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#pragma once
#include "diffvg.h"
#include "edge_query.h"
#include "scene.h"
#include "shape.h"
#include "solve.h"
#include "vector.h"
#include <cassert>
struct ClosestPointPathInfo {
int base_point_id;
int point_id;
float t_root;
};
DEVICE
inline
bool closest_point(const Circle &circle, const Vector2f &pt,
Vector2f *result) {
*result = circle.center + circle.radius * normalize(pt - circle.center);
return false;
}
DEVICE
inline
bool closest_point(const Path &path, const BVHNode *bvh_nodes, const Vector2f &pt, float max_radius,
ClosestPointPathInfo *path_info,
Vector2f *result) {
auto min_dist = max_radius;
auto ret_pt = Vector2f{0, 0};
auto found = false;
auto num_segments = path.num_base_points;
constexpr auto max_bvh_size = 128;
int bvh_stack[max_bvh_size];
auto stack_size = 0;
bvh_stack[stack_size++] = 2 * num_segments - 2;
while (stack_size > 0) {
const BVHNode &node = bvh_nodes[bvh_stack[--stack_size]];
if (node.child1 < 0) {
// leaf
auto base_point_id = node.child0;
auto point_id = - node.child1 - 1;
assert(base_point_id < num_segments);
assert(point_id < path.num_points);
auto dist = 0.f;
auto closest_pt = Vector2f{0, 0};
auto t_root = 0.f;
if (path.num_control_points[base_point_id] == 0) {
// Straight line
auto i0 = point_id;
auto i1 = (point_id + 1) % path.num_points;
auto p0 = Vector2f{path.points[2 * i0], path.points[2 * i0 + 1]};
auto p1 = Vector2f{path.points[2 * i1], path.points[2 * i1 + 1]};
// project pt to line
auto t = dot(pt - p0, p1 - p0) / dot(p1 - p0, p1 - p0);
if (t < 0) {
dist = distance(p0, pt);
closest_pt = p0;
t_root = 0;
} else if (t > 1) {
dist = distance(p1, pt);
closest_pt = p1;
t_root = 1;
} else {
dist = distance(p0 + t * (p1 - p0), pt);
closest_pt = p0 + t * (p1 - p0);
t_root = t;
}
} else if (path.num_control_points[base_point_id] == 1) {
// Quadratic Bezier curve
auto i0 = point_id;
auto i1 = point_id + 1;
auto i2 = (point_id + 2) % path.num_points;
auto p0 = Vector2f{path.points[2 * i0], path.points[2 * i0 + 1]};
auto p1 = Vector2f{path.points[2 * i1], path.points[2 * i1 + 1]};
auto p2 = Vector2f{path.points[2 * i2], path.points[2 * i2 + 1]};
if (path.use_distance_approx) {
closest_pt = quadratic_closest_pt_approx(p0, p1, p2, pt, &t_root);
dist = distance(closest_pt, pt);
} else {
auto eval = [&](float t) -> Vector2f {
auto tt = 1 - t;
return (tt*tt)*p0 + (2*tt*t)*p1 + (t*t)*p2;
};
auto pt0 = eval(0);
auto pt1 = eval(1);
auto dist0 = distance(pt0, pt);
auto dist1 = distance(pt1, pt);
{
dist = dist0;
closest_pt = pt0;
t_root = 0;
}
if (dist1 < dist) {
dist = dist1;
closest_pt = pt1;
t_root = 1;
}
// The curve is (1-t)^2p0 + 2(1-t)tp1 + t^2p2
// = (p0-2p1+p2)t^2+(-2p0+2p1)t+p0 = q
// Want to solve (q - pt) dot q' = 0
// q' = (p0-2p1+p2)t + (-p0+p1)
// Expanding (p0-2p1+p2)^2 t^3 +
// 3(p0-2p1+p2)(-p0+p1) t^2 +
// (2(-p0+p1)^2+(p0-2p1+p2)(p0-pt))t +
// (-p0+p1)(p0-pt) = 0
auto A = sum((p0-2*p1+p2)*(p0-2*p1+p2));
auto B = sum(3*(p0-2*p1+p2)*(-p0+p1));
auto C = sum(2*(-p0+p1)*(-p0+p1)+(p0-2*p1+p2)*(p0-pt));
auto D = sum((-p0+p1)*(p0-pt));
float t[3];
int num_sol = solve_cubic(A, B, C, D, t);
for (int j = 0; j < num_sol; j++) {
if (t[j] >= 0 && t[j] <= 1) {
auto p = eval(t[j]);
auto distp = distance(p, pt);
if (distp < dist) {
dist = distp;
closest_pt = p;
t_root = t[j];
}
}
}
}
} else if (path.num_control_points[base_point_id] == 2) {
// Cubic Bezier curve
auto i0 = point_id;
auto i1 = point_id + 1;
auto i2 = point_id + 2;
auto i3 = (point_id + 3) % path.num_points;
auto p0 = Vector2f{path.points[2 * i0], path.points[2 * i0 + 1]};
auto p1 = Vector2f{path.points[2 * i1], path.points[2 * i1 + 1]};
auto p2 = Vector2f{path.points[2 * i2], path.points[2 * i2 + 1]};
auto p3 = Vector2f{path.points[2 * i3], path.points[2 * i3 + 1]};
auto eval = [&](float t) -> Vector2f {
auto tt = 1 - t;
return (tt*tt*tt)*p0 + (3*tt*tt*t)*p1 + (3*tt*t*t)*p2 + (t*t*t)*p3;
};
auto pt0 = eval(0);
auto pt1 = eval(1);
auto dist0 = distance(pt0, pt);
auto dist1 = distance(pt1, pt);
{
dist = dist0;
closest_pt = pt0;
t_root = 0;
}
if (dist1 < dist) {
dist = dist1;
closest_pt = pt1;
t_root = 1;
}
// The curve is (1 - t)^3 p0 + 3 * (1 - t)^2 t p1 + 3 * (1 - t) t^2 p2 + t^3 p3
// = (-p0+3p1-3p2+p3) t^3 + (3p0-6p1+3p2) t^2 + (-3p0+3p1) t + p0
// Want to solve (q - pt) dot q' = 0
// q' = 3*(-p0+3p1-3p2+p3)t^2 + 2*(3p0-6p1+3p2)t + (-3p0+3p1)
// Expanding
// 3*(-p0+3p1-3p2+p3)^2 t^5
// 5*(-p0+3p1-3p2+p3)(3p0-6p1+3p2) t^4
// 4*(-p0+3p1-3p2+p3)(-3p0+3p1) + 2*(3p0-6p1+3p2)^2 t^3
// 3*(3p0-6p1+3p2)(-3p0+3p1) + 3*(-p0+3p1-3p2+p3)(p0-pt) t^2
// (-3p0+3p1)^2+2(p0-pt)(3p0-6p1+3p2) t
// (p0-pt)(-3p0+3p1)
double A = 3*sum((-p0+3*p1-3*p2+p3)*(-p0+3*p1-3*p2+p3));
double B = 5*sum((-p0+3*p1-3*p2+p3)*(3*p0-6*p1+3*p2));
double C = 4*sum((-p0+3*p1-3*p2+p3)*(-3*p0+3*p1)) + 2*sum((3*p0-6*p1+3*p2)*(3*p0-6*p1+3*p2));
double D = 3*(sum((3*p0-6*p1+3*p2)*(-3*p0+3*p1)) + sum((-p0+3*p1-3*p2+p3)*(p0-pt)));
double E = sum((-3*p0+3*p1)*(-3*p0+3*p1)) + 2*sum((p0-pt)*(3*p0-6*p1+3*p2));
double F = sum((p0-pt)*(-3*p0+3*p1));
// normalize the polynomial
B /= A;
C /= A;
D /= A;
E /= A;
F /= A;
// Isolator Polynomials:
// https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.133.2233&rep=rep1&type=pdf
// x/5 + B/25
// /-----------------------------------------------------
// 5x^4 + 4B x^3 + 3C x^2 + 2D x + E / x^5 + B x^4 + C x^3 + D x^2 + E x + F
// x^5 + 4B/5 x^4 + 3C/5 x^3 + 2D/5 x^2 + E/5 x
// ----------------------------------------------------
// B/5 x^4 + 2C/5 x^3 + 3D/5 x^2 + 4E/5 x + F
// B/5 x^4 + 4B^2/25 x^3 + 3BC/25 x^2 + 2BD/25 x + BE/25
// ----------------------------------------------------
// (2C/5 - 4B^2/25)x^3 + (3D/5-3BC/25)x^2 + (4E/5-2BD/25) + (F-BE/25)
auto p1A = ((2 / 5.f) * C - (4 / 25.f) * B * B);
auto p1B = ((3 / 5.f) * D - (3 / 25.f) * B * C);
auto p1C = ((4 / 5.f) * E - (2 / 25.f) * B * D);
auto p1D = F - B * E / 25.f;
// auto q1A = 1 / 5.f;
// auto q1B = B / 25.f;
// x/5 + B/25 = 0
// x = -B/5
auto q_root = -B/5.f;
double p_roots[3];
int num_sol = solve_cubic(p1A, p1B, p1C, p1D, p_roots);
float intervals[4];
if (q_root >= 0 && q_root <= 1) {
intervals[0] = q_root;
}
for (int j = 0; j < num_sol; j++) {
intervals[j + 1] = p_roots[j];
}
auto num_intervals = 1 + num_sol;
// sort intervals
for (int j = 1; j < num_intervals; j++) {
for (int k = j; k > 0 && intervals[k - 1] > intervals[k]; k--) {
auto tmp = intervals[k];
intervals[k] = intervals[k - 1];
intervals[k - 1] = tmp;
}
}
auto eval_polynomial = [&] (double t) {
return t*t*t*t*t+
B*t*t*t*t+
C*t*t*t+
D*t*t+
E*t+
F;
};
auto eval_polynomial_deriv = [&] (double t) {
return 5*t*t*t*t+
4*B*t*t*t+
3*C*t*t+
2*D*t+
E;
};
auto lower_bound = 0.f;
for (int j = 0; j < num_intervals + 1; j++) {
if (j < num_intervals && intervals[j] < 0.f) {
continue;
}
auto upper_bound = j < num_intervals ?
min(intervals[j], 1.f) : 1.f;
auto lb = lower_bound;
auto ub = upper_bound;
auto lb_eval = eval_polynomial(lb);
auto ub_eval = eval_polynomial(ub);
if (lb_eval * ub_eval > 0) {
// Doesn't have root
continue;
}
if (lb_eval > ub_eval) {
swap_(lb, ub);
}
auto t = 0.5f * (lb + ub);
auto num_iter = 20;
for (int it = 0; it < num_iter; it++) {
if (!(t >= lb && t <= ub)) {
t = 0.5f * (lb + ub);
}
auto value = eval_polynomial(t);
if (fabs(value) < 1e-5f || it == num_iter - 1) {
break;
}
// The derivative may not be entirely accurate,
// but the bisection is going to handle this
if (value > 0.f) {
ub = t;
} else {
lb = t;
}
auto derivative = eval_polynomial_deriv(t);
t -= value / derivative;
}
auto p = eval(t);
auto distp = distance(p, pt);
if (distp < dist) {
dist = distp;
closest_pt = p;
t_root = t;
}
if (upper_bound >= 1.f) {
break;
}
lower_bound = upper_bound;
}
} else {
assert(false);
}
if (dist < min_dist) {
min_dist = dist;
ret_pt = closest_pt;
path_info->base_point_id = base_point_id;
path_info->point_id = point_id;
path_info->t_root = t_root;
found = true;
}
} else {
assert(node.child0 >= 0 && node.child1 >= 0);
const AABB &b0 = bvh_nodes[node.child0].box;
if (within_distance(b0, pt, min_dist)) {
bvh_stack[stack_size++] = node.child0;
}
const AABB &b1 = bvh_nodes[node.child1].box;
if (within_distance(b1, pt, min_dist)) {
bvh_stack[stack_size++] = node.child1;
}
assert(stack_size <= max_bvh_size);
}
}
if (found) {
assert(path_info->base_point_id < num_segments);
}
*result = ret_pt;
return found;
}
DEVICE
inline
bool closest_point(const Rect &rect, const Vector2f &pt,
Vector2f *result) {
auto min_dist = 0.f;
auto closest_pt = Vector2f{0, 0};
auto update = [&](const Vector2f &p0, const Vector2f &p1, bool first) {
// project pt to line
auto t = dot(pt - p0, p1 - p0) / dot(p1 - p0, p1 - p0);
if (t < 0) {
auto d = distance(p0, pt);
if (first || d < min_dist) {
min_dist = d;
closest_pt = p0;
}
} else if (t > 1) {
auto d = distance(p1, pt);
if (first || d < min_dist) {
min_dist = d;
closest_pt = p1;
}
} else {
auto p = p0 + t * (p1 - p0);
auto d = distance(p, pt);
if (first || d < min_dist) {
min_dist = d;
closest_pt = p0;
}
}
};
auto left_top = rect.p_min;
auto right_top = Vector2f{rect.p_max.x, rect.p_min.y};
auto left_bottom = Vector2f{rect.p_min.x, rect.p_max.y};
auto right_bottom = rect.p_max;
update(left_top, left_bottom, true);
update(left_top, right_top, false);
update(right_top, right_bottom, false);
update(left_bottom, right_bottom, false);
*result = closest_pt;
return true;
}
DEVICE
inline
bool closest_point(const Shape &shape, const BVHNode *bvh_nodes, const Vector2f &pt, float max_radius,
ClosestPointPathInfo *path_info,
Vector2f *result) {
switch (shape.type) {
case ShapeType::Circle:
return closest_point(*(const Circle *)shape.ptr, pt, result);
case ShapeType::Ellipse:
// https://www.geometrictools.com/Documentation/DistancePointEllipseEllipsoid.pdf
assert(false);
return false;
case ShapeType::Path:
return closest_point(*(const Path *)shape.ptr, bvh_nodes, pt, max_radius, path_info, result);
case ShapeType::Rect:
return closest_point(*(const Rect *)shape.ptr, pt, result);
}
assert(false);
return false;
}
DEVICE
inline
bool compute_distance(const SceneData &scene,
int shape_group_id,
const Vector2f &pt,
float max_radius,
int *min_shape_id,
Vector2f *closest_pt_,
ClosestPointPathInfo *path_info,
float *result) {
const ShapeGroup &shape_group = scene.shape_groups[shape_group_id];
// pt is in canvas space, transform it to shape's local space
auto local_pt = xform_pt(shape_group.canvas_to_shape, pt);
constexpr auto max_bvh_stack_size = 64;
int bvh_stack[max_bvh_stack_size];
auto stack_size = 0;
bvh_stack[stack_size++] = 2 * shape_group.num_shapes - 2;
const auto &bvh_nodes = scene.shape_groups_bvh_nodes[shape_group_id];
auto min_dist = max_radius;
auto found = false;
while (stack_size > 0) {
const BVHNode &node = bvh_nodes[bvh_stack[--stack_size]];
if (node.child1 < 0) {
// leaf
auto shape_id = node.child0;
const auto &shape = scene.shapes[shape_id];
ClosestPointPathInfo local_path_info{-1, -1};
auto local_closest_pt = Vector2f{0, 0};
if (closest_point(shape, scene.path_bvhs[shape_id], local_pt, max_radius, &local_path_info, &local_closest_pt)) {
auto closest_pt = xform_pt(shape_group.shape_to_canvas, local_closest_pt);
auto dist = distance(closest_pt, pt);
if (!found || dist < min_dist) {
found = true;
min_dist = dist;
if (min_shape_id != nullptr) {
*min_shape_id = shape_id;
}
if (closest_pt_ != nullptr) {
*closest_pt_ = closest_pt;
}
if (path_info != nullptr) {
*path_info = local_path_info;
}
}
}
} else {
assert(node.child0 >= 0 && node.child1 >= 0);
const AABB &b0 = bvh_nodes[node.child0].box;
if (inside(b0, local_pt, max_radius)) {
bvh_stack[stack_size++] = node.child0;
}
const AABB &b1 = bvh_nodes[node.child1].box;
if (inside(b1, local_pt, max_radius)) {
bvh_stack[stack_size++] = node.child1;
}
assert(stack_size <= max_bvh_stack_size);
}
}
*result = min_dist;
return found;
}
DEVICE
inline
void d_closest_point(const Circle &circle,
const Vector2f &pt,
const Vector2f &d_closest_pt,
Circle &d_circle,
Vector2f &d_pt) {
// return circle.center + circle.radius * normalize(pt - circle.center);
auto d_center = d_closest_pt *
(1 + d_normalize(pt - circle.center, circle.radius * d_closest_pt));
atomic_add(&d_circle.center.x, d_center);
atomic_add(&d_circle.radius, dot(d_closest_pt, normalize(pt - circle.center)));
}
DEVICE
inline
void d_closest_point(const Path &path,
const Vector2f &pt,
const Vector2f &d_closest_pt,
const ClosestPointPathInfo &path_info,
Path &d_path,
Vector2f &d_pt) {
auto base_point_id = path_info.base_point_id;
auto point_id = path_info.point_id;
auto min_t_root = path_info.t_root;
if (path.num_control_points[base_point_id] == 0) {
// Straight line
auto i0 = point_id;
auto i1 = (point_id + 1) % path.num_points;
auto p0 = Vector2f{path.points[2 * i0], path.points[2 * i0 + 1]};
auto p1 = Vector2f{path.points[2 * i1], path.points[2 * i1 + 1]};
// project pt to line
auto t = dot(pt - p0, p1 - p0) / dot(p1 - p0, p1 - p0);
auto d_p0 = Vector2f{0, 0};
auto d_p1 = Vector2f{0, 0};
if (t < 0) {
d_p0 += d_closest_pt;
} else if (t > 1) {
d_p1 += d_closest_pt;
} else {
auto d_p = d_closest_pt;
// p = p0 + t * (p1 - p0)
d_p0 += d_p * (1 - t);
d_p1 += d_p * t;
}
atomic_add(d_path.points + 2 * i0, d_p0);
atomic_add(d_path.points + 2 * i1, d_p1);
} else if (path.num_control_points[base_point_id] == 1) {
// Quadratic Bezier curve
auto i0 = point_id;
auto i1 = point_id + 1;
auto i2 = (point_id + 2) % path.num_points;
auto p0 = Vector2f{path.points[2 * i0], path.points[2 * i0 + 1]};
auto p1 = Vector2f{path.points[2 * i1], path.points[2 * i1 + 1]};
auto p2 = Vector2f{path.points[2 * i2], path.points[2 * i2 + 1]};
// auto eval = [&](float t) -> Vector2f {
// auto tt = 1 - t;
// return (tt*tt)*p0 + (2*tt*t)*p1 + (t*t)*p2;
// };
// auto dist0 = distance(eval(0), pt);
// auto dist1 = distance(eval(1), pt);
auto d_p0 = Vector2f{0, 0};
auto d_p1 = Vector2f{0, 0};
auto d_p2 = Vector2f{0, 0};
auto t = min_t_root;
if (t == 0) {
d_p0 += d_closest_pt;
} else if (t == 1) {
d_p2 += d_closest_pt;
} else {
// The curve is (1-t)^2p0 + 2(1-t)tp1 + t^2p2
// = (p0-2p1+p2)t^2+(-2p0+2p1)t+p0 = q
// Want to solve (q - pt) dot q' = 0
// q' = (p0-2p1+p2)t + (-p0+p1)
// Expanding (p0-2p1+p2)^2 t^3 +
// 3(p0-2p1+p2)(-p0+p1) t^2 +
// (2(-p0+p1)^2+(p0-2p1+p2)(p0-pt))t +
// (-p0+p1)(p0-pt) = 0
auto A = sum((p0-2*p1+p2)*(p0-2*p1+p2));
auto B = sum(3*(p0-2*p1+p2)*(-p0+p1));
auto C = sum(2*(-p0+p1)*(-p0+p1)+(p0-2*p1+p2)*(p0-pt));
// auto D = sum((-p0+p1)*(p0-pt));
auto d_p = d_closest_pt;
// p = eval(t)
auto tt = 1 - t;
// (tt*tt)*p0 + (2*tt*t)*p1 + (t*t)*p2
auto d_tt = 2 * tt * dot(d_p, p0) + 2 * t * dot(d_p, p1);
auto d_t = -d_tt + 2 * tt * dot(d_p, p1) + 2 * t * dot(d_p, p2);
auto d_p0 = d_p * tt * tt;
auto d_p1 = 2 * d_p * tt * t;
auto d_p2 = d_p * t * t;
// implicit function theorem: dt/dA = -1/(p'(t)) * dp/dA
auto poly_deriv_t = 3 * A * t * t + 2 * B * t + C;
if (fabs(poly_deriv_t) > 1e-6f) {
auto d_A = - (d_t / poly_deriv_t) * t * t * t;
auto d_B = - (d_t / poly_deriv_t) * t * t;
auto d_C = - (d_t / poly_deriv_t) * t;
auto d_D = - (d_t / poly_deriv_t);
// A = sum((p0-2*p1+p2)*(p0-2*p1+p2))
// B = sum(3*(p0-2*p1+p2)*(-p0+p1))
// C = sum(2*(-p0+p1)*(-p0+p1)+(p0-2*p1+p2)*(p0-pt))
// D = sum((-p0+p1)*(p0-pt))
d_p0 += 2*d_A*(p0-2*p1+p2)+
3*d_B*((-p0+p1)-(p0-2*p1+p2))+
2*d_C*(-2*(-p0+p1))+
d_C*((p0-pt)+(p0-2*p1+p2))+
2*d_D*(-(p0-pt)+(-p0+p1));
d_p1 += (-2)*2*d_A*(p0-2*p1+p2)+
3*d_B*(-2*(-p0+p1)+(p0-2*p1+p2))+
2*d_C*(2*(-p0+p1))+
d_C*((-2)*(p0-pt))+
d_D*(p0-pt);
d_p2 += 2*d_A*(p0-2*p1+p2)+
3*d_B*(-p0+p1)+
d_C*(p0-pt);
d_pt += d_C*(-(p0-2*p1+p2))+
d_D*(-(-p0+p1));
}
}
atomic_add(d_path.points + 2 * i0, d_p0);
atomic_add(d_path.points + 2 * i1, d_p1);
atomic_add(d_path.points + 2 * i2, d_p2);
} else if (path.num_control_points[base_point_id] == 2) {
// Cubic Bezier curve
auto i0 = point_id;
auto i1 = point_id + 1;
auto i2 = point_id + 2;
auto i3 = (point_id + 3) % path.num_points;
auto p0 = Vector2f{path.points[2 * i0], path.points[2 * i0 + 1]};
auto p1 = Vector2f{path.points[2 * i1], path.points[2 * i1 + 1]};
auto p2 = Vector2f{path.points[2 * i2], path.points[2 * i2 + 1]};
auto p3 = Vector2f{path.points[2 * i3], path.points[2 * i3 + 1]};
// auto eval = [&](float t) -> Vector2f {
// auto tt = 1 - t;
// return (tt*tt*tt)*p0 + (3*tt*tt*t)*p1 + (3*tt*t*t)*p2 + (t*t*t)*p3;
// };
auto d_p0 = Vector2f{0, 0};
auto d_p1 = Vector2f{0, 0};
auto d_p2 = Vector2f{0, 0};
auto d_p3 = Vector2f{0, 0};
auto t = min_t_root;
if (t == 0) {
// closest_pt = p0
d_p0 += d_closest_pt;
} else if (t == 1) {
// closest_pt = p1
d_p3 += d_closest_pt;
} else {
// The curve is (1 - t)^3 p0 + 3 * (1 - t)^2 t p1 + 3 * (1 - t) t^2 p2 + t^3 p3
// = (-p0+3p1-3p2+p3) t^3 + (3p0-6p1+3p2) t^2 + (-3p0+3p1) t + p0
// Want to solve (q - pt) dot q' = 0
// q' = 3*(-p0+3p1-3p2+p3)t^2 + 2*(3p0-6p1+3p2)t + (-3p0+3p1)
// Expanding
// 3*(-p0+3p1-3p2+p3)^2 t^5
// 5*(-p0+3p1-3p2+p3)(3p0-6p1+3p2) t^4
// 4*(-p0+3p1-3p2+p3)(-3p0+3p1) + 2*(3p0-6p1+3p2)^2 t^3
// 3*(3p0-6p1+3p2)(-3p0+3p1) + 3*(-p0+3p1-3p2+p3)(p0-pt) t^2
// (-3p0+3p1)^2+2(p0-pt)(3p0-6p1+3p2) t
// (p0-pt)(-3p0+3p1)
double A = 3*sum((-p0+3*p1-3*p2+p3)*(-p0+3*p1-3*p2+p3));
double B = 5*sum((-p0+3*p1-3*p2+p3)*(3*p0-6*p1+3*p2));
double C = 4*sum((-p0+3*p1-3*p2+p3)*(-3*p0+3*p1)) + 2*sum((3*p0-6*p1+3*p2)*(3*p0-6*p1+3*p2));
double D = 3*(sum((3*p0-6*p1+3*p2)*(-3*p0+3*p1)) + sum((-p0+3*p1-3*p2+p3)*(p0-pt)));
double E = sum((-3*p0+3*p1)*(-3*p0+3*p1)) + 2*sum((p0-pt)*(3*p0-6*p1+3*p2));
double F = sum((p0-pt)*(-3*p0+3*p1));
B /= A;
C /= A;
D /= A;
E /= A;
F /= A;
// auto eval_polynomial = [&] (double t) {
// return t*t*t*t*t+
// B*t*t*t*t+
// C*t*t*t+
// D*t*t+
// E*t+
// F;
// };
auto eval_polynomial_deriv = [&] (double t) {
return 5*t*t*t*t+
4*B*t*t*t+
3*C*t*t+
2*D*t+
E;
};
// auto p = eval(t);
auto d_p = d_closest_pt;
// (tt*tt*tt)*p0 + (3*tt*tt*t)*p1 + (3*tt*t*t)*p2 + (t*t*t)*p3
auto tt = 1 - t;
auto d_tt = 3 * tt * tt * dot(d_p, p0) +
6 * tt * t * dot(d_p, p1) +
3 * t * t * dot(d_p, p2);
auto d_t = -d_tt +
3 * tt * tt * dot(d_p, p1) +
6 * tt * t * dot(d_p, p2) +
3 * t * t * dot(d_p, p3);
d_p0 += d_p * (tt * tt * tt);
d_p1 += d_p * (3 * tt * tt * t);
d_p2 += d_p * (3 * tt * t * t);
d_p3 += d_p * (t * t * t);
// implicit function theorem: dt/dA = -1/(p'(t)) * dp/dA
auto poly_deriv_t = eval_polynomial_deriv(t);
if (fabs(poly_deriv_t) > 1e-10f) {
auto d_B = -(d_t / poly_deriv_t) * t * t * t * t;
auto d_C = -(d_t / poly_deriv_t) * t * t * t;
auto d_D = -(d_t / poly_deriv_t) * t * t;
auto d_E = -(d_t / poly_deriv_t) * t;
auto d_F = -(d_t / poly_deriv_t);
// B = B' / A
// C = C' / A
// D = D' / A
// E = E' / A
// F = F' / A
auto d_A = -d_B * B / A
-d_C * C / A
-d_D * D / A
-d_E * E / A
-d_F * F / A;
d_B /= A;
d_C /= A;
d_D /= A;
d_E /= A;
d_F /= A;
{
double A = 3*sum((-p0+3*p1-3*p2+p3)*(-p0+3*p1-3*p2+p3)) + 1e-3;
double B = 5*sum((-p0+3*p1-3*p2+p3)*(3*p0-6*p1+3*p2));
double C = 4*sum((-p0+3*p1-3*p2+p3)*(-3*p0+3*p1)) + 2*sum((3*p0-6*p1+3*p2)*(3*p0-6*p1+3*p2));
double D = 3*(sum((3*p0-6*p1+3*p2)*(-3*p0+3*p1)) + sum((-p0+3*p1-3*p2+p3)*(p0-pt)));
double E = sum((-3*p0+3*p1)*(-3*p0+3*p1)) + 2*sum((p0-pt)*(3*p0-6*p1+3*p2));
double F = sum((p0-pt)*(-3*p0+3*p1));
B /= A;
C /= A;
D /= A;
E /= A;
F /= A;
auto eval_polynomial = [&] (double t) {
return t*t*t*t*t+
B*t*t*t*t+
C*t*t*t+
D*t*t+
E*t+
F;
};
auto eval_polynomial_deriv = [&] (double t) {
return 5*t*t*t*t+
4*B*t*t*t+
3*C*t*t+
2*D*t+
E;
};
auto lb = t - 1e-2f;
auto ub = t + 1e-2f;
auto lb_eval = eval_polynomial(lb);
auto ub_eval = eval_polynomial(ub);
if (lb_eval > ub_eval) {
swap_(lb, ub);
}
auto t_ = 0.5f * (lb + ub);
auto num_iter = 20;
for (int it = 0; it < num_iter; it++) {
if (!(t_ >= lb && t_ <= ub)) {
t_ = 0.5f * (lb + ub);
}
auto value = eval_polynomial(t_);
if (fabs(value) < 1e-5f || it == num_iter - 1) {
break;
}
// The derivative may not be entirely accurate,
// but the bisection is going to handle this
if (value > 0.f) {
ub = t_;
} else {
lb = t_;
}
auto derivative = eval_polynomial_deriv(t);
t_ -= value / derivative;
}
}
// A = 3*sum((-p0+3*p1-3*p2+p3)*(-p0+3*p1-3*p2+p3))
d_p0 += d_A * 3 * (-1) * 2 * (-p0+3*p1-3*p2+p3);
d_p1 += d_A * 3 * 3 * 2 * (-p0+3*p1-3*p2+p3);
d_p2 += d_A * 3 * (-3) * 2 * (-p0+3*p1-3*p2+p3);
d_p3 += d_A * 3 * 1 * 2 * (-p0+3*p1-3*p2+p3);
// B = 5*sum((-p0+3*p1-3*p2+p3)*(3*p0-6*p1+3*p2))
d_p0 += d_B * 5 * ((-1) * (3*p0-6*p1+3*p2) + 3 * (-p0+3*p1-3*p2+p3));
d_p1 += d_B * 5 * (3 * (3*p0-6*p1+3*p2) + (-6) * (-p0+3*p1-3*p2+p3));
d_p2 += d_B * 5 * ((-3) * (3*p0-6*p1+3*p2) + 3 * (-p0+3*p1-3*p2+p3));
d_p3 += d_B * 5 * (3*p0-6*p1+3*p2);
// C = 4*sum((-p0+3*p1-3*p2+p3)*(-3*p0+3*p1)) + 2*sum((3*p0-6*p1+3*p2)*(3*p0-6*p1+3*p2))
d_p0 += d_C * 4 * ((-1) * (-3*p0+3*p1) + (-3) * (-p0+3*p1-3*p2+p3)) +
d_C * 2 * (3 * 2 * (3*p0-6*p1+3*p2));
d_p1 += d_C * 4 * (3 * (-3*p0+3*p1) + 3 * (-p0+3*p1-3*p2+p3)) +
d_C * 2 * ((-6) * 2 * (3*p0-6*p1+3*p2));
d_p2 += d_C * 4 * ((-3) * (-3*p0+3*p1)) +
d_C * 2 * (3 * 2 * (3*p0-6*p1+3*p2));
d_p3 += d_C * 4 * (-3*p0+3*p1);
// D = 3*(sum((3*p0-6*p1+3*p2)*(-3*p0+3*p1)) + sum((-p0+3*p1-3*p2+p3)*(p0-pt)))
d_p0 += d_D * 3 * (3 * (-3*p0+3*p1) + (-3) * (3*p0-6*p1+3*p2)) +
d_D * 3 * ((-1) * (p0-pt) + 1 * (-p0+3*p1-3*p2+p3));
d_p1 += d_D * 3 * ((-6) * (-3*p0+3*p1) + (3) * (3*p0-6*p1+3*p2)) +
d_D * 3 * (3 * (p0-pt));
d_p2 += d_D * 3 * (3 * (-3*p0+3*p1)) +
d_D * 3 * ((-3) * (p0-pt));
d_pt += d_D * 3 * ((-1) * (-p0+3*p1-3*p2+p3));
// E = sum((-3*p0+3*p1)*(-3*p0+3*p1)) + 2*sum((p0-pt)*(3*p0-6*p1+3*p2))
d_p0 += d_E * ((-3) * 2 * (-3*p0+3*p1)) +
d_E * 2 * (1 * (3*p0-6*p1+3*p2) + 3 * (p0-pt));
d_p1 += d_E * ( 3 * 2 * (-3*p0+3*p1)) +
d_E * 2 * ((-6) * (p0-pt));
d_p2 += d_E * 2 * ( 3 * (p0-pt));
d_pt += d_E * 2 * ((-1) * (3*p0-6*p1+3*p2));
// F = sum((p0-pt)*(-3*p0+3*p1))
d_p0 += d_F * (1 * (-3*p0+3*p1)) +
d_F * ((-3) * (p0-pt));
d_p1 += d_F * (3 * (p0-pt));
d_pt += d_F * ((-1) * (-3*p0+3*p1));
}
}
atomic_add(d_path.points + 2 * i0, d_p0);
atomic_add(d_path.points + 2 * i1, d_p1);
atomic_add(d_path.points + 2 * i2, d_p2);
atomic_add(d_path.points + 2 * i3, d_p3);
} else {
assert(false);
}
}
DEVICE
inline
void d_closest_point(const Rect &rect,
const Vector2f &pt,
const Vector2f &d_closest_pt,
Rect &d_rect,
Vector2f &d_pt) {
auto dist = [&](const Vector2f &p0, const Vector2f &p1) -> float {
// project pt to line
auto t = dot(pt - p0, p1 - p0) / dot(p1 - p0, p1 - p0);
if (t < 0) {
return distance(p0, pt);
} else if (t > 1) {
return distance(p1, pt);
} else {
return distance(p0 + t * (p1 - p0), pt);
}
// return 0;
};
auto left_top = rect.p_min;
auto right_top = Vector2f{rect.p_max.x, rect.p_min.y};
auto left_bottom = Vector2f{rect.p_min.x, rect.p_max.y};
auto right_bottom = rect.p_max;
auto left_dist = dist(left_top, left_bottom);
auto top_dist = dist(left_top, right_top);
auto right_dist = dist(right_top, right_bottom);
auto bottom_dist = dist(left_bottom, right_bottom);
int min_id = 0;
auto min_dist = left_dist;
if (top_dist < min_dist) { min_dist = top_dist; min_id = 1; }
if (right_dist < min_dist) { min_dist = right_dist; min_id = 2; }
if (bottom_dist < min_dist) { min_dist = bottom_dist; min_id = 3; }
auto d_update = [&](const Vector2f &p0, const Vector2f &p1,
const Vector2f &d_closest_pt,
Vector2f &d_p0, Vector2f &d_p1) {
// project pt to line
auto t = dot(pt - p0, p1 - p0) / dot(p1 - p0, p1 - p0);
if (t < 0) {
d_p0 += d_closest_pt;
} else if (t > 1) {
d_p1 += d_closest_pt;
} else {
// p = p0 + t * (p1 - p0)
auto d_p = d_closest_pt;
d_p0 += d_p * (1 - t);
d_p1 += d_p * t;
auto d_t = sum(d_p * (p1 - p0));
// t = dot(pt - p0, p1 - p0) / dot(p1 - p0, p1 - p0)
auto d_numerator = d_t / dot(p1 - p0, p1 - p0);
auto d_denominator = d_t * (-t) / dot(p1 - p0, p1 - p0);
// numerator = dot(pt - p0, p1 - p0)
d_pt += (p1 - p0) * d_numerator;
d_p1 += (pt - p0) * d_numerator;
d_p0 += ((p0 - p1) + (p0 - pt)) * d_numerator;
// denominator = dot(p1 - p0, p1 - p0)
d_p1 += 2 * (p1 - p0) * d_denominator;
d_p0 += 2 * (p0 - p1) * d_denominator;
}
};
auto d_left_top = Vector2f{0, 0};
auto d_right_top = Vector2f{0, 0};
auto d_left_bottom = Vector2f{0, 0};
auto d_right_bottom = Vector2f{0, 0};
if (min_id == 0) {
d_update(left_top, left_bottom, d_closest_pt, d_left_top, d_left_bottom);
} else if (min_id == 1) {
d_update(left_top, right_top, d_closest_pt, d_left_top, d_right_top);
} else if (min_id == 2) {
d_update(right_top, right_bottom, d_closest_pt, d_right_top, d_right_bottom);
} else {
assert(min_id == 3);
d_update(left_bottom, right_bottom, d_closest_pt, d_left_bottom, d_right_bottom);
}
auto d_p_min = Vector2f{0, 0};
auto d_p_max = Vector2f{0, 0};
// left_top = rect.p_min
// right_top = Vector2f{rect.p_max.x, rect.p_min.y}
// left_bottom = Vector2f{rect.p_min.x, rect.p_max.y}
// right_bottom = rect.p_max
d_p_min += d_left_top;
d_p_max.x += d_right_top.x;
d_p_min.y += d_right_top.y;
d_p_min.x += d_left_bottom.x;
d_p_max.y += d_left_bottom.y;
d_p_max += d_right_bottom;
atomic_add(d_rect.p_min, d_p_min);
atomic_add(d_rect.p_max, d_p_max);
}
DEVICE
inline
void d_closest_point(const Shape &shape,
const Vector2f &pt,
const Vector2f &d_closest_pt,
const ClosestPointPathInfo &path_info,
Shape &d_shape,
Vector2f &d_pt) {
switch (shape.type) {
case ShapeType::Circle:
d_closest_point(*(const Circle *)shape.ptr,
pt,
d_closest_pt,
*(Circle *)d_shape.ptr,
d_pt);
break;
case ShapeType::Ellipse:
// https://www.geometrictools.com/Documentation/DistancePointEllipseEllipsoid.pdf
assert(false);
break;
case ShapeType::Path:
d_closest_point(*(const Path *)shape.ptr,
pt,
d_closest_pt,
path_info,
*(Path *)d_shape.ptr,
d_pt);
break;
case ShapeType::Rect:
d_closest_point(*(const Rect *)shape.ptr,
pt,
d_closest_pt,
*(Rect *)d_shape.ptr,
d_pt);
break;
}
}
DEVICE
inline
void d_compute_distance(const Matrix3x3f &canvas_to_shape,
const Matrix3x3f &shape_to_canvas,
const Shape &shape,
const Vector2f &pt,
const Vector2f &closest_pt,
const ClosestPointPathInfo &path_info,
float d_dist,
Matrix3x3f &d_shape_to_canvas,
Shape &d_shape,
float *d_translation) {
if (distance_squared(pt, closest_pt) < 1e-10f) {
// The derivative at distance=0 is undefined
return;
}
assert(isfinite(d_dist));
// pt is in canvas space, transform it to shape's local space
auto local_pt = xform_pt(canvas_to_shape, pt);
auto local_closest_pt = xform_pt(canvas_to_shape, closest_pt);
// auto local_closest_pt = closest_point(shape, local_pt);
// auto closest_pt = xform_pt(shape_group.shape_to_canvas, local_closest_pt);
// auto dist = distance(closest_pt, pt);
auto d_pt = Vector2f{0, 0};
auto d_closest_pt = Vector2f{0, 0};
d_distance(closest_pt, pt, d_dist, d_closest_pt, d_pt);
assert(isfinite(d_pt));
assert(isfinite(d_closest_pt));
// auto closest_pt = xform_pt(shape_group.shape_to_canvas, local_closest_pt);
auto d_local_closest_pt = Vector2f{0, 0};
auto d_shape_to_canvas_ = Matrix3x3f();
d_xform_pt(shape_to_canvas, local_closest_pt, d_closest_pt,
d_shape_to_canvas_, d_local_closest_pt);
assert(isfinite(d_local_closest_pt));
auto d_local_pt = Vector2f{0, 0};
d_closest_point(shape, local_pt, d_local_closest_pt, path_info, d_shape, d_local_pt);
assert(isfinite(d_local_pt));
auto d_canvas_to_shape = Matrix3x3f();
d_xform_pt(canvas_to_shape,
pt,
d_local_pt,
d_canvas_to_shape,
d_pt);
// http://jack.valmadre.net/notes/2016/09/04/back-prop-differentials/#back-propagation-using-differentials
auto tc2s = transpose(canvas_to_shape);
d_shape_to_canvas_ += -tc2s * d_canvas_to_shape * tc2s;
atomic_add(&d_shape_to_canvas(0, 0), d_shape_to_canvas_);
if (d_translation != nullptr) {
atomic_add(d_translation, -d_pt);
}
}
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