1#define_import_path bevy_render::maths
2
3const PI: f32 = 3.141592653589793;      // π
4const PI_2: f32 = 6.283185307179586;    // 2π
5const HALF_PI: f32 = 1.57079632679;     // π/2
6const FRAC_PI_3: f32 = 1.0471975512;    // π/3
7const E: f32 = 2.718281828459045;       // exp(1)
8
9fn affine2_to_square(affine: mat3x2<f32>) -> mat3x3<f32> {
10    return mat3x3<f32>(
11        vec3<f32>(affine[0].xy, 0.0),
12        vec3<f32>(affine[1].xy, 0.0),
13        vec3<f32>(affine[2].xy, 1.0),
14    );
15}
16
17fn affine3_to_square(affine: mat3x4<f32>) -> mat4x4<f32> {
18    return transpose(mat4x4<f32>(
19        affine[0],
20        affine[1],
21        affine[2],
22        vec4<f32>(0.0, 0.0, 0.0, 1.0),
23    ));
24}
25
26fn mat2x4_f32_to_mat3x3_unpack(
27    a: mat2x4<f32>,
28    b: f32,
29) -> mat3x3<f32> {
30    return mat3x3<f32>(
31        a[0].xyz,
32        vec3<f32>(a[0].w, a[1].xy),
33        vec3<f32>(a[1].zw, b),
34    );
35}
36
37// Extracts the square portion of an affine matrix: i.e. discards the
38// translation.
39fn affine3_to_mat3x3(affine: mat4x3<f32>) -> mat3x3<f32> {
40    return mat3x3<f32>(affine[0].xyz, affine[1].xyz, affine[2].xyz);
41}
42
43// Returns the inverse of a 3x3 matrix.
44fn inverse_mat3x3(matrix: mat3x3<f32>) -> mat3x3<f32> {
45    let tmp0 = cross(matrix[1], matrix[2]);
46    let tmp1 = cross(matrix[2], matrix[0]);
47    let tmp2 = cross(matrix[0], matrix[1]);
48    let inv_det = 1.0 / dot(matrix[2], tmp2);
49    return transpose(mat3x3<f32>(tmp0 * inv_det, tmp1 * inv_det, tmp2 * inv_det));
50}
51
52// Returns the inverse of an affine matrix.
53//
54// https://en.wikipedia.org/wiki/Affine_transformation#Groups
55fn inverse_affine3(affine: mat4x3<f32>) -> mat4x3<f32> {
56    let matrix3 = affine3_to_mat3x3(affine);
57    let inv_matrix3 = inverse_mat3x3(matrix3);
58    return mat4x3<f32>(inv_matrix3[0], inv_matrix3[1], inv_matrix3[2], -(inv_matrix3 * affine[3]));
59}
60
61// Extracts the upper 3x3 portion of a 4x4 matrix.
62fn mat4x4_to_mat3x3(m: mat4x4<f32>) -> mat3x3<f32> {
63    return mat3x3<f32>(m[0].xyz, m[1].xyz, m[2].xyz);
64}
65
66// Creates an orthonormal basis given a Z vector and an up vector (which becomes
67// Y after orthonormalization).
68//
69// The results are equivalent to the Gram-Schmidt process [1].
70//
71// [1]: https://math.stackexchange.com/a/1849294
72fn orthonormalize(z_unnormalized: vec3<f32>, up: vec3<f32>) -> mat3x3<f32> {
73    let z_basis = normalize(z_unnormalized);
74    let x_basis = normalize(cross(z_basis, up));
75    let y_basis = cross(z_basis, x_basis);
76    return mat3x3(x_basis, y_basis, z_basis);
77}
78
79// Returns true if any part of a sphere is on the positive side of a plane.
80//
81// `sphere_center.w` should be 1.0.
82//
83// This is used for frustum culling.
84fn sphere_intersects_plane_half_space(
85    plane: vec4<f32>,
86    sphere_center: vec4<f32>,
87    sphere_radius: f32
88) -> bool {
89    return dot(plane, sphere_center) + sphere_radius > 0.0;
90}
91
92// pow() but safe for NaNs/negatives
93fn powsafe(color: vec3<f32>, power: f32) -> vec3<f32> {
94    return pow(abs(color), vec3(power)) * sign(color);
95}
96
97// https://en.wikipedia.org/wiki/Vector_projection#Vector_projection_2
98fn project_onto(lhs: vec3<f32>, rhs: vec3<f32>) -> vec3<f32> {
99    let other_len_sq_rcp = 1.0 / dot(rhs, rhs);
100    return rhs * dot(lhs, rhs) * other_len_sq_rcp;
101}
102
103// Below are fast approximations of common irrational and trig functions. These
104// are likely most useful when raymarching, for example, where complete numeric
105// accuracy can be sacrificed for greater sample count.
106
107fn fast_sqrt(x: f32) -> f32 {
108    let n = bitcast<f32>(0x1fbd1df5 + (bitcast<i32>(x) >> 1u));
109    // One Newton's method iteration for better precision
110    return 0.5 * (n + x / n);
111}
112
113// Slightly less accurate than fast_acos_4, but much simpler.
114fn fast_acos(in_x: f32) -> f32 {
115    let x = abs(in_x);
116    var res = -0.156583 * x + HALF_PI;
117    res *= fast_sqrt(1.0 - x);
118    return select(PI - res, res, in_x >= 0.0);
119}
120
121// 4th order polynomial approximation
122// 4 VGRP, 16 ALU Full Rate
123// 7 * 10^-5 radians precision
124// Reference : Handbook of Mathematical Functions (chapter : Elementary Transcendental Functions), M. Abramowitz and I.A. Stegun, Ed.
125fn fast_acos_4(x: f32) -> f32 {
126    let x1 = abs(x);
127    let x2 = x1 * x1;
128    let x3 = x2 * x1;
129    var s: f32;
130
131    s = -0.2121144 * x1 + 1.5707288;
132    s = 0.0742610 * x2 + s;
133    s = -0.0187293 * x3 + s;
134    s = fast_sqrt(1.0 - x1) * s;
135
136	// acos function mirroring
137    return select(PI - s, s, x >= 0.0);
138}
139
140fn fast_atan2(y: f32, x: f32) -> f32 {
141    var t0 = max(abs(x), abs(y));
142    var t1 = min(abs(x), abs(y));
143    var t3 = t1 / t0;
144    var t4 = t3 * t3;
145
146    t0 = 0.0872929;
147    t0 = t0 * t4 - 0.301895;
148    t0 = t0 * t4 + 1.0;
149    t3 = t0 * t3;
150
151    t3 = select(t3, (0.5 * PI) - t3, abs(y) > abs(x));
152    t3 = select(t3, PI - t3, x < 0);
153    t3 = select(-t3, t3, y > 0);
154
155    return t3;
156}