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Ray-caster: avoid roundoff error in quadratic equation

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This commit is contained in:
Aaron Fenyes 2024-09-08 23:00:28 -07:00
parent ab830b194e
commit 163361184b
1 changed files with 23 additions and 9 deletions
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32
app-proto/inversive-display/src/inversive.frag
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@ -110,23 +110,37 @@ taggedFrag sphere_shading(vecInv v, vec3 pt, vec3 base_color, int id) {
// --- ray-casting ---
// if `a/b` is less than this threshold, we approximate `a*u^2 + b*u + c` by
// the linear function `b*u + c`
const float DEG_THRESHOLD = 1e-9;
// the depths, represented as multiples of `dir`, where the line generated by
// `dir` hits the sphere represented by `v`. if both depths are positive, the
// smaller one is returned in the first component. if only one depth is
// positive, it could be returned in either component
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)
);
// as long as `b` is non-zero, the linear approximation of
//
// a*u^2 + b*u + c
//
// at `u = 0` will reach zero at a finite depth `u_lin`. the root of the
// quadratic adjacent to `u_lin` is stored in `lin_root`. if both roots
// have the same sign, `lin_root` will be the one closer to `u = 0`
float square_rect_ratio = 1. + sqrt(1. - adjust);
float lin_root = -(2.*c)/b / square_rect_ratio;
if (abs(a) > DEG_THRESHOLD * abs(b)) {
return vec2(lin_root, -b/(2.*a) * square_rect_ratio);
} else {
return vec2(lin_root, -1.);
}
} else {
// these parameters describe points behind the camera, so the
// corresponding fragments won't be drawn
// the line through `dir` misses the sphere completely
return vec2(-1., -1.);
}
}