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/*************************************************************************
** Bezier.cpp **
** **
** This file is part of dvisvgm -- the DVI to SVG converter **
** Copyright (C) 2005-2015 Martin Gieseking <martin.gieseking@uos.de> **
** **
** 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 <http://www.gnu.org/licenses/>. **
*************************************************************************/
#include <algorithm>
#include <utility>
#include "Bezier.h"
using namespace std;
Bezier::Bezier () {
_points[0] = _points[1] = _points[2] = _points[3] = 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, 0);
else if (t1 == 1)
source.subdivide(t0, 0, this);
else {
Bezier subcurve;
source.subdivide(t0, 0, &subcurve);
subcurve.subdivide((t1-t0)/(1-t0), this, 0);
}
}
}
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<DPair> &p, vector<double> *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<DPair> &p, vector<double> *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<DPair> &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]);
}
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