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- vec2 xs_on_clean_parabola(vec3 b0, vec3 b1, vec3 b2){
- /*
- Given three control points for a quadratic bezier,
- this returns the two values (x0, x2) such that the
- section of the parabola y = x^2 between those values
- is isometric to the given quadratic bezier.
- Adapated from https://raphlinus.github.io/graphics/curves/2019/12/23/flatten-quadbez.html
- */
- vec3 dd = 2 * b1 - b0 - b2;
- float u0 = dot(b1 - b0, dd);
- float u2 = dot(b2 - b1, dd);
- float cp = length(cross(b2 - b0, dd));
- return vec2(u0 / cp, u2 / cp);
- }
- mat4 map_triangles(vec3 src0, vec3 src1, vec3 src2, vec3 dst0, vec3 dst1, vec3 dst2){
- /*
- Return an affine transform which maps the triangle (src0, src1, src2)
- onto the triangle (dst0, dst1, dst2)
- */
- mat4 src_mat = mat4(
- src0, 1.0,
- src1, 1.0,
- src2, 1.0,
- vec4(1.0)
- );
- mat4 dst_mat = mat4(
- dst0, 1.0,
- dst1, 1.0,
- dst2, 1.0,
- vec4(1.0)
- );
- return dst_mat * inverse(src_mat);
- }
- mat4 rotation(vec3 axis, float cos_angle){
- float c = cos_angle;
- float s = sqrt(1 - c * c); // Sine of the angle
- float oc = 1.0 - c;
- float ax = axis.x;
- float ay = axis.y;
- float az = axis.z;
- return mat4(
- oc * ax * ax + c, oc * ax * ay + az * s, oc * az * ax - ay * s, 0.0,
- oc * ax * ay - az * s, oc * ay * ay + c, oc * ay * az + ax * s, 0.0,
- oc * az * ax + ay * s, oc * ay * az - ax * s, oc * az * az + c, 0.0,
- 0.0, 0.0, 0.0, 1.0
- );
- }
- mat4 map_onto_x_axis(vec3 src0, vec3 src1){
- mat4 shift = mat4(1.0);
- shift[3].xyz = -src0;
- // Find rotation matrix between unit vectors in each direction
- vec3 vect = normalize(src1 - src0);
- // No rotation needed
- if(vect.x > 1 - 1e-6) return shift;
- // Equivalent to cross(vect, vec3(1, 0, 0))
- vec3 axis = normalize(vec3(0.0, vect.z, -vect.y));
- mat4 rotate = rotation(axis, vect.x);
- return rotate * shift;
- }
- mat4 get_xyz_to_uv(
- vec3 b0, vec3 b1, vec3 b2,
- float threshold,
- out bool exceeds_threshold
- ){
- /*
- Populates the matrix `result` with an affine transformation which maps a set of
- quadratic bezier controls points into a new coordinate system such that the bezier
- curve coincides with y = x^2.
- If the x-range under this part of the curve exceeds `threshold`, this returns false
- and populates result a matrix mapping b0 and b2 onto the x-axis
- */
- vec2 xs = xs_on_clean_parabola(b0, b1, b2);
- float x0 = xs[0];
- float x1 = 0.5 * (xs[0] + xs[1]);
- float x2 = xs[1];
- // Portions of the parabola y = x^2 where abs(x) exceeds
- // this value are treated as straight lines.
- exceeds_threshold = (min(x0, x2) > threshold || max(x0, x2) < -threshold);
- if(exceeds_threshold){
- return map_onto_x_axis(b0, b2);
- }
- // This triangle on the xy plane should be isometric
- // to (b0, b1, b2), and it should define a quadratic
- // bezier segment aligned with y = x^2
- vec3 dst0 = vec3(x0, x0 * x0, 0.0);
- vec3 dst1 = vec3(x1, x0 * x2, 0.0);
- vec3 dst2 = vec3(x2, x2 * x2, 0.0);
- return map_triangles(b0, b1, b2, dst0, dst1, dst2);
- }
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