#include "edge-segments.h" #include "arithmetics.hpp" #include "equation-solver.h" namespace msdfgen { void EdgeSegment::distanceToPseudoDistance(SignedDistance &distance, Point2 origin, double param) const { if (param < 0) { Vector2 dir = direction(0).normalize(); Vector2 aq = origin-point(0); double ts = dotProduct(aq, dir); if (ts < 0) { double pseudoDistance = crossProduct(aq, dir); if (fabs(pseudoDistance) <= fabs(distance.distance)) { distance.distance = pseudoDistance; distance.dot = 0; } } } else if (param > 1) { Vector2 dir = direction(1).normalize(); Vector2 bq = origin-point(1); double ts = dotProduct(bq, dir); if (ts > 0) { double pseudoDistance = crossProduct(bq, dir); if (fabs(pseudoDistance) <= fabs(distance.distance)) { distance.distance = pseudoDistance; distance.dot = 0; } } } } LinearSegment::LinearSegment(Point2 p0, Point2 p1, EdgeColor edgeColor) : EdgeSegment(edgeColor) { p[0] = p0; p[1] = p1; } QuadraticSegment::QuadraticSegment(Point2 p0, Point2 p1, Point2 p2, EdgeColor edgeColor) : EdgeSegment(edgeColor) { if (p1 == p0 || p1 == p2) p1 = 0.5*(p0+p2); p[0] = p0; p[1] = p1; p[2] = p2; } CubicSegment::CubicSegment(Point2 p0, Point2 p1, Point2 p2, Point2 p3, EdgeColor edgeColor) : EdgeSegment(edgeColor) { if ((p1 == p0 || p1 == p3) && (p2 == p0 || p2 == p3)) { p1 = mix(p0, p3, 1/3.); p2 = mix(p0, p3, 2/3.); } p[0] = p0; p[1] = p1; p[2] = p2; p[3] = p3; } LinearSegment *LinearSegment::clone() const { return new LinearSegment(p[0], p[1], color); } QuadraticSegment *QuadraticSegment::clone() const { return new QuadraticSegment(p[0], p[1], p[2], color); } CubicSegment *CubicSegment::clone() const { return new CubicSegment(p[0], p[1], p[2], p[3], color); } int LinearSegment::type() const { return (int) EDGE_TYPE; } int QuadraticSegment::type() const { return (int) EDGE_TYPE; } int CubicSegment::type() const { return (int) EDGE_TYPE; } const Point2 *LinearSegment::controlPoints() const { return p; } const Point2 *QuadraticSegment::controlPoints() const { return p; } const Point2 *CubicSegment::controlPoints() const { return p; } Point2 LinearSegment::point(double param) const { return mix(p[0], p[1], param); } Point2 QuadraticSegment::point(double param) const { return mix(mix(p[0], p[1], param), mix(p[1], p[2], param), param); } Point2 CubicSegment::point(double param) const { Vector2 p12 = mix(p[1], p[2], param); return mix(mix(mix(p[0], p[1], param), p12, param), mix(p12, mix(p[2], p[3], param), param), param); } Vector2 LinearSegment::direction(double param) const { return p[1]-p[0]; } Vector2 QuadraticSegment::direction(double param) const { Vector2 tangent = mix(p[1]-p[0], p[2]-p[1], param); if (!tangent) return p[2]-p[0]; return tangent; } Vector2 CubicSegment::direction(double param) const { Vector2 tangent = mix(mix(p[1]-p[0], p[2]-p[1], param), mix(p[2]-p[1], p[3]-p[2], param), param); if (!tangent) { if (param == 0) return p[2]-p[0]; if (param == 1) return p[3]-p[1]; } return tangent; } Vector2 LinearSegment::directionChange(double param) const { return Vector2(); } Vector2 QuadraticSegment::directionChange(double param) const { return (p[2]-p[1])-(p[1]-p[0]); } Vector2 CubicSegment::directionChange(double param) const { return mix((p[2]-p[1])-(p[1]-p[0]), (p[3]-p[2])-(p[2]-p[1]), param); } double LinearSegment::length() const { return (p[1]-p[0]).length(); } double QuadraticSegment::length() const { Vector2 ab = p[1]-p[0]; Vector2 br = p[2]-p[1]-ab; double abab = dotProduct(ab, ab); double abbr = dotProduct(ab, br); double brbr = dotProduct(br, br); double abLen = sqrt(abab); double brLen = sqrt(brbr); double crs = crossProduct(ab, br); double h = sqrt(abab+abbr+abbr+brbr); return ( brLen*((abbr+brbr)*h-abbr*abLen)+ crs*crs*log((brLen*h+abbr+brbr)/(brLen*abLen+abbr)) )/(brbr*brLen); } SignedDistance LinearSegment::signedDistance(Point2 origin, double ¶m) const { Vector2 aq = origin-p[0]; Vector2 ab = p[1]-p[0]; param = dotProduct(aq, ab)/dotProduct(ab, ab); Vector2 eq = p[param > .5]-origin; double endpointDistance = eq.length(); if (param > 0 && param < 1) { double orthoDistance = dotProduct(ab.getOrthonormal(false), aq); if (fabs(orthoDistance) < endpointDistance) return SignedDistance(orthoDistance, 0); } return SignedDistance(nonZeroSign(crossProduct(aq, ab))*endpointDistance, fabs(dotProduct(ab.normalize(), eq.normalize()))); } SignedDistance QuadraticSegment::signedDistance(Point2 origin, double ¶m) const { Vector2 qa = p[0]-origin; Vector2 ab = p[1]-p[0]; Vector2 br = p[2]-p[1]-ab; double a = dotProduct(br, br); double b = 3*dotProduct(ab, br); double c = 2*dotProduct(ab, ab)+dotProduct(qa, br); double d = dotProduct(qa, ab); double t[3]; int solutions = solveCubic(t, a, b, c, d); Vector2 epDir = direction(0); double minDistance = nonZeroSign(crossProduct(epDir, qa))*qa.length(); // distance from A param = -dotProduct(qa, epDir)/dotProduct(epDir, epDir); { epDir = direction(1); double distance = (p[2]-origin).length(); // distance from B if (distance < fabs(minDistance)) { minDistance = nonZeroSign(crossProduct(epDir, p[2]-origin))*distance; param = dotProduct(origin-p[1], epDir)/dotProduct(epDir, epDir); } } for (int i = 0; i < solutions; ++i) { if (t[i] > 0 && t[i] < 1) { Point2 qe = qa+2*t[i]*ab+t[i]*t[i]*br; double distance = qe.length(); if (distance <= fabs(minDistance)) { minDistance = nonZeroSign(crossProduct(ab+t[i]*br, qe))*distance; param = t[i]; } } } if (param >= 0 && param <= 1) return SignedDistance(minDistance, 0); if (param < .5) return SignedDistance(minDistance, fabs(dotProduct(direction(0).normalize(), qa.normalize()))); else return SignedDistance(minDistance, fabs(dotProduct(direction(1).normalize(), (p[2]-origin).normalize()))); } SignedDistance CubicSegment::signedDistance(Point2 origin, double ¶m) const { Vector2 qa = p[0]-origin; Vector2 ab = p[1]-p[0]; Vector2 br = p[2]-p[1]-ab; Vector2 as = (p[3]-p[2])-(p[2]-p[1])-br; Vector2 epDir = direction(0); double minDistance = nonZeroSign(crossProduct(epDir, qa))*qa.length(); // distance from A param = -dotProduct(qa, epDir)/dotProduct(epDir, epDir); { epDir = direction(1); double distance = (p[3]-origin).length(); // distance from B if (distance < fabs(minDistance)) { minDistance = nonZeroSign(crossProduct(epDir, p[3]-origin))*distance; param = dotProduct(epDir-(p[3]-origin), epDir)/dotProduct(epDir, epDir); } } // Iterative minimum distance search for (int i = 0; i <= MSDFGEN_CUBIC_SEARCH_STARTS; ++i) { double t = (double) i/MSDFGEN_CUBIC_SEARCH_STARTS; Vector2 qe = qa+3*t*ab+3*t*t*br+t*t*t*as; for (int step = 0; step < MSDFGEN_CUBIC_SEARCH_STEPS; ++step) { // Improve t Vector2 d1 = 3*ab+6*t*br+3*t*t*as; Vector2 d2 = 6*br+6*t*as; t -= dotProduct(qe, d1)/(dotProduct(d1, d1)+dotProduct(qe, d2)); if (t <= 0 || t >= 1) break; qe = qa+3*t*ab+3*t*t*br+t*t*t*as; double distance = qe.length(); if (distance < fabs(minDistance)) { minDistance = nonZeroSign(crossProduct(d1, qe))*distance; param = t; } } } if (param >= 0 && param <= 1) return SignedDistance(minDistance, 0); if (param < .5) return SignedDistance(minDistance, fabs(dotProduct(direction(0).normalize(), qa.normalize()))); else return SignedDistance(minDistance, fabs(dotProduct(direction(1).normalize(), (p[3]-origin).normalize()))); } int LinearSegment::scanlineIntersections(double x[3], int dy[3], double y) const { if ((y >= p[0].y && y < p[1].y) || (y >= p[1].y && y < p[0].y)) { double param = (y-p[0].y)/(p[1].y-p[0].y); x[0] = mix(p[0].x, p[1].x, param); dy[0] = sign(p[1].y-p[0].y); return 1; } return 0; } int QuadraticSegment::scanlineIntersections(double x[3], int dy[3], double y) const { int total = 0; int nextDY = y > p[0].y ? 1 : -1; x[total] = p[0].x; if (p[0].y == y) { if (p[0].y < p[1].y || (p[0].y == p[1].y && p[0].y < p[2].y)) dy[total++] = 1; else nextDY = 1; } { Vector2 ab = p[1]-p[0]; Vector2 br = p[2]-p[1]-ab; double t[2]; int solutions = solveQuadratic(t, br.y, 2*ab.y, p[0].y-y); // Sort solutions double tmp; if (solutions >= 2 && t[0] > t[1]) tmp = t[0], t[0] = t[1], t[1] = tmp; for (int i = 0; i < solutions && total < 2; ++i) { if (t[i] >= 0 && t[i] <= 1) { x[total] = p[0].x+2*t[i]*ab.x+t[i]*t[i]*br.x; if (nextDY*(ab.y+t[i]*br.y) >= 0) { dy[total++] = nextDY; nextDY = -nextDY; } } } } if (p[2].y == y) { if (nextDY > 0 && total > 0) { --total; nextDY = -1; } if ((p[2].y < p[1].y || (p[2].y == p[1].y && p[2].y < p[0].y)) && total < 2) { x[total] = p[2].x; if (nextDY < 0) { dy[total++] = -1; nextDY = 1; } } } if (nextDY != (y >= p[2].y ? 1 : -1)) { if (total > 0) --total; else { if (fabs(p[2].y-y) < fabs(p[0].y-y)) x[total] = p[2].x; dy[total++] = nextDY; } } return total; } int CubicSegment::scanlineIntersections(double x[3], int dy[3], double y) const { int total = 0; int nextDY = y > p[0].y ? 1 : -1; x[total] = p[0].x; if (p[0].y == y) { if (p[0].y < p[1].y || (p[0].y == p[1].y && (p[0].y < p[2].y || (p[0].y == p[2].y && p[0].y < p[3].y)))) dy[total++] = 1; else nextDY = 1; } { Vector2 ab = p[1]-p[0]; Vector2 br = p[2]-p[1]-ab; Vector2 as = (p[3]-p[2])-(p[2]-p[1])-br; double t[3]; int solutions = solveCubic(t, as.y, 3*br.y, 3*ab.y, p[0].y-y); // Sort solutions double tmp; if (solutions >= 2) { if (t[0] > t[1]) tmp = t[0], t[0] = t[1], t[1] = tmp; if (solutions >= 3 && t[1] > t[2]) { tmp = t[1], t[1] = t[2], t[2] = tmp; if (t[0] > t[1]) tmp = t[0], t[0] = t[1], t[1] = tmp; } } for (int i = 0; i < solutions && total < 3; ++i) { if (t[i] >= 0 && t[i] <= 1) { x[total] = p[0].x+3*t[i]*ab.x+3*t[i]*t[i]*br.x+t[i]*t[i]*t[i]*as.x; if (nextDY*(ab.y+2*t[i]*br.y+t[i]*t[i]*as.y) >= 0) { dy[total++] = nextDY; nextDY = -nextDY; } } } } if (p[3].y == y) { if (nextDY > 0 && total > 0) { --total; nextDY = -1; } if ((p[3].y < p[2].y || (p[3].y == p[2].y && (p[3].y < p[1].y || (p[3].y == p[1].y && p[3].y < p[0].y)))) && total < 3) { x[total] = p[3].x; if (nextDY < 0) { dy[total++] = -1; nextDY = 1; } } } if (nextDY != (y >= p[3].y ? 1 : -1)) { if (total > 0) --total; else { if (fabs(p[3].y-y) < fabs(p[0].y-y)) x[total] = p[3].x; dy[total++] = nextDY; } } return total; } static void pointBounds(Point2 p, double &l, double &b, double &r, double &t) { if (p.x < l) l = p.x; if (p.y < b) b = p.y; if (p.x > r) r = p.x; if (p.y > t) t = p.y; } void LinearSegment::bound(double &l, double &b, double &r, double &t) const { pointBounds(p[0], l, b, r, t); pointBounds(p[1], l, b, r, t); } void QuadraticSegment::bound(double &l, double &b, double &r, double &t) const { pointBounds(p[0], l, b, r, t); pointBounds(p[2], l, b, r, t); Vector2 bot = (p[1]-p[0])-(p[2]-p[1]); if (bot.x) { double param = (p[1].x-p[0].x)/bot.x; if (param > 0 && param < 1) pointBounds(point(param), l, b, r, t); } if (bot.y) { double param = (p[1].y-p[0].y)/bot.y; if (param > 0 && param < 1) pointBounds(point(param), l, b, r, t); } } void CubicSegment::bound(double &l, double &b, double &r, double &t) const { pointBounds(p[0], l, b, r, t); pointBounds(p[3], l, b, r, t); Vector2 a0 = p[1]-p[0]; Vector2 a1 = 2*(p[2]-p[1]-a0); Vector2 a2 = p[3]-3*p[2]+3*p[1]-p[0]; double params[2]; int solutions; solutions = solveQuadratic(params, a2.x, a1.x, a0.x); for (int i = 0; i < solutions; ++i) if (params[i] > 0 && params[i] < 1) pointBounds(point(params[i]), l, b, r, t); solutions = solveQuadratic(params, a2.y, a1.y, a0.y); for (int i = 0; i < solutions; ++i) if (params[i] > 0 && params[i] < 1) pointBounds(point(params[i]), l, b, r, t); } void LinearSegment::reverse() { Point2 tmp = p[0]; p[0] = p[1]; p[1] = tmp; } void QuadraticSegment::reverse() { Point2 tmp = p[0]; p[0] = p[2]; p[2] = tmp; } void CubicSegment::reverse() { Point2 tmp = p[0]; p[0] = p[3]; p[3] = tmp; tmp = p[1]; p[1] = p[2]; p[2] = tmp; } void LinearSegment::moveStartPoint(Point2 to) { p[0] = to; } void QuadraticSegment::moveStartPoint(Point2 to) { Vector2 origSDir = p[0]-p[1]; Point2 origP1 = p[1]; p[1] += crossProduct(p[0]-p[1], to-p[0])/crossProduct(p[0]-p[1], p[2]-p[1])*(p[2]-p[1]); p[0] = to; if (dotProduct(origSDir, p[0]-p[1]) < 0) p[1] = origP1; } void CubicSegment::moveStartPoint(Point2 to) { p[1] += to-p[0]; p[0] = to; } void LinearSegment::moveEndPoint(Point2 to) { p[1] = to; } void QuadraticSegment::moveEndPoint(Point2 to) { Vector2 origEDir = p[2]-p[1]; Point2 origP1 = p[1]; p[1] += crossProduct(p[2]-p[1], to-p[2])/crossProduct(p[2]-p[1], p[0]-p[1])*(p[0]-p[1]); p[2] = to; if (dotProduct(origEDir, p[2]-p[1]) < 0) p[1] = origP1; } void CubicSegment::moveEndPoint(Point2 to) { p[2] += to-p[3]; p[3] = to; } void LinearSegment::splitInThirds(EdgeSegment *&part1, EdgeSegment *&part2, EdgeSegment *&part3) const { part1 = new LinearSegment(p[0], point(1/3.), color); part2 = new LinearSegment(point(1/3.), point(2/3.), color); part3 = new LinearSegment(point(2/3.), p[1], color); } void QuadraticSegment::splitInThirds(EdgeSegment *&part1, EdgeSegment *&part2, EdgeSegment *&part3) const { part1 = new QuadraticSegment(p[0], mix(p[0], p[1], 1/3.), point(1/3.), color); part2 = new QuadraticSegment(point(1/3.), mix(mix(p[0], p[1], 5/9.), mix(p[1], p[2], 4/9.), .5), point(2/3.), color); part3 = new QuadraticSegment(point(2/3.), mix(p[1], p[2], 2/3.), p[2], color); } void CubicSegment::splitInThirds(EdgeSegment *&part1, EdgeSegment *&part2, EdgeSegment *&part3) const { part1 = new CubicSegment(p[0], p[0] == p[1] ? p[0] : mix(p[0], p[1], 1/3.), mix(mix(p[0], p[1], 1/3.), mix(p[1], p[2], 1/3.), 1/3.), point(1/3.), color); part2 = new CubicSegment(point(1/3.), mix(mix(mix(p[0], p[1], 1/3.), mix(p[1], p[2], 1/3.), 1/3.), mix(mix(p[1], p[2], 1/3.), mix(p[2], p[3], 1/3.), 1/3.), 2/3.), mix(mix(mix(p[0], p[1], 2/3.), mix(p[1], p[2], 2/3.), 2/3.), mix(mix(p[1], p[2], 2/3.), mix(p[2], p[3], 2/3.), 2/3.), 1/3.), point(2/3.), color); part3 = new CubicSegment(point(2/3.), mix(mix(p[1], p[2], 2/3.), mix(p[2], p[3], 2/3.), 2/3.), p[2] == p[3] ? p[3] : mix(p[2], p[3], 2/3.), p[3], color); } EdgeSegment *QuadraticSegment::convertToCubic() const { return new CubicSegment(p[0], mix(p[0], p[1], 2/3.), mix(p[1], p[2], 1/3.), p[2], color); } void CubicSegment::deconverge(int param, double amount) { Vector2 dir = direction(param); Vector2 normal = dir.getOrthonormal(); double h = dotProduct(directionChange(param)-dir, normal); switch (param) { case 0: p[1] += amount*(dir+sign(h)*sqrt(fabs(h))*normal); break; case 1: p[2] -= amount*(dir-sign(h)*sqrt(fabs(h))*normal); break; } } }