Comments on Plain spheres

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Plain spheres

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Given a list of spheres, output an image.

Produce a greyscale orthographic (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 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 $1024$.






       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:

$$a=\sqrt{(x_{ray}-x_{sphere})^2+(y_{ray}-y_{sphere})^2}$$

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:

$$d=z_{sphere}-\sqrt{r_{sphere}^2-a^2}$$

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.






    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.

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.

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.

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:

[
    [0, 0, 0, 1, 128],
]

Output:

Two separated spheres

Input:

[
    [0, -0.55, 0, 0.4, 100],
    [0, 0.55, 0, 0.4, 200],
]

Output:

Ring of spheres

Input:

[
    [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:

Beach ball

Input:

[
    [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:

Football

Input:

[
    [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:

Eight ball

Input:

[
    [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:

Star ball

Input:

[
    [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:

Combination scene

Input:

[
    [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:

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:

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:

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.

Animations

TODO

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.

posted 5 months ago

CC BY-SA 4.0

2mo ago

trichoplax‭

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5 comment threads

What should happen when two spheres are hit at the same distance? (5 comments)

What is he range of z values? (2 comments)

Loosen up the colour inputs (2 comments)

Input mechanics? (6 comments)

Can we assume the existence of a graphics library? (3 comments)

trichoplax‭ wrote 2 months ago

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Good question.

Short answer:
I should make explicit that the code can choose either. I will edit accordingly.

Waffle:
The only case where I'd expect this to make a noticeable difference to the resulting image is when two spheres are identical in both centre and radius, which could cause only one of their colours to show, or a mottling of both, depending on how this decision is made. In practice, this doesn't matter because none of the test cases have such conditions.

In all other cases, the only places where the surfaces of two spheres will coincide is a point or circle of intersection, which will be infinitely thin in theory, and mostly thinner than a pixel in practice. I imagine a worst case could be constructed with two spheres very close to having identical centre and radius, so that the region where they show as being hit at the same distance according to the floating point precision is much larger, but I've designed the test cases to avoid this.

trichoplax‭ wrote 2 months ago

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When 2 spheres A and B overlap, there are 3 types of overlapping points:

The surface of A is inside the surface of B (so the nearest intersection will be B).
The surface of B is inside the surface of A (so the nearest intersection will be A).
The 2 surfaces meet along a circle (infinitely thin) where the intersection with A will be just as near as the intersection with B. The rules now specify that these points can show either colour - that of A or B. In most cases (and all of the test cases) the difference will be a single pixel wide.

Communities

Comments on Plain ​​spheres

Plain ​​spheres

Input

Output

Calculating a pixel value

Example implementations

Test cases

Single sphere

Two separated spheres

Ring of spheres

Beach ball

Football

Eight ball

Star ball

Combination scene

Accuracy

Explanatory animations

Scoring

5 comment threads

Comments on Plain spheres

Plain spheres