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