/************************************************************************* ** Bezier.cpp ** ** ** ** This file is part of dvisvgm -- a fast DVI to SVG converter ** ** Copyright (C) 2005-2019 Martin Gieseking ** ** ** ** 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 3 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. ** ** ** ** You should have received a copy of the GNU General Public License ** ** along with this program; if not, see . ** *************************************************************************/ #include #include #include "Bezier.hpp" using namespace std; Bezier::Bezier () { _points[0] = _points[1] = _points[2] = _points[3] = DPair(0); } /** Creates a quadratic Bézier curve. internally, it's represented as a cubic one. */ Bezier::Bezier (const DPair &p0, const DPair &p1, const DPair &p2) { setPoints(p0, p0+(p1-p0)*2.0/3.0, p2+(p1-p2)*2.0/3.0, p2); } Bezier::Bezier (const DPair &p0, const DPair &p1, const DPair &p2, const DPair &p3) { setPoints(p0, p1, p2, p3); } /** Creates a subcurve of a given Bézier curve. * @param[in] source original curve to be clipped * @param[in] t0 'time' parameter \f$\in[0,1]\f$ of source curve where the subcurve starts * @param[in] t1 'time' parameter \f$\in[0,1]\f$ of source curve where the subcurve ends */ Bezier::Bezier (const Bezier &source, double t0, double t1) { if (t0 == t1) _points[0] = _points[1] = _points[2] = _points[3] = source.valueAt(t0); else { if (t0 > t1) swap(t0, t1); if (t0 == 0) source.subdivide(t1, this, nullptr); else if (t1 == 1) source.subdivide(t0, nullptr, this); else { Bezier subcurve; source.subdivide(t0, nullptr, &subcurve); subcurve.subdivide((t1-t0)/(1-t0), this, nullptr); } } } void Bezier::setPoints(const DPair &p0, const DPair &p1, const DPair &p2, const DPair &p3) { _points[0] = p0; _points[1] = p1; _points[2] = p2; _points[3] = p3; } void Bezier::reverse() { swap(_points[0], _points[3]); swap(_points[1], _points[2]); } DPair Bezier::valueAt (double t) const { const double s = 1-t; return _points[0]*s*s*s + _points[1]*3.0*s*s*t + _points[2]*3.0*s*t*t + _points[3]*t*t*t; } /** Returns a value of the Bézier curve's blossom representation. */ DPair Bezier::blossomValue (double u, double v, double w) const { const double uv = u*v; const double uw = u*w; const double vw = v*w; const double uvw = u*v*w; return _points[0]*(1.0-u-v-w+uv+uw+vw-uvw) +_points[1]*(u+v+w-2.0*(uv+uw+vw)+3.0*uvw) +_points[2]*(uv+uw+vw-3.0*uvw) +_points[3]*uvw; } /** Splits the curve at t into two sub-curves. */ void Bezier::subdivide (double t, Bezier *bezier1, Bezier *bezier2) const { const double s = 1-t; DPair p01 = _points[0]*s + _points[1]*t; DPair p12 = _points[1]*s + _points[2]*t; DPair p23 = _points[2]*s + _points[3]*t; DPair p012 = p01*s + p12*t; DPair p123 = p12*s + p23*t; DPair p0123 = p012*s + p123*t; if (bezier1) bezier1->setPoints(_points[0], p01, p012, p0123); if (bezier2) bezier2->setPoints(p0123, p123, p23, _points[3]); } /** Approximates the current Bézier curve by a sequence of line segments. * This is done by subdividing the curve several times using De Casteljau's algorithm. * If a sub-curve is almost flat, i.e. \f$\sum\limits_{k=0}^2 |p_{k+1}-p_k| - |p_3-p_0| < \delta\f$, * the curve is not further subdivided. * @param[in] delta threshold where to stop further subdivisions (see description above) * @param[out] p the resulting sequence of points defining the start/end points of the line segments * @param[out] t corresponding curve parameters of the approximated points p: \f$b(t_i)=p_i \f$ * @return number of points in vector p */ int Bezier::approximate (double delta, std::vector &p, vector *t) const { p.push_back(_points[0]); if (t) t->push_back(0); return approximate(delta, 0, 1, p, t); } int Bezier::approximate (double delta, double t0, double t1, vector &p, vector *t) const { // compute distance of adjacent control points const double l01 = (_points[1]-_points[0]).length(); const double l12 = (_points[2]-_points[1]).length(); const double l23 = (_points[3]-_points[2]).length(); const double l03 = (_points[3]-_points[0]).length(); if (l01+l12+l23-l03 < delta) { // is curve flat enough? p.push_back(_points[3]); // => store endpoint if (t) t->push_back(t1); } else { // subdivide curve at b(0.5) and approximate the resulting parts separately Bezier b1, b2; subdivide(0.5, &b1, &b2); double tmid = (t0+t1)/2; b1.approximate(delta, t0, tmid, p, t); b2.approximate(delta, tmid, t1, p, t); } return p.size(); } /** Returns the signed area of the triangle (p1, p2, p3). */ static inline double signed_area (const DPair &p1, const DPair &p2, const DPair &p3) { return ((p2.x()-p1.x())*(p3.y()-p1.y()) - (p3.x()-p1.x())*(p2.y()-p1.y()))/2.0; } static inline double dot_prod (const DPair &p1, const DPair &p2) { return p1.x()*p2.x() + p1.y()*p2.y(); } /** Returns true if p3 is located between p1 and p2, i.e. p3 lays almost on the line * between p1 and p2. */ static bool between (const DPair &p1, const DPair &p2, const DPair &p3, double delta) { double sqr_dist = dot_prod(p2-p1, p2-p1); double factor = sqr_dist == 0.0 ? 1.0 : sqr_dist; double area2 = fabs(signed_area(p1, p2, p3)); return area2*area2/factor < delta // does p3 lay almost on the line through p1 and p2... && min(p1.x(), p2.x()) <= p3.x() // ...and on or inside the rectangle spanned by p1 and p2? && max(p1.x(), p2.x()) >= p3.x() && min(p1.y(), p2.y()) <= p3.y() && max(p1.y(), p2.y()) >= p3.y(); } static inline bool near (const DPair &p1, const DPair &p2, double delta) { DPair diff = p2-p1; return fabs(diff.x()) < delta && fabs(diff.y()) < delta; } /** Tries to reduce the degree of the Bézier curve. This only works if the number of * control points can be reduces without changing the shape of the curve significantly. * @param[in] delta deviation tolerance * @param[in] p control points of the reduced curve * @return degree of the reduced curve */ int Bezier::reduceDegree (double delta, vector &p) const { p.clear(); if (near(_points[0], _points[1], delta) && near(_points[0], _points[2], delta) && near(_points[0], _points[3], delta)) p.push_back(_points[0]); else if (between(_points[0], _points[3], _points[1], delta) && between(_points[0], _points[3], _points[2], delta)) { p.push_back(_points[0]); p.push_back(_points[3]); } else if (near((_points[1]-_points[0])*1.5+_points[0], (_points[2]-_points[3])*1.5+_points[3], delta)) { p.push_back(_points[0]); p.push_back((_points[1]-_points[0])*1.5 + _points[0]); p.push_back(_points[3]); } else { p.resize(4); for (int i=0; i < 4; i++) p[i] = _points[i]; } return p.size()-1; } /** Try to solve the quadratic equation ax^2 + bx + c = 0. */ static bool solve_quadratic_equation (double a, double b, double c, double &x1, double &x2) { if (a == 0) { if (b == 0) return false; x1 = x2 = -c/b; } else { double discr = b*b - 4*a*c; if (discr < 0) return false; double p = -b/a/2; double r = sqrt(discr)/a/2; x1 = p+r; x2 = p-r; } return true; } /** Returns a tight bounding box parallel to the x- and y-axis. */ void Bezier::getBBox (BoundingBox &bbox) const { bbox.invalidate(); // coefficients of the derivative DPair pa = _points[3] - _points[2]*3.0 + _points[1]*3.0 - _points[0]; DPair pb = (_points[2]-_points[1]*2.0+_points[0])*2.0; DPair pc = _points[1]-_points[0]; // compute extrema for t > 0 and t < 1 double t1, t2; if (solve_quadratic_equation(pa.x(), pb.x(), pc.x(), t1, t2)) { if (t1 > 0.001 && t1 < 0.999) bbox.embed(valueAt(t1)); if (t1 != t2 && t2 > 0.001 && t2 < 0.999) bbox.embed(valueAt(t2)); } if (solve_quadratic_equation(pa.y(), pb.y(), pc.y(), t1, t2)) { if (t1 > 0.001 && t1 < 0.999) bbox.embed(valueAt(t1)); if (t1 != t2 && t2 > 0.001 && t2 < 0.999) bbox.embed(valueAt(t2)); } bbox.embed(_points[0]); bbox.embed(_points[3]); }