// -*- related-file-name: "../include/lcdf/bezier.hh" -*- /* bezier.{cc,hh} -- cubic Bezier curves * * Copyright (c) 1998-2006 Eddie Kohler * * This program is free software; you can redistribute it and/or modify it * under the terms of the GNU General Public License as published by the Free * Software Foundation; either version 2 of the License, or (at your option) * any later version. This program is distributed in the hope that it will be * useful, but WITHOUT ANY WARRANTY; without even the implied warranty of * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General * Public License for more details. */ #ifdef HAVE_CONFIG_H # include #endif #include // // bounding box // void Bezier::make_bb() const throw () { _bb = 0; for (int i = 1; i < 4; i++) { if (_p[i].x > bb_right_x()) _bb = (_bb & ~0x03) | (i << 0); else if (_p[i].x < bb_left_x()) _bb = (_bb & ~0x0C) | (i << 2); if (_p[i].y > bb_top_x()) _bb = (_bb & ~0x30) | (i << 4); else if (_p[i].y < bb_bottom_x()) _bb = (_bb & ~0xC0) | (i << 6); } } // // is_flat, eval // bool Bezier::is_flat(double t) const throw () { return (_p[2].on_segment(_p[0], _p[3], t) && _p[1].on_segment(_p[0], _p[3], t)); } static Point eval_bezier(Point *b_in, int degree, double u) { assert(degree < 4); Point b[4]; for (int i = 0; i <= degree; i++) b[i] = b_in[i]; double m = 1.0 - u; for (int i = 1; i <= degree; i++) for (int j = 0; j <= degree - i; j++) b[j] = b[j]*m + b[j+1]*u; return b[0]; } Point Bezier::eval(double u) const throw () { Bezier b = *this; double m = 1.0 - u; for (int i = 1; i < 4; i++) for (int j = 0; j < 4 - i; j++) b._p[j] = m * b._p[j] + u * b._p[j+1]; return b._p[0]; } // // halve // void Bezier::halve(Bezier &l, Bezier &r) const throw () { Point half = Point::midpoint(_p[1], _p[2]); l._p[0] = _p[0]; l._p[1] = Point::midpoint(_p[0], _p[1]); l._p[2] = Point::midpoint(l._p[1], half); r._p[3] = _p[3]; r._p[2] = Point::midpoint(_p[2], _p[3]); r._p[1] = Point::midpoint(r._p[2], half); r._p[0] = l._p[3] = Point::midpoint(l._p[2], r._p[1]); } // // hit testing // bool Bezier::in_bb(const Point &p, double tolerance) const throw () { ensure_bb(); if (bb_right() + tolerance < p.x || bb_left() - tolerance > p.x || bb_top() + tolerance < p.y || bb_bottom() - tolerance > p.y) return false; else return true; } double Bezier::hit_recurse(const Point &p, double tolerance, double leftd, double rightd, double leftt, double rightt) const throw () { Bezier left, right; double middled, resultt; if (is_flat(tolerance)) { if (p.on_segment(_p[0], _p[3], tolerance)) return (leftt + rightt) / 2; else return -1; } if (leftd < tolerance * tolerance) return leftt; if (rightd < tolerance * tolerance) return rightt; if (!in_bb(p, tolerance)) return -1; halve(left, right); middled = (right._p[0] - p).squared_length(); resultt = left.hit_recurse (p, tolerance, leftd, middled, leftt, (leftt + rightt) / 2); if (resultt >= 0) return resultt; return right.hit_recurse (p, tolerance, middled, rightd, (leftt + rightt) / 2, rightt); } bool Bezier::hit(const Point &p, double tolerance) const throw () { double leftd = (_p[0] - p).squared_length(); double rightd = (_p[3] - p).squared_length(); double resultt = hit_recurse(p, tolerance, leftd, rightd, 0, 1); return resultt >= 0; } // // segmentize to list of points // // uses recursive subdivision // void Bezier::segmentize(Vector &v, bool first) const { if (is_flat(0.5)) { if (first) v.push_back(_p[0]); v.push_back(_p[3]); } else { Bezier left, right; halve(left, right); left.segmentize(v, first); right.segmentize(v, false); } } // // curve fitting // // code after Philip J. Schneider's algorithm described, with code, in the // first Graphics Gems // static void chord_length_parameterize(const Point *d, int nd, Vector &result) { assert(result.size() == 0); result.reserve(nd); result.push_back(0); for (int i = 1; i < nd; i++) result.push_back(result.back() + Point::distance(d[i-1], d[i])); double last_dist = result.back(); for (int i = 1; i < nd; i++) result[i] /= last_dist; } static inline double B0(double u) { double m = 1.0 - u; return m*m*m; } static inline double B1(double u) { double m = 1.0 - u; return 3*m*m*u; } static inline double B2(double u) { double m = 1.0 - u; return 3*m*u*u; } static inline double B3(double u) { return u*u*u; } static Bezier generate_bezier(const Point *d, int nd, const Vector ¶meters, const Point &left_tangent, const Point &right_tangent) { Point *a0 = new Point[nd]; Point *a1 = new Point[nd]; for (int i = 0; i < nd; i++) { a0[i] = left_tangent * B1(parameters[i]); a1[i] = right_tangent * B2(parameters[i]); } double c[2][2], x[2]; c[0][0] = c[0][1] = c[1][0] = c[1][1] = x[0] = x[1] = 0.0; int last = nd - 1; for (int i = 0; i < nd; i++) { c[0][0] += Point::dot(a0[i], a0[i]); c[0][1] += Point::dot(a0[i], a1[i]); c[1][1] += Point::dot(a1[i], a1[i]); Point tmp = d[i] - (d[0] * (B0(parameters[i]) + B1(parameters[i])) + d[last] * (B2(parameters[i]) + B3(parameters[i]))); x[0] += Point::dot(a0[i], tmp); x[1] += Point::dot(a1[i], tmp); } c[1][0] = c[0][1]; // compute determinants double det_c0_c1 = c[0][0]*c[1][1] - c[1][0]*c[0][1]; double det_c0_x = c[0][0]*x[1] - c[0][1]*x[0]; double det_x_c1 = x[0]*c[1][1] - x[1]*c[0][1]; // finally, derive alpha values if (det_c0_c1 == 0.0) det_c0_c1 = c[0][0]*c[1][1] * 10e-12; double alpha_l = det_x_c1 / det_c0_c1; double alpha_r = det_c0_x / det_c0_c1; // if alpha negative, use the Wu/Barsky heuristic if (alpha_l < 0.0 || alpha_r < 0.0) { double distance = Point::distance(d[0], d[last]) / 3; return Bezier(d[0], d[0] + left_tangent*distance, d[last] + right_tangent*distance, d[last]); } else return Bezier(d[0], d[0] + left_tangent*alpha_l, d[last] + right_tangent*alpha_r, d[last]); } static double newton_raphson_root_find(const Bezier &b, const Point &p, double u) { const Point *b_pts = b.points(); Point b_det[3]; for (int i = 0; i < 3; i++) b_det[i] = (b_pts[i+1] - b_pts[i]) * 3; Point b_det_det[2]; for (int i = 0; i < 2; i++) b_det_det[i] = (b_det[i+1] - b_det[i]) * 2; Point b_u = b.eval(u); Point b_det_u = eval_bezier(b_det, 2, u); Point b_det_det_u = eval_bezier(b_det_det, 1, u); double numerator = Point::dot(b_u - p, b_det_u); double denominator = Point::dot(b_det_u, b_det_u) + Point::dot(b_u - p, b_det_det_u); return u - numerator/denominator; } static void reparameterize(const Point *d, int nd, Vector ¶meters, const Bezier &b) { for (int i = 0; i < nd; i++) parameters[i] = newton_raphson_root_find(b, d[i], parameters[i]); } static double compute_max_error(const Point *d, int nd, const Bezier &b, const Vector ¶meters, int *split_point) { *split_point = nd/2; double max_dist = 0.0; for (int i = 1; i < nd - 1; i++) { double dist = (b.eval(parameters[i]) - d[i]).squared_length(); if (dist >= max_dist) { max_dist = dist; *split_point = i; } } return max_dist; } static void fit0(const Point *d, int nd, Point left_tangent, Point right_tangent, double error, Vector &result) { // Use a heuristic for small regions (only two points) if (nd == 2) { double dist = Point::distance(d[0], d[1]) / 3; result.push_back(Bezier(d[0], d[0] + dist*left_tangent, d[1] + dist*right_tangent, d[1])); return; } // Parameterize points and attempt to fit curve Vector parameters; chord_length_parameterize(d, nd, parameters); Bezier b = generate_bezier(d, nd, parameters, left_tangent, right_tangent); // find max error int split_point; double max_error = compute_max_error(d, nd, b, parameters, &split_point); if (max_error < error) { result.push_back(b); return; } // if error not too large, try iteration and reparameterization if (max_error < error*error) for (int i = 0; i < 4; i++) { reparameterize(d, nd, parameters, b); b = generate_bezier(d, nd, parameters, left_tangent, right_tangent); max_error = compute_max_error(d, nd, b, parameters, &split_point); if (max_error < error) { result.push_back(b); return; } } // fitting failed -- split at max error point and fit again Point center_tangent = ((d[split_point-1] - d[split_point+1])/2).normal(); fit0(d, split_point+1, left_tangent, center_tangent, error, result); fit0(d+split_point, nd-split_point, -center_tangent, right_tangent, error, result); } void Bezier::fit(const Vector &points, double error, Vector &result) { int npoints = points.size(); Point left_tangent = (points[1] - points[0]).normal(); Point right_tangent = (points[npoints-2] - points[npoints-1]).normal(); fit0(&points[0], npoints, left_tangent, right_tangent, error, result); }