, 0.0, 0.25, 64.0],
[0.53033, 0.375, -0.375, 0.25, 185.0],
[0.0, 0.53033, -0.53033, 0.25, 64.0],
[-0.53033, 0.375, -0.375, 0.25, 185.0],
[-0.75, 0.0, 0.0, 0.25, 64.0],
[-0.53033, -0.375, 0.375, 0.25, 185.0],
[0.0, -0.53033, 0.53033, 0.25, 64.0],
[0.53033, -0.375, 0.375, 0.25, 185.0],
],
vec![
[0.001, 0.0, 0.0, 1.0, 64.0],
[0.000707, 0.000707, 0.0, 1.0, 85.0],
[0.0, 0.001, 0.0, 1.0, 106.0],
[-0.000707, 0.000707, 0.0, 1.0, 127.0],
[-0.001, 0.0, 0.0, 1.0, 148.0],
[-0.000707, -0.000707, 0.0, 1.0, 169.0],
[0.0, -0.001, 0.0, 1.0, 190.0],
[0.000707, -0.000707, 0.0, 1.0, 211.0],
],
vec![
[0.0, 0.000058, 0.001947, 1.0, 0.0],
[0.0, -0.001715, 0.000923, 1.0, 0.0],
[0.0, 0.001715, -0.000923, 1.0, 0.0],
[0.0, -0.000058, -0.001947, 1.0, 0.0],
[0.001657, -0.000512, 0.000887, 1.0, 0.0],
[-0.001657, -0.000512, 0.000887, 1.0, 0.0],
[0.001657, 0.000512, -0.000887, 1.0, 0.0],
[-0.001657, 0.000512, -0.000887, 1.0, 0.0],
[0.001024, 0.001435, 0.000829, 1.0, 0.0],
[-0.001024, 0.001435, 0.000829, 1.0, 0.0],
[0.001024, -0.001435, -0.000829, 1.0, 0.0],
[-0.001024, -0.001435, -0.000829, 1.0, 0.0],
[0.001868, 0.000618, 0.000357, 1.0, 255.0],
[-0.001868, 0.000618, 0.000357, 1.0, 255.0],
[0.001868, -0.000618, -0.000357, 1.0, 255.0],
[-0.001868, -0.000618, -0.000357, 1.0, 255.0],
[0.000714, -0.000934, 0.001618, 1.0, 255.0],
[-0.000714, -0.000934, 0.001618, 1.0, 255.0],
[0.000714, 0.000934, -0.001618, 1.0, 255.0],
[-0.000714, 0.000934, -0.001618, 1.0, 255.0],
[0.0, -0.001975, -0.000316, 1.0, 255.0],
[0.0, 0.001261, 0.001552, 1.0, 255.0],
[0.0, -0.001261, -0.001552, 1.0, 255.0],
[0.0, 0.001975, 0.000316, 1.0, 255.0],
[0.001155, 0.000423, 0.001577, 1.0, 255.0],
[-0.001155, 0.000423, 0.001577, 1.0, 255.0],
[0.001155, -0.001577, 0.000423, 1.0, 255.0],
[-0.001155, -0.001577, 0.000423, 1.0, 255.0],
[0.001155, 0.001577, -0.000423, 1.0, 255.0],
[-0.001155, 0.001577, -0.000423, 1.0, 255.0],
[0.001155, -0.000423, -0.001577, 1.0, 255.0],
[-0.001155, -0.000423, -0.001577, 1.0, 255.0],
],
vec![
[0.0, 0.0, 1000.0, 2.0, 128.0],
[0.0, 0.0, 0.003125, 1.0, 0.0],
[0.0, 0.0, 0.0, 0.9975, 255.0],
[0.0, 0.000547, -0.002573, 0.995, 0.0],
[0.0, -0.000547, -0.002573, 0.995, 0.0],
[0.0, 0.001071, -0.005037, 0.9925, 255.0],
[0.0, -0.001071, -0.005037, 0.9925, 255.0],
],
vec![
[0.0, 0.0, 0.0, 1.0, 210.0],
[0.0, 0.0, 0.0008, 1.0, 210.0],
[0.0, 0.0, 0.0004, 1.00012, 50.0],
[0.012969, 0.0, -0.042045, 0.96, 100.0],
[0.016397, 0.011913, -0.039054, 0.96, 210.0],
[0.004008, 0.012334, -0.042045, 0.96, 100.0],
[-0.006263, 0.019276, -0.039054, 0.96, 210.0],
[-0.010492, 0.007623, -0.042045, 0.96, 100.0],
[-0.020268, 0.0, -0.039054, 0.96, 210.0],
[-0.010492, -0.007623, -0.042045, 0.96, 100.0],
[-0.006263, -0.019276, -0.039054, 0.96, 210.0],
[0.004008, -0.012334, -0.042045, 0.96, 100.0],
[0.016397, -0.011913, -0.039054, 0.96, 210.0],
],
vec![
[0.0, 0.0, 1000.0, 2.0, 192.0],
[-2.0, 2.0, 100.0, 3.0, 98.0],
[2.0, 2.0, 100.0, 3.0, 98.0],
[0.5, -0.5, 500.0, 0.2, 255.0],
[-0.349469, -0.1, 20.000279, 0.6, 64.0],
[-0.349775, -0.100273, 20.000485, 0.6, 85.0],
[-0.350214, -0.100386, 20.000407, 0.6, 106.0],
[-0.350527, -0.100273, 20.000091, 0.6, 127.0],
[-0.350531, -0.1, 19.999721, 0.6, 148.0],
[-0.350225, -0.099727, 19.999515, 0.6, 169.0],
[-0.349786, -0.099614, 19.999593, 0.6, 190.0],
[-0.349473, -0.099727, 19.999909, 0.6, 211.0],
[0.20032, 0.349824, 15.000456, 0.3, 0.0],
[0.20032, 0.349517, 14.999924, 0.3, 0.0],
[0.19968, 0.350483, 15.000076, 0.3, 0.0],
[0.19968, 0.350176, 14.999544, 0.3, 0.0],
[0.200578, 0.350073, 14.999958, 0.3, 0.0],
[0.199817, 0.349519, 15.000277, 0.3, 0.0],
[0.200183, 0.350481, 14.999723, 0.3, 0.0],
[0.199422, 0.349927, 15.000042, 0.3, 0.0],
[0.200235, 0.35042, 15.000332, 0.3, 0.0],
[0.199765, 0.350078, 15.000529, 0.3, 0.0],
[0.200235, 0.349922, 14.999471, 0.3, 0.0],
[0.199765, 0.34958, 14.999668, 0.3, 0.0],
[0.200429, 0.350419, 15.000005, 0.3, 255.0],
[0.199571, 0.349795, 15.000366, 0.3, 255.0],
[0.200429, 0.350205, 14.999634, 0.3, 255.0],
[0.199571, 0.349581, 14.999995, 0.3, 255.0],
[0.200524, 0.349747, 15.000146, 0.3, 255.0],
[0.200196, 0.349509, 15.000283, 0.3, 255.0],
[0.199804, 0.350491, 14.999717, 0.3, 255.0],
[0.199476, 0.350253, 14.999854, 0.3, 255.0],
[0.200138, 0.349578, 14.999597, 0.3, 255.0],
[0.200138, 0.350138, 15.000567, 0.3, 255.0],
[0.199862, 0.349862, 14.999433, 0.3, 255.0],
[0.199862, 0.350422, 15.000403, 0.3, 255.0],
[0.200488, 0.350136, 15.000321, 0.3, 255.0],
[0.199957, 0.349751, 15.000544, 0.3, 255.0],
[0.200488, 0.34979, 14.999721, 0.3, 255.0],
[0.199957, 0.349404, 14.999944, 0.3, 255.0],
[0.200043, 0.350596, 15.000056, 0.3, 255.0],
[0.199512, 0.35021, 15.000279, 0.3, 255.0],
[0.200043, 0.350249, 14.999456, 0.3, 255.0],
[0.199512, 0.349864, 14.999679, 0.3, 255.0],
[-0.5, 0.500442, 10.000442, 0.2, 0.0],
[-0.5, 0.5, 10.0, 0.1995, 255.0],
[-0.499966, 0.49971, 9.999563, 0.199, 0.0],
[-0.500034, 0.499563, 9.99971, 0.199, 0.0],
[-0.499934, 0.499432, 9.999144, 0.1985, 255.0],
[-0.500066, 0.499144, 9.999432, 0.1985, 255.0],
[0.6, 0.5, 10.0, 0.25, 210.0],
[0.600122, 0.4999, 10.000122, 0.25, 210.0],
[0.600061, 0.49995, 10.000061, 0.25003, 50.0],
[0.595856, 0.505256, 9.991271, 0.24, 100.0],
[0.597973, 0.507461, 9.992175, 0.24, 210.0],
[0.595362, 0.507926, 9.993945, 0.24, 100.0],
[0.594618, 0.509055, 9.996832, 0.24, 210.0],
[0.592382, 0.506906, 9.996092, 0.24, 100.0],
[0.590438, 0.504882, 9.997604, 0.24, 210.0],
[0.591035, 0.503605, 9.994744, 0.24, 100.0],
[0.59121, 0.500708, 9.993425, 0.24, 210.0],
[0.593181, 0.502585, 9.991764, 0.24, 100.0],
[0.595867, 0.502302, 9.990069, 0.24, 210.0],
],
];
let scene_names = [
"single_sphere",
"two_spheres",
"ring_of_spheres",
"beachball",
"football",
"eightball",
"starball",
"combination_scene",
];
for i in 0..8 {
eprintln!("Generating {}", scene_names[i]);
let values = grey_values(&test_cases[i]);
let filename = format!("{}.png", scene_names[i]);
let path = std::path::Path::new(&filename);
let file = std::fs::File::create(path).unwrap();
let w = &mut std::io::BufWriter::new(file);
let mut encoder = png::Encoder::new(w, 1024, 1024);
encoder.set_color(png::ColorType::Grayscale);
encoder.set_depth(png::BitDepth::Eight);
let mut writer = encoder.write_header().unwrap();
writer.write_image_data(&values).unwrap();
}
}
```
## Test cases
The images may be scaled down. Clicking or tapping on an image will open it full size.
### Single sphere
Input:
```text
TODO
```
Output:
```text
TODO
```
### Two separated spheres
Input:
```text
TODO
```
Output:
```text
TODO
```
### Eight spheres in a ring
Input:
```text
TODO
```
Output:
```text
TODO
```
### Beach ball
Input:
```text
TODO
```
Output:
```text
TODO
```
### Football
Input:
```text
TODO
```
Output:
```text
TODO
```
### Eight ball
Input:
```text
TODO
```
Output:
```text
TODO
```
### Star ball
Input:
```text
TODO
```
Output:
```text
TODO
```
### Combination scene
Input:
```text
TODO
```
Output:
```text
TODO
```
## Explanatory animations
The example images were produced from plain grey spheres with no surface pattern or texture. The following animations show how they break down into their component plain spheres.
This section is not necessary to write valid code for this challenge. If your code accepts the parameters of plain grey spheres and renders them correctly, all of the test cases will look right without needing to understand how they were constructed. This section is mainly out of interest, and to make clear that your code *only* needs to render plain grey spheres.
```text
CONSIDER HOSTING THESE OFF SITE AS THEY ARE NOT REQUIRED FOR THE CHALLENGE
AND EVEN THE SCALED DOWN VERSIONS ARE UNNECESSARY DATA USE FOR ANYONE
CHOOSING NOT TO VIEW THEM.
```
<details>
<summary>Animations</summary>
<details>
<summary>Beach ball</summary>
This animation is reduced to 256 by 256 pixels. Clicking or tapping on the animation will open it at full size (______MB).
```text
TODO
```
</details>
<details>
<summary>Football</summary>
This animation is reduced to 256 by 256 pixels. Clicking or tapping on the animation will open it at full size (______MB).
```text
TODO
```
</details>
<details>
<summary>Eight ball</summary>
This animation is reduced to 256 by 256 pixels. Clicking or tapping on the animation will open it at full size (______MB).
```text
TODO
```
</details>
<details>
<summary>Star ball</summary>
This animation is reduced to 256 by 256 pixels. Clicking or tapping on the animation will open it at full size (______MB).
```text
TODO
```
</details>
</details>
## Scoring
This is a [code golf challenge]. Your score is the number of bytes in your code. Lowest score for each language wins.
> Explanations are optional, but I'm more likely to upvote answers that have one.
[code golf challenge]: https://codegolf.codidact.com/categories/49/tags/4274 "The code-golf tag"
[PGM]: https://en.wikipedia.org/wiki/Netpbm#PGM_example

#5: Post edited by

trichoplax‭ · 2024-07-26T10:55:28Z (5 months ago)
Make Python implementation avoid horizontal scroll bars

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Raw

Markdown

Given a list of spheres, output an image of them.
The challenge is to produce a greyscale [orthographic](https://en.wikipedia.org/wiki/Parallel_projection#Orthographic_projection) (parallel projection - no perspective) ray caster, that renders only plain grey spheres with no lighting, no shading, no shadows, no surface patterns or textures, no reflections, and no transparency.
## Input
- A list of spheres, each consisting of:
- The coordinates of its centre $(x, y, z)$.
- Its radius $r$.
- Its greyscale value $g$, an integer from $0$ to $255$ (both inclusive).
- The input may include up to $100$ spheres.
- Only the greyscale value will be restricted to integer values. The coordinates and radius may have fractional values.
- Here "integer" refers to the values the greyscale value make take. You are not required to use an integer variable type.[]()
## Output
- A single greyscale image:
- $1024$ by $1024$ pixels.
- Each pixel has a single greyscale value from $0$ to $255$.
- Horizontally, the image covers x values from $-1$ to $1$.
- Vertically, the image covers y values from $-1$ to $1$.
- The image is orthographic, so there is no perspective. Changing the $z$ coordinate of a sphere does not change its size in the image, instead only affecting whether it is in front of or behind other spheres.
- A sphere's greyscale value is applied to every visible part of it. There is no variation over the surface of the sphere, so it appears flat like a disk.
- The point of the challenge is producing the pixel values, not manipulating image file formats, so you may output whatever image format is convenient, including simple text formats such as [PGM] or even just a list of pixel greyscale values. If you want to include an example of your output as an image in your answer, the image formats that Codidact supports are PNG, JPEG, and GIF. You could output PGM and then convert to PNG to upload to your answer. If you output just the greyscale values as a list of numbers from $0$ to $255$, I've made a `[ TODO page to convert these to greyscale PNG]`.
- The test cases, which match the reference implementations, happen to have the x axis increasing from left to right, and the y axis increasing from top to bottom. You are free to instead interpret these as increasing from right to left for x or from bottom to top for y, resulting in a reflected image. You are also free to output an image rotated by a multiple of 90 degrees, allowing for swapping the x and y axes. However, the z axis is required to be increasing in the forwards direction (into the image) otherwise some of the test cases will look incorrect.
## Calculating a pixel value
Imagine a ray passing into the image through a pixel, perpendicular to the image plane. If it hits a sphere, it shows the greyscale value of the first sphere it hits. Otherwise, it shows a value of zero (black). One possible way of calculating this is as follows:
For each pixel:
- The pixel $x, y$ coordinates will each be from $0$ to $1023$. Scale these to the range $-1$ to $1$ (for example, divide by $512$ and subtract $1$).
- If these scaled $x, y$ coordinates are within the radius $r$ of the $x, y$ coordinates of a sphere's centre, the ray intersects that sphere (the $z$ coordinate is not relevant to this test because the image is an orthographic projection - the rays are all parallel to the $z$ axis). The distance $a$ of the ray from the sphere's centre can be calculated as follows:
$$a=\sqrt{(x_{ray}-x_{sphere})^2+(y_{ray}-y_{sphere})^2}$$
```text
CONSIDER A DIAGRAM HERE
```
- For each sphere that the ray intersects, find the $z$ coordinate of the nearest intersection (there will often be $2$ per sphere, $1$ on the front of the sphere and $1$ on the back of the sphere, as the ray passes through - only the nearest is relevant). This can be calculated as follows:
- The distance $d$ to the nearest intersection point can be calculated from the $z$ coordinate of the sphere's centre, the radius $r$, and the distance $a$ from the $x, y$ coordinates of the sphere's centre to the scaled $x, y$ coordinates of the pixel:
$$d=z_{sphere}-\sqrt{r_{sphere}^2-a^2}$$
```text
CONSIDER A DIAGRAM HERE
```
- Of all the spheres that the ray intersects, the sphere with the intersection point with the smallest $z$ coordinate $d$ is the one to display. Set the pixel to the greyscale value of this sphere.
- If the ray did not intersect any spheres, set the pixel to zero (black).
## Example implementations
These are ungolfed implementations designed to supplement understanding of the requirements. You are free to golf these or to implement your own approach that matches the test cases up to reflection and rotation.
The Python implementation uses 64 bit floating point giving accurate images. The Rust implementation deliberately uses 32 bit floating point to show the difference in output. I've tried to optimise the test cases for compatibility with 32 bit floating point, so the differences are restricted to being a pixel out in some places.
### Python
The function `grey_values` would be sufficient as an entry to this challenge. The rest of the code saves the outputs as PNG files. The function itself has no dependence on the `import png` so it would not be required in an entry.
```python
def grey_values(spheres):
values = []
for y_pixel in range(1024):
for x_pixel in range(1024):
x_ray, y_ray = x_pixel / 512 - 1, y_pixel / 512 - 1
min_distance, hit_grey_value = None, 0
for sphere in spheres:
~~x_sphere, y_sphere, z_sphere, radius, grey_value = sphere~~
~~a = ((x_ray-x_sphere) ** 2 + (y_ray-y_sphere) ** 2) ** 0.5~~
if a < radius:
distance = z_sphere - (radius ** 2 - a ** 2) ** 0.5
if min_distance is None or distance < min_distance:
min_distance = distance
hit_grey_value = grey_value
values.append(hit_grey_value)
return values
import png
test_cases = [
[
[0, 0, 0, 1, 128],
],
[
[0, -0.55, 0, 0.4, 100],
[0, 0.55, 0, 0.4, 200],
],
[
[0.75, 0, 0, 0.25, 64],
[0.53033, 0.375, -0.375, 0.25, 185],
[0, 0.53033, -0.53033, 0.25, 64],
[-0.53033, 0.375, -0.375, 0.25, 185],
[-0.75, 0, -0, 0.25, 64],
[-0.53033, -0.375, 0.375, 0.25, 185],
[-0, -0.53033, 0.53033, 0.25, 64],
[0.53033, -0.375, 0.375, 0.25, 185],
],
[
[0.001, 0, 0, 1, 64],
[0.000707, 0.000707, -0, 1, 85],
[0, 0.001, -0, 1, 106],
[-0.000707, 0.000707, -0, 1, 127],
[-0.001, 0, -0, 1, 148],
[-0.000707, -0.000707, 0, 1, 169],
[-0, -0.001, 0, 1, 190],
[0.000707, -0.000707, 0, 1, 211],
],
[
[0, 0.000058, 0.001947, 1, 0],
[0, -0.001715, 0.000923, 1, 0],
[0, 0.001715, -0.000923, 1, 0],
[0, -0.000058, -0.001947, 1, 0],
[0.001657, -0.000512, 0.000887, 1, 0],
[-0.001657, -0.000512, 0.000887, 1, 0],
[0.001657, 0.000512, -0.000887, 1, 0],
[-0.001657, 0.000512, -0.000887, 1, 0],
[0.001024, 0.001435, 0.000829, 1, 0],
[-0.001024, 0.001435, 0.000829, 1, 0],
[0.001024, -0.001435, -0.000829, 1, 0],
[-0.001024, -0.001435, -0.000829, 1, 0],
[0.001868, 0.000618, 0.000357, 1, 255],
[-0.001868, 0.000618, 0.000357, 1, 255],
[0.001868, -0.000618, -0.000357, 1, 255],
[-0.001868, -0.000618, -0.000357, 1, 255],
[0.000714, -0.000934, 0.001618, 1, 255],
[-0.000714, -0.000934, 0.001618, 1, 255],
[0.000714, 0.000934, -0.001618, 1, 255],
[-0.000714, 0.000934, -0.001618, 1, 255],
[0, -0.001975, -0.000316, 1, 255],
[0, 0.001261, 0.001552, 1, 255],
[0, -0.001261, -0.001552, 1, 255],
[0, 0.001975, 0.000316, 1, 255],
[0.001155, 0.000423, 0.001577, 1, 255],
[-0.001155, 0.000423, 0.001577, 1, 255],
[0.001155, -0.001577, 0.000423, 1, 255],
[-0.001155, -0.001577, 0.000423, 1, 255],
[0.001155, 0.001577, -0.000423, 1, 255],
[-0.001155, 0.001577, -0.000423, 1, 255],
[0.001155, -0.000423, -0.001577, 1, 255],
[-0.001155, -0.000423, -0.001577, 1, 255],
],
[
[0, 0, 1000, 2, 128],
[0, 0, 0.003125, 1, 0],
[0, 0, 0, 0.9975, 255],
[0, 0.000547, -0.002573, 0.995, 0],
[-0, -0.000547, -0.002573, 0.995, 0],
[0, 0.001071, -0.005037, 0.9925, 255],
[-0, -0.001071, -0.005037, 0.9925, 255],
],
[
[0, 0, 0, 1, 210],
[0, 0, 0.0008, 1, 210],
[0, 0, 0.0004, 1.00012, 50],
[0.012969, -0, -0.042045, 0.96, 100],
[0.016397, 0.011913, -0.039054, 0.96, 210],
[0.004008, 0.012334, -0.042045, 0.96, 100],
[-0.006263, 0.019276, -0.039054, 0.96, 210],
[-0.010492, 0.007623, -0.042045, 0.96, 100],
[-0.020268, 0, -0.039054, 0.96, 210],
[-0.010492, -0.007623, -0.042045, 0.96, 100],
[-0.006263, -0.019276, -0.039054, 0.96, 210],
[0.004008, -0.012334, -0.042045, 0.96, 100],
[0.016397, -0.011913, -0.039054, 0.96, 210],
],
[
[0, 0, 1000, 2, 192],
[-2, 2, 100, 3, 98],
[2, 2, 100, 3, 98],
[0.5, -0.5, 500, 0.2, 255],
[-0.349469, -0.1, 20.000279, 0.6, 64],
[-0.349775, -0.100273, 20.000485, 0.6, 85],
[-0.350214, -0.100386, 20.000407, 0.6, 106],
[-0.350527, -0.100273, 20.000091, 0.6, 127],
[-0.350531, -0.1, 19.999721, 0.6, 148],
[-0.350225, -0.099727, 19.999515, 0.6, 169],
[-0.349786, -0.099614, 19.999593, 0.6, 190],
[-0.349473, -0.099727, 19.999909, 0.6, 211],
[0.20032, 0.349824, 15.000456, 0.3, 0],
[0.20032, 0.349517, 14.999924, 0.3, 0],
[0.19968, 0.350483, 15.000076, 0.3, 0],
[0.19968, 0.350176, 14.999544, 0.3, 0],
[0.200578, 0.350073, 14.999958, 0.3, 0],
[0.199817, 0.349519, 15.000277, 0.3, 0],
[0.200183, 0.350481, 14.999723, 0.3, 0],
[0.199422, 0.349927, 15.000042, 0.3, 0],
[0.200235, 0.35042, 15.000332, 0.3, 0],
[0.199765, 0.350078, 15.000529, 0.3, 0],
[0.200235, 0.349922, 14.999471, 0.3, 0],
[0.199765, 0.34958, 14.999668, 0.3, 0],
[0.200429, 0.350419, 15.000005, 0.3, 255],
[0.199571, 0.349795, 15.000366, 0.3, 255],
[0.200429, 0.350205, 14.999634, 0.3, 255],
[0.199571, 0.349581, 14.999995, 0.3, 255],
[0.200524, 0.349747, 15.000146, 0.3, 255],
[0.200196, 0.349509, 15.000283, 0.3, 255],
[0.199804, 0.350491, 14.999717, 0.3, 255],
[0.199476, 0.350253, 14.999854, 0.3, 255],
[0.200138, 0.349578, 14.999597, 0.3, 255],
[0.200138, 0.350138, 15.000567, 0.3, 255],
[0.199862, 0.349862, 14.999433, 0.3, 255],
[0.199862, 0.350422, 15.000403, 0.3, 255],
[0.200488, 0.350136, 15.000321, 0.3, 255],
[0.199957, 0.349751, 15.000544, 0.3, 255],
[0.200488, 0.34979, 14.999721, 0.3, 255],
[0.199957, 0.349404, 14.999944, 0.3, 255],
[0.200043, 0.350596, 15.000056, 0.3, 255],
[0.199512, 0.35021, 15.000279, 0.3, 255],
[0.200043, 0.350249, 14.999456, 0.3, 255],
[0.199512, 0.349864, 14.999679, 0.3, 255],
[-0.5, 0.500442, 10.000442, 0.2, 0],
[-0.5, 0.5, 10, 0.1995, 255],
[-0.499966, 0.49971, 9.999563, 0.199, 0],
[-0.500034, 0.499563, 9.99971, 0.199, 0],
[-0.499934, 0.499432, 9.999144, 0.1985, 255],
[-0.500066, 0.499144, 9.999432, 0.1985, 255],
[0.6, 0.5, 10, 0.25, 210],
[0.600122, 0.4999, 10.000122, 0.25, 210],
[0.600061, 0.49995, 10.000061, 0.25003, 50],
[0.595856, 0.505256, 9.991271, 0.24, 100],
[0.597973, 0.507461, 9.992175, 0.24, 210],
[0.595362, 0.507926, 9.993945, 0.24, 100],
[0.594618, 0.509055, 9.996832, 0.24, 210],
[0.592382, 0.506906, 9.996092, 0.24, 100],
[0.590438, 0.504882, 9.997604, 0.24, 210],
[0.591035, 0.503605, 9.994744, 0.24, 100],
[0.59121, 0.500708, 9.993425, 0.24, 210],
[0.593181, 0.502585, 9.991764, 0.24, 100],
[0.595867, 0.502302, 9.990069, 0.24, 210],
],
]
scene_names = (
"single_sphere",
"two_spheres",
"ring_of_spheres",
"beachball",
"football",
"eightball",
"starball",
"combination_scene",
)
for i in range(8):
print(f'Generating {scene_names[i]}.')
values = grey_values(test_cases[i])
rows = [values[row * 1024:row * 1024 + 1024] for row in range(1024)]
with open(f'{scene_names[i]}.png', 'wb') as file:
w = png.Writer(1024, 1024, greyscale=True)
w.write(file, rows)
```
### Rust
The function `grey_values` would be sufficient as an entry to this challenge. The rest of the code saves the outputs as PNG files.
```rust
fn grey_values(spheres: &Vec<[f32; 5]>) -> Vec<u8> {
let mut values = vec![];
for y_pixel in 0..1024 {
for x_pixel in 0..1024 {
let x_ray = x_pixel as f32 / 512.0 - 1.0;
let y_ray = y_pixel as f32 / 512.0 - 1.0;
let mut min_distance = f32::INFINITY;
let mut hit_grey_value = 0;
for sphere in spheres {
let [
x_sphere, y_sphere, z_sphere, radius, grey
] = *sphere;
let a = (
(x_ray - x_sphere).powf(2.0)
+ (y_ray - y_sphere).powf(2.0)
).sqrt();
if a < radius {
let distance = z_sphere - (
radius.powf(2.0) - a.powf(2.0)
).sqrt();
if distance < min_distance {
min_distance = distance;
hit_grey_value = grey as u8;
}
}
}
values.push(hit_grey_value);
}
}
values
}
fn main() {
let test_cases: Vec<Vec<[f32; 5]>> = vec![
vec![
[0.0, 0.0, 0.0, 1.0, 128.0]
],
vec![
[0.0, -0.55, 0.0, 0.4, 100.0],
[0.0, 0.55, 0.0, 0.4, 200.0]
],
vec![
[0.75, 0.0, 0.0, 0.25, 64.0],
[0.53033, 0.375, -0.375, 0.25, 185.0],
[0.0, 0.53033, -0.53033, 0.25, 64.0],
[-0.53033, 0.375, -0.375, 0.25, 185.0],
[-0.75, 0.0, 0.0, 0.25, 64.0],
[-0.53033, -0.375, 0.375, 0.25, 185.0],
[0.0, -0.53033, 0.53033, 0.25, 64.0],
[0.53033, -0.375, 0.375, 0.25, 185.0],
],
vec![
[0.001, 0.0, 0.0, 1.0, 64.0],
[0.000707, 0.000707, 0.0, 1.0, 85.0],
[0.0, 0.001, 0.0, 1.0, 106.0],
[-0.000707, 0.000707, 0.0, 1.0, 127.0],
[-0.001, 0.0, 0.0, 1.0, 148.0],
[-0.000707, -0.000707, 0.0, 1.0, 169.0],
[0.0, -0.001, 0.0, 1.0, 190.0],
[0.000707, -0.000707, 0.0, 1.0, 211.0],
],
vec![
[0.0, 0.000058, 0.001947, 1.0, 0.0],
[0.0, -0.001715, 0.000923, 1.0, 0.0],
[0.0, 0.001715, -0.000923, 1.0, 0.0],
[0.0, -0.000058, -0.001947, 1.0, 0.0],
[0.001657, -0.000512, 0.000887, 1.0, 0.0],
[-0.001657, -0.000512, 0.000887, 1.0, 0.0],
[0.001657, 0.000512, -0.000887, 1.0, 0.0],
[-0.001657, 0.000512, -0.000887, 1.0, 0.0],
[0.001024, 0.001435, 0.000829, 1.0, 0.0],
[-0.001024, 0.001435, 0.000829, 1.0, 0.0],
[0.001024, -0.001435, -0.000829, 1.0, 0.0],
[-0.001024, -0.001435, -0.000829, 1.0, 0.0],
[0.001868, 0.000618, 0.000357, 1.0, 255.0],
[-0.001868, 0.000618, 0.000357, 1.0, 255.0],
[0.001868, -0.000618, -0.000357, 1.0, 255.0],
[-0.001868, -0.000618, -0.000357, 1.0, 255.0],
[0.000714, -0.000934, 0.001618, 1.0, 255.0],
[-0.000714, -0.000934, 0.001618, 1.0, 255.0],
[0.000714, 0.000934, -0.001618, 1.0, 255.0],
[-0.000714, 0.000934, -0.001618, 1.0, 255.0],
[0.0, -0.001975, -0.000316, 1.0, 255.0],
[0.0, 0.001261, 0.001552, 1.0, 255.0],
[0.0, -0.001261, -0.001552, 1.0, 255.0],
[0.0, 0.001975, 0.000316, 1.0, 255.0],
[0.001155, 0.000423, 0.001577, 1.0, 255.0],
[-0.001155, 0.000423, 0.001577, 1.0, 255.0],
[0.001155, -0.001577, 0.000423, 1.0, 255.0],
[-0.001155, -0.001577, 0.000423, 1.0, 255.0],
[0.001155, 0.001577, -0.000423, 1.0, 255.0],
[-0.001155, 0.001577, -0.000423, 1.0, 255.0],
[0.001155, -0.000423, -0.001577, 1.0, 255.0],
[-0.001155, -0.000423, -0.001577, 1.0, 255.0],
],
vec![
[0.0, 0.0, 1000.0, 2.0, 128.0],
[0.0, 0.0, 0.003125, 1.0, 0.0],
[0.0, 0.0, 0.0, 0.9975, 255.0],
[0.0, 0.000547, -0.002573, 0.995, 0.0],
[0.0, -0.000547, -0.002573, 0.995, 0.0],
[0.0, 0.001071, -0.005037, 0.9925, 255.0],
[0.0, -0.001071, -0.005037, 0.9925, 255.0],
],
vec![
[0.0, 0.0, 0.0, 1.0, 210.0],
[0.0, 0.0, 0.0008, 1.0, 210.0],
[0.0, 0.0, 0.0004, 1.00012, 50.0],
[0.012969, 0.0, -0.042045, 0.96, 100.0],
[0.016397, 0.011913, -0.039054, 0.96, 210.0],
[0.004008, 0.012334, -0.042045, 0.96, 100.0],
[-0.006263, 0.019276, -0.039054, 0.96, 210.0],
[-0.010492, 0.007623, -0.042045, 0.96, 100.0],
[-0.020268, 0.0, -0.039054, 0.96, 210.0],
[-0.010492, -0.007623, -0.042045, 0.96, 100.0],
[-0.006263, -0.019276, -0.039054, 0.96, 210.0],
[0.004008, -0.012334, -0.042045, 0.96, 100.0],
[0.016397, -0.011913, -0.039054, 0.96, 210.0],
],
vec![
[0.0, 0.0, 1000.0, 2.0, 192.0],
[-2.0, 2.0, 100.0, 3.0, 98.0],
[2.0, 2.0, 100.0, 3.0, 98.0],
[0.5, -0.5, 500.0, 0.2, 255.0],
[-0.349469, -0.1, 20.000279, 0.6, 64.0],
[-0.349775, -0.100273, 20.000485, 0.6, 85.0],
[-0.350214, -0.100386, 20.000407, 0.6, 106.0],
[-0.350527, -0.100273, 20.000091, 0.6, 127.0],
[-0.350531, -0.1, 19.999721, 0.6, 148.0],
[-0.350225, -0.099727, 19.999515, 0.6, 169.0],
[-0.349786, -0.099614, 19.999593, 0.6, 190.0],
[-0.349473, -0.099727, 19.999909, 0.6, 211.0],
[0.20032, 0.349824, 15.000456, 0.3, 0.0],
[0.20032, 0.349517, 14.999924, 0.3, 0.0],
[0.19968, 0.350483, 15.000076, 0.3, 0.0],
[0.19968, 0.350176, 14.999544, 0.3, 0.0],
[0.200578, 0.350073, 14.999958, 0.3, 0.0],
[0.199817, 0.349519, 15.000277, 0.3, 0.0],
[0.200183, 0.350481, 14.999723, 0.3, 0.0],
[0.199422, 0.349927, 15.000042, 0.3, 0.0],
[0.200235, 0.35042, 15.000332, 0.3, 0.0],
[0.199765, 0.350078, 15.000529, 0.3, 0.0],
[0.200235, 0.349922, 14.999471, 0.3, 0.0],
[0.199765, 0.34958, 14.999668, 0.3, 0.0],
[0.200429, 0.350419, 15.000005, 0.3, 255.0],
[0.199571, 0.349795, 15.000366, 0.3, 255.0],
[0.200429, 0.350205, 14.999634, 0.3, 255.0],
[0.199571, 0.349581, 14.999995, 0.3, 255.0],
[0.200524, 0.349747, 15.000146, 0.3, 255.0],
[0.200196, 0.349509, 15.000283, 0.3, 255.0],
[0.199804, 0.350491, 14.999717, 0.3, 255.0],
[0.199476, 0.350253, 14.999854, 0.3, 255.0],
[0.200138, 0.349578, 14.999597, 0.3, 255.0],
[0.200138, 0.350138, 15.000567, 0.3, 255.0],
[0.199862, 0.349862, 14.999433, 0.3, 255.0],
[0.199862, 0.350422, 15.000403, 0.3, 255.0],
[0.200488, 0.350136, 15.000321, 0.3, 255.0],
[0.199957, 0.349751, 15.000544, 0.3, 255.0],
[0.200488, 0.34979, 14.999721, 0.3, 255.0],
[0.199957, 0.349404, 14.999944, 0.3, 255.0],
[0.200043, 0.350596, 15.000056, 0.3, 255.0],
[0.199512, 0.35021, 15.000279, 0.3, 255.0],
[0.200043, 0.350249, 14.999456, 0.3, 255.0],
[0.199512, 0.349864, 14.999679, 0.3, 255.0],
[-0.5, 0.500442, 10.000442, 0.2, 0.0],
[-0.5, 0.5, 10.0, 0.1995, 255.0],
[-0.499966, 0.49971, 9.999563, 0.199, 0.0],
[-0.500034, 0.499563, 9.99971, 0.199, 0.0],
[-0.499934, 0.499432, 9.999144, 0.1985, 255.0],
[-0.500066, 0.499144, 9.999432, 0.1985, 255.0],
[0.6, 0.5, 10.0, 0.25, 210.0],
[0.600122, 0.4999, 10.000122, 0.25, 210.0],
[0.600061, 0.49995, 10.000061, 0.25003, 50.0],
[0.595856, 0.505256, 9.991271, 0.24, 100.0],
[0.597973, 0.507461, 9.992175, 0.24, 210.0],
[0.595362, 0.507926, 9.993945, 0.24, 100.0],
[0.594618, 0.509055, 9.996832, 0.24, 210.0],
[0.592382, 0.506906, 9.996092, 0.24, 100.0],
[0.590438, 0.504882, 9.997604, 0.24, 210.0],
[0.591035, 0.503605, 9.994744, 0.24, 100.0],
[0.59121, 0.500708, 9.993425, 0.24, 210.0],
[0.593181, 0.502585, 9.991764, 0.24, 100.0],
[0.595867, 0.502302, 9.990069, 0.24, 210.0],
],
];
let scene_names = [
"single_sphere",
"two_spheres",
"ring_of_spheres",
"beachball",
"football",
"eightball",
"starball",
"combination_scene",
];
for i in 0..8 {
eprintln!("Generating {}", scene_names[i]);
let values = grey_values(&test_cases[i]);
let filename = format!("{}.png", scene_names[i]);
let path = std::path::Path::new(&filename);
let file = std::fs::File::create(path).unwrap();
let w = &mut std::io::BufWriter::new(file);
let mut encoder = png::Encoder::new(w, 1024, 1024);
encoder.set_color(png::ColorType::Grayscale);
encoder.set_depth(png::BitDepth::Eight);
let mut writer = encoder.write_header().unwrap();
writer.write_image_data(&values).unwrap();
}
}
```
## Test cases
The images may be scaled down. Clicking or tapping on an image will open it full size.
### Single sphere
Input:
```text
TODO
```
Output:
```text
TODO
```
### Two separated spheres
Input:
```text
TODO
```
Output:
```text
TODO
```
### Eight spheres in a ring
Input:
```text
TODO
```
Output:
```text
TODO
```
### Beach ball
Input:
```text
TODO
```
Output:
```text
TODO
```
### Football
Input:
```text
TODO
```
Output:
```text
TODO
```
### Eight ball
Input:
```text
TODO
```
Output:
```text
TODO
```
### Star ball
Input:
```text
TODO
```
Output:
```text
TODO
```
### Combination scene
Input:
```text
TODO
```
Output:
```text
TODO
```
## Explanatory animations
The example images were produced from plain grey spheres with no surface pattern or texture. The following animations show how they break down into their component plain spheres.
This section is not necessary to write valid code for this challenge. If your code accepts the parameters of plain grey spheres and renders them correctly, all of the test cases will look right without needing to understand how they were constructed. This section is mainly out of interest, and to make clear that your code *only* needs to render plain grey spheres.
```text
CONSIDER HOSTING THESE OFF SITE AS THEY ARE NOT REQUIRED FOR THE CHALLENGE
AND EVEN THE SCALED DOWN VERSIONS ARE UNNECESSARY DATA USE FOR ANYONE
CHOOSING NOT TO VIEW THEM.
```
<details>
<summary>Animations</summary>
<details>
<summary>Beach ball</summary>
This animation is reduced to 256 by 256 pixels. Clicking or tapping on the animation will open it at full size (______MB).
```text
TODO
```
</details>
<details>
<summary>Football</summary>
This animation is reduced to 256 by 256 pixels. Clicking or tapping on the animation will open it at full size (______MB).
```text
TODO
```
</details>
<details>
<summary>Eight ball</summary>
This animation is reduced to 256 by 256 pixels. Clicking or tapping on the animation will open it at full size (______MB).
```text
TODO
```
</details>
<details>
<summary>Star ball</summary>
This animation is reduced to 256 by 256 pixels. Clicking or tapping on the animation will open it at full size (______MB).
```text
TODO
```
</details>
</details>
## Scoring
This is a [code golf challenge]. Your score is the number of bytes in your code. Lowest score for each language wins.
> Explanations are optional, but I'm more likely to upvote answers that have one.
[code golf challenge]: https://codegolf.codidact.com/categories/49/tags/4274 "The code-golf tag"
[PGM]: https://en.wikipedia.org/wiki/Netpbm#PGM_example

Given a list of spheres, output an image of them.
The challenge is to produce a greyscale [orthographic](https://en.wikipedia.org/wiki/Parallel_projection#Orthographic_projection) (parallel projection - no perspective) ray caster, that renders only plain grey spheres with no lighting, no shading, no shadows, no surface patterns or textures, no reflections, and no transparency.
## Input
- A list of spheres, each consisting of:
- The coordinates of its centre $(x, y, z)$.
- Its radius $r$.
- Its greyscale value $g$, an integer from $0$ to $255$ (both inclusive).
- The input may include up to $100$ spheres.
- Only the greyscale value will be restricted to integer values. The coordinates and radius may have fractional values.
- Here "integer" refers to the values the greyscale value make take. You are not required to use an integer variable type.[]()
## Output
- A single greyscale image:
- $1024$ by $1024$ pixels.
- Each pixel has a single greyscale value from $0$ to $255$.
- Horizontally, the image covers x values from $-1$ to $1$.
- Vertically, the image covers y values from $-1$ to $1$.
- The image is orthographic, so there is no perspective. Changing the $z$ coordinate of a sphere does not change its size in the image, instead only affecting whether it is in front of or behind other spheres.
- A sphere's greyscale value is applied to every visible part of it. There is no variation over the surface of the sphere, so it appears flat like a disk.
- The point of the challenge is producing the pixel values, not manipulating image file formats, so you may output whatever image format is convenient, including simple text formats such as [PGM] or even just a list of pixel greyscale values. If you want to include an example of your output as an image in your answer, the image formats that Codidact supports are PNG, JPEG, and GIF. You could output PGM and then convert to PNG to upload to your answer. If you output just the greyscale values as a list of numbers from $0$ to $255$, I've made a `[ TODO page to convert these to greyscale PNG]`.
- The test cases, which match the reference implementations, happen to have the x axis increasing from left to right, and the y axis increasing from top to bottom. You are free to instead interpret these as increasing from right to left for x or from bottom to top for y, resulting in a reflected image. You are also free to output an image rotated by a multiple of 90 degrees, allowing for swapping the x and y axes. However, the z axis is required to be increasing in the forwards direction (into the image) otherwise some of the test cases will look incorrect.
## Calculating a pixel value
Imagine a ray passing into the image through a pixel, perpendicular to the image plane. If it hits a sphere, it shows the greyscale value of the first sphere it hits. Otherwise, it shows a value of zero (black). One possible way of calculating this is as follows:
For each pixel:
- The pixel $x, y$ coordinates will each be from $0$ to $1023$. Scale these to the range $-1$ to $1$ (for example, divide by $512$ and subtract $1$).
- If these scaled $x, y$ coordinates are within the radius $r$ of the $x, y$ coordinates of a sphere's centre, the ray intersects that sphere (the $z$ coordinate is not relevant to this test because the image is an orthographic projection - the rays are all parallel to the $z$ axis). The distance $a$ of the ray from the sphere's centre can be calculated as follows:
$$a=\sqrt{(x_{ray}-x_{sphere})^2+(y_{ray}-y_{sphere})^2}$$
```text
CONSIDER A DIAGRAM HERE
```
- For each sphere that the ray intersects, find the $z$ coordinate of the nearest intersection (there will often be $2$ per sphere, $1$ on the front of the sphere and $1$ on the back of the sphere, as the ray passes through - only the nearest is relevant). This can be calculated as follows:
- The distance $d$ to the nearest intersection point can be calculated from the $z$ coordinate of the sphere's centre, the radius $r$, and the distance $a$ from the $x, y$ coordinates of the sphere's centre to the scaled $x, y$ coordinates of the pixel:
$$d=z_{sphere}-\sqrt{r_{sphere}^2-a^2}$$
```text
CONSIDER A DIAGRAM HERE
```
- Of all the spheres that the ray intersects, the sphere with the intersection point with the smallest $z$ coordinate $d$ is the one to display. Set the pixel to the greyscale value of this sphere.
- If the ray did not intersect any spheres, set the pixel to zero (black).
## Example implementations
These are ungolfed implementations designed to supplement understanding of the requirements. You are free to golf these or to implement your own approach that matches the test cases up to reflection and rotation.
The Python implementation uses 64 bit floating point giving accurate images. The Rust implementation deliberately uses 32 bit floating point to show the difference in output. I've tried to optimise the test cases for compatibility with 32 bit floating point, so the differences are restricted to being a pixel out in some places.
### Python
The function `grey_values` would be sufficient as an entry to this challenge. The rest of the code saves the outputs as PNG files. The function itself has no dependence on the `import png` so it would not be required in an entry.
```python
def grey_values(spheres):
values = []
for y_pixel in range(1024):
for x_pixel in range(1024):
x_ray, y_ray = x_pixel / 512 - 1, y_pixel / 512 - 1
min_distance, hit_grey_value = None, 0
for sphere in spheres:
(
x_sphere, y_sphere, z_sphere, radius, grey_value
) = sphere
a = (
(x_ray - x_sphere) ** 2 + (y_ray - y_sphere) ** 2
) ** 0.5
if a < radius:
distance = z_sphere - (radius ** 2 - a ** 2) ** 0.5
if min_distance is None or distance < min_distance:
min_distance = distance
hit_grey_value = grey_value
values.append(hit_grey_value)
return values
import png
test_cases = [
[
[0, 0, 0, 1, 128],
],
[
[0, -0.55, 0, 0.4, 100],
[0, 0.55, 0, 0.4, 200],
],
[
[0.75, 0, 0, 0.25, 64],
[0.53033, 0.375, -0.375, 0.25, 185],
[0, 0.53033, -0.53033, 0.25, 64],
[-0.53033, 0.375, -0.375, 0.25, 185],
[-0.75, 0, -0, 0.25, 64],
[-0.53033, -0.375, 0.375, 0.25, 185],
[-0, -0.53033, 0.53033, 0.25, 64],
[0.53033, -0.375, 0.375, 0.25, 185],
],
[
[0.001, 0, 0, 1, 64],
[0.000707, 0.000707, -0, 1, 85],
[0, 0.001, -0, 1, 106],
[-0.000707, 0.000707, -0, 1, 127],
[-0.001, 0, -0, 1, 148],
[-0.000707, -0.000707, 0, 1, 169],
[-0, -0.001, 0, 1, 190],
[0.000707, -0.000707, 0, 1, 211],
],
[
[0, 0.000058, 0.001947, 1, 0],
[0, -0.001715, 0.000923, 1, 0],
[0, 0.001715, -0.000923, 1, 0],
[0, -0.000058, -0.001947, 1, 0],
[0.001657, -0.000512, 0.000887, 1, 0],
[-0.001657, -0.000512, 0.000887, 1, 0],
[0.001657, 0.000512, -0.000887, 1, 0],
[-0.001657, 0.000512, -0.000887, 1, 0],
[0.001024, 0.001435, 0.000829, 1, 0],
[-0.001024, 0.001435, 0.000829, 1, 0],
[0.001024, -0.001435, -0.000829, 1, 0],
[-0.001024, -0.001435, -0.000829, 1, 0],
[0.001868, 0.000618, 0.000357, 1, 255],
[-0.001868, 0.000618, 0.000357, 1, 255],
[0.001868, -0.000618, -0.000357, 1, 255],
[-0.001868, -0.000618, -0.000357, 1, 255],
[0.000714, -0.000934, 0.001618, 1, 255],
[-0.000714, -0.000934, 0.001618, 1, 255],
[0.000714, 0.000934, -0.001618, 1, 255],
[-0.000714, 0.000934, -0.001618, 1, 255],
[0, -0.001975, -0.000316, 1, 255],
[0, 0.001261, 0.001552, 1, 255],
[0, -0.001261, -0.001552, 1, 255],
[0, 0.001975, 0.000316, 1, 255],
[0.001155, 0.000423, 0.001577, 1, 255],
[-0.001155, 0.000423, 0.001577, 1, 255],
[0.001155, -0.001577, 0.000423, 1, 255],
[-0.001155, -0.001577, 0.000423, 1, 255],
[0.001155, 0.001577, -0.000423, 1, 255],
[-0.001155, 0.001577, -0.000423, 1, 255],
[0.001155, -0.000423, -0.001577, 1, 255],
[-0.001155, -0.000423, -0.001577, 1, 255],
],
[
[0, 0, 1000, 2, 128],
[0, 0, 0.003125, 1, 0],
[0, 0, 0, 0.9975, 255],
[0, 0.000547, -0.002573, 0.995, 0],
[-0, -0.000547, -0.002573, 0.995, 0],
[0, 0.001071, -0.005037, 0.9925, 255],
[-0, -0.001071, -0.005037, 0.9925, 255],
],
[
[0, 0, 0, 1, 210],
[0, 0, 0.0008, 1, 210],
[0, 0, 0.0004, 1.00012, 50],
[0.012969, -0, -0.042045, 0.96, 100],
[0.016397, 0.011913, -0.039054, 0.96, 210],
[0.004008, 0.012334, -0.042045, 0.96, 100],
[-0.006263, 0.019276, -0.039054, 0.96, 210],
[-0.010492, 0.007623, -0.042045, 0.96, 100],
[-0.020268, 0, -0.039054, 0.96, 210],
[-0.010492, -0.007623, -0.042045, 0.96, 100],
[-0.006263, -0.019276, -0.039054, 0.96, 210],
[0.004008, -0.012334, -0.042045, 0.96, 100],
[0.016397, -0.011913, -0.039054, 0.96, 210],
],
[
[0, 0, 1000, 2, 192],
[-2, 2, 100, 3, 98],
[2, 2, 100, 3, 98],
[0.5, -0.5, 500, 0.2, 255],
[-0.349469, -0.1, 20.000279, 0.6, 64],
[-0.349775, -0.100273, 20.000485, 0.6, 85],
[-0.350214, -0.100386, 20.000407, 0.6, 106],
[-0.350527, -0.100273, 20.000091, 0.6, 127],
[-0.350531, -0.1, 19.999721, 0.6, 148],
[-0.350225, -0.099727, 19.999515, 0.6, 169],
[-0.349786, -0.099614, 19.999593, 0.6, 190],
[-0.349473, -0.099727, 19.999909, 0.6, 211],
[0.20032, 0.349824, 15.000456, 0.3, 0],
[0.20032, 0.349517, 14.999924, 0.3, 0],
[0.19968, 0.350483, 15.000076, 0.3, 0],
[0.19968, 0.350176, 14.999544, 0.3, 0],
[0.200578, 0.350073, 14.999958, 0.3, 0],
[0.199817, 0.349519, 15.000277, 0.3, 0],
[0.200183, 0.350481, 14.999723, 0.3, 0],
[0.199422, 0.349927, 15.000042, 0.3, 0],
[0.200235, 0.35042, 15.000332, 0.3, 0],
[0.199765, 0.350078, 15.000529, 0.3, 0],
[0.200235, 0.349922, 14.999471, 0.3, 0],
[0.199765, 0.34958, 14.999668, 0.3, 0],
[0.200429, 0.350419, 15.000005, 0.3, 255],
[0.199571, 0.349795, 15.000366, 0.3, 255],
[0.200429, 0.350205, 14.999634, 0.3, 255],
[0.199571, 0.349581, 14.999995, 0.3, 255],
[0.200524, 0.349747, 15.000146, 0.3, 255],
[0.200196, 0.349509, 15.000283, 0.3, 255],
[0.199804, 0.350491, 14.999717, 0.3, 255],
[0.199476, 0.350253, 14.999854, 0.3, 255],
[0.200138, 0.349578, 14.999597, 0.3, 255],
[0.200138, 0.350138, 15.000567, 0.3, 255],
[0.199862, 0.349862, 14.999433, 0.3, 255],
[0.199862, 0.350422, 15.000403, 0.3, 255],
[0.200488, 0.350136, 15.000321, 0.3, 255],
[0.199957, 0.349751, 15.000544, 0.3, 255],
[0.200488, 0.34979, 14.999721, 0.3, 255],
[0.199957, 0.349404, 14.999944, 0.3, 255],
[0.200043, 0.350596, 15.000056, 0.3, 255],
[0.199512, 0.35021, 15.000279, 0.3, 255],
[0.200043, 0.350249, 14.999456, 0.3, 255],
[0.199512, 0.349864, 14.999679, 0.3, 255],
[-0.5, 0.500442, 10.000442, 0.2, 0],
[-0.5, 0.5, 10, 0.1995, 255],
[-0.499966, 0.49971, 9.999563, 0.199, 0],
[-0.500034, 0.499563, 9.99971, 0.199, 0],
[-0.499934, 0.499432, 9.999144, 0.1985, 255],
[-0.500066, 0.499144, 9.999432, 0.1985, 255],
[0.6, 0.5, 10, 0.25, 210],
[0.600122, 0.4999, 10.000122, 0.25, 210],
[0.600061, 0.49995, 10.000061, 0.25003, 50],
[0.595856, 0.505256, 9.991271, 0.24, 100],
[0.597973, 0.507461, 9.992175, 0.24, 210],
[0.595362, 0.507926, 9.993945, 0.24, 100],
[0.594618, 0.509055, 9.996832, 0.24, 210],
[0.592382, 0.506906, 9.996092, 0.24, 100],
[0.590438, 0.504882, 9.997604, 0.24, 210],
[0.591035, 0.503605, 9.994744, 0.24, 100],
[0.59121, 0.500708, 9.993425, 0.24, 210],
[0.593181, 0.502585, 9.991764, 0.24, 100],
[0.595867, 0.502302, 9.990069, 0.24, 210],
],
]
scene_names = (
"single_sphere",
"two_spheres",
"ring_of_spheres",
"beachball",
"football",
"eightball",
"starball",
"combination_scene",
)
for i in range(8):
print(f'Generating {scene_names[i]}.')
values = grey_values(test_cases[i])
rows = [values[row * 1024:row * 1024 + 1024] for row in range(1024)]
with open(f'{scene_names[i]}.png', 'wb') as file:
w = png.Writer(1024, 1024, greyscale=True)
w.write(file, rows)
```
### Rust
The function `grey_values` would be sufficient as an entry to this challenge. The rest of the code saves the outputs as PNG files.
```rust
fn grey_values(spheres: &Vec<[f32; 5]>) -> Vec<u8> {
let mut values = vec![];
for y_pixel in 0..1024 {
for x_pixel in 0..1024 {
let x_ray = x_pixel as f32 / 512.0 - 1.0;
let y_ray = y_pixel as f32 / 512.0 - 1.0;
let mut min_distance = f32::INFINITY;
let mut hit_grey_value = 0;
for sphere in spheres {
let [
x_sphere, y_sphere, z_sphere, radius, grey
] = *sphere;
let a = (
(x_ray - x_sphere).powf(2.0)
+ (y_ray - y_sphere).powf(2.0)
).sqrt();
if a < radius {
let distance = z_sphere - (
radius.powf(2.0) - a.powf(2.0)
).sqrt();
if distance < min_distance {
min_distance = distance;
hit_grey_value = grey as u8;
}
}
}
values.push(hit_grey_value);
}
}
values
}
fn main() {
let test_cases: Vec<Vec<[f32; 5]>> = vec![
vec![
[0.0, 0.0, 0.0, 1.0, 128.0]
],
vec![
[0.0, -0.55, 0.0, 0.4, 100.0],
[0.0, 0.55, 0.0, 0.4, 200.0]
],
vec![
[0.75, 0.0, 0.0, 0.25, 64.0],
[0.53033, 0.375, -0.375, 0.25, 185.0],
[0.0, 0.53033, -0.53033, 0.25, 64.0],
[-0.53033, 0.375, -0.375, 0.25, 185.0],
[-0.75, 0.0, 0.0, 0.25, 64.0],
[-0.53033, -0.375, 0.375, 0.25, 185.0],
[0.0, -0.53033, 0.53033, 0.25, 64.0],
[0.53033, -0.375, 0.375, 0.25, 185.0],
],
vec![
[0.001, 0.0, 0.0, 1.0, 64.0],
[0.000707, 0.000707, 0.0, 1.0, 85.0],
[0.0, 0.001, 0.0, 1.0, 106.0],
[-0.000707, 0.000707, 0.0, 1.0, 127.0],
[-0.001, 0.0, 0.0, 1.0, 148.0],
[-0.000707, -0.000707, 0.0, 1.0, 169.0],
[0.0, -0.001, 0.0, 1.0, 190.0],
[0.000707, -0.000707, 0.0, 1.0, 211.0],
],
vec![
[0.0, 0.000058, 0.001947, 1.0, 0.0],
[0.0, -0.001715, 0.000923, 1.0, 0.0],
[0.0, 0.001715, -0.000923, 1.0, 0.0],
[0.0, -0.000058, -0.001947, 1.0, 0.0],
[0.001657, -0.000512, 0.000887, 1.0, 0.0],
[-0.001657, -0.000512, 0.000887, 1.0, 0.0],
[0.001657, 0.000512, -0.000887, 1.0, 0.0],
[-0.001657, 0.000512, -0.000887, 1.0, 0.0],
[0.001024, 0.001435, 0.000829, 1.0, 0.0],
[-0.001024, 0.001435, 0.000829, 1.0, 0.0],
[0.001024, -0.001435, -0.000829, 1.0, 0.0],
[-0.001024, -0.001435, -0.000829, 1.0, 0.0],
[0.001868, 0.000618, 0.000357, 1.0, 255.0],
[-0.001868, 0.000618, 0.000357, 1.0, 255.0],
[0.001868, -0.000618, -0.000357, 1.0, 255.0],
[-0.001868, -0.000618, -0.000357, 1.0, 255.0],
[0.000714, -0.000934, 0.001618, 1.0, 255.0],
[-0.000714, -0.000934, 0.001618, 1.0, 255.0],
[0.000714, 0.000934, -0.001618, 1.0, 255.0],
[-0.000714, 0.000934, -0.001618, 1.0, 255.0],
[0.0, -0.001975, -0.000316, 1.0, 255.0],
[0.0, 0.001261, 0.001552, 1.0, 255.0],
[0.0, -0.001261, -0.001552, 1.0, 255.0],
[0.0, 0.001975, 0.000316, 1.0, 255.0],
[0.001155, 0.000423, 0.001577, 1.0, 255.0],
[-0.001155, 0.000423, 0.001577, 1.0, 255.0],
[0.001155, -0.001577, 0.000423, 1.0, 255.0],
[-0.001155, -0.001577, 0.000423, 1.0, 255.0],
[0.001155, 0.001577, -0.000423, 1.0, 255.0],
[-0.001155, 0.001577, -0.000423, 1.0, 255.0],
[0.001155, -0.000423, -0.001577, 1.0, 255.0],
[-0.001155, -0.000423, -0.001577, 1.0, 255.0],
],
vec![
[0.0, 0.0, 1000.0, 2.0, 128.0],
[0.0, 0.0, 0.003125, 1.0, 0.0],
[0.0, 0.0, 0.0, 0.9975, 255.0],
[0.0, 0.000547, -0.002573, 0.995, 0.0],
[0.0, -0.000547, -0.002573, 0.995, 0.0],
[0.0, 0.001071, -0.005037, 0.9925, 255.0],
[0.0, -0.001071, -0.005037, 0.9925, 255.0],
],
vec![
[0.0, 0.0, 0.0, 1.0, 210.0],
[0.0, 0.0, 0.0008, 1.0, 210.0],
[0.0, 0.0, 0.0004, 1.00012, 50.0],
[0.012969, 0.0, -0.042045, 0.96, 100.0],
[0.016397, 0.011913, -0.039054, 0.96, 210.0],
[0.004008, 0.012334, -0.042045, 0.96, 100.0],
[-0.006263, 0.019276, -0.039054, 0.96, 210.0],
[-0.010492, 0.007623, -0.042045, 0.96, 100.0],
[-0.020268, 0.0, -0.039054, 0.96, 210.0],
[-0.010492, -0.007623, -0.042045, 0.96, 100.0],
[-0.006263, -0.019276, -0.039054, 0.96, 210.0],
[0.004008, -0.012334, -0.042045, 0.96, 100.0],
[0.016397, -0.011913, -0.039054, 0.96, 210.0],
],
vec![
[0.0, 0.0, 1000.0, 2.0, 192.0],
[-2.0, 2.0, 100.0, 3.0, 98.0],
[2.0, 2.0, 100.0, 3.0, 98.0],
[0.5, -0.5, 500.0, 0.2, 255.0],
[-0.349469, -0.1, 20.000279, 0.6, 64.0],
[-0.349775, -0.100273, 20.000485, 0.6, 85.0],
[-0.350214, -0.100386, 20.000407, 0.6, 106.0],
[-0.350527, -0.100273, 20.000091, 0.6, 127.0],
[-0.350531, -0.1, 19.999721, 0.6, 148.0],
[-0.350225, -0.099727, 19.999515, 0.6, 169.0],
[-0.349786, -0.099614, 19.999593, 0.6, 190.0],
[-0.349473, -0.099727, 19.999909, 0.6, 211.0],
[0.20032, 0.349824, 15.000456, 0.3, 0.0],
[0.20032, 0.349517, 14.999924, 0.3, 0.0],
[0.19968, 0.350483, 15.000076, 0.3, 0.0],
[0.19968, 0.350176, 14.999544, 0.3, 0.0],
[0.200578, 0.350073, 14.999958, 0.3, 0.0],
[0.199817, 0.349519, 15.000277, 0.3, 0.0],
[0.200183, 0.350481, 14.999723, 0.3, 0.0],
[0.199422, 0.349927, 15.000042, 0.3, 0.0],
[0.200235, 0.35042, 15.000332, 0.3, 0.0],
[0.199765, 0.350078, 15.000529, 0.3, 0.0],
[0.200235, 0.349922, 14.999471, 0.3, 0.0],
[0.199765, 0.34958, 14.999668, 0.3, 0.0],
[0.200429, 0.350419, 15.000005, 0.3, 255.0],
[0.199571, 0.349795, 15.000366, 0.3, 255.0],
[0.200429, 0.350205, 14.999634, 0.3, 255.0],
[0.199571, 0.349581, 14.999995, 0.3, 255.0],
[0.200524, 0.349747, 15.000146, 0.3, 255.0],
[0.200196, 0.349509, 15.000283, 0.3, 255.0],
[0.199804, 0.350491, 14.999717, 0.3, 255.0],
[0.199476, 0.350253, 14.999854, 0.3, 255.0],
[0.200138, 0.349578, 14.999597, 0.3, 255.0],
[0.200138, 0.350138, 15.000567, 0.3, 255.0],
[0.199862, 0.349862, 14.999433, 0.3, 255.0],
[0.199862, 0.350422, 15.000403, 0.3, 255.0],
[0.200488, 0.350136, 15.000321, 0.3, 255.0],
[0.199957, 0.349751, 15.000544, 0.3, 255.0],
[0.200488, 0.34979, 14.999721, 0.3, 255.0],
[0.199957, 0.349404, 14.999944, 0.3, 255.0],
[0.200043, 0.350596, 15.000056, 0.3, 255.0],
[0.199512, 0.35021, 15.000279, 0.3, 255.0],
[0.200043, 0.350249, 14.999456, 0.3, 255.0],
[0.199512, 0.349864, 14.999679, 0.3, 255.0],
[-0.5, 0.500442, 10.000442, 0.2, 0.0],
[-0.5, 0.5, 10.0, 0.1995, 255.0],
[-0.499966, 0.49971, 9.999563, 0.199, 0.0],
[-0.500034, 0.499563, 9.99971, 0.199, 0.0],
[-0.499934, 0.499432, 9.999144, 0.1985, 255.0],
[-0.500066, 0.499144, 9.999432, 0.1985, 255.0],
[0.6, 0.5, 10.0, 0.25, 210.0],
[0.600122, 0.4999, 10.000122, 0.25, 210.0],
[0.600061, 0.49995, 10.000061, 0.25003, 50.0],
[0.595856, 0.505256, 9.991271, 0.24, 100.0],
[0.597973, 0.507461, 9.992175, 0.24, 210.0],
[0.595362, 0.507926, 9.993945, 0.24, 100.0],
[0.594618, 0.509055, 9.996832, 0.24, 210.0],
[0.592382, 0.506906, 9.996092, 0.24, 100.0],
[0.590438, 0.504882, 9.997604, 0.24, 210.0],
[0.591035, 0.503605, 9.994744, 0.24, 100.0],
[0.59121, 0.500708, 9.993425, 0.24, 210.0],
[0.593181, 0.502585, 9.991764, 0.24, 100.0],
[0.595867, 0.502302, 9.990069, 0.24, 210.0],
],
];
let scene_names = [
"single_sphere",
"two_spheres",
"ring_of_spheres",
"beachball",
"football",
"eightball",
"starball",
"combination_scene",
];
for i in 0..8 {
eprintln!("Generating {}", scene_names[i]);
let values = grey_values(&test_cases[i]);
let filename = format!("{}.png", scene_names[i]);
let path = std::path::Path::new(&filename);
let file = std::fs::File::create(path).unwrap();
let w = &mut std::io::BufWriter::new(file);
let mut encoder = png::Encoder::new(w, 1024, 1024);
encoder.set_color(png::ColorType::Grayscale);
encoder.set_depth(png::BitDepth::Eight);
let mut writer = encoder.write_header().unwrap();
writer.write_image_data(&values).unwrap();
}
}
```
## Test cases
The images may be scaled down. Clicking or tapping on an image will open it full size.
### Single sphere
Input:
```text
TODO
```
Output:
```text
TODO
```
### Two separated spheres
Input:
```text
TODO
```
Output:
```text
TODO
```
### Eight spheres in a ring
Input:
```text
TODO
```
Output:
```text
TODO
```
### Beach ball
Input:
```text
TODO
```
Output:
```text
TODO
```
### Football
Input:
```text
TODO
```
Output:
```text
TODO
```
### Eight ball
Input:
```text
TODO
```
Output:
```text
TODO
```
### Star ball
Input:
```text
TODO
```
Output:
```text
TODO
```
### Combination scene
Input:
```text
TODO
```
Output:
```text
TODO
```
## Explanatory animations
The example images were produced from plain grey spheres with no surface pattern or texture. The following animations show how they break down into their component plain spheres.
This section is not necessary to write valid code for this challenge. If your code accepts the parameters of plain grey spheres and renders them correctly, all of the test cases will look right without needing to understand how they were constructed. This section is mainly out of interest, and to make clear that your code *only* needs to render plain grey spheres.
```text
CONSIDER HOSTING THESE OFF SITE AS THEY ARE NOT REQUIRED FOR THE CHALLENGE
AND EVEN THE SCALED DOWN VERSIONS ARE UNNECESSARY DATA USE FOR ANYONE
CHOOSING NOT TO VIEW THEM.
```
<details>
<summary>Animations</summary>
<details>
<summary>Beach ball</summary>
This animation is reduced to 256 by 256 pixels. Clicking or tapping on the animation will open it at full size (______MB).
```text
TODO
```
</details>
<details>
<summary>Football</summary>
This animation is reduced to 256 by 256 pixels. Clicking or tapping on the animation will open it at full size (______MB).
```text
TODO
```
</details>
<details>
<summary>Eight ball</summary>
This animation is reduced to 256 by 256 pixels. Clicking or tapping on the animation will open it at full size (______MB).
```text
TODO
```
</details>
<details>
<summary>Star ball</summary>
This animation is reduced to 256 by 256 pixels. Clicking or tapping on the animation will open it at full size (______MB).
```text
TODO
```
</details>
</details>
## Scoring
This is a [code golf challenge]. Your score is the number of bytes in your code. Lowest score for each language wins.
> Explanations are optional, but I'm more likely to upvote answers that have one.
[code golf challenge]: https://codegolf.codidact.com/categories/49/tags/4274 "The code-golf tag"
[PGM]: https://en.wikipedia.org/wiki/Netpbm#PGM_example

#4: Post edited by

trichoplax‭ · 2024-07-26T10:49:20Z (5 months ago)
Make Rust test cases avoid horizontal scroll bars

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Given a list of spheres, output an image of them.
The challenge is to produce a greyscale [orthographic](https://en.wikipedia.org/wiki/Parallel_projection#Orthographic_projection) (parallel projection - no perspective) ray caster, that renders only plain grey spheres with no lighting, no shading, no shadows, no surface patterns or textures, no reflections, and no transparency.
## Input
- A list of spheres, each consisting of:
- The coordinates of its centre $(x, y, z)$.
- Its radius $r$.
- Its greyscale value $g$, an integer from $0$ to $255$ (both inclusive).
- The input may include up to $100$ spheres.
- Only the greyscale value will be restricted to integer values. The coordinates and radius may have fractional values.
- Here "integer" refers to the values the greyscale value make take. You are not required to use an integer variable type.[]()
## Output
- A single greyscale image:
- $1024$ by $1024$ pixels.
- Each pixel has a single greyscale value from $0$ to $255$.
- Horizontally, the image covers x values from $-1$ to $1$.
- Vertically, the image covers y values from $-1$ to $1$.
- The image is orthographic, so there is no perspective. Changing the $z$ coordinate of a sphere does not change its size in the image, instead only affecting whether it is in front of or behind other spheres.
- A sphere's greyscale value is applied to every visible part of it. There is no variation over the surface of the sphere, so it appears flat like a disk.
- The point of the challenge is producing the pixel values, not manipulating image file formats, so you may output whatever image format is convenient, including simple text formats such as [PGM] or even just a list of pixel greyscale values. If you want to include an example of your output as an image in your answer, the image formats that Codidact supports are PNG, JPEG, and GIF. You could output PGM and then convert to PNG to upload to your answer. If you output just the greyscale values as a list of numbers from $0$ to $255$, I've made a `[ TODO page to convert these to greyscale PNG]`.
- The test cases, which match the reference implementations, happen to have the x axis increasing from left to right, and the y axis increasing from top to bottom. You are free to instead interpret these as increasing from right to left for x or from bottom to top for y, resulting in a reflected image. You are also free to output an image rotated by a multiple of 90 degrees, allowing for swapping the x and y axes. However, the z axis is required to be increasing in the forwards direction (into the image) otherwise some of the test cases will look incorrect.
## Calculating a pixel value
Imagine a ray passing into the image through a pixel, perpendicular to the image plane. If it hits a sphere, it shows the greyscale value of the first sphere it hits. Otherwise, it shows a value of zero (black). One possible way of calculating this is as follows:
For each pixel:
- The pixel $x, y$ coordinates will each be from $0$ to $1023$. Scale these to the range $-1$ to $1$ (for example, divide by $512$ and subtract $1$).
- If these scaled $x, y$ coordinates are within the radius $r$ of the $x, y$ coordinates of a sphere's centre, the ray intersects that sphere (the $z$ coordinate is not relevant to this test because the image is an orthographic projection - the rays are all parallel to the $z$ axis). The distance $a$ of the ray from the sphere's centre can be calculated as follows:
$$a=\sqrt{(x_{ray}-x_{sphere})^2+(y_{ray}-y_{sphere})^2}$$
```text
CONSIDER A DIAGRAM HERE
```
- For each sphere that the ray intersects, find the $z$ coordinate of the nearest intersection (there will often be $2$ per sphere, $1$ on the front of the sphere and $1$ on the back of the sphere, as the ray passes through - only the nearest is relevant). This can be calculated as follows:
- The distance $d$ to the nearest intersection point can be calculated from the $z$ coordinate of the sphere's centre, the radius $r$, and the distance $a$ from the $x, y$ coordinates of the sphere's centre to the scaled $x, y$ coordinates of the pixel:
$$d=z_{sphere}-\sqrt{r_{sphere}^2-a^2}$$
```text
CONSIDER A DIAGRAM HERE
```
- Of all the spheres that the ray intersects, the sphere with the intersection point with the smallest $z$ coordinate $d$ is the one to display. Set the pixel to the greyscale value of this sphere.
- If the ray did not intersect any spheres, set the pixel to zero (black).
## Example implementations
These are ungolfed implementations designed to supplement understanding of the requirements. You are free to golf these or to implement your own approach that matches the test cases up to reflection and rotation.
The Python implementation uses 64 bit floating point giving accurate images. The Rust implementation deliberately uses 32 bit floating point to show the difference in output. I've tried to optimise the test cases for compatibility with 32 bit floating point, so the differences are restricted to being a pixel out in some places.
### Python
The function `grey_values` would be sufficient as an entry to this challenge. The rest of the code saves the outputs as PNG files. The function itself has no dependence on the `import png` so it would not be required in an entry.
```python
def grey_values(spheres):
values = []
for y_pixel in range(1024):
for x_pixel in range(1024):
x_ray, y_ray = x_pixel / 512 - 1, y_pixel / 512 - 1
min_distance, hit_grey_value = None, 0
for sphere in spheres:
x_sphere, y_sphere, z_sphere, radius, grey_value = sphere
a = ((x_ray-x_sphere) ** 2 + (y_ray-y_sphere) ** 2) ** 0.5
if a < radius:
distance = z_sphere - (radius ** 2 - a ** 2) ** 0.5
if min_distance is None or distance < min_distance:
min_distance = distance
hit_grey_value = grey_value
values.append(hit_grey_value)
return values
import png
test_cases = [
[
[0, 0, 0, 1, 128],
],
[
[0, -0.55, 0, 0.4, 100],
[0, 0.55, 0, 0.4, 200],
],
[
[0.75, 0, 0, 0.25, 64],
[0.53033, 0.375, -0.375, 0.25, 185],
[0, 0.53033, -0.53033, 0.25, 64],
[-0.53033, 0.375, -0.375, 0.25, 185],
[-0.75, 0, -0, 0.25, 64],
[-0.53033, -0.375, 0.375, 0.25, 185],
[-0, -0.53033, 0.53033, 0.25, 64],
[0.53033, -0.375, 0.375, 0.25, 185],
],
[
[0.001, 0, 0, 1, 64],
[0.000707, 0.000707, -0, 1, 85],
[0, 0.001, -0, 1, 106],
[-0.000707, 0.000707, -0, 1, 127],
[-0.001, 0, -0, 1, 148],
[-0.000707, -0.000707, 0, 1, 169],
[-0, -0.001, 0, 1, 190],
[0.000707, -0.000707, 0, 1, 211],
],
[
[0, 0.000058, 0.001947, 1, 0],
[0, -0.001715, 0.000923, 1, 0],
[0, 0.001715, -0.000923, 1, 0],
[0, -0.000058, -0.001947, 1, 0],
[0.001657, -0.000512, 0.000887, 1, 0],
[-0.001657, -0.000512, 0.000887, 1, 0],
[0.001657, 0.000512, -0.000887, 1, 0],
[-0.001657, 0.000512, -0.000887, 1, 0],
[0.001024, 0.001435, 0.000829, 1, 0],
[-0.001024, 0.001435, 0.000829, 1, 0],
[0.001024, -0.001435, -0.000829, 1, 0],
[-0.001024, -0.001435, -0.000829, 1, 0],
[0.001868, 0.000618, 0.000357, 1, 255],
[-0.001868, 0.000618, 0.000357, 1, 255],
[0.001868, -0.000618, -0.000357, 1, 255],
[-0.001868, -0.000618, -0.000357, 1, 255],
[0.000714, -0.000934, 0.001618, 1, 255],
[-0.000714, -0.000934, 0.001618, 1, 255],
[0.000714, 0.000934, -0.001618, 1, 255],
[-0.000714, 0.000934, -0.001618, 1, 255],
[0, -0.001975, -0.000316, 1, 255],
[0, 0.001261, 0.001552, 1, 255],
[0, -0.001261, -0.001552, 1, 255],
[0, 0.001975, 0.000316, 1, 255],
[0.001155, 0.000423, 0.001577, 1, 255],
[-0.001155, 0.000423, 0.001577, 1, 255],
[0.001155, -0.001577, 0.000423, 1, 255],
[-0.001155, -0.001577, 0.000423, 1, 255],
[0.001155, 0.001577, -0.000423, 1, 255],
[-0.001155, 0.001577, -0.000423, 1, 255],
[0.001155, -0.000423, -0.001577, 1, 255],
[-0.001155, -0.000423, -0.001577, 1, 255],
],
[
[0, 0, 1000, 2, 128],
[0, 0, 0.003125, 1, 0],
[0, 0, 0, 0.9975, 255],
[0, 0.000547, -0.002573, 0.995, 0],
[-0, -0.000547, -0.002573, 0.995, 0],
[0, 0.001071, -0.005037, 0.9925, 255],
[-0, -0.001071, -0.005037, 0.9925, 255],
],
[
[0, 0, 0, 1, 210],
[0, 0, 0.0008, 1, 210],
[0, 0, 0.0004, 1.00012, 50],
[0.012969, -0, -0.042045, 0.96, 100],
[0.016397, 0.011913, -0.039054, 0.96, 210],
[0.004008, 0.012334, -0.042045, 0.96, 100],
[-0.006263, 0.019276, -0.039054, 0.96, 210],
[-0.010492, 0.007623, -0.042045, 0.96, 100],
[-0.020268, 0, -0.039054, 0.96, 210],
[-0.010492, -0.007623, -0.042045, 0.96, 100],
[-0.006263, -0.019276, -0.039054, 0.96, 210],
[0.004008, -0.012334, -0.042045, 0.96, 100],
[0.016397, -0.011913, -0.039054, 0.96, 210],
],
[
[0, 0, 1000, 2, 192],
[-2, 2, 100, 3, 98],
[2, 2, 100, 3, 98],
[0.5, -0.5, 500, 0.2, 255],
[-0.349469, -0.1, 20.000279, 0.6, 64],
[-0.349775, -0.100273, 20.000485, 0.6, 85],
[-0.350214, -0.100386, 20.000407, 0.6, 106],
[-0.350527, -0.100273, 20.000091, 0.6, 127],
[-0.350531, -0.1, 19.999721, 0.6, 148],
[-0.350225, -0.099727, 19.999515, 0.6, 169],
[-0.349786, -0.099614, 19.999593, 0.6, 190],
[-0.349473, -0.099727, 19.999909, 0.6, 211],
[0.20032, 0.349824, 15.000456, 0.3, 0],
[0.20032, 0.349517, 14.999924, 0.3, 0],
[0.19968, 0.350483, 15.000076, 0.3, 0],
[0.19968, 0.350176, 14.999544, 0.3, 0],
[0.200578, 0.350073, 14.999958, 0.3, 0],
[0.199817, 0.349519, 15.000277, 0.3, 0],
[0.200183, 0.350481, 14.999723, 0.3, 0],
[0.199422, 0.349927, 15.000042, 0.3, 0],
[0.200235, 0.35042, 15.000332, 0.3, 0],
[0.199765, 0.350078, 15.000529, 0.3, 0],
[0.200235, 0.349922, 14.999471, 0.3, 0],
[0.199765, 0.34958, 14.999668, 0.3, 0],
[0.200429, 0.350419, 15.000005, 0.3, 255],
[0.199571, 0.349795, 15.000366, 0.3, 255],
[0.200429, 0.350205, 14.999634, 0.3, 255],
[0.199571, 0.349581, 14.999995, 0.3, 255],
[0.200524, 0.349747, 15.000146, 0.3, 255],
[0.200196, 0.349509, 15.000283, 0.3, 255],
[0.199804, 0.350491, 14.999717, 0.3, 255],
[0.199476, 0.350253, 14.999854, 0.3, 255],
[0.200138, 0.349578, 14.999597, 0.3, 255],
[0.200138, 0.350138, 15.000567, 0.3, 255],
[0.199862, 0.349862, 14.999433, 0.3, 255],
[0.199862, 0.350422, 15.000403, 0.3, 255],
[0.200488, 0.350136, 15.000321, 0.3, 255],
[0.199957, 0.349751, 15.000544, 0.3, 255],
[0.200488, 0.34979, 14.999721, 0.3, 255],
[0.199957, 0.349404, 14.999944, 0.3, 255],
[0.200043, 0.350596, 15.000056, 0.3, 255],
[0.199512, 0.35021, 15.000279, 0.3, 255],
[0.200043, 0.350249, 14.999456, 0.3, 255],
[0.199512, 0.349864, 14.999679, 0.3, 255],
[-0.5, 0.500442, 10.000442, 0.2, 0],
[-0.5, 0.5, 10, 0.1995, 255],
[-0.499966, 0.49971, 9.999563, 0.199, 0],
[-0.500034, 0.499563, 9.99971, 0.199, 0],
[-0.499934, 0.499432, 9.999144, 0.1985, 255],
[-0.500066, 0.499144, 9.999432, 0.1985, 255],
[0.6, 0.5, 10, 0.25, 210],
[0.600122, 0.4999, 10.000122, 0.25, 210],
[0.600061, 0.49995, 10.000061, 0.25003, 50],
[0.595856, 0.505256, 9.991271, 0.24, 100],
[0.597973, 0.507461, 9.992175, 0.24, 210],
[0.595362, 0.507926, 9.993945, 0.24, 100],
[0.594618, 0.509055, 9.996832, 0.24, 210],
[0.592382, 0.506906, 9.996092, 0.24, 100],
[0.590438, 0.504882, 9.997604, 0.24, 210],
[0.591035, 0.503605, 9.994744, 0.24, 100],
[0.59121, 0.500708, 9.993425, 0.24, 210],
[0.593181, 0.502585, 9.991764, 0.24, 100],
[0.595867, 0.502302, 9.990069, 0.24, 210],
],
]
scene_names = (
"single_sphere",
"two_spheres",
"ring_of_spheres",
"beachball",
"football",
"eightball",
"starball",
"combination_scene",
)
for i in range(8):
print(f'Generating {scene_names[i]}.')
values = grey_values(test_cases[i])
rows = [values[row * 1024:row * 1024 + 1024] for row in range(1024)]
with open(f'{scene_names[i]}.png', 'wb') as file:
w = png.Writer(1024, 1024, greyscale=True)
w.write(file, rows)
```
### Rust
The function `grey_values` would be sufficient as an entry to this challenge. The rest of the code saves the outputs as PNG files.
```rust
fn grey_values(spheres: &Vec<[f32; 5]>) -> Vec<u8> {
let mut values = vec![];
for y_pixel in 0..1024 {
for x_pixel in 0..1024 {
let x_ray = x_pixel as f32 / 512.0 - 1.0;
let y_ray = y_pixel as f32 / 512.0 - 1.0;
let mut min_distance = f32::INFINITY;
let mut hit_grey_value = 0;
for sphere in spheres {
let [
x_sphere, y_sphere, z_sphere, radius, grey
] = *sphere;
let a = (
(x_ray - x_sphere).powf(2.0)
+ (y_ray - y_sphere).powf(2.0)
).sqrt();
if a < radius {
let distance = z_sphere - (
radius.powf(2.0) - a.powf(2.0)
).sqrt();
if distance < min_distance {
min_distance = distance;
hit_grey_value = grey as u8;
}
}
}
values.push(hit_grey_value);
}
}
values
}
fn main() {
let test_cases: Vec<Vec<[f32; 5]>> = vec![
~~vec![[0.0, 0.0, 0.0, 1.0, 128.0]],~~
~~vec![[0.0, -0.55, 0.0, 0.4, 100.0], [0.0, 0.55, 0.0, 0.4, 200.0]],~~
vec![
[0.75, 0.0, 0.0, 0.25, 64.0],
[0.53033, 0.375, -0.375, 0.25, 185.0],
[0.0, 0.53033, -0.53033, 0.25, 64.0],
[-0.53033, 0.375, -0.375, 0.25, 185.0],
[-0.75, 0.0, 0.0, 0.25, 64.0],
[-0.53033, -0.375, 0.375, 0.25, 185.0],
[0.0, -0.53033, 0.53033, 0.25, 64.0],
[0.53033, -0.375, 0.375, 0.25, 185.0],
],
vec![
[0.001, 0.0, 0.0, 1.0, 64.0],
[0.000707, 0.000707, 0.0, 1.0, 85.0],
[0.0, 0.001, 0.0, 1.0, 106.0],
[-0.000707, 0.000707, 0.0, 1.0, 127.0],
[-0.001, 0.0, 0.0, 1.0, 148.0],
[-0.000707, -0.000707, 0.0, 1.0, 169.0],
[0.0, -0.001, 0.0, 1.0, 190.0],
[0.000707, -0.000707, 0.0, 1.0, 211.0],
],
vec![
[0.0, 0.000058, 0.001947, 1.0, 0.0],
[0.0, -0.001715, 0.000923, 1.0, 0.0],
[0.0, 0.001715, -0.000923, 1.0, 0.0],
[0.0, -0.000058, -0.001947, 1.0, 0.0],
[0.001657, -0.000512, 0.000887, 1.0, 0.0],
[-0.001657, -0.000512, 0.000887, 1.0, 0.0],
[0.001657, 0.000512, -0.000887, 1.0, 0.0],
[-0.001657, 0.000512, -0.000887, 1.0, 0.0],
[0.001024, 0.001435, 0.000829, 1.0, 0.0],
[-0.001024, 0.001435, 0.000829, 1.0, 0.0],
[0.001024, -0.001435, -0.000829, 1.0, 0.0],
[-0.001024, -0.001435, -0.000829, 1.0, 0.0],
[0.001868, 0.000618, 0.000357, 1.0, 255.0],
[-0.001868, 0.000618, 0.000357, 1.0, 255.0],
[0.001868, -0.000618, -0.000357, 1.0, 255.0],
[-0.001868, -0.000618, -0.000357, 1.0, 255.0],
[0.000714, -0.000934, 0.001618, 1.0, 255.0],
[-0.000714, -0.000934, 0.001618, 1.0, 255.0],
[0.000714, 0.000934, -0.001618, 1.0, 255.0],
[-0.000714, 0.000934, -0.001618, 1.0, 255.0],
[0.0, -0.001975, -0.000316, 1.0, 255.0],
[0.0, 0.001261, 0.001552, 1.0, 255.0],
[0.0, -0.001261, -0.001552, 1.0, 255.0],
[0.0, 0.001975, 0.000316, 1.0, 255.0],
[0.001155, 0.000423, 0.001577, 1.0, 255.0],
[-0.001155, 0.000423, 0.001577, 1.0, 255.0],
[0.001155, -0.001577, 0.000423, 1.0, 255.0],
[-0.001155, -0.001577, 0.000423, 1.0, 255.0],
[0.001155, 0.001577, -0.000423, 1.0, 255.0],
[-0.001155, 0.001577, -0.000423, 1.0, 255.0],
[0.001155, -0.000423, -0.001577, 1.0, 255.0],
[-0.001155, -0.000423, -0.001577, 1.0, 255.0],
],
vec![
[0.0, 0.0, 1000.0, 2.0, 128.0],
[0.0, 0.0, 0.003125, 1.0, 0.0],
[0.0, 0.0, 0.0, 0.9975, 255.0],
[0.0, 0.000547, -0.002573, 0.995, 0.0],
[0.0, -0.000547, -0.002573, 0.995, 0.0],
[0.0, 0.001071, -0.005037, 0.9925, 255.0],
[0.0, -0.001071, -0.005037, 0.9925, 255.0],
],
vec![
[0.0, 0.0, 0.0, 1.0, 210.0],
[0.0, 0.0, 0.0008, 1.0, 210.0],
[0.0, 0.0, 0.0004, 1.00012, 50.0],
[0.012969, 0.0, -0.042045, 0.96, 100.0],
[0.016397, 0.011913, -0.039054, 0.96, 210.0],
[0.004008, 0.012334, -0.042045, 0.96, 100.0],
[-0.006263, 0.019276, -0.039054, 0.96, 210.0],
[-0.010492, 0.007623, -0.042045, 0.96, 100.0],
[-0.020268, 0.0, -0.039054, 0.96, 210.0],
[-0.010492, -0.007623, -0.042045, 0.96, 100.0],
[-0.006263, -0.019276, -0.039054, 0.96, 210.0],
[0.004008, -0.012334, -0.042045, 0.96, 100.0],
[0.016397, -0.011913, -0.039054, 0.96, 210.0],
],
vec![
[0.0, 0.0, 1000.0, 2.0, 192.0],
[-2.0, 2.0, 100.0, 3.0, 98.0],
[2.0, 2.0, 100.0, 3.0, 98.0],
[0.5, -0.5, 500.0, 0.2, 255.0],
[-0.349469, -0.1, 20.000279, 0.6, 64.0],
[-0.349775, -0.100273, 20.000485, 0.6, 85.0],
[-0.350214, -0.100386, 20.000407, 0.6, 106.0],
[-0.350527, -0.100273, 20.000091, 0.6, 127.0],
[-0.350531, -0.1, 19.999721, 0.6, 148.0],
[-0.350225, -0.099727, 19.999515, 0.6, 169.0],
[-0.349786, -0.099614, 19.999593, 0.6, 190.0],
[-0.349473, -0.099727, 19.999909, 0.6, 211.0],
[0.20032, 0.349824, 15.000456, 0.3, 0.0],
[0.20032, 0.349517, 14.999924, 0.3, 0.0],
[0.19968, 0.350483, 15.000076, 0.3, 0.0],
[0.19968, 0.350176, 14.999544, 0.3, 0.0],
[0.200578, 0.350073, 14.999958, 0.3, 0.0],
[0.199817, 0.349519, 15.000277, 0.3, 0.0],
[0.200183, 0.350481, 14.999723, 0.3, 0.0],
[0.199422, 0.349927, 15.000042, 0.3, 0.0],
[0.200235, 0.35042, 15.000332, 0.3, 0.0],
[0.199765, 0.350078, 15.000529, 0.3, 0.0],
[0.200235, 0.349922, 14.999471, 0.3, 0.0],
[0.199765, 0.34958, 14.999668, 0.3, 0.0],
[0.200429, 0.350419, 15.000005, 0.3, 255.0],
[0.199571, 0.349795, 15.000366, 0.3, 255.0],
[0.200429, 0.350205, 14.999634, 0.3, 255.0],
[0.199571, 0.349581, 14.999995, 0.3, 255.0],
[0.200524, 0.349747, 15.000146, 0.3, 255.0],
[0.200196, 0.349509, 15.000283, 0.3, 255.0],
[0.199804, 0.350491, 14.999717, 0.3, 255.0],
[0.199476, 0.350253, 14.999854, 0.3, 255.0],
[0.200138, 0.349578, 14.999597, 0.3, 255.0],
[0.200138, 0.350138, 15.000567, 0.3, 255.0],
[0.199862, 0.349862, 14.999433, 0.3, 255.0],
[0.199862, 0.350422, 15.000403, 0.3, 255.0],
[0.200488, 0.350136, 15.000321, 0.3, 255.0],
[0.199957, 0.349751, 15.000544, 0.3, 255.0],
[0.200488, 0.34979, 14.999721, 0.3, 255.0],
[0.199957, 0.349404, 14.999944, 0.3, 255.0],
[0.200043, 0.350596, 15.000056, 0.3, 255.0],
[0.199512, 0.35021, 15.000279, 0.3, 255.0],
[0.200043, 0.350249, 14.999456, 0.3, 255.0],
[0.199512, 0.349864, 14.999679, 0.3, 255.0],
[-0.5, 0.500442, 10.000442, 0.2, 0.0],
[-0.5, 0.5, 10.0, 0.1995, 255.0],
[-0.499966, 0.49971, 9.999563, 0.199, 0.0],
[-0.500034, 0.499563, 9.99971, 0.199, 0.0],
[-0.499934, 0.499432, 9.999144, 0.1985, 255.0],
[-0.500066, 0.499144, 9.999432, 0.1985, 255.0],
[0.6, 0.5, 10.0, 0.25, 210.0],
[0.600122, 0.4999, 10.000122, 0.25, 210.0],
[0.600061, 0.49995, 10.000061, 0.25003, 50.0],
[0.595856, 0.505256, 9.991271, 0.24, 100.0],
[0.597973, 0.507461, 9.992175, 0.24, 210.0],
[0.595362, 0.507926, 9.993945, 0.24, 100.0],
[0.594618, 0.509055, 9.996832, 0.24, 210.0],
[0.592382, 0.506906, 9.996092, 0.24, 100.0],
[0.590438, 0.504882, 9.997604, 0.24, 210.0],
[0.591035, 0.503605, 9.994744, 0.24, 100.0],
[0.59121, 0.500708, 9.993425, 0.24, 210.0],
[0.593181, 0.502585, 9.991764, 0.24, 100.0],
[0.595867, 0.502302, 9.990069, 0.24, 210.0],
],
];
let scene_names = [
"single_sphere",
"two_spheres",
"ring_of_spheres",
"beachball",
"football",
"eightball",
"starball",
"combination_scene",
];
for i in 0..8 {
eprintln!("Generating {}", scene_names[i]);
let values = grey_values(&test_cases[i]);
let filename = format!("{}.png", scene_names[i]);
let path = std::path::Path::new(&filename);
let file = std::fs::File::create(path).unwrap();
let w = &mut std::io::BufWriter::new(file);
let mut encoder = png::Encoder::new(w, 1024, 1024);
encoder.set_color(png::ColorType::Grayscale);
encoder.set_depth(png::BitDepth::Eight);
let mut writer = encoder.write_header().unwrap();
writer.write_image_data(&values).unwrap();
}
}
```
## Test cases
The images may be scaled down. Clicking or tapping on an image will open it full size.
### Single sphere
Input:
```text
TODO
```
Output:
```text
TODO
```
### Two separated spheres
Input:
```text
TODO
```
Output:
```text
TODO
```
### Eight spheres in a ring
Input:
```text
TODO
```
Output:
```text
TODO
```
### Beach ball
Input:
```text
TODO
```
Output:
```text
TODO
```
### Football
Input:
```text
TODO
```
Output:
```text
TODO
```
### Eight ball
Input:
```text
TODO
```
Output:
```text
TODO
```
### Star ball
Input:
```text
TODO
```
Output:
```text
TODO
```
### Combination scene
Input:
```text
TODO
```
Output:
```text
TODO
```
## Explanatory animations
The example images were produced from plain grey spheres with no surface pattern or texture. The following animations show how they break down into their component plain spheres.
This section is not necessary to write valid code for this challenge. If your code accepts the parameters of plain grey spheres and renders them correctly, all of the test cases will look right without needing to understand how they were constructed. This section is mainly out of interest, and to make clear that your code *only* needs to render plain grey spheres.
```text
CONSIDER HOSTING THESE OFF SITE AS THEY ARE NOT REQUIRED FOR THE CHALLENGE
AND EVEN THE SCALED DOWN VERSIONS ARE UNNECESSARY DATA USE FOR ANYONE
CHOOSING NOT TO VIEW THEM.
```
<details>
<summary>Animations</summary>
<details>
<summary>Beach ball</summary>
This animation is reduced to 256 by 256 pixels. Clicking or tapping on the animation will open it at full size (______MB).
```text
TODO
```
</details>
<details>
<summary>Football</summary>
This animation is reduced to 256 by 256 pixels. Clicking or tapping on the animation will open it at full size (______MB).
```text
TODO
```
</details>
<details>
<summary>Eight ball</summary>
This animation is reduced to 256 by 256 pixels. Clicking or tapping on the animation will open it at full size (______MB).
```text
TODO
```
</details>
<details>
<summary>Star ball</summary>
This animation is reduced to 256 by 256 pixels. Clicking or tapping on the animation will open it at full size (______MB).
```text
TODO
```
</details>
</details>
## Scoring
This is a [code golf challenge]. Your score is the number of bytes in your code. Lowest score for each language wins.
> Explanations are optional, but I'm more likely to upvote answers that have one.
[code golf challenge]: https://codegolf.codidact.com/categories/49/tags/4274 "The code-golf tag"
[PGM]: https://en.wikipedia.org/wiki/Netpbm#PGM_example

Given a list of spheres, output an image of them.
The challenge is to produce a greyscale [orthographic](https://en.wikipedia.org/wiki/Parallel_projection#Orthographic_projection) (parallel projection - no perspective) ray caster, that renders only plain grey spheres with no lighting, no shading, no shadows, no surface patterns or textures, no reflections, and no transparency.
## Input
- A list of spheres, each consisting of:
- The coordinates of its centre $(x, y, z)$.
- Its radius $r$.
- Its greyscale value $g$, an integer from $0$ to $255$ (both inclusive).
- The input may include up to $100$ spheres.
- Only the greyscale value will be restricted to integer values. The coordinates and radius may have fractional values.
- Here "integer" refers to the values the greyscale value make take. You are not required to use an integer variable type.[]()
## Output
- A single greyscale image:
- $1024$ by $1024$ pixels.
- Each pixel has a single greyscale value from $0$ to $255$.
- Horizontally, the image covers x values from $-1$ to $1$.
- Vertically, the image covers y values from $-1$ to $1$.
- The image is orthographic, so there is no perspective. Changing the $z$ coordinate of a sphere does not change its size in the image, instead only affecting whether it is in front of or behind other spheres.
- A sphere's greyscale value is applied to every visible part of it. There is no variation over the surface of the sphere, so it appears flat like a disk.
- The point of the challenge is producing the pixel values, not manipulating image file formats, so you may output whatever image format is convenient, including simple text formats such as [PGM] or even just a list of pixel greyscale values. If you want to include an example of your output as an image in your answer, the image formats that Codidact supports are PNG, JPEG, and GIF. You could output PGM and then convert to PNG to upload to your answer. If you output just the greyscale values as a list of numbers from $0$ to $255$, I've made a `[ TODO page to convert these to greyscale PNG]`.
- The test cases, which match the reference implementations, happen to have the x axis increasing from left to right, and the y axis increasing from top to bottom. You are free to instead interpret these as increasing from right to left for x or from bottom to top for y, resulting in a reflected image. You are also free to output an image rotated by a multiple of 90 degrees, allowing for swapping the x and y axes. However, the z axis is required to be increasing in the forwards direction (into the image) otherwise some of the test cases will look incorrect.
## Calculating a pixel value
Imagine a ray passing into the image through a pixel, perpendicular to the image plane. If it hits a sphere, it shows the greyscale value of the first sphere it hits. Otherwise, it shows a value of zero (black). One possible way of calculating this is as follows:
For each pixel:
- The pixel $x, y$ coordinates will each be from $0$ to $1023$. Scale these to the range $-1$ to $1$ (for example, divide by $512$ and subtract $1$).
- If these scaled $x, y$ coordinates are within the radius $r$ of the $x, y$ coordinates of a sphere's centre, the ray intersects that sphere (the $z$ coordinate is not relevant to this test because the image is an orthographic projection - the rays are all parallel to the $z$ axis). The distance $a$ of the ray from the sphere's centre can be calculated as follows:
$$a=\sqrt{(x_{ray}-x_{sphere})^2+(y_{ray}-y_{sphere})^2}$$
```text
CONSIDER A DIAGRAM HERE
```
- For each sphere that the ray intersects, find the $z$ coordinate of the nearest intersection (there will often be $2$ per sphere, $1$ on the front of the sphere and $1$ on the back of the sphere, as the ray passes through - only the nearest is relevant). This can be calculated as follows:
- The distance $d$ to the nearest intersection point can be calculated from the $z$ coordinate of the sphere's centre, the radius $r$, and the distance $a$ from the $x, y$ coordinates of the sphere's centre to the scaled $x, y$ coordinates of the pixel:
$$d=z_{sphere}-\sqrt{r_{sphere}^2-a^2}$$
```text
CONSIDER A DIAGRAM HERE
```
- Of all the spheres that the ray intersects, the sphere with the intersection point with the smallest $z$ coordinate $d$ is the one to display. Set the pixel to the greyscale value of this sphere.
- If the ray did not intersect any spheres, set the pixel to zero (black).
## Example implementations
These are ungolfed implementations designed to supplement understanding of the requirements. You are free to golf these or to implement your own approach that matches the test cases up to reflection and rotation.
The Python implementation uses 64 bit floating point giving accurate images. The Rust implementation deliberately uses 32 bit floating point to show the difference in output. I've tried to optimise the test cases for compatibility with 32 bit floating point, so the differences are restricted to being a pixel out in some places.
### Python
The function `grey_values` would be sufficient as an entry to this challenge. The rest of the code saves the outputs as PNG files. The function itself has no dependence on the `import png` so it would not be required in an entry.
```python
def grey_values(spheres):
values = []
for y_pixel in range(1024):
for x_pixel in range(1024):
x_ray, y_ray = x_pixel / 512 - 1, y_pixel / 512 - 1
min_distance, hit_grey_value = None, 0
for sphere in spheres:
x_sphere, y_sphere, z_sphere, radius, grey_value = sphere
a = ((x_ray-x_sphere) ** 2 + (y_ray-y_sphere) ** 2) ** 0.5
if a < radius:
distance = z_sphere - (radius ** 2 - a ** 2) ** 0.5
if min_distance is None or distance < min_distance:
min_distance = distance
hit_grey_value = grey_value
values.append(hit_grey_value)
return values
import png
test_cases = [
[
[0, 0, 0, 1, 128],
],
[
[0, -0.55, 0, 0.4, 100],
[0, 0.55, 0, 0.4, 200],
],
[
[0.75, 0, 0, 0.25, 64],
[0.53033, 0.375, -0.375, 0.25, 185],
[0, 0.53033, -0.53033, 0.25, 64],
[-0.53033, 0.375, -0.375, 0.25, 185],
[-0.75, 0, -0, 0.25, 64],
[-0.53033, -0.375, 0.375, 0.25, 185],
[-0, -0.53033, 0.53033, 0.25, 64],
[0.53033, -0.375, 0.375, 0.25, 185],
],
[
[0.001, 0, 0, 1, 64],
[0.000707, 0.000707, -0, 1, 85],
[0, 0.001, -0, 1, 106],
[-0.000707, 0.000707, -0, 1, 127],
[-0.001, 0, -0, 1, 148],
[-0.000707, -0.000707, 0, 1, 169],
[-0, -0.001, 0, 1, 190],
[0.000707, -0.000707, 0, 1, 211],
],
[
[0, 0.000058, 0.001947, 1, 0],
[0, -0.001715, 0.000923, 1, 0],
[0, 0.001715, -0.000923, 1, 0],
[0, -0.000058, -0.001947, 1, 0],
[0.001657, -0.000512, 0.000887, 1, 0],
[-0.001657, -0.000512, 0.000887, 1, 0],
[0.001657, 0.000512, -0.000887, 1, 0],
[-0.001657, 0.000512, -0.000887, 1, 0],
[0.001024, 0.001435, 0.000829, 1, 0],
[-0.001024, 0.001435, 0.000829, 1, 0],
[0.001024, -0.001435, -0.000829, 1, 0],
[-0.001024, -0.001435, -0.000829, 1, 0],
[0.001868, 0.000618, 0.000357, 1, 255],
[-0.001868, 0.000618, 0.000357, 1, 255],
[0.001868, -0.000618, -0.000357, 1, 255],
[-0.001868, -0.000618, -0.000357, 1, 255],
[0.000714, -0.000934, 0.001618, 1, 255],
[-0.000714, -0.000934, 0.001618, 1, 255],
[0.000714, 0.000934, -0.001618, 1, 255],
[-0.000714, 0.000934, -0.001618, 1, 255],
[0, -0.001975, -0.000316, 1, 255],
[0, 0.001261, 0.001552, 1, 255],
[0, -0.001261, -0.001552, 1, 255],
[0, 0.001975, 0.000316, 1, 255],
[0.001155, 0.000423, 0.001577, 1, 255],
[-0.001155, 0.000423, 0.001577, 1, 255],
[0.001155, -0.001577, 0.000423, 1, 255],
[-0.001155, -0.001577, 0.000423, 1, 255],
[0.001155, 0.001577, -0.000423, 1, 255],
[-0.001155, 0.001577, -0.000423, 1, 255],
[0.001155, -0.000423, -0.001577, 1, 255],
[-0.001155, -0.000423, -0.001577, 1, 255],
],
[
[0, 0, 1000, 2, 128],
[0, 0, 0.003125, 1, 0],
[0, 0, 0, 0.9975, 255],
[0, 0.000547, -0.002573, 0.995, 0],
[-0, -0.000547, -0.002573, 0.995, 0],
[0, 0.001071, -0.005037, 0.9925, 255],
[-0, -0.001071, -0.005037, 0.9925, 255],
],
[
[0, 0, 0, 1, 210],
[0, 0, 0.0008, 1, 210],
[0, 0, 0.0004, 1.00012, 50],
[0.012969, -0, -0.042045, 0.96, 100],
[0.016397, 0.011913, -0.039054, 0.96, 210],
[0.004008, 0.012334, -0.042045, 0.96, 100],
[-0.006263, 0.019276, -0.039054, 0.96, 210],
[-0.010492, 0.007623, -0.042045, 0.96, 100],
[-0.020268, 0, -0.039054, 0.96, 210],
[-0.010492, -0.007623, -0.042045, 0.96, 100],
[-0.006263, -0.019276, -0.039054, 0.96, 210],
[0.004008, -0.012334, -0.042045, 0.96, 100],
[0.016397, -0.011913, -0.039054, 0.96, 210],
],
[
[0, 0, 1000, 2, 192],
[-2, 2, 100, 3, 98],
[2, 2, 100, 3, 98],
[0.5, -0.5, 500, 0.2, 255],
[-0.349469, -0.1, 20.000279, 0.6, 64],
[-0.349775, -0.100273, 20.000485, 0.6, 85],
[-0.350214, -0.100386, 20.000407, 0.6, 106],
[-0.350527, -0.100273, 20.000091, 0.6, 127],
[-0.350531, -0.1, 19.999721, 0.6, 148],
[-0.350225, -0.099727, 19.999515, 0.6, 169],
[-0.349786, -0.099614, 19.999593, 0.6, 190],
[-0.349473, -0.099727, 19.999909, 0.6, 211],
[0.20032, 0.349824, 15.000456, 0.3, 0],
[0.20032, 0.349517, 14.999924, 0.3, 0],
[0.19968, 0.350483, 15.000076, 0.3, 0],
[0.19968, 0.350176, 14.999544, 0.3, 0],
[0.200578, 0.350073, 14.999958, 0.3, 0],
[0.199817, 0.349519, 15.000277, 0.3, 0],
[0.200183, 0.350481, 14.999723, 0.3, 0],
[0.199422, 0.349927, 15.000042, 0.3, 0],
[0.200235, 0.35042, 15.000332, 0.3, 0],
[0.199765, 0.350078, 15.000529, 0.3, 0],
[0.200235, 0.349922, 14.999471, 0.3, 0],
[0.199765, 0.34958, 14.999668, 0.3, 0],
[0.200429, 0.350419, 15.000005, 0.3, 255],
[0.199571, 0.349795, 15.000366, 0.3, 255],
[0.200429, 0.350205, 14.999634, 0.3, 255],
[0.199571, 0.349581, 14.999995, 0.3, 255],
[0.200524, 0.349747, 15.000146, 0.3, 255],
[0.200196, 0.349509, 15.000283, 0.3, 255],
[0.199804, 0.350491, 14.999717, 0.3, 255],
[0.199476, 0.350253, 14.999854, 0.3, 255],
[0.200138, 0.349578, 14.999597, 0.3, 255],
[0.200138, 0.350138, 15.000567, 0.3, 255],
[0.199862, 0.349862, 14.999433, 0.3, 255],
[0.199862, 0.350422, 15.000403, 0.3, 255],
[0.200488, 0.350136, 15.000321, 0.3, 255],
[0.199957, 0.349751, 15.000544, 0.3, 255],
[0.200488, 0.34979, 14.999721, 0.3, 255],
[0.199957, 0.349404, 14.999944, 0.3, 255],
[0.200043, 0.350596, 15.000056, 0.3, 255],
[0.199512, 0.35021, 15.000279, 0.3, 255],
[0.200043, 0.350249, 14.999456, 0.3, 255],
[0.199512, 0.349864, 14.999679, 0.3, 255],
[-0.5, 0.500442, 10.000442, 0.2, 0],
[-0.5, 0.5, 10, 0.1995, 255],
[-0.499966, 0.49971, 9.999563, 0.199, 0],
[-0.500034, 0.499563, 9.99971, 0.199, 0],
[-0.499934, 0.499432, 9.999144, 0.1985, 255],
[-0.500066, 0.499144, 9.999432, 0.1985, 255],
[0.6, 0.5, 10, 0.25, 210],
[0.600122, 0.4999, 10.000122, 0.25, 210],
[0.600061, 0.49995, 10.000061, 0.25003, 50],
[0.595856, 0.505256, 9.991271, 0.24, 100],
[0.597973, 0.507461, 9.992175, 0.24, 210],
[0.595362, 0.507926, 9.993945, 0.24, 100],
[0.594618, 0.509055, 9.996832, 0.24, 210],
[0.592382, 0.506906, 9.996092, 0.24, 100],
[0.590438, 0.504882, 9.997604, 0.24, 210],
[0.591035, 0.503605, 9.994744, 0.24, 100],
[0.59121, 0.500708, 9.993425, 0.24, 210],
[0.593181, 0.502585, 9.991764, 0.24, 100],
[0.595867, 0.502302, 9.990069, 0.24, 210],
],
]
scene_names = (
"single_sphere",
"two_spheres",
"ring_of_spheres",
"beachball",
"football",
"eightball",
"starball",
"combination_scene",
)
for i in range(8):
print(f'Generating {scene_names[i]}.')
values = grey_values(test_cases[i])
rows = [values[row * 1024:row * 1024 + 1024] for row in range(1024)]
with open(f'{scene_names[i]}.png', 'wb') as file:
w = png.Writer(1024, 1024, greyscale=True)
w.write(file, rows)
```
### Rust
The function `grey_values` would be sufficient as an entry to this challenge. The rest of the code saves the outputs as PNG files.
```rust
fn grey_values(spheres: &Vec<[f32; 5]>) -> Vec<u8> {
let mut values = vec![];
for y_pixel in 0..1024 {
for x_pixel in 0..1024 {
let x_ray = x_pixel as f32 / 512.0 - 1.0;
let y_ray = y_pixel as f32 / 512.0 - 1.0;
let mut min_distance = f32::INFINITY;
let mut hit_grey_value = 0;
for sphere in spheres {
let [
x_sphere, y_sphere, z_sphere, radius, grey
] = *sphere;
let a = (
(x_ray - x_sphere).powf(2.0)
+ (y_ray - y_sphere).powf(2.0)
).sqrt();
if a < radius {
let distance = z_sphere - (
radius.powf(2.0) - a.powf(2.0)
).sqrt();
if distance < min_distance {
min_distance = distance;
hit_grey_value = grey as u8;
}
}
}
values.push(hit_grey_value);
}
}
values
}
fn main() {
let test_cases: Vec<Vec<[f32; 5]>> = vec![
vec![
[0.0, 0.0, 0.0, 1.0, 128.0]
],
vec![
[0.0, -0.55, 0.0, 0.4, 100.0],
[0.0, 0.55, 0.0, 0.4, 200.0]
],
vec![
[0.75, 0.0, 0.0, 0.25, 64.0],
[0.53033, 0.375, -0.375, 0.25, 185.0],
[0.0, 0.53033, -0.53033, 0.25, 64.0],
[-0.53033, 0.375, -0.375, 0.25, 185.0],
[-0.75, 0.0, 0.0, 0.25, 64.0],
[-0.53033, -0.375, 0.375, 0.25, 185.0],
[0.0, -0.53033, 0.53033, 0.25, 64.0],
[0.53033, -0.375, 0.375, 0.25, 185.0],
],
vec![
[0.001, 0.0, 0.0, 1.0, 64.0],
[0.000707, 0.000707, 0.0, 1.0, 85.0],
[0.0, 0.001, 0.0, 1.0, 106.0],
[-0.000707, 0.000707, 0.0, 1.0, 127.0],
[-0.001, 0.0, 0.0, 1.0, 148.0],
[-0.000707, -0.000707, 0.0, 1.0, 169.0],
[0.0, -0.001, 0.0, 1.0, 190.0],
[0.000707, -0.000707, 0.0, 1.0, 211.0],
],
vec![
[0.0, 0.000058, 0.001947, 1.0, 0.0],
[0.0, -0.001715, 0.000923, 1.0, 0.0],
[0.0, 0.001715, -0.000923, 1.0, 0.0],
[0.0, -0.000058, -0.001947, 1.0, 0.0],
[0.001657, -0.000512, 0.000887, 1.0, 0.0],
[-0.001657, -0.000512, 0.000887, 1.0, 0.0],
[0.001657, 0.000512, -0.000887, 1.0, 0.0],
[-0.001657, 0.000512, -0.000887, 1.0, 0.0],
[0.001024, 0.001435, 0.000829, 1.0, 0.0],
[-0.001024, 0.001435, 0.000829, 1.0, 0.0],
[0.001024, -0.001435, -0.000829, 1.0, 0.0],
[-0.001024, -0.001435, -0.000829, 1.0, 0.0],
[0.001868, 0.000618, 0.000357, 1.0, 255.0],
[-0.001868, 0.000618, 0.000357, 1.0, 255.0],
[0.001868, -0.000618, -0.000357, 1.0, 255.0],
[-0.001868, -0.000618, -0.000357, 1.0, 255.0],
[0.000714, -0.000934, 0.001618, 1.0, 255.0],
[-0.000714, -0.000934, 0.001618, 1.0, 255.0],
[0.000714, 0.000934, -0.001618, 1.0, 255.0],
[-0.000714, 0.000934, -0.001618, 1.0, 255.0],
[0.0, -0.001975, -0.000316, 1.0, 255.0],
[0.0, 0.001261, 0.001552, 1.0, 255.0],
[0.0, -0.001261, -0.001552, 1.0, 255.0],
[0.0, 0.001975, 0.000316, 1.0, 255.0],
[0.001155, 0.000423, 0.001577, 1.0, 255.0],
[-0.001155, 0.000423, 0.001577, 1.0, 255.0],
[0.001155, -0.001577, 0.000423, 1.0, 255.0],
[-0.001155, -0.001577, 0.000423, 1.0, 255.0],
[0.001155, 0.001577, -0.000423, 1.0, 255.0],
[-0.001155, 0.001577, -0.000423, 1.0, 255.0],
[0.001155, -0.000423, -0.001577, 1.0, 255.0],
[-0.001155, -0.000423, -0.001577, 1.0, 255.0],
],
vec![
[0.0, 0.0, 1000.0, 2.0, 128.0],
[0.0, 0.0, 0.003125, 1.0, 0.0],
[0.0, 0.0, 0.0, 0.9975, 255.0],
[0.0, 0.000547, -0.002573, 0.995, 0.0],
[0.0, -0.000547, -0.002573, 0.995, 0.0],
[0.0, 0.001071, -0.005037, 0.9925, 255.0],
[0.0, -0.001071, -0.005037, 0.9925, 255.0],
],
vec![
[0.0, 0.0, 0.0, 1.0, 210.0],
[0.0, 0.0, 0.0008, 1.0, 210.0],
[0.0, 0.0, 0.0004, 1.00012, 50.0],
[0.012969, 0.0, -0.042045, 0.96, 100.0],
[0.016397, 0.011913, -0.039054, 0.96, 210.0],
[0.004008, 0.012334, -0.042045, 0.96, 100.0],
[-0.006263, 0.019276, -0.039054, 0.96, 210.0],
[-0.010492, 0.007623, -0.042045, 0.96, 100.0],
[-0.020268, 0.0, -0.039054, 0.96, 210.0],
[-0.010492, -0.007623, -0.042045, 0.96, 100.0],
[-0.006263, -0.019276, -0.039054, 0.96, 210.0],
[0.004008, -0.012334, -0.042045, 0.96, 100.0],
[0.016397, -0.011913, -0.039054, 0.96, 210.0],
],
vec![
[0.0, 0.0, 1000.0, 2.0, 192.0],
[-2.0, 2.0, 100.0, 3.0, 98.0],
[2.0, 2.0, 100.0, 3.0, 98.0],
[0.5, -0.5, 500.0, 0.2, 255.0],
[-0.349469, -0.1, 20.000279, 0.6, 64.0],
[-0.349775, -0.100273, 20.000485, 0.6, 85.0],
[-0.350214, -0.100386, 20.000407, 0.6, 106.0],
[-0.350527, -0.100273, 20.000091, 0.6, 127.0],
[-0.350531, -0.1, 19.999721, 0.6, 148.0],
[-0.350225, -0.099727, 19.999515, 0.6, 169.0],
[-0.349786, -0.099614, 19.999593, 0.6, 190.0],
[-0.349473, -0.099727, 19.999909, 0.6, 211.0],
[0.20032, 0.349824, 15.000456, 0.3, 0.0],
[0.20032, 0.349517, 14.999924, 0.3, 0.0],
[0.19968, 0.350483, 15.000076, 0.3, 0.0],
[0.19968, 0.350176, 14.999544, 0.3, 0.0],
[0.200578, 0.350073, 14.999958, 0.3, 0.0],
[0.199817, 0.349519, 15.000277, 0.3, 0.0],
[0.200183, 0.350481, 14.999723, 0.3, 0.0],
[0.199422, 0.349927, 15.000042, 0.3, 0.0],
[0.200235, 0.35042, 15.000332, 0.3, 0.0],
[0.199765, 0.350078, 15.000529, 0.3, 0.0],
[0.200235, 0.349922, 14.999471, 0.3, 0.0],
[0.199765, 0.34958, 14.999668, 0.3, 0.0],
[0.200429, 0.350419, 15.000005, 0.3, 255.0],
[0.199571, 0.349795, 15.000366, 0.3, 255.0],
[0.200429, 0.350205, 14.999634, 0.3, 255.0],
[0.199571, 0.349581, 14.999995, 0.3, 255.0],
[0.200524, 0.349747, 15.000146, 0.3, 255.0],
[0.200196, 0.349509, 15.000283, 0.3, 255.0],
[0.199804, 0.350491, 14.999717, 0.3, 255.0],
[0.199476, 0.350253, 14.999854, 0.3, 255.0],
[0.200138, 0.349578, 14.999597, 0.3, 255.0],
[0.200138, 0.350138, 15.000567, 0.3, 255.0],
[0.199862, 0.349862, 14.999433, 0.3, 255.0],
[0.199862, 0.350422, 15.000403, 0.3, 255.0],
[0.200488, 0.350136, 15.000321, 0.3, 255.0],
[0.199957, 0.349751, 15.000544, 0.3, 255.0],
[0.200488, 0.34979, 14.999721, 0.3, 255.0],
[0.199957, 0.349404, 14.999944, 0.3, 255.0],
[0.200043, 0.350596, 15.000056, 0.3, 255.0],
[0.199512, 0.35021, 15.000279, 0.3, 255.0],
[0.200043, 0.350249, 14.999456, 0.3, 255.0],
[0.199512, 0.349864, 14.999679, 0.3, 255.0],
[-0.5, 0.500442, 10.000442, 0.2, 0.0],
[-0.5, 0.5, 10.0, 0.1995, 255.0],
[-0.499966, 0.49971, 9.999563, 0.199, 0.0],
[-0.500034, 0.499563, 9.99971, 0.199, 0.0],
[-0.499934, 0.499432, 9.999144, 0.1985, 255.0],
[-0.500066, 0.499144, 9.999432, 0.1985, 255.0],
[0.6, 0.5, 10.0, 0.25, 210.0],
[0.600122, 0.4999, 10.000122, 0.25, 210.0],
[0.600061, 0.49995, 10.000061, 0.25003, 50.0],
[0.595856, 0.505256, 9.991271, 0.24, 100.0],
[0.597973, 0.507461, 9.992175, 0.24, 210.0],
[0.595362, 0.507926, 9.993945, 0.24, 100.0],
[0.594618, 0.509055, 9.996832, 0.24, 210.0],
[0.592382, 0.506906, 9.996092, 0.24, 100.0],
[0.590438, 0.504882, 9.997604, 0.24, 210.0],
[0.591035, 0.503605, 9.994744, 0.24, 100.0],
[0.59121, 0.500708, 9.993425, 0.24, 210.0],
[0.593181, 0.502585, 9.991764, 0.24, 100.0],
[0.595867, 0.502302, 9.990069, 0.24, 210.0],
],
];
let scene_names = [
"single_sphere",
"two_spheres",
"ring_of_spheres",
"beachball",
"football",
"eightball",
"starball",
"combination_scene",
];
for i in 0..8 {
eprintln!("Generating {}", scene_names[i]);
let values = grey_values(&test_cases[i]);
let filename = format!("{}.png", scene_names[i]);
let path = std::path::Path::new(&filename);
let file = std::fs::File::create(path).unwrap();
let w = &mut std::io::BufWriter::new(file);
let mut encoder = png::Encoder::new(w, 1024, 1024);
encoder.set_color(png::ColorType::Grayscale);
encoder.set_depth(png::BitDepth::Eight);
let mut writer = encoder.write_header().unwrap();
writer.write_image_data(&values).unwrap();
}
}
```
## Test cases
The images may be scaled down. Clicking or tapping on an image will open it full size.
### Single sphere
Input:
```text
TODO
```
Output:
```text
TODO
```
### Two separated spheres
Input:
```text
TODO
```
Output:
```text
TODO
```
### Eight spheres in a ring
Input:
```text
TODO
```
Output:
```text
TODO
```
### Beach ball
Input:
```text
TODO
```
Output:
```text
TODO
```
### Football
Input:
```text
TODO
```
Output:
```text
TODO
```
### Eight ball
Input:
```text
TODO
```
Output:
```text
TODO
```
### Star ball
Input:
```text
TODO
```
Output:
```text
TODO
```
### Combination scene
Input:
```text
TODO
```
Output:
```text
TODO
```
## Explanatory animations
The example images were produced from plain grey spheres with no surface pattern or texture. The following animations show how they break down into their component plain spheres.
This section is not necessary to write valid code for this challenge. If your code accepts the parameters of plain grey spheres and renders them correctly, all of the test cases will look right without needing to understand how they were constructed. This section is mainly out of interest, and to make clear that your code *only* needs to render plain grey spheres.
```text
CONSIDER HOSTING THESE OFF SITE AS THEY ARE NOT REQUIRED FOR THE CHALLENGE
AND EVEN THE SCALED DOWN VERSIONS ARE UNNECESSARY DATA USE FOR ANYONE
CHOOSING NOT TO VIEW THEM.
```
<details>
<summary>Animations</summary>
<details>
<summary>Beach ball</summary>
This animation is reduced to 256 by 256 pixels. Clicking or tapping on the animation will open it at full size (______MB).
```text
TODO
```
</details>
<details>
<summary>Football</summary>
This animation is reduced to 256 by 256 pixels. Clicking or tapping on the animation will open it at full size (______MB).
```text
TODO
```
</details>
<details>
<summary>Eight ball</summary>
This animation is reduced to 256 by 256 pixels. Clicking or tapping on the animation will open it at full size (______MB).
```text
TODO
```
</details>
<details>
<summary>Star ball</summary>
This animation is reduced to 256 by 256 pixels. Clicking or tapping on the animation will open it at full size (______MB).
```text
TODO
```
</details>
</details>
## Scoring
This is a [code golf challenge]. Your score is the number of bytes in your code. Lowest score for each language wins.
> Explanations are optional, but I'm more likely to upvote answers that have one.
[code golf challenge]: https://codegolf.codidact.com/categories/49/tags/4274 "The code-golf tag"
[PGM]: https://en.wikipedia.org/wiki/Netpbm#PGM_example

#3: Post edited by

trichoplax‭ · 2024-07-26T10:48:08Z (5 months ago)
Make Rust implementation avoid horizontal scroll bars

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Raw

Markdown

Given a list of spheres, output an image of them.
The challenge is to produce a greyscale [orthographic](https://en.wikipedia.org/wiki/Parallel_projection#Orthographic_projection) (parallel projection - no perspective) ray caster, that renders only plain grey spheres with no lighting, no shading, no shadows, no surface patterns or textures, no reflections, and no transparency.
## Input
- A list of spheres, each consisting of:
- The coordinates of its centre $(x, y, z)$.
- Its radius $r$.
- Its greyscale value $g$, an integer from $0$ to $255$ (both inclusive).
- The input may include up to $100$ spheres.
- Only the greyscale value will be restricted to integer values. The coordinates and radius may have fractional values.
- Here "integer" refers to the values the greyscale value make take. You are not required to use an integer variable type.[]()
## Output
- A single greyscale image:
- $1024$ by $1024$ pixels.
- Each pixel has a single greyscale value from $0$ to $255$.
- Horizontally, the image covers x values from $-1$ to $1$.
- Vertically, the image covers y values from $-1$ to $1$.
- The image is orthographic, so there is no perspective. Changing the $z$ coordinate of a sphere does not change its size in the image, instead only affecting whether it is in front of or behind other spheres.
- A sphere's greyscale value is applied to every visible part of it. There is no variation over the surface of the sphere, so it appears flat like a disk.
- The point of the challenge is producing the pixel values, not manipulating image file formats, so you may output whatever image format is convenient, including simple text formats such as [PGM] or even just a list of pixel greyscale values. If you want to include an example of your output as an image in your answer, the image formats that Codidact supports are PNG, JPEG, and GIF. You could output PGM and then convert to PNG to upload to your answer. If you output just the greyscale values as a list of numbers from $0$ to $255$, I've made a `[ TODO page to convert these to greyscale PNG]`.
- The test cases, which match the reference implementations, happen to have the x axis increasing from left to right, and the y axis increasing from top to bottom. You are free to instead interpret these as increasing from right to left for x or from bottom to top for y, resulting in a reflected image. You are also free to output an image rotated by a multiple of 90 degrees, allowing for swapping the x and y axes. However, the z axis is required to be increasing in the forwards direction (into the image) otherwise some of the test cases will look incorrect.
## Calculating a pixel value
Imagine a ray passing into the image through a pixel, perpendicular to the image plane. If it hits a sphere, it shows the greyscale value of the first sphere it hits. Otherwise, it shows a value of zero (black). One possible way of calculating this is as follows:
For each pixel:
- The pixel $x, y$ coordinates will each be from $0$ to $1023$. Scale these to the range $-1$ to $1$ (for example, divide by $512$ and subtract $1$).
- If these scaled $x, y$ coordinates are within the radius $r$ of the $x, y$ coordinates of a sphere's centre, the ray intersects that sphere (the $z$ coordinate is not relevant to this test because the image is an orthographic projection - the rays are all parallel to the $z$ axis). The distance $a$ of the ray from the sphere's centre can be calculated as follows:
$$a=\sqrt{(x_{ray}-x_{sphere})^2+(y_{ray}-y_{sphere})^2}$$
```text
CONSIDER A DIAGRAM HERE
```
- For each sphere that the ray intersects, find the $z$ coordinate of the nearest intersection (there will often be $2$ per sphere, $1$ on the front of the sphere and $1$ on the back of the sphere, as the ray passes through - only the nearest is relevant). This can be calculated as follows:
- The distance $d$ to the nearest intersection point can be calculated from the $z$ coordinate of the sphere's centre, the radius $r$, and the distance $a$ from the $x, y$ coordinates of the sphere's centre to the scaled $x, y$ coordinates of the pixel:
$$d=z_{sphere}-\sqrt{r_{sphere}^2-a^2}$$
```text
CONSIDER A DIAGRAM HERE
```
- Of all the spheres that the ray intersects, the sphere with the intersection point with the smallest $z$ coordinate $d$ is the one to display. Set the pixel to the greyscale value of this sphere.
- If the ray did not intersect any spheres, set the pixel to zero (black).
## Example implementations
These are ungolfed implementations designed to supplement understanding of the requirements. You are free to golf these or to implement your own approach that matches the test cases up to reflection and rotation.
The Python implementation uses 64 bit floating point giving accurate images. The Rust implementation deliberately uses 32 bit floating point to show the difference in output. I've tried to optimise the test cases for compatibility with 32 bit floating point, so the differences are restricted to being a pixel out in some places.
### Python
The function `grey_values` would be sufficient as an entry to this challenge. The rest of the code saves the outputs as PNG files. The function itself has no dependence on the `import png` so it would not be required in an entry.
```python
def grey_values(spheres):
values = []
for y_pixel in range(1024):
for x_pixel in range(1024):
x_ray, y_ray = x_pixel / 512 - 1, y_pixel / 512 - 1
min_distance, hit_grey_value = None, 0
for sphere in spheres:
x_sphere, y_sphere, z_sphere, radius, grey_value = sphere
a = ((x_ray-x_sphere) ** 2 + (y_ray-y_sphere) ** 2) ** 0.5
if a < radius:
distance = z_sphere - (radius ** 2 - a ** 2) ** 0.5
if min_distance is None or distance < min_distance:
min_distance = distance
hit_grey_value = grey_value
values.append(hit_grey_value)
return values
import png
test_cases = [
[
[0, 0, 0, 1, 128],
],
[
[0, -0.55, 0, 0.4, 100],
[0, 0.55, 0, 0.4, 200],
],
[
[0.75, 0, 0, 0.25, 64],
[0.53033, 0.375, -0.375, 0.25, 185],
[0, 0.53033, -0.53033, 0.25, 64],
[-0.53033, 0.375, -0.375, 0.25, 185],
[-0.75, 0, -0, 0.25, 64],
[-0.53033, -0.375, 0.375, 0.25, 185],
[-0, -0.53033, 0.53033, 0.25, 64],
[0.53033, -0.375, 0.375, 0.25, 185],
],
[
[0.001, 0, 0, 1, 64],
[0.000707, 0.000707, -0, 1, 85],
[0, 0.001, -0, 1, 106],
[-0.000707, 0.000707, -0, 1, 127],
[-0.001, 0, -0, 1, 148],
[-0.000707, -0.000707, 0, 1, 169],
[-0, -0.001, 0, 1, 190],
[0.000707, -0.000707, 0, 1, 211],
],
[
[0, 0.000058, 0.001947, 1, 0],
[0, -0.001715, 0.000923, 1, 0],
[0, 0.001715, -0.000923, 1, 0],
[0, -0.000058, -0.001947, 1, 0],
[0.001657, -0.000512, 0.000887, 1, 0],
[-0.001657, -0.000512, 0.000887, 1, 0],
[0.001657, 0.000512, -0.000887, 1, 0],
[-0.001657, 0.000512, -0.000887, 1, 0],
[0.001024, 0.001435, 0.000829, 1, 0],
[-0.001024, 0.001435, 0.000829, 1, 0],
[0.001024, -0.001435, -0.000829, 1, 0],
[-0.001024, -0.001435, -0.000829, 1, 0],
[0.001868, 0.000618, 0.000357, 1, 255],
[-0.001868, 0.000618, 0.000357, 1, 255],
[0.001868, -0.000618, -0.000357, 1, 255],
[-0.001868, -0.000618, -0.000357, 1, 255],
[0.000714, -0.000934, 0.001618, 1, 255],
[-0.000714, -0.000934, 0.001618, 1, 255],
[0.000714, 0.000934, -0.001618, 1, 255],
[-0.000714, 0.000934, -0.001618, 1, 255],
[0, -0.001975, -0.000316, 1, 255],
[0, 0.001261, 0.001552, 1, 255],
[0, -0.001261, -0.001552, 1, 255],
[0, 0.001975, 0.000316, 1, 255],
[0.001155, 0.000423, 0.001577, 1, 255],
[-0.001155, 0.000423, 0.001577, 1, 255],
[0.001155, -0.001577, 0.000423, 1, 255],
[-0.001155, -0.001577, 0.000423, 1, 255],
[0.001155, 0.001577, -0.000423, 1, 255],
[-0.001155, 0.001577, -0.000423, 1, 255],
[0.001155, -0.000423, -0.001577, 1, 255],
[-0.001155, -0.000423, -0.001577, 1, 255],
],
[
[0, 0, 1000, 2, 128],
[0, 0, 0.003125, 1, 0],
[0, 0, 0, 0.9975, 255],
[0, 0.000547, -0.002573, 0.995, 0],
[-0, -0.000547, -0.002573, 0.995, 0],
[0, 0.001071, -0.005037, 0.9925, 255],
[-0, -0.001071, -0.005037, 0.9925, 255],
],
[
[0, 0, 0, 1, 210],
[0, 0, 0.0008, 1, 210],
[0, 0, 0.0004, 1.00012, 50],
[0.012969, -0, -0.042045, 0.96, 100],
[0.016397, 0.011913, -0.039054, 0.96, 210],
[0.004008, 0.012334, -0.042045, 0.96, 100],
[-0.006263, 0.019276, -0.039054, 0.96, 210],
[-0.010492, 0.007623, -0.042045, 0.96, 100],
[-0.020268, 0, -0.039054, 0.96, 210],
[-0.010492, -0.007623, -0.042045, 0.96, 100],
[-0.006263, -0.019276, -0.039054, 0.96, 210],
[0.004008, -0.012334, -0.042045, 0.96, 100],
[0.016397, -0.011913, -0.039054, 0.96, 210],
],
[
[0, 0, 1000, 2, 192],
[-2, 2, 100, 3, 98],
[2, 2, 100, 3, 98],
[0.5, -0.5, 500, 0.2, 255],
[-0.349469, -0.1, 20.000279, 0.6, 64],
[-0.349775, -0.100273, 20.000485, 0.6, 85],
[-0.350214, -0.100386, 20.000407, 0.6, 106],
[-0.350527, -0.100273, 20.000091, 0.6, 127],
[-0.350531, -0.1, 19.999721, 0.6, 148],
[-0.350225, -0.099727, 19.999515, 0.6, 169],
[-0.349786, -0.099614, 19.999593, 0.6, 190],
[-0.349473, -0.099727, 19.999909, 0.6, 211],
[0.20032, 0.349824, 15.000456, 0.3, 0],
[0.20032, 0.349517, 14.999924, 0.3, 0],
[0.19968, 0.350483, 15.000076, 0.3, 0],
[0.19968, 0.350176, 14.999544, 0.3, 0],
[0.200578, 0.350073, 14.999958, 0.3, 0],
[0.199817, 0.349519, 15.000277, 0.3, 0],
[0.200183, 0.350481, 14.999723, 0.3, 0],
[0.199422, 0.349927, 15.000042, 0.3, 0],
[0.200235, 0.35042, 15.000332, 0.3, 0],
[0.199765, 0.350078, 15.000529, 0.3, 0],
[0.200235, 0.349922, 14.999471, 0.3, 0],
[0.199765, 0.34958, 14.999668, 0.3, 0],
[0.200429, 0.350419, 15.000005, 0.3, 255],
[0.199571, 0.349795, 15.000366, 0.3, 255],
[0.200429, 0.350205, 14.999634, 0.3, 255],
[0.199571, 0.349581, 14.999995, 0.3, 255],
[0.200524, 0.349747, 15.000146, 0.3, 255],
[0.200196, 0.349509, 15.000283, 0.3, 255],
[0.199804, 0.350491, 14.999717, 0.3, 255],
[0.199476, 0.350253, 14.999854, 0.3, 255],
[0.200138, 0.349578, 14.999597, 0.3, 255],
[0.200138, 0.350138, 15.000567, 0.3, 255],
[0.199862, 0.349862, 14.999433, 0.3, 255],
[0.199862, 0.350422, 15.000403, 0.3, 255],
[0.200488, 0.350136, 15.000321, 0.3, 255],
[0.199957, 0.349751, 15.000544, 0.3, 255],
[0.200488, 0.34979, 14.999721, 0.3, 255],
[0.199957, 0.349404, 14.999944, 0.3, 255],
[0.200043, 0.350596, 15.000056, 0.3, 255],
[0.199512, 0.35021, 15.000279, 0.3, 255],
[0.200043, 0.350249, 14.999456, 0.3, 255],
[0.199512, 0.349864, 14.999679, 0.3, 255],
[-0.5, 0.500442, 10.000442, 0.2, 0],
[-0.5, 0.5, 10, 0.1995, 255],
[-0.499966, 0.49971, 9.999563, 0.199, 0],
[-0.500034, 0.499563, 9.99971, 0.199, 0],
[-0.499934, 0.499432, 9.999144, 0.1985, 255],
[-0.500066, 0.499144, 9.999432, 0.1985, 255],
[0.6, 0.5, 10, 0.25, 210],
[0.600122, 0.4999, 10.000122, 0.25, 210],
[0.600061, 0.49995, 10.000061, 0.25003, 50],
[0.595856, 0.505256, 9.991271, 0.24, 100],
[0.597973, 0.507461, 9.992175, 0.24, 210],
[0.595362, 0.507926, 9.993945, 0.24, 100],
[0.594618, 0.509055, 9.996832, 0.24, 210],
[0.592382, 0.506906, 9.996092, 0.24, 100],
[0.590438, 0.504882, 9.997604, 0.24, 210],
[0.591035, 0.503605, 9.994744, 0.24, 100],
[0.59121, 0.500708, 9.993425, 0.24, 210],
[0.593181, 0.502585, 9.991764, 0.24, 100],
[0.595867, 0.502302, 9.990069, 0.24, 210],
],
]
scene_names = (
"single_sphere",
"two_spheres",
"ring_of_spheres",
"beachball",
"football",
"eightball",
"starball",
"combination_scene",
)
for i in range(8):
print(f'Generating {scene_names[i]}.')
values = grey_values(test_cases[i])
rows = [values[row * 1024:row * 1024 + 1024] for row in range(1024)]
with open(f'{scene_names[i]}.png', 'wb') as file:
w = png.Writer(1024, 1024, greyscale=True)
w.write(file, rows)
```
### Rust
The function `grey_values` would be sufficient as an entry to this challenge. The rest of the code saves the outputs as PNG files.
```rust
fn grey_values(spheres: &Vec<[f32; 5]>) -> Vec<u8> {
let mut values = vec![];
for y_pixel in 0..1024 {
for x_pixel in 0..1024 {
let x_ray = x_pixel as f32 / 512.0 - 1.0;
let y_ray = y_pixel as f32 / 512.0 - 1.0;
let mut min_distance = f32::INFINITY;
let mut hit_grey_value = 0;
for sphere in spheres {
~~let [x_sphere, y_sphere, z_sphere, radius, grey] = *sphere;~~
let a = (
~~(x_ray-x_sphere).powf(2.0) + (y_ray-y_sphere).powf(2.0)~~
).sqrt();
if a < radius {
let distance = z_sphere - (
radius.powf(2.0) - a.powf(2.0)
).sqrt();
if distance < min_distance {
min_distance = distance;
hit_grey_value = grey as u8;
}
}
}
values.push(hit_grey_value);
}
}
values
}
fn main() {
let test_cases: Vec<Vec<[f32; 5]>> = vec![
vec![[0.0, 0.0, 0.0, 1.0, 128.0]],
vec![[0.0, -0.55, 0.0, 0.4, 100.0], [0.0, 0.55, 0.0, 0.4, 200.0]],
vec![
[0.75, 0.0, 0.0, 0.25, 64.0],
[0.53033, 0.375, -0.375, 0.25, 185.0],
[0.0, 0.53033, -0.53033, 0.25, 64.0],
[-0.53033, 0.375, -0.375, 0.25, 185.0],
[-0.75, 0.0, 0.0, 0.25, 64.0],
[-0.53033, -0.375, 0.375, 0.25, 185.0],
[0.0, -0.53033, 0.53033, 0.25, 64.0],
[0.53033, -0.375, 0.375, 0.25, 185.0],
],
vec![
[0.001, 0.0, 0.0, 1.0, 64.0],
[0.000707, 0.000707, 0.0, 1.0, 85.0],
[0.0, 0.001, 0.0, 1.0, 106.0],
[-0.000707, 0.000707, 0.0, 1.0, 127.0],
[-0.001, 0.0, 0.0, 1.0, 148.0],
[-0.000707, -0.000707, 0.0, 1.0, 169.0],
[0.0, -0.001, 0.0, 1.0, 190.0],
[0.000707, -0.000707, 0.0, 1.0, 211.0],
],
vec![
[0.0, 0.000058, 0.001947, 1.0, 0.0],
[0.0, -0.001715, 0.000923, 1.0, 0.0],
[0.0, 0.001715, -0.000923, 1.0, 0.0],
[0.0, -0.000058, -0.001947, 1.0, 0.0],
[0.001657, -0.000512, 0.000887, 1.0, 0.0],
[-0.001657, -0.000512, 0.000887, 1.0, 0.0],
[0.001657, 0.000512, -0.000887, 1.0, 0.0],
[-0.001657, 0.000512, -0.000887, 1.0, 0.0],
[0.001024, 0.001435, 0.000829, 1.0, 0.0],
[-0.001024, 0.001435, 0.000829, 1.0, 0.0],
[0.001024, -0.001435, -0.000829, 1.0, 0.0],
[-0.001024, -0.001435, -0.000829, 1.0, 0.0],
[0.001868, 0.000618, 0.000357, 1.0, 255.0],
[-0.001868, 0.000618, 0.000357, 1.0, 255.0],
[0.001868, -0.000618, -0.000357, 1.0, 255.0],
[-0.001868, -0.000618, -0.000357, 1.0, 255.0],
[0.000714, -0.000934, 0.001618, 1.0, 255.0],
[-0.000714, -0.000934, 0.001618, 1.0, 255.0],
[0.000714, 0.000934, -0.001618, 1.0, 255.0],
[-0.000714, 0.000934, -0.001618, 1.0, 255.0],
[0.0, -0.001975, -0.000316, 1.0, 255.0],
[0.0, 0.001261, 0.001552, 1.0, 255.0],
[0.0, -0.001261, -0.001552, 1.0, 255.0],
[0.0, 0.001975, 0.000316, 1.0, 255.0],
[0.001155, 0.000423, 0.001577, 1.0, 255.0],
[-0.001155, 0.000423, 0.001577, 1.0, 255.0],
[0.001155, -0.001577, 0.000423, 1.0, 255.0],
[-0.001155, -0.001577, 0.000423, 1.0, 255.0],
[0.001155, 0.001577, -0.000423, 1.0, 255.0],
[-0.001155, 0.001577, -0.000423, 1.0, 255.0],
[0.001155, -0.000423, -0.001577, 1.0, 255.0],
[-0.001155, -0.000423, -0.001577, 1.0, 255.0],
],
vec![
[0.0, 0.0, 1000.0, 2.0, 128.0],
[0.0, 0.0, 0.003125, 1.0, 0.0],
[0.0, 0.0, 0.0, 0.9975, 255.0],
[0.0, 0.000547, -0.002573, 0.995, 0.0],
[0.0, -0.000547, -0.002573, 0.995, 0.0],
[0.0, 0.001071, -0.005037, 0.9925, 255.0],
[0.0, -0.001071, -0.005037, 0.9925, 255.0],
],
vec![
[0.0, 0.0, 0.0, 1.0, 210.0],
[0.0, 0.0, 0.0008, 1.0, 210.0],
[0.0, 0.0, 0.0004, 1.00012, 50.0],
[0.012969, 0.0, -0.042045, 0.96, 100.0],
[0.016397, 0.011913, -0.039054, 0.96, 210.0],
[0.004008, 0.012334, -0.042045, 0.96, 100.0],
[-0.006263, 0.019276, -0.039054, 0.96, 210.0],
[-0.010492, 0.007623, -0.042045, 0.96, 100.0],
[-0.020268, 0.0, -0.039054, 0.96, 210.0],
[-0.010492, -0.007623, -0.042045, 0.96, 100.0],
[-0.006263, -0.019276, -0.039054, 0.96, 210.0],
[0.004008, -0.012334, -0.042045, 0.96, 100.0],
[0.016397, -0.011913, -0.039054, 0.96, 210.0],
],
vec![
[0.0, 0.0, 1000.0, 2.0, 192.0],
[-2.0, 2.0, 100.0, 3.0, 98.0],
[2.0, 2.0, 100.0, 3.0, 98.0],
[0.5, -0.5, 500.0, 0.2, 255.0],
[-0.349469, -0.1, 20.000279, 0.6, 64.0],
[-0.349775, -0.100273, 20.000485, 0.6, 85.0],
[-0.350214, -0.100386, 20.000407, 0.6, 106.0],
[-0.350527, -0.100273, 20.000091, 0.6, 127.0],
[-0.350531, -0.1, 19.999721, 0.6, 148.0],
[-0.350225, -0.099727, 19.999515, 0.6, 169.0],
[-0.349786, -0.099614, 19.999593, 0.6, 190.0],
[-0.349473, -0.099727, 19.999909, 0.6, 211.0],
[0.20032, 0.349824, 15.000456, 0.3, 0.0],
[0.20032, 0.349517, 14.999924, 0.3, 0.0],
[0.19968, 0.350483, 15.000076, 0.3, 0.0],
[0.19968, 0.350176, 14.999544, 0.3, 0.0],
[0.200578, 0.350073, 14.999958, 0.3, 0.0],
[0.199817, 0.349519, 15.000277, 0.3, 0.0],
[0.200183, 0.350481, 14.999723, 0.3, 0.0],
[0.199422, 0.349927, 15.000042, 0.3, 0.0],
[0.200235, 0.35042, 15.000332, 0.3, 0.0],
[0.199765, 0.350078, 15.000529, 0.3, 0.0],
[0.200235, 0.349922, 14.999471, 0.3, 0.0],
[0.199765, 0.34958, 14.999668, 0.3, 0.0],
[0.200429, 0.350419, 15.000005, 0.3, 255.0],
[0.199571, 0.349795, 15.000366, 0.3, 255.0],
[0.200429, 0.350205, 14.999634, 0.3, 255.0],
[0.199571, 0.349581, 14.999995, 0.3, 255.0],
[0.200524, 0.349747, 15.000146, 0.3, 255.0],
[0.200196, 0.349509, 15.000283, 0.3, 255.0],
[0.199804, 0.350491, 14.999717, 0.3, 255.0],
[0.199476, 0.350253, 14.999854, 0.3, 255.0],
[0.200138, 0.349578, 14.999597, 0.3, 255.0],
[0.200138, 0.350138, 15.000567, 0.3, 255.0],
[0.199862, 0.349862, 14.999433, 0.3, 255.0],
[0.199862, 0.350422, 15.000403, 0.3, 255.0],
[0.200488, 0.350136, 15.000321, 0.3, 255.0],
[0.199957, 0.349751, 15.000544, 0.3, 255.0],
[0.200488, 0.34979, 14.999721, 0.3, 255.0],
[0.199957, 0.349404, 14.999944, 0.3, 255.0],
[0.200043, 0.350596, 15.000056, 0.3, 255.0],
[0.199512, 0.35021, 15.000279, 0.3, 255.0],
[0.200043, 0.350249, 14.999456, 0.3, 255.0],
[0.199512, 0.349864, 14.999679, 0.3, 255.0],
[-0.5, 0.500442, 10.000442, 0.2, 0.0],
[-0.5, 0.5, 10.0, 0.1995, 255.0],
[-0.499966, 0.49971, 9.999563, 0.199, 0.0],
[-0.500034, 0.499563, 9.99971, 0.199, 0.0],
[-0.499934, 0.499432, 9.999144, 0.1985, 255.0],
[-0.500066, 0.499144, 9.999432, 0.1985, 255.0],
[0.6, 0.5, 10.0, 0.25, 210.0],
[0.600122, 0.4999, 10.000122, 0.25, 210.0],
[0.600061, 0.49995, 10.000061, 0.25003, 50.0],
[0.595856, 0.505256, 9.991271, 0.24, 100.0],
[0.597973, 0.507461, 9.992175, 0.24, 210.0],
[0.595362, 0.507926, 9.993945, 0.24, 100.0],
[0.594618, 0.509055, 9.996832, 0.24, 210.0],
[0.592382, 0.506906, 9.996092, 0.24, 100.0],
[0.590438, 0.504882, 9.997604, 0.24, 210.0],
[0.591035, 0.503605, 9.994744, 0.24, 100.0],
[0.59121, 0.500708, 9.993425, 0.24, 210.0],
[0.593181, 0.502585, 9.991764, 0.24, 100.0],
[0.595867, 0.502302, 9.990069, 0.24, 210.0],
],
];
let scene_names = [
"single_sphere",
"two_spheres",
"ring_of_spheres",
"beachball",
"football",
"eightball",
"starball",
"combination_scene",
];
for i in 0..8 {
eprintln!("Generating {}", scene_names[i]);
let values = grey_values(&test_cases[i]);
let filename = format!("{}.png", scene_names[i]);
let path = std::path::Path::new(&filename);
let file = std::fs::File::create(path).unwrap();
let w = &mut std::io::BufWriter::new(file);
let mut encoder = png::Encoder::new(w, 1024, 1024);
encoder.set_color(png::ColorType::Grayscale);
encoder.set_depth(png::BitDepth::Eight);
let mut writer = encoder.write_header().unwrap();
writer.write_image_data(&values).unwrap();
}
}
```
## Test cases
The images may be scaled down. Clicking or tapping on an image will open it full size.
### Single sphere
Input:
```text
TODO
```
Output:
```text
TODO
```
### Two separated spheres
Input:
```text
TODO
```
Output:
```text
TODO
```
### Eight spheres in a ring
Input:
```text
TODO
```
Output:
```text
TODO
```
### Beach ball
Input:
```text
TODO
```
Output:
```text
TODO
```
### Football
Input:
```text
TODO
```
Output:
```text
TODO
```
### Eight ball
Input:
```text
TODO
```
Output:
```text
TODO
```
### Star ball
Input:
```text
TODO
```
Output:
```text
TODO
```
### Combination scene
Input:
```text
TODO
```
Output:
```text
TODO
```
## Explanatory animations
The example images were produced from plain grey spheres with no surface pattern or texture. The following animations show how they break down into their component plain spheres.
This section is not necessary to write valid code for this challenge. If your code accepts the parameters of plain grey spheres and renders them correctly, all of the test cases will look right without needing to understand how they were constructed. This section is mainly out of interest, and to make clear that your code *only* needs to render plain grey spheres.
```text
CONSIDER HOSTING THESE OFF SITE AS THEY ARE NOT REQUIRED FOR THE CHALLENGE
AND EVEN THE SCALED DOWN VERSIONS ARE UNNECESSARY DATA USE FOR ANYONE
CHOOSING NOT TO VIEW THEM.
```
<details>
<summary>Animations</summary>
<details>
<summary>Beach ball</summary>
This animation is reduced to 256 by 256 pixels. Clicking or tapping on the animation will open it at full size (______MB).
```text
TODO
```
</details>
<details>
<summary>Football</summary>
This animation is reduced to 256 by 256 pixels. Clicking or tapping on the animation will open it at full size (______MB).
```text
TODO
```
</details>
<details>
<summary>Eight ball</summary>
This animation is reduced to 256 by 256 pixels. Clicking or tapping on the animation will open it at full size (______MB).
```text
TODO
```
</details>
<details>
<summary>Star ball</summary>
This animation is reduced to 256 by 256 pixels. Clicking or tapping on the animation will open it at full size (______MB).
```text
TODO
```
</details>
</details>
## Scoring
This is a [code golf challenge]. Your score is the number of bytes in your code. Lowest score for each language wins.
> Explanations are optional, but I'm more likely to upvote answers that have one.
[code golf challenge]: https://codegolf.codidact.com/categories/49/tags/4274 "The code-golf tag"
[PGM]: https://en.wikipedia.org/wiki/Netpbm#PGM_example

Given a list of spheres, output an image of them.
The challenge is to produce a greyscale [orthographic](https://en.wikipedia.org/wiki/Parallel_projection#Orthographic_projection) (parallel projection - no perspective) ray caster, that renders only plain grey spheres with no lighting, no shading, no shadows, no surface patterns or textures, no reflections, and no transparency.
## Input
- A list of spheres, each consisting of:
- The coordinates of its centre $(x, y, z)$.
- Its radius $r$.
- Its greyscale value $g$, an integer from $0$ to $255$ (both inclusive).
- The input may include up to $100$ spheres.
- Only the greyscale value will be restricted to integer values. The coordinates and radius may have fractional values.
- Here "integer" refers to the values the greyscale value make take. You are not required to use an integer variable type.[]()
## Output
- A single greyscale image:
- $1024$ by $1024$ pixels.
- Each pixel has a single greyscale value from $0$ to $255$.
- Horizontally, the image covers x values from $-1$ to $1$.
- Vertically, the image covers y values from $-1$ to $1$.
- The image is orthographic, so there is no perspective. Changing the $z$ coordinate of a sphere does not change its size in the image, instead only affecting whether it is in front of or behind other spheres.
- A sphere's greyscale value is applied to every visible part of it. There is no variation over the surface of the sphere, so it appears flat like a disk.
- The point of the challenge is producing the pixel values, not manipulating image file formats, so you may output whatever image format is convenient, including simple text formats such as [PGM] or even just a list of pixel greyscale values. If you want to include an example of your output as an image in your answer, the image formats that Codidact supports are PNG, JPEG, and GIF. You could output PGM and then convert to PNG to upload to your answer. If you output just the greyscale values as a list of numbers from $0$ to $255$, I've made a `[ TODO page to convert these to greyscale PNG]`.
- The test cases, which match the reference implementations, happen to have the x axis increasing from left to right, and the y axis increasing from top to bottom. You are free to instead interpret these as increasing from right to left for x or from bottom to top for y, resulting in a reflected image. You are also free to output an image rotated by a multiple of 90 degrees, allowing for swapping the x and y axes. However, the z axis is required to be increasing in the forwards direction (into the image) otherwise some of the test cases will look incorrect.
## Calculating a pixel value
Imagine a ray passing into the image through a pixel, perpendicular to the image plane. If it hits a sphere, it shows the greyscale value of the first sphere it hits. Otherwise, it shows a value of zero (black). One possible way of calculating this is as follows:
For each pixel:
- The pixel $x, y$ coordinates will each be from $0$ to $1023$. Scale these to the range $-1$ to $1$ (for example, divide by $512$ and subtract $1$).
- If these scaled $x, y$ coordinates are within the radius $r$ of the $x, y$ coordinates of a sphere's centre, the ray intersects that sphere (the $z$ coordinate is not relevant to this test because the image is an orthographic projection - the rays are all parallel to the $z$ axis). The distance $a$ of the ray from the sphere's centre can be calculated as follows:
$$a=\sqrt{(x_{ray}-x_{sphere})^2+(y_{ray}-y_{sphere})^2}$$
```text
CONSIDER A DIAGRAM HERE
```
- For each sphere that the ray intersects, find the $z$ coordinate of the nearest intersection (there will often be $2$ per sphere, $1$ on the front of the sphere and $1$ on the back of the sphere, as the ray passes through - only the nearest is relevant). This can be calculated as follows:
- The distance $d$ to the nearest intersection point can be calculated from the $z$ coordinate of the sphere's centre, the radius $r$, and the distance $a$ from the $x, y$ coordinates of the sphere's centre to the scaled $x, y$ coordinates of the pixel:
$$d=z_{sphere}-\sqrt{r_{sphere}^2-a^2}$$
```text
CONSIDER A DIAGRAM HERE
```
- Of all the spheres that the ray intersects, the sphere with the intersection point with the smallest $z$ coordinate $d$ is the one to display. Set the pixel to the greyscale value of this sphere.
- If the ray did not intersect any spheres, set the pixel to zero (black).
## Example implementations
These are ungolfed implementations designed to supplement understanding of the requirements. You are free to golf these or to implement your own approach that matches the test cases up to reflection and rotation.
The Python implementation uses 64 bit floating point giving accurate images. The Rust implementation deliberately uses 32 bit floating point to show the difference in output. I've tried to optimise the test cases for compatibility with 32 bit floating point, so the differences are restricted to being a pixel out in some places.
### Python
The function `grey_values` would be sufficient as an entry to this challenge. The rest of the code saves the outputs as PNG files. The function itself has no dependence on the `import png` so it would not be required in an entry.
```python
def grey_values(spheres):
values = []
for y_pixel in range(1024):
for x_pixel in range(1024):
x_ray, y_ray = x_pixel / 512 - 1, y_pixel / 512 - 1
min_distance, hit_grey_value = None, 0
for sphere in spheres:
x_sphere, y_sphere, z_sphere, radius, grey_value = sphere
a = ((x_ray-x_sphere) ** 2 + (y_ray-y_sphere) ** 2) ** 0.5
if a < radius:
distance = z_sphere - (radius ** 2 - a ** 2) ** 0.5
if min_distance is None or distance < min_distance:
min_distance = distance
hit_grey_value = grey_value
values.append(hit_grey_value)
return values
import png
test_cases = [
[
[0, 0, 0, 1, 128],
],
[
[0, -0.55, 0, 0.4, 100],
[0, 0.55, 0, 0.4, 200],
],
[
[0.75, 0, 0, 0.25, 64],
[0.53033, 0.375, -0.375, 0.25, 185],
[0, 0.53033, -0.53033, 0.25, 64],
[-0.53033, 0.375, -0.375, 0.25, 185],
[-0.75, 0, -0, 0.25, 64],
[-0.53033, -0.375, 0.375, 0.25, 185],
[-0, -0.53033, 0.53033, 0.25, 64],
[0.53033, -0.375, 0.375, 0.25, 185],
],
[
[0.001, 0, 0, 1, 64],
[0.000707, 0.000707, -0, 1, 85],
[0, 0.001, -0, 1, 106],
[-0.000707, 0.000707, -0, 1, 127],
[-0.001, 0, -0, 1, 148],
[-0.000707, -0.000707, 0, 1, 169],
[-0, -0.001, 0, 1, 190],
[0.000707, -0.000707, 0, 1, 211],
],
[
[0, 0.000058, 0.001947, 1, 0],
[0, -0.001715, 0.000923, 1, 0],
[0, 0.001715, -0.000923, 1, 0],
[0, -0.000058, -0.001947, 1, 0],
[0.001657, -0.000512, 0.000887, 1, 0],
[-0.001657, -0.000512, 0.000887, 1, 0],
[0.001657, 0.000512, -0.000887, 1, 0],
[-0.001657, 0.000512, -0.000887, 1, 0],
[0.001024, 0.001435, 0.000829, 1, 0],
[-0.001024, 0.001435, 0.000829, 1, 0],
[0.001024, -0.001435, -0.000829, 1, 0],
[-0.001024, -0.001435, -0.000829, 1, 0],
[0.001868, 0.000618, 0.000357, 1, 255],
[-0.001868, 0.000618, 0.000357, 1, 255],
[0.001868, -0.000618, -0.000357, 1, 255],
[-0.001868, -0.000618, -0.000357, 1, 255],
[0.000714, -0.000934, 0.001618, 1, 255],
[-0.000714, -0.000934, 0.001618, 1, 255],
[0.000714, 0.000934, -0.001618, 1, 255],
[-0.000714, 0.000934, -0.001618, 1, 255],
[0, -0.001975, -0.000316, 1, 255],
[0, 0.001261, 0.001552, 1, 255],
[0, -0.001261, -0.001552, 1, 255],
[0, 0.001975, 0.000316, 1, 255],
[0.001155, 0.000423, 0.001577, 1, 255],
[-0.001155, 0.000423, 0.001577, 1, 255],
[0.001155, -0.001577, 0.000423, 1, 255],
[-0.001155, -0.001577, 0.000423, 1, 255],
[0.001155, 0.001577, -0.000423, 1, 255],
[-0.001155, 0.001577, -0.000423, 1, 255],
[0.001155, -0.000423, -0.001577, 1, 255],
[-0.001155, -0.000423, -0.001577, 1, 255],
],
[
[0, 0, 1000, 2, 128],
[0, 0, 0.003125, 1, 0],
[0, 0, 0, 0.9975, 255],
[0, 0.000547, -0.002573, 0.995, 0],
[-0, -0.000547, -0.002573, 0.995, 0],
[0, 0.001071, -0.005037, 0.9925, 255],
[-0, -0.001071, -0.005037, 0.9925, 255],
],
[
[0, 0, 0, 1, 210],
[0, 0, 0.0008, 1, 210],
[0, 0, 0.0004, 1.00012, 50],
[0.012969, -0, -0.042045, 0.96, 100],
[0.016397, 0.011913, -0.039054, 0.96, 210],
[0.004008, 0.012334, -0.042045, 0.96, 100],
[-0.006263, 0.019276, -0.039054, 0.96, 210],
[-0.010492, 0.007623, -0.042045, 0.96, 100],
[-0.020268, 0, -0.039054, 0.96, 210],
[-0.010492, -0.007623, -0.042045, 0.96, 100],
[-0.006263, -0.019276, -0.039054, 0.96, 210],
[0.004008, -0.012334, -0.042045, 0.96, 100],
[0.016397, -0.011913, -0.039054, 0.96, 210],
],
[
[0, 0, 1000, 2, 192],
[-2, 2, 100, 3, 98],
[2, 2, 100, 3, 98],
[0.5, -0.5, 500, 0.2, 255],
[-0.349469, -0.1, 20.000279, 0.6, 64],
[-0.349775, -0.100273, 20.000485, 0.6, 85],
[-0.350214, -0.100386, 20.000407, 0.6, 106],
[-0.350527, -0.100273, 20.000091, 0.6, 127],
[-0.350531, -0.1, 19.999721, 0.6, 148],
[-0.350225, -0.099727, 19.999515, 0.6, 169],
[-0.349786, -0.099614, 19.999593, 0.6, 190],
[-0.349473, -0.099727, 19.999909, 0.6, 211],
[0.20032, 0.349824, 15.000456, 0.3, 0],
[0.20032, 0.349517, 14.999924, 0.3, 0],
[0.19968, 0.350483, 15.000076, 0.3, 0],
[0.19968, 0.350176, 14.999544, 0.3, 0],
[0.200578, 0.350073, 14.999958, 0.3, 0],
[0.199817, 0.349519, 15.000277, 0.3, 0],
[0.200183, 0.350481, 14.999723, 0.3, 0],
[0.199422, 0.349927, 15.000042, 0.3, 0],
[0.200235, 0.35042, 15.000332, 0.3, 0],
[0.199765, 0.350078, 15.000529, 0.3, 0],
[0.200235, 0.349922, 14.999471, 0.3, 0],
[0.199765, 0.34958, 14.999668, 0.3, 0],
[0.200429, 0.350419, 15.000005, 0.3, 255],
[0.199571, 0.349795, 15.000366, 0.3, 255],
[0.200429, 0.350205, 14.999634, 0.3, 255],
[0.199571, 0.349581, 14.999995, 0.3, 255],
[0.200524, 0.349747, 15.000146, 0.3, 255],
[0.200196, 0.349509, 15.000283, 0.3, 255],
[0.199804, 0.350491, 14.999717, 0.3, 255],
[0.199476, 0.350253, 14.999854, 0.3, 255],
[0.200138, 0.349578, 14.999597, 0.3, 255],
[0.200138, 0.350138, 15.000567, 0.3, 255],
[0.199862, 0.349862, 14.999433, 0.3, 255],
[0.199862, 0.350422, 15.000403, 0.3, 255],
[0.200488, 0.350136, 15.000321, 0.3, 255],
[0.199957, 0.349751, 15.000544, 0.3, 255],
[0.200488, 0.34979, 14.999721, 0.3, 255],
[0.199957, 0.349404, 14.999944, 0.3, 255],
[0.200043, 0.350596, 15.000056, 0.3, 255],
[0.199512, 0.35021, 15.000279, 0.3, 255],
[0.200043, 0.350249, 14.999456, 0.3, 255],
[0.199512, 0.349864, 14.999679, 0.3, 255],
[-0.5, 0.500442, 10.000442, 0.2, 0],
[-0.5, 0.5, 10, 0.1995, 255],
[-0.499966, 0.49971, 9.999563, 0.199, 0],
[-0.500034, 0.499563, 9.99971, 0.199, 0],
[-0.499934, 0.499432, 9.999144, 0.1985, 255],
[-0.500066, 0.499144, 9.999432, 0.1985, 255],
[0.6, 0.5, 10, 0.25, 210],
[0.600122, 0.4999, 10.000122, 0.25, 210],
[0.600061, 0.49995, 10.000061, 0.25003, 50],
[0.595856, 0.505256, 9.991271, 0.24, 100],
[0.597973, 0.507461, 9.992175, 0.24, 210],
[0.595362, 0.507926, 9.993945, 0.24, 100],
[0.594618, 0.509055, 9.996832, 0.24, 210],
[0.592382, 0.506906, 9.996092, 0.24, 100],
[0.590438, 0.504882, 9.997604, 0.24, 210],
[0.591035, 0.503605, 9.994744, 0.24, 100],
[0.59121, 0.500708, 9.993425, 0.24, 210],
[0.593181, 0.502585, 9.991764, 0.24, 100],
[0.595867, 0.502302, 9.990069, 0.24, 210],
],
]
scene_names = (
"single_sphere",
"two_spheres",
"ring_of_spheres",
"beachball",
"football",
"eightball",
"starball",
"combination_scene",
)
for i in range(8):
print(f'Generating {scene_names[i]}.')
values = grey_values(test_cases[i])
rows = [values[row * 1024:row * 1024 + 1024] for row in range(1024)]
with open(f'{scene_names[i]}.png', 'wb') as file:
w = png.Writer(1024, 1024, greyscale=True)
w.write(file, rows)
```
### Rust
The function `grey_values` would be sufficient as an entry to this challenge. The rest of the code saves the outputs as PNG files.
```rust
fn grey_values(spheres: &Vec<[f32; 5]>) -> Vec<u8> {
let mut values = vec![];
for y_pixel in 0..1024 {
for x_pixel in 0..1024 {
let x_ray = x_pixel as f32 / 512.0 - 1.0;
let y_ray = y_pixel as f32 / 512.0 - 1.0;
let mut min_distance = f32::INFINITY;
let mut hit_grey_value = 0;
for sphere in spheres {
let [
x_sphere, y_sphere, z_sphere, radius, grey
] = *sphere;
let a = (
(x_ray - x_sphere).powf(2.0)
+ (y_ray - y_sphere).powf(2.0)
).sqrt();
if a < radius {
let distance = z_sphere - (
radius.powf(2.0) - a.powf(2.0)
).sqrt();
if distance < min_distance {
min_distance = distance;
hit_grey_value = grey as u8;
}
}
}
values.push(hit_grey_value);
}
}
values
}
fn main() {
let test_cases: Vec<Vec<[f32; 5]>> = vec![
vec![[0.0, 0.0, 0.0, 1.0, 128.0]],
vec![[0.0, -0.55, 0.0, 0.4, 100.0], [0.0, 0.55, 0.0, 0.4, 200.0]],
vec![
[0.75, 0.0, 0.0, 0.25, 64.0],
[0.53033, 0.375, -0.375, 0.25, 185.0],
[0.0, 0.53033, -0.53033, 0.25, 64.0],
[-0.53033, 0.375, -0.375, 0.25, 185.0],
[-0.75, 0.0, 0.0, 0.25, 64.0],
[-0.53033, -0.375, 0.375, 0.25, 185.0],
[0.0, -0.53033, 0.53033, 0.25, 64.0],
[0.53033, -0.375, 0.375, 0.25, 185.0],
],
vec![
[0.001, 0.0, 0.0, 1.0, 64.0],
[0.000707, 0.000707, 0.0, 1.0, 85.0],
[0.0, 0.001, 0.0, 1.0, 106.0],
[-0.000707, 0.000707, 0.0, 1.0, 127.0],
[-0.001, 0.0, 0.0, 1.0, 148.0],
[-0.000707, -0.000707, 0.0, 1.0, 169.0],
[0.0, -0.001, 0.0, 1.0, 190.0],
[0.000707, -0.000707, 0.0, 1.0, 211.0],
],
vec![
[0.0, 0.000058, 0.001947, 1.0, 0.0],
[0.0, -0.001715, 0.000923, 1.0, 0.0],
[0.0, 0.001715, -0.000923, 1.0, 0.0],
[0.0, -0.000058, -0.001947, 1.0, 0.0],
[0.001657, -0.000512, 0.000887, 1.0, 0.0],
[-0.001657, -0.000512, 0.000887, 1.0, 0.0],
[0.001657, 0.000512, -0.000887, 1.0, 0.0],
[-0.001657, 0.000512, -0.000887, 1.0, 0.0],
[0.001024, 0.001435, 0.000829, 1.0, 0.0],
[-0.001024, 0.001435, 0.000829, 1.0, 0.0],
[0.001024, -0.001435, -0.000829, 1.0, 0.0],
[-0.001024, -0.001435, -0.000829, 1.0, 0.0],
[0.001868, 0.000618, 0.000357, 1.0, 255.0],
[-0.001868, 0.000618, 0.000357, 1.0, 255.0],
[0.001868, -0.000618, -0.000357, 1.0, 255.0],
[-0.001868, -0.000618, -0.000357, 1.0, 255.0],
[0.000714, -0.000934, 0.001618, 1.0, 255.0],
[-0.000714, -0.000934, 0.001618, 1.0, 255.0],
[0.000714, 0.000934, -0.001618, 1.0, 255.0],
[-0.000714, 0.000934, -0.001618, 1.0, 255.0],
[0.0, -0.001975, -0.000316, 1.0, 255.0],
[0.0, 0.001261, 0.001552, 1.0, 255.0],
[0.0, -0.001261, -0.001552, 1.0, 255.0],
[0.0, 0.001975, 0.000316, 1.0, 255.0],
[0.001155, 0.000423, 0.001577, 1.0, 255.0],
[-0.001155, 0.000423, 0.001577, 1.0, 255.0],
[0.001155, -0.001577, 0.000423, 1.0, 255.0],
[-0.001155, -0.001577, 0.000423, 1.0, 255.0],
[0.001155, 0.001577, -0.000423, 1.0, 255.0],
[-0.001155, 0.001577, -0.000423, 1.0, 255.0],
[0.001155, -0.000423, -0.001577, 1.0, 255.0],
[-0.001155, -0.000423, -0.001577, 1.0, 255.0],
],
vec![
[0.0, 0.0, 1000.0, 2.0, 128.0],
[0.0, 0.0, 0.003125, 1.0, 0.0],
[0.0, 0.0, 0.0, 0.9975, 255.0],
[0.0, 0.000547, -0.002573, 0.995, 0.0],
[0.0, -0.000547, -0.002573, 0.995, 0.0],
[0.0, 0.001071, -0.005037, 0.9925, 255.0],
[0.0, -0.001071, -0.005037, 0.9925, 255.0],
],
vec![
[0.0, 0.0, 0.0, 1.0, 210.0],
[0.0, 0.0, 0.0008, 1.0, 210.0],
[0.0, 0.0, 0.0004, 1.00012, 50.0],
[0.012969, 0.0, -0.042045, 0.96, 100.0],
[0.016397, 0.011913, -0.039054, 0.96, 210.0],
[0.004008, 0.012334, -0.042045, 0.96, 100.0],
[-0.006263, 0.019276, -0.039054, 0.96, 210.0],
[-0.010492, 0.007623, -0.042045, 0.96, 100.0],
[-0.020268, 0.0, -0.039054, 0.96, 210.0],
[-0.010492, -0.007623, -0.042045, 0.96, 100.0],
[-0.006263, -0.019276, -0.039054, 0.96, 210.0],
[0.004008, -0.012334, -0.042045, 0.96, 100.0],
[0.016397, -0.011913, -0.039054, 0.96, 210.0],
],
vec![
[0.0, 0.0, 1000.0, 2.0, 192.0],
[-2.0, 2.0, 100.0, 3.0, 98.0],
[2.0, 2.0, 100.0, 3.0, 98.0],
[0.5, -0.5, 500.0, 0.2, 255.0],
[-0.349469, -0.1, 20.000279, 0.6, 64.0],
[-0.349775, -0.100273, 20.000485, 0.6, 85.0],
[-0.350214, -0.100386, 20.000407, 0.6, 106.0],
[-0.350527, -0.100273, 20.000091, 0.6, 127.0],
[-0.350531, -0.1, 19.999721, 0.6, 148.0],
[-0.350225, -0.099727, 19.999515, 0.6, 169.0],
[-0.349786, -0.099614, 19.999593, 0.6, 190.0],
[-0.349473, -0.099727, 19.999909, 0.6, 211.0],
[0.20032, 0.349824, 15.000456, 0.3, 0.0],
[0.20032, 0.349517, 14.999924, 0.3, 0.0],
[0.19968, 0.350483, 15.000076, 0.3, 0.0],
[0.19968, 0.350176, 14.999544, 0.3, 0.0],
[0.200578, 0.350073, 14.999958, 0.3, 0.0],
[0.199817, 0.349519, 15.000277, 0.3, 0.0],
[0.200183, 0.350481, 14.999723, 0.3, 0.0],
[0.199422, 0.349927, 15.000042, 0.3, 0.0],
[0.200235, 0.35042, 15.000332, 0.3, 0.0],
[0.199765, 0.350078, 15.000529, 0.3, 0.0],
[0.200235, 0.349922, 14.999471, 0.3, 0.0],
[0.199765, 0.34958, 14.999668, 0.3, 0.0],
[0.200429, 0.350419, 15.000005, 0.3, 255.0],
[0.199571, 0.349795, 15.000366, 0.3, 255.0],
[0.200429, 0.350205, 14.999634, 0.3, 255.0],
[0.199571, 0.349581, 14.999995, 0.3, 255.0],
[0.200524, 0.349747, 15.000146, 0.3, 255.0],
[0.200196, 0.349509, 15.000283, 0.3, 255.0],
[0.199804, 0.350491, 14.999717, 0.3, 255.0],
[0.199476, 0.350253, 14.999854, 0.3, 255.0],
[0.200138, 0.349578, 14.999597, 0.3, 255.0],
[0.200138, 0.350138, 15.000567, 0.3, 255.0],
[0.199862, 0.349862, 14.999433, 0.3, 255.0],
[0.199862, 0.350422, 15.000403, 0.3, 255.0],
[0.200488, 0.350136, 15.000321, 0.3, 255.0],
[0.199957, 0.349751, 15.000544, 0.3, 255.0],
[0.200488, 0.34979, 14.999721, 0.3, 255.0],
[0.199957, 0.349404, 14.999944, 0.3, 255.0],
[0.200043, 0.350596, 15.000056, 0.3, 255.0],
[0.199512, 0.35021, 15.000279, 0.3, 255.0],
[0.200043, 0.350249, 14.999456, 0.3, 255.0],
[0.199512, 0.349864, 14.999679, 0.3, 255.0],
[-0.5, 0.500442, 10.000442, 0.2, 0.0],
[-0.5, 0.5, 10.0, 0.1995, 255.0],
[-0.499966, 0.49971, 9.999563, 0.199, 0.0],
[-0.500034, 0.499563, 9.99971, 0.199, 0.0],
[-0.499934, 0.499432, 9.999144, 0.1985, 255.0],
[-0.500066, 0.499144, 9.999432, 0.1985, 255.0],
[0.6, 0.5, 10.0, 0.25, 210.0],
[0.600122, 0.4999, 10.000122, 0.25, 210.0],
[0.600061, 0.49995, 10.000061, 0.25003, 50.0],
[0.595856, 0.505256, 9.991271, 0.24, 100.0],
[0.597973, 0.507461, 9.992175, 0.24, 210.0],
[0.595362, 0.507926, 9.993945, 0.24, 100.0],
[0.594618, 0.509055, 9.996832, 0.24, 210.0],
[0.592382, 0.506906, 9.996092, 0.24, 100.0],
[0.590438, 0.504882, 9.997604, 0.24, 210.0],
[0.591035, 0.503605, 9.994744, 0.24, 100.0],
[0.59121, 0.500708, 9.993425, 0.24, 210.0],
[0.593181, 0.502585, 9.991764, 0.24, 100.0],
[0.595867, 0.502302, 9.990069, 0.24, 210.0],
],
];
let scene_names = [
"single_sphere",
"two_spheres",
"ring_of_spheres",
"beachball",
"football",
"eightball",
"starball",
"combination_scene",
];
for i in 0..8 {
eprintln!("Generating {}", scene_names[i]);
let values = grey_values(&test_cases[i]);
let filename = format!("{}.png", scene_names[i]);
let path = std::path::Path::new(&filename);
let file = std::fs::File::create(path).unwrap();
let w = &mut std::io::BufWriter::new(file);
let mut encoder = png::Encoder::new(w, 1024, 1024);
encoder.set_color(png::ColorType::Grayscale);
encoder.set_depth(png::BitDepth::Eight);
let mut writer = encoder.write_header().unwrap();
writer.write_image_data(&values).unwrap();
}
}
```
## Test cases
The images may be scaled down. Clicking or tapping on an image will open it full size.
### Single sphere
Input:
```text
TODO
```
Output:
```text
TODO
```
### Two separated spheres
Input:
```text
TODO
```
Output:
```text
TODO
```
### Eight spheres in a ring
Input:
```text
TODO
```
Output:
```text
TODO
```
### Beach ball
Input:
```text
TODO
```
Output:
```text
TODO
```
### Football
Input:
```text
TODO
```
Output:
```text
TODO
```
### Eight ball
Input:
```text
TODO
```
Output:
```text
TODO
```
### Star ball
Input:
```text
TODO
```
Output:
```text
TODO
```
### Combination scene
Input:
```text
TODO
```
Output:
```text
TODO
```
## Explanatory animations
The example images were produced from plain grey spheres with no surface pattern or texture. The following animations show how they break down into their component plain spheres.
This section is not necessary to write valid code for this challenge. If your code accepts the parameters of plain grey spheres and renders them correctly, all of the test cases will look right without needing to understand how they were constructed. This section is mainly out of interest, and to make clear that your code *only* needs to render plain grey spheres.
```text
CONSIDER HOSTING THESE OFF SITE AS THEY ARE NOT REQUIRED FOR THE CHALLENGE
AND EVEN THE SCALED DOWN VERSIONS ARE UNNECESSARY DATA USE FOR ANYONE
CHOOSING NOT TO VIEW THEM.
```
<details>
<summary>Animations</summary>
<details>
<summary>Beach ball</summary>
This animation is reduced to 256 by 256 pixels. Clicking or tapping on the animation will open it at full size (______MB).
```text
TODO
```
</details>
<details>
<summary>Football</summary>
This animation is reduced to 256 by 256 pixels. Clicking or tapping on the animation will open it at full size (______MB).
```text
TODO
```
</details>
<details>
<summary>Eight ball</summary>
This animation is reduced to 256 by 256 pixels. Clicking or tapping on the animation will open it at full size (______MB).
```text
TODO
```
</details>
<details>
<summary>Star ball</summary>
This animation is reduced to 256 by 256 pixels. Clicking or tapping on the animation will open it at full size (______MB).
```text
TODO
```
</details>
</details>
## Scoring
This is a [code golf challenge]. Your score is the number of bytes in your code. Lowest score for each language wins.
> Explanations are optional, but I'm more likely to upvote answers that have one.
[code golf challenge]: https://codegolf.codidact.com/categories/49/tags/4274 "The code-golf tag"
[PGM]: https://en.wikipedia.org/wiki/Netpbm#PGM_example

#2: Post edited by

trichoplax‭ · 2024-07-26T10:39:54Z (5 months ago)
Add example implementations

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Given a list of spheres, output an image of them.
The challenge is to produce a greyscale [orthographic](https://en.wikipedia.org/wiki/Parallel_projection#Orthographic_projection) (parallel projection - no perspective) ray caster, that renders only plain grey spheres with no lighting, no shading, no shadows, no surface patterns or textures, no reflections, and no transparency.
## Input
- A list of spheres, each consisting of:
- The coordinates of its centre $(x, y, z)$.
- Its radius $r$.
- Its greyscale value $g$, an integer from $0$ to $255$ (both inclusive).
- The input may include up to $100$ spheres.
- Only the greyscale value will be restricted to integer values. The coordinates and radius may have fractional values.
- Here "integer" refers to the values the greyscale value make take. You are not required to use an integer variable type.[]()
## Output
- A single greyscale image:
- $1024$ by $1024$ pixels.
- Each pixel has a single greyscale value from $0$ to $255$.
- Horizontally, the image covers x values from $-1$ to $1$.
- Vertically, the image covers y values from $-1$ to $1$.
- The image is orthographic, so there is no perspective. Changing the $z$ coordinate of a sphere does not change its size in the image, instead only affecting whether it is in front of or behind other spheres.
- A sphere's greyscale value is applied to every visible part of it. There is no variation over the surface of the sphere, so it appears flat like a disk.
- The point of the challenge is producing the pixel values, not manipulating image file formats, so you may output whatever image format is convenient, including simple text formats such as [PGM] or even just a list of pixel greyscale values. If you want to include an example of your output as an image in your answer, the image formats that Codidact supports are PNG, JPEG, and GIF. You could output PGM and then convert to PNG to upload to your answer. If you output just the greyscale values as a list of numbers from $0$ to $255$, I've made a `[ TODO page to convert these to greyscale PNG]`.
- The test cases, which match the reference implementations, happen to have the x axis increasing from left to right, and the y axis increasing from top to bottom. You are free to instead interpret these as increasing from right to left for x or from bottom to top for y, resulting in a reflected image. You are also free to output an image rotated by a multiple of 90 degrees, allowing for swapping the x and y axes. However, the z axis is required to be increasing in the forwards direction (into the image) otherwise some of the test cases will look incorrect.
## Calculating a pixel value
Imagine a ray passing into the image through a pixel, perpendicular to the image plane. If it hits a sphere, it shows the greyscale value of the first sphere it hits. Otherwise, it shows a value of zero (black). One possible way of calculating this is as follows:
For each pixel:
- The pixel $x, y$ coordinates will each be from $0$ to $1023$. Scale these to the range $-1$ to $1$ (for example, divide by $512$ and subtract $1$).
- If these scaled $x, y$ coordinates are within the radius $r$ of the $x, y$ coordinates of a sphere's centre, the ray intersects that sphere (the $z$ coordinate is not relevant to this test because the image is an orthographic projection - the rays are all parallel to the $z$ axis). The distance $a$ of the ray from the sphere's centre can be calculated as follows:
$$a=\sqrt{(x_{ray}-x_{sphere})^2+(y_{ray}-y_{sphere})^2}$$
```text
CONSIDER A DIAGRAM HERE
```
- For each sphere that the ray intersects, find the $z$ coordinate of the nearest intersection (there will often be $2$ per sphere, $1$ on the front of the sphere and $1$ on the back of the sphere, as the ray passes through - only the nearest is relevant). This can be calculated as follows:
- The distance $d$ to the nearest intersection point can be calculated from the $z$ coordinate of the sphere's centre, the radius $r$, and the distance $a$ from the $x, y$ coordinates of the sphere's centre to the scaled $x, y$ coordinates of the pixel:
$$d=z_{sphere}-\sqrt{r_{sphere}^2-a^2}$$
```text
CONSIDER A DIAGRAM HERE
```
- Of all the spheres that the ray intersects, the sphere with the intersection point with the smallest $z$ coordinate $d$ is the one to display. Set the pixel to the greyscale value of this sphere.
- If the ray did not intersect any spheres, set the pixel to zero (black).
~~## Reference implementations~~
These are ungolfed implementations designed to supplement understanding of the requirements. You are free to golf these or to implement your own approach that matches the test cases up to reflection and rotation.
### Python
The function `greyscale_values` would be sufficient as an entry to this challenge. The rest of the code saves these as a PNG file. The function has no dependence on the imports so they would not be required in an entry.
```python
~~TODO~~
```
### Rust
The function `greyscale_values` would be sufficient as an entry to this challenge. The rest of the code saves these as a PNG file. The function has no dependence on the imports so they would not be required in an entry.
```rust
~~TODO~~
```
## Test cases
The images may be scaled down. Clicking or tapping on an image will open it full size.
### Single sphere
Input:
```text
TODO
```
Output:
```text
TODO
```
### Two separated spheres
Input:
```text
TODO
```
Output:
```text
TODO
```
### Eight spheres in a ring
Input:
```text
TODO
```
Output:
```text
TODO
```
### Beach ball
Input:
```text
TODO
```
Output:
```text
TODO
```
### Football
Input:
```text
TODO
```
Output:
```text
TODO
```
### Eight ball
Input:
```text
TODO
```
Output:
```text
TODO
```
### Star ball
Input:
```text
TODO
```
Output:
```text
TODO
```
### Combination scene
Input:
```text
TODO
```
Output:
```text
TODO
```
## Explanatory animations
The example images were produced from plain grey spheres with no surface pattern or texture. The following animations show how they break down into their component plain spheres.
This section is not necessary to write valid code for this challenge. If your code accepts the parameters of plain grey spheres and renders them correctly, all of the test cases will look right without needing to understand how they were constructed. This section is mainly out of interest, and to make clear that your code *only* needs to render plain grey spheres.
```text
CONSIDER HOSTING THESE OFF SITE AS THEY ARE NOT REQUIRED FOR THE CHALLENGE
AND EVEN THE SCALED DOWN VERSIONS ARE UNNECESSARY DATA USE FOR ANYONE
CHOOSING NOT TO VIEW THEM.
```
<details>
<summary>Animations</summary>
<details>
<summary>Beach ball</summary>
This animation is reduced to 256 by 256 pixels. Clicking or tapping on the animation will open it at full size (______MB).
```text
TODO
```
</details>
<details>
<summary>Football</summary>
This animation is reduced to 256 by 256 pixels. Clicking or tapping on the animation will open it at full size (______MB).
```text
TODO
```
</details>
<details>
<summary>Eight ball</summary>
This animation is reduced to 256 by 256 pixels. Clicking or tapping on the animation will open it at full size (______MB).
```text
TODO
```
</details>
<details>
<summary>Star ball</summary>
This animation is reduced to 256 by 256 pixels. Clicking or tapping on the animation will open it at full size (______MB).
```text
TODO
```
</details>
</details>
## Scoring
This is a [code golf challenge]. Your score is the number of bytes in your code. Lowest score for each language wins.
> Explanations are optional, but I'm more likely to upvote answers that have one.
[code golf challenge]: https://codegolf.codidact.com/categories/49/tags/4274 "The code-golf tag"
[PGM]: https://en.wikipedia.org/wiki/Netpbm#PGM_example

Given a list of spheres, output an image of them.
The challenge is to produce a greyscale [orthographic](https://en.wikipedia.org/wiki/Parallel_projection#Orthographic_projection) (parallel projection - no perspective) ray caster, that renders only plain grey spheres with no lighting, no shading, no shadows, no surface patterns or textures, no reflections, and no transparency.
## Input
- A list of spheres, each consisting of:
- The coordinates of its centre $(x, y, z)$.
- Its radius $r$.
- Its greyscale value $g$, an integer from $0$ to $255$ (both inclusive).
- The input may include up to $100$ spheres.
- Only the greyscale value will be restricted to integer values. The coordinates and radius may have fractional values.
- Here "integer" refers to the values the greyscale value make take. You are not required to use an integer variable type.[]()
## Output
- A single greyscale image:
- $1024$ by $1024$ pixels.
- Each pixel has a single greyscale value from $0$ to $255$.
- Horizontally, the image covers x values from $-1$ to $1$.
- Vertically, the image covers y values from $-1$ to $1$.
- The image is orthographic, so there is no perspective. Changing the $z$ coordinate of a sphere does not change its size in the image, instead only affecting whether it is in front of or behind other spheres.
- A sphere's greyscale value is applied to every visible part of it. There is no variation over the surface of the sphere, so it appears flat like a disk.
- The point of the challenge is producing the pixel values, not manipulating image file formats, so you may output whatever image format is convenient, including simple text formats such as [PGM] or even just a list of pixel greyscale values. If you want to include an example of your output as an image in your answer, the image formats that Codidact supports are PNG, JPEG, and GIF. You could output PGM and then convert to PNG to upload to your answer. If you output just the greyscale values as a list of numbers from $0$ to $255$, I've made a `[ TODO page to convert these to greyscale PNG]`.
- The test cases, which match the reference implementations, happen to have the x axis increasing from left to right, and the y axis increasing from top to bottom. You are free to instead interpret these as increasing from right to left for x or from bottom to top for y, resulting in a reflected image. You are also free to output an image rotated by a multiple of 90 degrees, allowing for swapping the x and y axes. However, the z axis is required to be increasing in the forwards direction (into the image) otherwise some of the test cases will look incorrect.
## Calculating a pixel value
Imagine a ray passing into the image through a pixel, perpendicular to the image plane. If it hits a sphere, it shows the greyscale value of the first sphere it hits. Otherwise, it shows a value of zero (black). One possible way of calculating this is as follows:
For each pixel:
- The pixel $x, y$ coordinates will each be from $0$ to $1023$. Scale these to the range $-1$ to $1$ (for example, divide by $512$ and subtract $1$).
- If these scaled $x, y$ coordinates are within the radius $r$ of the $x, y$ coordinates of a sphere's centre, the ray intersects that sphere (the $z$ coordinate is not relevant to this test because the image is an orthographic projection - the rays are all parallel to the $z$ axis). The distance $a$ of the ray from the sphere's centre can be calculated as follows:
$$a=\sqrt{(x_{ray}-x_{sphere})^2+(y_{ray}-y_{sphere})^2}$$
```text
CONSIDER A DIAGRAM HERE
```
- For each sphere that the ray intersects, find the $z$ coordinate of the nearest intersection (there will often be $2$ per sphere, $1$ on the front of the sphere and $1$ on the back of the sphere, as the ray passes through - only the nearest is relevant). This can be calculated as follows:
- The distance $d$ to the nearest intersection point can be calculated from the $z$ coordinate of the sphere's centre, the radius $r$, and the distance $a$ from the $x, y$ coordinates of the sphere's centre to the scaled $x, y$ coordinates of the pixel:
$$d=z_{sphere}-\sqrt{r_{sphere}^2-a^2}$$
```text
CONSIDER A DIAGRAM HERE
```
- Of all the spheres that the ray intersects, the sphere with the intersection point with the smallest $z$ coordinate $d$ is the one to display. Set the pixel to the greyscale value of this sphere.
- If the ray did not intersect any spheres, set the pixel to zero (black).
## Example implementations
These are ungolfed implementations designed to supplement understanding of the requirements. You are free to golf these or to implement your own approach that matches the test cases up to reflection and rotation.
The Python implementation uses 64 bit floating point giving accurate images. The Rust implementation deliberately uses 32 bit floating point to show the difference in output. I've tried to optimise the test cases for compatibility with 32 bit floating point, so the differences are restricted to being a pixel out in some places.
### Python
The function `grey_values` would be sufficient as an entry to this challenge. The rest of the code saves the outputs as PNG files. The function itself has no dependence on the `import png` so it would not be required in an entry.
```python
def grey_values(spheres):
values = []
for y_pixel in range(1024):
for x_pixel in range(1024):
x_ray, y_ray = x_pixel / 512 - 1, y_pixel / 512 - 1
min_distance, hit_grey_value = None, 0
for sphere in spheres:
x_sphere, y_sphere, z_sphere, radius, grey_value = sphere
a = ((x_ray-x_sphere) ** 2 + (y_ray-y_sphere) ** 2) ** 0.5
if a < radius:
distance = z_sphere - (radius ** 2 - a ** 2) ** 0.5
if min_distance is None or distance < min_distance:
min_distance = distance
hit_grey_value = grey_value
values.append(hit_grey_value)
return values
import png
test_cases = [
[
[0, 0, 0, 1, 128],
],
[
[0, -0.55, 0, 0.4, 100],
[0, 0.55, 0, 0.4, 200],
],
[
[0.75, 0, 0, 0.25, 64],
[0.53033, 0.375, -0.375, 0.25, 185],
[0, 0.53033, -0.53033, 0.25, 64],
[-0.53033, 0.375, -0.375, 0.25, 185],
[-0.75, 0, -0, 0.25, 64],
[-0.53033, -0.375, 0.375, 0.25, 185],
[-0, -0.53033, 0.53033, 0.25, 64],
[0.53033, -0.375, 0.375, 0.25, 185],
],
[
[0.001, 0, 0, 1, 64],
[0.000707, 0.000707, -0, 1, 85],
[0, 0.001, -0, 1, 106],
[-0.000707, 0.000707, -0, 1, 127],
[-0.001, 0, -0, 1, 148],
[-0.000707, -0.000707, 0, 1, 169],
[-0, -0.001, 0, 1, 190],
[0.000707, -0.000707, 0, 1, 211],
],
[
[0, 0.000058, 0.001947, 1, 0],
[0, -0.001715, 0.000923, 1, 0],
[0, 0.001715, -0.000923, 1, 0],
[0, -0.000058, -0.001947, 1, 0],
[0.001657, -0.000512, 0.000887, 1, 0],
[-0.001657, -0.000512, 0.000887, 1, 0],
[0.001657, 0.000512, -0.000887, 1, 0],
[-0.001657, 0.000512, -0.000887, 1, 0],
[0.001024, 0.001435, 0.000829, 1, 0],
[-0.001024, 0.001435, 0.000829, 1, 0],
[0.001024, -0.001435, -0.000829, 1, 0],
[-0.001024, -0.001435, -0.000829, 1, 0],
[0.001868, 0.000618, 0.000357, 1, 255],
[-0.001868, 0.000618, 0.000357, 1, 255],
[0.001868, -0.000618, -0.000357, 1, 255],
[-0.001868, -0.000618, -0.000357, 1, 255],
[0.000714, -0.000934, 0.001618, 1, 255],
[-0.000714, -0.000934, 0.001618, 1, 255],
[0.000714, 0.000934, -0.001618, 1, 255],
[-0.000714, 0.000934, -0.001618, 1, 255],
[0, -0.001975, -0.000316, 1, 255],
[0, 0.001261, 0.001552, 1, 255],
[0, -0.001261, -0.001552, 1, 255],
[0, 0.001975, 0.000316, 1, 255],
[0.001155, 0.000423, 0.001577, 1, 255],
[-0.001155, 0.000423, 0.001577, 1, 255],
[0.001155, -0.001577, 0.000423, 1, 255],
[-0.001155, -0.001577, 0.000423, 1, 255],
[0.001155, 0.001577, -0.000423, 1, 255],
[-0.001155, 0.001577, -0.000423, 1, 255],
[0.001155, -0.000423, -0.001577, 1, 255],
[-0.001155, -0.000423, -0.001577, 1, 255],
],
[
[0, 0, 1000, 2, 128],
[0, 0, 0.003125, 1, 0],
[0, 0, 0, 0.9975, 255],
[0, 0.000547, -0.002573, 0.995, 0],
[-0, -0.000547, -0.002573, 0.995, 0],
[0, 0.001071, -0.005037, 0.9925, 255],
[-0, -0.001071, -0.005037, 0.9925, 255],
],
[
[0, 0, 0, 1, 210],
[0, 0, 0.0008, 1, 210],
[0, 0, 0.0004, 1.00012, 50],
[0.012969, -0, -0.042045, 0.96, 100],
[0.016397, 0.011913, -0.039054, 0.96, 210],
[0.004008, 0.012334, -0.042045, 0.96, 100],
[-0.006263, 0.019276, -0.039054, 0.96, 210],
[-0.010492, 0.007623, -0.042045, 0.96, 100],
[-0.020268, 0, -0.039054, 0.96, 210],
[-0.010492, -0.007623, -0.042045, 0.96, 100],
[-0.006263, -0.019276, -0.039054, 0.96, 210],
[0.004008, -0.012334, -0.042045, 0.96, 100],
[0.016397, -0.011913, -0.039054, 0.96, 210],
],
[
[0, 0, 1000, 2, 192],
[-2, 2, 100, 3, 98],
[2, 2, 100, 3, 98],
[0.5, -0.5, 500, 0.2, 255],
[-0.349469, -0.1, 20.000279, 0.6, 64],
[-0.349775, -0.100273, 20.000485, 0.6, 85],
[-0.350214, -0.100386, 20.000407, 0.6, 106],
[-0.350527, -0.100273, 20.000091, 0.6, 127],
[-0.350531, -0.1, 19.999721, 0.6, 148],
[-0.350225, -0.099727, 19.999515, 0.6, 169],
[-0.349786, -0.099614, 19.999593, 0.6, 190],
[-0.349473, -0.099727, 19.999909, 0.6, 211],
[0.20032, 0.349824, 15.000456, 0.3, 0],
[0.20032, 0.349517, 14.999924, 0.3, 0],
[0.19968, 0.350483, 15.000076, 0.3, 0],
[0.19968, 0.350176, 14.999544, 0.3, 0],
[0.200578, 0.350073, 14.999958, 0.3, 0],
[0.199817, 0.349519, 15.000277, 0.3, 0],
[0.200183, 0.350481, 14.999723, 0.3, 0],
[0.199422, 0.349927, 15.000042, 0.3, 0],
[0.200235, 0.35042, 15.000332, 0.3, 0],
[0.199765, 0.350078, 15.000529, 0.3, 0],
[0.200235, 0.349922, 14.999471, 0.3, 0],
[0.199765, 0.34958, 14.999668, 0.3, 0],
[0.200429, 0.350419, 15.000005, 0.3, 255],
[0.199571, 0.349795, 15.000366, 0.3, 255],
[0.200429, 0.350205, 14.999634, 0.3, 255],
[0.199571, 0.349581, 14.999995, 0.3, 255],
[0.200524, 0.349747, 15.000146, 0.3, 255],
[0.200196, 0.349509, 15.000283, 0.3, 255],
[0.199804, 0.350491, 14.999717, 0.3, 255],
[0.199476, 0.350253, 14.999854, 0.3, 255],
[0.200138, 0.349578, 14.999597, 0.3, 255],
[0.200138, 0.350138, 15.000567, 0.3, 255],
[0.199862, 0.349862, 14.999433, 0.3, 255],
[0.199862, 0.350422, 15.000403, 0.3, 255],
[0.200488, 0.350136, 15.000321, 0.3, 255],
[0.199957, 0.349751, 15.000544, 0.3, 255],
[0.200488, 0.34979, 14.999721, 0.3, 255],
[0.199957, 0.349404, 14.999944, 0.3, 255],
[0.200043, 0.350596, 15.000056, 0.3, 255],
[0.199512, 0.35021, 15.000279, 0.3, 255],
[0.200043, 0.350249, 14.999456, 0.3, 255],
[0.199512, 0.349864, 14.999679, 0.3, 255],
[-0.5, 0.500442, 10.000442, 0.2, 0],
[-0.5, 0.5, 10, 0.1995, 255],
[-0.499966, 0.49971, 9.999563, 0.199, 0],
[-0.500034, 0.499563, 9.99971, 0.199, 0],
[-0.499934, 0.499432, 9.999144, 0.1985, 255],
[-0.500066, 0.499144, 9.999432, 0.1985, 255],
[0.6, 0.5, 10, 0.25, 210],
[0.600122, 0.4999, 10.000122, 0.25, 210],
[0.600061, 0.49995, 10.000061, 0.25003, 50],
[0.595856, 0.505256, 9.991271, 0.24, 100],
[0.597973, 0.507461, 9.992175, 0.24, 210],
[0.595362, 0.507926, 9.993945, 0.24, 100],
[0.594618, 0.509055, 9.996832, 0.24, 210],
[0.592382, 0.506906, 9.996092, 0.24, 100],
[0.590438, 0.504882, 9.997604, 0.24, 210],
[0.591035, 0.503605, 9.994744, 0.24, 100],
[0.59121, 0.500708, 9.993425, 0.24, 210],
[0.593181, 0.502585, 9.991764, 0.24, 100],
[0.595867, 0.502302, 9.990069, 0.24, 210],
],
]
scene_names = (
"single_sphere",
"two_spheres",
"ring_of_spheres",
"beachball",
"football",
"eightball",
"starball",
"combination_scene",
)
for i in range(8):
print(f'Generating {scene_names[i]}.')
values = grey_values(test_cases[i])
rows = [values[row * 1024:row * 1024 + 1024] for row in range(1024)]
with open(f'{scene_names[i]}.png', 'wb') as file:
w = png.Writer(1024, 1024, greyscale=True)
w.write(file, rows)
```
### Rust
The function `grey_values` would be sufficient as an entry to this challenge. The rest of the code saves the outputs as PNG files.
```rust
fn grey_values(spheres: &Vec<[f32; 5]>) -> Vec<u8> {
let mut values = vec![];
for y_pixel in 0..1024 {
for x_pixel in 0..1024 {
let x_ray = x_pixel as f32 / 512.0 - 1.0;
let y_ray = y_pixel as f32 / 512.0 - 1.0;
let mut min_distance = f32::INFINITY;
let mut hit_grey_value = 0;
for sphere in spheres {
let [x_sphere, y_sphere, z_sphere, radius, grey] = *sphere;
let a = (
(x_ray-x_sphere).powf(2.0) + (y_ray-y_sphere).powf(2.0)
).sqrt();
if a < radius {
let distance = z_sphere - (
radius.powf(2.0) - a.powf(2.0)
).sqrt();
if distance < min_distance {
min_distance = distance;
hit_grey_value = grey as u8;
}
}
}
values.push(hit_grey_value);
}
}
values
}
fn main() {
let test_cases: Vec<Vec<[f32; 5]>> = vec![
vec![[0.0, 0.0, 0.0, 1.0, 128.0]],
vec![[0.0, -0.55, 0.0, 0.4, 100.0], [0.0, 0.55, 0.0, 0.4, 200.0]],
vec![
[0.75, 0.0, 0.0, 0.25, 64.0],
[0.53033, 0.375, -0.375, 0.25, 185.0],
[0.0, 0.53033, -0.53033, 0.25, 64.0],
[-0.53033, 0.375, -0.375, 0.25, 185.0],
[-0.75, 0.0, 0.0, 0.25, 64.0],
[-0.53033, -0.375, 0.375, 0.25, 185.0],
[0.0, -0.53033, 0.53033, 0.25, 64.0],
[0.53033, -0.375, 0.375, 0.25, 185.0],
],
vec![
[0.001, 0.0, 0.0, 1.0, 64.0],
[0.000707, 0.000707, 0.0, 1.0, 85.0],
[0.0, 0.001, 0.0, 1.0, 106.0],
[-0.000707, 0.000707, 0.0, 1.0, 127.0],
[-0.001, 0.0, 0.0, 1.0, 148.0],
[-0.000707, -0.000707, 0.0, 1.0, 169.0],
[0.0, -0.001, 0.0, 1.0, 190.0],
[0.000707, -0.000707, 0.0, 1.0, 211.0],
],
vec![
[0.0, 0.000058, 0.001947, 1.0, 0.0],
[0.0, -0.001715, 0.000923, 1.0, 0.0],
[0.0, 0.001715, -0.000923, 1.0, 0.0],
[0.0, -0.000058, -0.001947, 1.0, 0.0],
[0.001657, -0.000512, 0.000887, 1.0, 0.0],
[-0.001657, -0.000512, 0.000887, 1.0, 0.0],
[0.001657, 0.000512, -0.000887, 1.0, 0.0],
[-0.001657, 0.000512, -0.000887, 1.0, 0.0],
[0.001024, 0.001435, 0.000829, 1.0, 0.0],
[-0.001024, 0.001435, 0.000829, 1.0, 0.0],
[0.001024, -0.001435, -0.000829, 1.0, 0.0],
[-0.001024, -0.001435, -0.000829, 1.0, 0.0],
[0.001868, 0.000618, 0.000357, 1.0, 255.0],
[-0.001868, 0.000618, 0.000357, 1.0, 255.0],
[0.001868, -0.000618, -0.000357, 1.0, 255.0],
[-0.001868, -0.000618, -0.000357, 1.0, 255.0],
[0.000714, -0.000934, 0.001618, 1.0, 255.0],
[-0.000714, -0.000934, 0.001618, 1.0, 255.0],
[0.000714, 0.000934, -0.001618, 1.0, 255.0],
[-0.000714, 0.000934, -0.001618, 1.0, 255.0],
[0.0, -0.001975, -0.000316, 1.0, 255.0],
[0.0, 0.001261, 0.001552, 1.0, 255.0],
[0.0, -0.001261, -0.001552, 1.0, 255.0],
[0.0, 0.001975, 0.000316, 1.0, 255.0],
[0.001155, 0.000423, 0.001577, 1.0, 255.0],
[-0.001155, 0.000423, 0.001577, 1.0, 255.0],
[0.001155, -0.001577, 0.000423, 1.0, 255.0],
[-0.001155, -0.001577, 0.000423, 1.0, 255.0],
[0.001155, 0.001577, -0.000423, 1.0, 255.0],
[-0.001155, 0.001577, -0.000423, 1.0, 255.0],
[0.001155, -0.000423, -0.001577, 1.0, 255.0],
[-0.001155, -0.000423, -0.001577, 1.0, 255.0],
],
vec![
[0.0, 0.0, 1000.0, 2.0, 128.0],
[0.0, 0.0, 0.003125, 1.0, 0.0],
[0.0, 0.0, 0.0, 0.9975, 255.0],
[0.0, 0.000547, -0.002573, 0.995, 0.0],
[0.0, -0.000547, -0.002573, 0.995, 0.0],
[0.0, 0.001071, -0.005037, 0.9925, 255.0],
[0.0, -0.001071, -0.005037, 0.9925, 255.0],
],
vec![
[0.0, 0.0, 0.0, 1.0, 210.0],
[0.0, 0.0, 0.0008, 1.0, 210.0],
[0.0, 0.0, 0.0004, 1.00012, 50.0],
[0.012969, 0.0, -0.042045, 0.96, 100.0],
[0.016397, 0.011913, -0.039054, 0.96, 210.0],
[0.004008, 0.012334, -0.042045, 0.96, 100.0],
[-0.006263, 0.019276, -0.039054, 0.96, 210.0],
[-0.010492, 0.007623, -0.042045, 0.96, 100.0],
[-0.020268, 0.0, -0.039054, 0.96, 210.0],
[-0.010492, -0.007623, -0.042045, 0.96, 100.0],
[-0.006263, -0.019276, -0.039054, 0.96, 210.0],
[0.004008, -0.012334, -0.042045, 0.96, 100.0],
[0.016397, -0.011913, -0.039054, 0.96, 210.0],
],
vec![
[0.0, 0.0, 1000.0, 2.0, 192.0],
[-2.0, 2.0, 100.0, 3.0, 98.0],
[2.0, 2.0, 100.0, 3.0, 98.0],
[0.5, -0.5, 500.0, 0.2, 255.0],
[-0.349469, -0.1, 20.000279, 0.6, 64.0],
[-0.349775, -0.100273, 20.000485, 0.6, 85.0],
[-0.350214, -0.100386, 20.000407, 0.6, 106.0],
[-0.350527, -0.100273, 20.000091, 0.6, 127.0],
[-0.350531, -0.1, 19.999721, 0.6, 148.0],
[-0.350225, -0.099727, 19.999515, 0.6, 169.0],
[-0.349786, -0.099614, 19.999593, 0.6, 190.0],
[-0.349473, -0.099727, 19.999909, 0.6, 211.0],
[0.20032, 0.349824, 15.000456, 0.3, 0.0],
[0.20032, 0.349517, 14.999924, 0.3, 0.0],
[0.19968, 0.350483, 15.000076, 0.3, 0.0],
[0.19968, 0.350176, 14.999544, 0.3, 0.0],
[0.200578, 0.350073, 14.999958, 0.3, 0.0],
[0.199817, 0.349519, 15.000277, 0.3, 0.0],
[0.200183, 0.350481, 14.999723, 0.3, 0.0],
[0.199422, 0.349927, 15.000042, 0.3, 0.0],
[0.200235, 0.35042, 15.000332, 0.3, 0.0],
[0.199765, 0.350078, 15.000529, 0.3, 0.0],
[0.200235, 0.349922, 14.999471, 0.3, 0.0],
[0.199765, 0.34958, 14.999668, 0.3, 0.0],
[0.200429, 0.350419, 15.000005, 0.3, 255.0],
[0.199571, 0.349795, 15.000366, 0.3, 255.0],
[0.200429, 0.350205, 14.999634, 0.3, 255.0],
[0.199571, 0.349581, 14.999995, 0.3, 255.0],
[0.200524, 0.349747, 15.000146, 0.3, 255.0],
[0.200196, 0.349509, 15.000283, 0.3, 255.0],
[0.199804, 0.350491, 14.999717, 0.3, 255.0],
[0.199476, 0.350253, 14.999854, 0.3, 255.0],
[0.200138, 0.349578, 14.999597, 0.3, 255.0],
[0.200138, 0.350138, 15.000567, 0.3, 255.0],
[0.199862, 0.349862, 14.999433, 0.3, 255.0],
[0.199862, 0.350422, 15.000403, 0.3, 255.0],
[0.200488, 0.350136, 15.000321, 0.3, 255.0],
[0.199957, 0.349751, 15.000544, 0.3, 255.0],
[0.200488, 0.34979, 14.999721, 0.3, 255.0],
[0.199957, 0.349404, 14.999944, 0.3, 255.0],
[0.200043, 0.350596, 15.000056, 0.3, 255.0],
[0.199512, 0.35021, 15.000279, 0.3, 255.0],
[0.200043, 0.350249, 14.999456, 0.3, 255.0],
[0.199512, 0.349864, 14.999679, 0.3, 255.0],
[-0.5, 0.500442, 10.000442, 0.2, 0.0],
[-0.5, 0.5, 10.0, 0.1995, 255.0],
[-0.499966, 0.49971, 9.999563, 0.199, 0.0],
[-0.500034, 0.499563, 9.99971, 0.199, 0.0],
[-0.499934, 0.499432, 9.999144, 0.1985, 255.0],
[-0.500066, 0.499144, 9.999432, 0.1985, 255.0],
[0.6, 0.5, 10.0, 0.25, 210.0],
[0.600122, 0.4999, 10.000122, 0.25, 210.0],
[0.600061, 0.49995, 10.000061, 0.25003, 50.0],
[0.595856, 0.505256, 9.991271, 0.24, 100.0],
[0.597973, 0.507461, 9.992175, 0.24, 210.0],
[0.595362, 0.507926, 9.993945, 0.24, 100.0],
[0.594618, 0.509055, 9.996832, 0.24, 210.0],
[0.592382, 0.506906, 9.996092, 0.24, 100.0],
[0.590438, 0.504882, 9.997604, 0.24, 210.0],
[0.591035, 0.503605, 9.994744, 0.24, 100.0],
[0.59121, 0.500708, 9.993425, 0.24, 210.0],
[0.593181, 0.502585, 9.991764, 0.24, 100.0],
[0.595867, 0.502302, 9.990069, 0.24, 210.0],
],
];
let scene_names = [
"single_sphere",
"two_spheres",
"ring_of_spheres",
"beachball",
"football",
"eightball",
"starball",
"combination_scene",
];
for i in 0..8 {
eprintln!("Generating {}", scene_names[i]);
let values = grey_values(&test_cases[i]);
let filename = format!("{}.png", scene_names[i]);
let path = std::path::Path::new(&filename);
let file = std::fs::File::create(path).unwrap();
let w = &mut std::io::BufWriter::new(file);
let mut encoder = png::Encoder::new(w, 1024, 1024);
encoder.set_color(png::ColorType::Grayscale);
encoder.set_depth(png::BitDepth::Eight);
let mut writer = encoder.write_header().unwrap();
writer.write_image_data(&values).unwrap();
}
}
```
## Test cases
The images may be scaled down. Clicking or tapping on an image will open it full size.
### Single sphere
Input:
```text
TODO
```
Output:
```text
TODO
```
### Two separated spheres
Input:
```text
TODO
```
Output:
```text
TODO
```
### Eight spheres in a ring
Input:
```text
TODO
```
Output:
```text
TODO
```
### Beach ball
Input:
```text
TODO
```
Output:
```text
TODO
```
### Football
Input:
```text
TODO
```
Output:
```text
TODO
```
### Eight ball
Input:
```text
TODO
```
Output:
```text
TODO
```
### Star ball
Input:
```text
TODO
```
Output:
```text
TODO
```
### Combination scene
Input:
```text
TODO
```
Output:
```text
TODO
```
## Explanatory animations
The example images were produced from plain grey spheres with no surface pattern or texture. The following animations show how they break down into their component plain spheres.
This section is not necessary to write valid code for this challenge. If your code accepts the parameters of plain grey spheres and renders them correctly, all of the test cases will look right without needing to understand how they were constructed. This section is mainly out of interest, and to make clear that your code *only* needs to render plain grey spheres.
```text
CONSIDER HOSTING THESE OFF SITE AS THEY ARE NOT REQUIRED FOR THE CHALLENGE
AND EVEN THE SCALED DOWN VERSIONS ARE UNNECESSARY DATA USE FOR ANYONE
CHOOSING NOT TO VIEW THEM.
```
<details>
<summary>Animations</summary>
<details>
<summary>Beach ball</summary>
This animation is reduced to 256 by 256 pixels. Clicking or tapping on the animation will open it at full size (______MB).
```text
TODO
```
</details>
<details>
<summary>Football</summary>
This animation is reduced to 256 by 256 pixels. Clicking or tapping on the animation will open it at full size (______MB).
```text
TODO
```
</details>
<details>
<summary>Eight ball</summary>
This animation is reduced to 256 by 256 pixels. Clicking or tapping on the animation will open it at full size (______MB).
```text
TODO
```
</details>
<details>
<summary>Star ball</summary>
This animation is reduced to 256 by 256 pixels. Clicking or tapping on the animation will open it at full size (______MB).
```text
TODO
```
</details>
</details>
## Scoring
This is a [code golf challenge]. Your score is the number of bytes in your code. Lowest score for each language wins.
> Explanations are optional, but I'm more likely to upvote answers that have one.
[code golf challenge]: https://codegolf.codidact.com/categories/49/tags/4274 "The code-golf tag"
[PGM]: https://en.wikipedia.org/wiki/Netpbm#PGM_example

#1: Post edited by

trichoplax‭ · 2024-07-22T20:56:07Z (5 months ago)
Clarify orthographic

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Raw

Markdown

Given a list of spheres, output an image of them.
The challenge is to produce a greyscale [orthographic](https://en.wikipedia.org/wiki/Parallel_projection#Orthographic_projection) ray caster, that renders only plain grey spheres with no lighting, no shading, no shadows, no surface patterns or textures, no reflections, and no transparency.
## Input
- A list of spheres, each consisting of:
- The coordinates of its centre $(x, y, z)$.
- Its radius $r$.
- Its greyscale value $g$, an integer from $0$ to $255$ (both inclusive).
- The input may include up to $100$ spheres.
- Only the greyscale value will be restricted to integer values. The coordinates and radius may have fractional values.
- Here "integer" refers to the values the greyscale value make take. You are not required to use an integer variable type.[]()
## Output
- A single greyscale image:
- $1024$ by $1024$ pixels.
- Each pixel has a single greyscale value from $0$ to $255$.
- Horizontally, the image covers x values from $-1$ to $1$.
- Vertically, the image covers y values from $-1$ to $1$.
- The image is orthographic, so there is no perspective. Changing the $z$ coordinate of a sphere does not change its size in the image, instead only affecting whether it is in front of or behind other spheres.
- A sphere's greyscale value is applied to every visible part of it. There is no variation over the surface of the sphere, so it appears flat like a disk.
- The point of the challenge is producing the pixel values, not manipulating image file formats, so you may output whatever image format is convenient, including simple text formats such as [PGM] or even just a list of pixel greyscale values. If you want to include an example of your output as an image in your answer, the image formats that Codidact supports are PNG, JPEG, and GIF. You could output PGM and then convert to PNG to upload to your answer. If you output just the greyscale values as a list of numbers from $0$ to $255$, I've made a `[ TODO page to convert these to greyscale PNG]`.
- The test cases, which match the reference implementations, happen to have the x axis increasing from left to right, and the y axis increasing from top to bottom. You are free to instead interpret these as increasing from right to left for x or from bottom to top for y, resulting in a reflected image. You are also free to output an image rotated by a multiple of 90 degrees, allowing for swapping the x and y axes. However, the z axis is required to be increasing in the forwards direction (into the image) otherwise some of the test cases will look incorrect.
## Calculating a pixel value
Imagine a ray passing into the image through a pixel, perpendicular to the image plane. If it hits a sphere, it shows the greyscale value of the first sphere it hits. Otherwise, it shows a value of zero (black). One possible way of calculating this is as follows:
For each pixel:
- The pixel $x, y$ coordinates will each be from $0$ to $1023$. Scale these to the range $-1$ to $1$ (for example, divide by $512$ and subtract $1$).
- If these scaled $x, y$ coordinates are within the radius $r$ of the $x, y$ coordinates of a sphere's centre, the ray intersects that sphere (the $z$ coordinate is not relevant to this test because the image is an orthographic projection - the rays are all parallel to the $z$ axis). The distance $a$ of the ray from the sphere's centre can be calculated as follows:
$$a=\sqrt{(x_{ray}-x_{sphere})^2+(y_{ray}-y_{sphere})^2}$$
```text
CONSIDER A DIAGRAM HERE
```
- For each sphere that the ray intersects, find the $z$ coordinate of the nearest intersection (there will often be $2$ per sphere, $1$ on the front of the sphere and $1$ on the back of the sphere, as the ray passes through - only the nearest is relevant). This can be calculated as follows:
- The distance $d$ to the nearest intersection point can be calculated from the $z$ coordinate of the sphere's centre, the radius $r$, and the distance $a$ from the $x, y$ coordinates of the sphere's centre to the scaled $x, y$ coordinates of the pixel:
$$d=z_{sphere}-\sqrt{r_{sphere}^2-a^2}$$
```text
CONSIDER A DIAGRAM HERE
```
- Of all the spheres that the ray intersects, the sphere with the intersection point with the smallest $z$ coordinate $d$ is the one to display. Set the pixel to the greyscale value of this sphere.
- If the ray did not intersect any spheres, set the pixel to zero (black).
## Reference implementations
These are ungolfed implementations designed to supplement understanding of the requirements. You are free to golf these or to implement your own approach that matches the test cases up to reflection and rotation.
### Python
The function `greyscale_values` would be sufficient as an entry to this challenge. The rest of the code saves these as a PNG file. The function has no dependence on the imports so they would not be required in an entry.
```python
TODO
```
### Rust
The function `greyscale_values` would be sufficient as an entry to this challenge. The rest of the code saves these as a PNG file. The function has no dependence on the imports so they would not be required in an entry.
```rust
TODO
```
## Test cases
The images may be scaled down. Clicking or tapping on an image will open it full size.
### Single sphere
Input:
```text
TODO
```
Output:
```text
TODO
```
### Two separated spheres
Input:
```text
TODO
```
Output:
```text
TODO
```
### Eight spheres in a ring
Input:
```text
TODO
```
Output:
```text
TODO
```
### Beach ball
Input:
```text
TODO
```
Output:
```text
TODO
```
### Football
Input:
```text
TODO
```
Output:
```text
TODO
```
### Eight ball
Input:
```text
TODO
```
Output:
```text
TODO
```
### Star ball
Input:
```text
TODO
```
Output:
```text
TODO
```
### Combination scene
Input:
```text
TODO
```
Output:
```text
TODO
```
## Explanatory animations
The example images were produced from plain grey spheres with no surface pattern or texture. The following animations show how they break down into their component plain spheres.
This section is not necessary to write valid code for this challenge. If your code accepts the parameters of plain grey spheres and renders them correctly, all of the test cases will look right without needing to understand how they were constructed. This section is mainly out of interest, and to make clear that your code *only* needs to render plain grey spheres.
```text
CONSIDER HOSTING THESE OFF SITE AS THEY ARE NOT REQUIRED FOR THE CHALLENGE
AND EVEN THE SCALED DOWN VERSIONS ARE UNNECESSARY DATA USE FOR ANYONE
CHOOSING NOT TO VIEW THEM.
```
<details>
<summary>Animations</summary>
<details>
<summary>Beach ball</summary>
This animation is reduced to 256 by 256 pixels. Clicking or tapping on the animation will open it at full size (______MB).
```text
TODO
```
</details>
<details>
<summary>Football</summary>
This animation is reduced to 256 by 256 pixels. Clicking or tapping on the animation will open it at full size (______MB).
```text
TODO
```
</details>
<details>
<summary>Eight ball</summary>
This animation is reduced to 256 by 256 pixels. Clicking or tapping on the animation will open it at full size (______MB).
```text
TODO
```
</details>
<details>
<summary>Star ball</summary>
This animation is reduced to 256 by 256 pixels. Clicking or tapping on the animation will open it at full size (______MB).
```text
TODO
```
</details>
</details>
## Scoring
This is a [code golf challenge]. Your score is the number of bytes in your code. Lowest score for each language wins.
> Explanations are optional, but I'm more likely to upvote answers that have one.
[code golf challenge]: https://codegolf.codidact.com/categories/49/tags/4274 "The code-golf tag"
[PGM]: https://en.wikipedia.org/wiki/Netpbm#PGM_example

Given a list of spheres, output an image of them.
The challenge is to produce a greyscale [orthographic](https://en.wikipedia.org/wiki/Parallel_projection#Orthographic_projection) (parallel projection - no perspective) ray caster, that renders only plain grey spheres with no lighting, no shading, no shadows, no surface patterns or textures, no reflections, and no transparency.
## Input
- A list of spheres, each consisting of:
- The coordinates of its centre $(x, y, z)$.
- Its radius $r$.
- Its greyscale value $g$, an integer from $0$ to $255$ (both inclusive).
- The input may include up to $100$ spheres.
- Only the greyscale value will be restricted to integer values. The coordinates and radius may have fractional values.
- Here "integer" refers to the values the greyscale value make take. You are not required to use an integer variable type.[]()
## Output
- A single greyscale image:
- $1024$ by $1024$ pixels.
- Each pixel has a single greyscale value from $0$ to $255$.
- Horizontally, the image covers x values from $-1$ to $1$.
- Vertically, the image covers y values from $-1$ to $1$.
- The image is orthographic, so there is no perspective. Changing the $z$ coordinate of a sphere does not change its size in the image, instead only affecting whether it is in front of or behind other spheres.
- A sphere's greyscale value is applied to every visible part of it. There is no variation over the surface of the sphere, so it appears flat like a disk.
- The point of the challenge is producing the pixel values, not manipulating image file formats, so you may output whatever image format is convenient, including simple text formats such as [PGM] or even just a list of pixel greyscale values. If you want to include an example of your output as an image in your answer, the image formats that Codidact supports are PNG, JPEG, and GIF. You could output PGM and then convert to PNG to upload to your answer. If you output just the greyscale values as a list of numbers from $0$ to $255$, I've made a `[ TODO page to convert these to greyscale PNG]`.
- The test cases, which match the reference implementations, happen to have the x axis increasing from left to right, and the y axis increasing from top to bottom. You are free to instead interpret these as increasing from right to left for x or from bottom to top for y, resulting in a reflected image. You are also free to output an image rotated by a multiple of 90 degrees, allowing for swapping the x and y axes. However, the z axis is required to be increasing in the forwards direction (into the image) otherwise some of the test cases will look incorrect.
## Calculating a pixel value
Imagine a ray passing into the image through a pixel, perpendicular to the image plane. If it hits a sphere, it shows the greyscale value of the first sphere it hits. Otherwise, it shows a value of zero (black). One possible way of calculating this is as follows:
For each pixel:
- The pixel $x, y$ coordinates will each be from $0$ to $1023$. Scale these to the range $-1$ to $1$ (for example, divide by $512$ and subtract $1$).
- If these scaled $x, y$ coordinates are within the radius $r$ of the $x, y$ coordinates of a sphere's centre, the ray intersects that sphere (the $z$ coordinate is not relevant to this test because the image is an orthographic projection - the rays are all parallel to the $z$ axis). The distance $a$ of the ray from the sphere's centre can be calculated as follows:
$$a=\sqrt{(x_{ray}-x_{sphere})^2+(y_{ray}-y_{sphere})^2}$$
```text
CONSIDER A DIAGRAM HERE
```
- For each sphere that the ray intersects, find the $z$ coordinate of the nearest intersection (there will often be $2$ per sphere, $1$ on the front of the sphere and $1$ on the back of the sphere, as the ray passes through - only the nearest is relevant). This can be calculated as follows:
- The distance $d$ to the nearest intersection point can be calculated from the $z$ coordinate of the sphere's centre, the radius $r$, and the distance $a$ from the $x, y$ coordinates of the sphere's centre to the scaled $x, y$ coordinates of the pixel:
$$d=z_{sphere}-\sqrt{r_{sphere}^2-a^2}$$
```text
CONSIDER A DIAGRAM HERE
```
- Of all the spheres that the ray intersects, the sphere with the intersection point with the smallest $z$ coordinate $d$ is the one to display. Set the pixel to the greyscale value of this sphere.
- If the ray did not intersect any spheres, set the pixel to zero (black).
## Reference implementations
These are ungolfed implementations designed to supplement understanding of the requirements. You are free to golf these or to implement your own approach that matches the test cases up to reflection and rotation.
### Python
The function `greyscale_values` would be sufficient as an entry to this challenge. The rest of the code saves these as a PNG file. The function has no dependence on the imports so they would not be required in an entry.
```python
TODO
```
### Rust
The function `greyscale_values` would be sufficient as an entry to this challenge. The rest of the code saves these as a PNG file. The function has no dependence on the imports so they would not be required in an entry.
```rust
TODO
```
## Test cases
The images may be scaled down. Clicking or tapping on an image will open it full size.
### Single sphere
Input:
```text
TODO
```
Output:
```text
TODO
```
### Two separated spheres
Input:
```text
TODO
```
Output:
```text
TODO
```
### Eight spheres in a ring
Input:
```text
TODO
```
Output:
```text
TODO
```
### Beach ball
Input:
```text
TODO
```
Output:
```text
TODO
```
### Football
Input:
```text
TODO
```
Output:
```text
TODO
```
### Eight ball
Input:
```text
TODO
```
Output:
```text
TODO
```
### Star ball
Input:
```text
TODO
```
Output:
```text
TODO
```
### Combination scene
Input:
```text
TODO
```
Output:
```text
TODO
```
## Explanatory animations
The example images were produced from plain grey spheres with no surface pattern or texture. The following animations show how they break down into their component plain spheres.
This section is not necessary to write valid code for this challenge. If your code accepts the parameters of plain grey spheres and renders them correctly, all of the test cases will look right without needing to understand how they were constructed. This section is mainly out of interest, and to make clear that your code *only* needs to render plain grey spheres.
```text
CONSIDER HOSTING THESE OFF SITE AS THEY ARE NOT REQUIRED FOR THE CHALLENGE
AND EVEN THE SCALED DOWN VERSIONS ARE UNNECESSARY DATA USE FOR ANYONE
CHOOSING NOT TO VIEW THEM.
```
<details>
<summary>Animations</summary>
<details>
<summary>Beach ball</summary>
This animation is reduced to 256 by 256 pixels. Clicking or tapping on the animation will open it at full size (______MB).
```text
TODO
```
</details>
<details>
<summary>Football</summary>
This animation is reduced to 256 by 256 pixels. Clicking or tapping on the animation will open it at full size (______MB).
```text
TODO
```
</details>
<details>
<summary>Eight ball</summary>
This animation is reduced to 256 by 256 pixels. Clicking or tapping on the animation will open it at full size (______MB).
```text
TODO
```
</details>
<details>
<summary>Star ball</summary>
This animation is reduced to 256 by 256 pixels. Clicking or tapping on the animation will open it at full size (______MB).
```text
TODO
```
</details>
</details>
## Scoring
This is a [code golf challenge]. Your score is the number of bytes in your code. Lowest score for each language wins.
> Explanations are optional, but I'm more likely to upvote answers that have one.
[code golf challenge]: https://codegolf.codidact.com/categories/49/tags/4274 "The code-golf tag"
[PGM]: https://en.wikipedia.org/wiki/Netpbm#PGM_example

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