/************************************************************************* ** Bezier.cpp ** ** ** ** This file is part of dvisvgm -- a fast DVI to SVG converter ** ** Copyright (C) 2005-2024 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" #include "Matrix.hpp" #include "utility.hpp" using namespace std; QuadBezier::QuadBezier () { _points[0] = _points[1] = _points[2] = DPair(0, 0); } QuadBezier::QuadBezier (const DPair &p0, const DPair &p1, const DPair &p2) { setPoints(p0, p1, p2); } void QuadBezier::setPoints(const DPair &p0, const DPair &p1, const DPair &p2) { _points[0] = p0; _points[1] = p1; _points[2] = p2; } /** Returns the value (curve point) at t. */ DPair QuadBezier::valueAt (double t) const { const double s = 1-t; return _points[0]*s*s + _points[1]*2.0*s*t + _points[2]*t*t; } /** Returns the value of the first derivative of the curve at t. */ DPair QuadBezier::derivativeAt (double t) const { return _points[0]*(2*t-2) + _points[1]*(2-4*t) + _points[2]*(2*t); } /** Returns the arc length of the curve from 0 to t. */ double QuadBezier::arclen (double t) const { return math::integral(0, t, 20, [this](double t) -> double { DPair deriv = derivativeAt(t); return sqrt(deriv.x()*deriv.x() + deriv.y()*deriv.y()); }); } //////////////////////////////////////////////////////////////////////////////// CubicBezier::CubicBezier () { _points[0] = _points[1] = _points[2] = _points[3] = DPair(0, 0); } CubicBezier::CubicBezier (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 */ CubicBezier::CubicBezier (const CubicBezier &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 { CubicBezier subcurve; source.subdivide(t0, nullptr, &subcurve); subcurve.subdivide((t1-t0)/(1-t0), this, nullptr); } } } /** Creates a cubic Bézier from a quadratic one. */ CubicBezier::CubicBezier (const QuadBezier &qbezier) { const DPair &p0 = qbezier.point(0); const DPair &p1 = qbezier.point(1); const DPair &p2 = qbezier.point(2); setPoints(p0, p0+(p1-p0)*2.0/3.0, p2+(p1-p2)*2.0/3.0, p2); } void CubicBezier::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 CubicBezier::reverse() { swap(_points[0], _points[3]); swap(_points[1], _points[2]); } DPair CubicBezier::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 CubicBezier::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 CubicBezier::subdivide (double t, CubicBezier *bezier1, CubicBezier *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 CubicBezier::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 CubicBezier::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 CubicBezier 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 static_cast(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 = abs(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 abs(diff.x()) < delta && abs(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 CubicBezier::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 static_cast(p.size()-1); } /** Approximates the cubic Bézier curve by a sequence of quadratic ones. * @param[in] precision specifies the precision of the approximation * @return map containing the split parameters t_n together with the qudratic curves */ vector CubicBezier::toQuadBeziers (double precision, vector *splitParams) const { vector qbeziers; toQuadBeziers(0, 1, precision, qbeziers, splitParams); return qbeziers; } /** Returns the "mid-point approximation" of this cubic Bézier. */ QuadBezier CubicBezier::midpointApproximation () const { DPair p0 = (_points[1]*3.0 - _points[0])/2.0; DPair p1 = (_points[2]*3.0 - _points[3])/2.0; return {_points[0], (p0+p1)/2.0, _points[3]}; } /** Approximates a segment of a cubic Bézier curve by a sequence of quadratic curves. * The quadratic segments are computed by adaptive subdivision of the cubic curve * as described at http://www.caffeineowl.com/graphics/2d/vectorial/cubic2quad01.html * @param[in] t0 curve parameter of the segment's start point * @param[in] t1 curve parameter of the segment's end point * @param[in] precision maximum allowed distance between the curve points b(t) and the * corresponding approximated point c(t) on the quadratic curve * @param[out] qbeziers the resulting quadratic curves sorted in ascending order by their split points * @param[out] startParams the start parameters t[k] of the curves qbeziers[k] relative to *this. */ void CubicBezier::toQuadBeziers (double t0, double t1, double precision, vector &qbeziers, vector *startParams) const { // If -p0+3p1-3p2+p3 = 0, the degree of the curve is <= 2 and it's not necessary do any approximation. // In this case, the control point of the quadratic Bézier curve is (-p0+3p1)/2 = (3p2-p3)/2. // Otherwise, the distance d between q1:=(-p0+3p1)/2 and q2:=(3p2-p3)/2 is != 0. // Now we compute the quadratic Bézier with start point p0, end point p3, and control point (q1+q2)/2, // the "mid-point approximation" (MPA) of b. // The maximal distance between the points of the original curve b and the corresponding ones // on the MPA is d*sqrt(3)/18. The same computations can be done for the cubic curves we get // when subdividing b at a parameter t. The maximal distance of these curves to their MPA // is t^3*d*sqrt(3)/18. // Based on the formula for this distance we compute the split point tmax for a given precision // and check 3 cases: // * tmax >= 1: curve is quadratic, return MPA of b // * tmax >= 0.5: split b at t=0.5, return the MPAs of both segments // * tmax < 0.5: split b at tmax and 1-tmax, return the MPAs of the first and third segment, // recurse the algorithm for the middle segment DPair q1 = (_points[1]*3.0 - _points[0])/2.0; DPair q2 = (_points[2]*3.0 - _points[3])/2.0; double dist = (q2-q1).length(); double tmax3 = 18.0/sqrt(3.0)*precision/dist; // the cube of tmax if (tmax3 >= 1.0) { // curve is already quadratic, no subdivision necessary, return MPA qbeziers.emplace_back(_points[0], (q1+q2)/2.0, _points[3]); if (startParams) startParams->push_back(t0); } else if (tmax3 >= 0.125) { // tmax >= 0.5 // split the curve at t=0.5 and compute the MPA for both segments CubicBezier cbezier1, cbezier2; subdivide(0.5, &cbezier1, &cbezier2); qbeziers.emplace_back(cbezier1.midpointApproximation()); qbeziers.emplace_back(cbezier2.midpointApproximation()); if (startParams) { startParams->push_back(t0); startParams->push_back((t0+t1)/2); } } else { // tmax < 0.5 double tmax = cbrt(tmax3); double smax = 1.0-tmax; double dt = t1-t0; // first segment can be approximated by its MPA qbeziers.emplace_back(CubicBezier(*this, 0, tmax).midpointApproximation()); if (startParams) startParams->push_back(t0); // recurse for middle segment CubicBezier(*this, tmax, smax).toQuadBeziers(t0+tmax*dt, t0+smax*dt, precision, qbeziers, startParams); // third segment can be approximated by its MPA qbeziers.emplace_back(CubicBezier(*this, smax, 1).midpointApproximation()); if (startParams) startParams->push_back(smax); } } /** 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. */ BoundingBox CubicBezier::getBBox () const { BoundingBox bbox; // 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]); return bbox; } CubicBezier& CubicBezier::transform (const Matrix &matrix) { for (auto &point : _points) point = matrix * point; return *this; }