dyna3/app-proto/inversive-display/src/inversive.frag

#version 300 es

precision highp float;

out vec4 outColor;

// --- inversive geometry ---

struct vecInv {
    vec3 sp;
    vec2 lt;
};

vecInv sphere(vec3 center, float radius) {
    return vecInv(
        center / radius,
        vec2(
            0.5 / radius,
            0.5 * (dot(center, center) / radius - radius)
        )
    );
}

// --- uniforms ---

// construction. the SPHERE_MAX array size seems to affect frame rate a lot,
// even though we should only be using the first few elements of each array
const int SPHERE_MAX_UNIFORM = 12;
uniform int sphere_cnt;
uniform vecInv sphere_list[SPHERE_MAX_UNIFORM];
uniform vec3 color_list[SPHERE_MAX_UNIFORM];

// view
uniform vec2 resolution;
uniform float shortdim;

// controls
uniform vec2 ctrl;
uniform vec2 radius;
uniform float opacity;
uniform float highlight;
uniform int layer_threshold;
uniform bool debug_mode;

// light and camera
const float focal_slope = 0.3;
const vec3 light_dir = normalize(vec3(2., 2., 1.));
const float ixn_threshold = 0.005;

// --- sRGB ---

// map colors from RGB space to sRGB space, as specified in the sRGB standard
// (IEC 61966-2-1:1999)
//
//   https://www.color.org/sRGB.pdf
//   https://www.color.org/chardata/rgb/srgb.xalter
//
// in RGB space, color value is proportional to light intensity, so linear
// color-vector interpolation corresponds to physical light mixing. in sRGB
// space, the color encoding used by many monitors, we use more of the value
// interval to represent low intensities, and less of the interval to represent
// high intensities. this improves color quantization

float sRGB(float t) {
    if (t <= 0.0031308) {
        return 12.92*t;
    } else {
        return 1.055*pow(t, 5./12.) - 0.055;
    }
}

vec3 sRGB(vec3 color) {
    return vec3(sRGB(color.r), sRGB(color.g), sRGB(color.b));
}

// --- shading ---

struct taggedFrag {
    int id;
    vec4 color;
    vec3 pt;
    vec3 normal;
};

taggedFrag[2] sort(taggedFrag a, taggedFrag b) {
    taggedFrag[2] result;
    if (a.pt.z > b.pt.z) {
        result[0] = a;
        result[1] = b;
    } else {
        result[0] = b;
        result[1] = a;
    }
    return result;
}

taggedFrag sphere_shading(vecInv v, vec3 pt, vec3 base_color, int id) {
    // the expression for normal needs to be checked. it's supposed to give the
    // negative gradient of the lorentz product between the impact point vector
    // and the sphere vector with respect to the coordinates of the impact
    // point. i calculated it in my head and decided that the result looked good
    // enough for now
    vec3 normal = normalize(-v.sp + 2.*v.lt.s*pt);
    
    float incidence = dot(normal, light_dir);
    float illum = mix(0.4, 1.0, max(incidence, 0.0));
    return taggedFrag(id, vec4(illum * base_color, opacity), pt, normal);
}

// --- ray-casting ---

vec2 sphere_cast(vecInv v, vec3 dir) {
    float a = -v.lt.s * dot(dir, dir);
    float b = dot(v.sp, dir);
    float c = -v.lt.t;
    
    float scale = -b/(2.*a);
    float adjust = 4.*a*c/(b*b);
    
    if (adjust < 1.) {
        float offset = sqrt(1. - adjust);
        return vec2(
            scale * (1. - offset),
            scale * (1. + offset)
        );
    } else {
        // these parameters describe points behind the camera, so the
        // corresponding fragments won't be drawn
        return vec2(-1., -1.);
    }
}

void main() {
    vec2 scr = (2.*gl_FragCoord.xy - resolution) / shortdim;
    vec3 dir = vec3(focal_slope * scr, -1.);
    
    // cast rays through the spheres
    const int SPHERE_MAX_INTERNAL = 6;
    taggedFrag frags [2*SPHERE_MAX_INTERNAL];
    int layer_cnt = 0;
    for (int id = 0; id < sphere_cnt; ++id) {
        // find out where the ray hits the sphere
        vec2 hit_depths = sphere_cast(sphere_list[id], dir);
        
        // insertion-sort the fragments we hit into the fragment list
        for (int side = 0; side < 2; ++side) {
            if (hit_depths[side] > 0.) {
                for (int layer = layer_cnt; layer >= 0; --layer) {
                    if (layer < 1 || frags[layer-1].pt.z >= -hit_depths[side]) {
                        // we're not as close to the screen as the fragment
                        // before the empty slot, so insert here
                        frags[layer] = sphere_shading(
                            sphere_list[id],
                            hit_depths[side] * dir,
                            color_list[id],
                            id
                        );
                        break;
                    } else {
                        // we're closer to the screen than the fragment before
                        // the empty slot, so move that fragment into the empty
                        // slot
                        frags[layer] = frags[layer-1];
                    }
                }
                ++layer_cnt;
            }
        }
    }
    
    /* DEBUG */
    // in debug mode, show the layer count instead of the shaded image
    if (debug_mode) {
        // at the bottom of the screen, show the color scale instead of the
        // layer count
        if (gl_FragCoord.y < 10.) layer_cnt = int(8. * gl_FragCoord.x / resolution.x);
        
        // convert number to color
        ivec3 bits = layer_cnt / ivec3(1, 2, 4);
        outColor = vec4(mod(vec3(bits), 2.), 1.);
        return;
    }
    
    // highlight intersections and cusps
    for (int i = layer_cnt-1; i >= 1; --i) {
        // intersections
        taggedFrag frag0 = frags[i];
        taggedFrag frag1 = frags[i-1];
        float ixn_sin = length(cross(frag0.normal, frag1.normal));
        vec3 disp = frag0.pt - frag1.pt;
        float ixn_dist = max(
            abs(dot(frag1.normal, disp)),
            abs(dot(frag0.normal, disp))
        ) / ixn_sin;
        float ixn_highlight = 0.5 * highlight * (1. - smoothstep(2./3.*ixn_threshold, 1.5*ixn_threshold, ixn_dist));
        frags[i].color = mix(frags[i].color, vec4(1.), ixn_highlight);
        frags[i-1].color = mix(frags[i-1].color, vec4(1.), ixn_highlight);
        
        // cusps
        float cusp_cos = abs(dot(dir, frag0.normal));
        float cusp_threshold = 2.*sqrt(ixn_threshold * sphere_list[frag0.id].lt.s);
        float cusp_highlight = highlight * (1. - smoothstep(2./3.*cusp_threshold, 1.5*cusp_threshold, cusp_cos));
        frags[i].color = mix(frags[i].color, vec4(1.), cusp_highlight);
    }
    
    // composite the sphere fragments
    vec3 color = vec3(0.);
    for (int i = layer_cnt-1; i >= layer_threshold; --i) {
        if (frags[i].pt.z < 0.) {
            vec4 frag_color = frags[i].color;
            color = mix(color, frag_color.rgb, frag_color.a);
        }
    }
    outColor = vec4(sRGB(color), 1.);
}