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 /* * Copyright 2012 Google Inc. * * Use of this source code is governed by a BSD-style license that can be * found in the LICENSE file. */ #include "src/pathops/SkIntersections.h" #include "src/pathops/SkPathOpsCurve.h" #include "src/pathops/SkPathOpsLine.h" #include "src/pathops/SkPathOpsQuad.h" /* Find the interection of a line and quadratic by solving for valid t values. From http://stackoverflow.com/questions/1853637/how-to-find-the-mathematical-function-defining-a-bezier-curve "A Bezier curve is a parametric function. A quadratic Bezier curve (i.e. three control points) can be expressed as: F(t) = A(1 - t)^2 + B(1 - t)t + Ct^2 where A, B and C are points and t goes from zero to one. This will give you two equations: x = a(1 - t)^2 + b(1 - t)t + ct^2 y = d(1 - t)^2 + e(1 - t)t + ft^2 If you add for instance the line equation (y = kx + m) to that, you'll end up with three equations and three unknowns (x, y and t)." Similar to above, the quadratic is represented as x = a(1-t)^2 + 2b(1-t)t + ct^2 y = d(1-t)^2 + 2e(1-t)t + ft^2 and the line as y = g*x + h Using Mathematica, solve for the values of t where the quadratic intersects the line: (in) t1 = Resultant[a*(1 - t)^2 + 2*b*(1 - t)*t + c*t^2 - x, d*(1 - t)^2 + 2*e*(1 - t)*t + f*t^2 - g*x - h, x] (out) -d + h + 2 d t - 2 e t - d t^2 + 2 e t^2 - f t^2 + g (a - 2 a t + 2 b t + a t^2 - 2 b t^2 + c t^2) (in) Solve[t1 == 0, t] (out) { {t -> (-2 d + 2 e + 2 a g - 2 b g - Sqrt[(2 d - 2 e - 2 a g + 2 b g)^2 - 4 (-d + 2 e - f + a g - 2 b g + c g) (-d + a g + h)]) / (2 (-d + 2 e - f + a g - 2 b g + c g)) }, {t -> (-2 d + 2 e + 2 a g - 2 b g + Sqrt[(2 d - 2 e - 2 a g + 2 b g)^2 - 4 (-d + 2 e - f + a g - 2 b g + c g) (-d + a g + h)]) / (2 (-d + 2 e - f + a g - 2 b g + c g)) } } Using the results above (when the line tends towards horizontal) A = (-(d - 2*e + f) + g*(a - 2*b + c) ) B = 2*( (d - e ) - g*(a - b ) ) C = (-(d ) + g*(a ) + h ) If g goes to infinity, we can rewrite the line in terms of x. x = g'*y + h' And solve accordingly in Mathematica: (in) t2 = Resultant[a*(1 - t)^2 + 2*b*(1 - t)*t + c*t^2 - g'*y - h', d*(1 - t)^2 + 2*e*(1 - t)*t + f*t^2 - y, y] (out) a - h' - 2 a t + 2 b t + a t^2 - 2 b t^2 + c t^2 - g' (d - 2 d t + 2 e t + d t^2 - 2 e t^2 + f t^2) (in) Solve[t2 == 0, t] (out) { {t -> (2 a - 2 b - 2 d g' + 2 e g' - Sqrt[(-2 a + 2 b + 2 d g' - 2 e g')^2 - 4 (a - 2 b + c - d g' + 2 e g' - f g') (a - d g' - h')]) / (2 (a - 2 b + c - d g' + 2 e g' - f g')) }, {t -> (2 a - 2 b - 2 d g' + 2 e g' + Sqrt[(-2 a + 2 b + 2 d g' - 2 e g')^2 - 4 (a - 2 b + c - d g' + 2 e g' - f g') (a - d g' - h')])/ (2 (a - 2 b + c - d g' + 2 e g' - f g')) } } Thus, if the slope of the line tends towards vertical, we use: A = ( (a - 2*b + c) - g'*(d - 2*e + f) ) B = 2*(-(a - b ) + g'*(d - e ) ) C = ( (a ) - g'*(d ) - h' ) */ class LineQuadraticIntersections { public: enum PinTPoint { kPointUninitialized, kPointInitialized }; LineQuadraticIntersections(const SkDQuad& q, const SkDLine& l, SkIntersections* i) : fQuad(q) , fLine(&l) , fIntersections(i) , fAllowNear(true) { i->setMax(5); // allow short partial coincidence plus discrete intersections } LineQuadraticIntersections(const SkDQuad& q) : fQuad(q) SkDEBUGPARAMS(fLine(nullptr)) SkDEBUGPARAMS(fIntersections(nullptr)) SkDEBUGPARAMS(fAllowNear(false)) { } void allowNear(bool allow) { fAllowNear = allow; } void checkCoincident() { int last = fIntersections->used() - 1; for (int index = 0; index < last; ) { double quadMidT = ((*fIntersections)[0][index] + (*fIntersections)[0][index + 1]) / 2; SkDPoint quadMidPt = fQuad.ptAtT(quadMidT); double t = fLine->nearPoint(quadMidPt, nullptr); if (t < 0) { ++index; continue; } if (fIntersections->isCoincident(index)) { fIntersections->removeOne(index); --last; } else if (fIntersections->isCoincident(index + 1)) { fIntersections->removeOne(index + 1); --last; } else { fIntersections->setCoincident(index++); } fIntersections->setCoincident(index); } } int intersectRay(double roots[2]) { /* solve by rotating line+quad so line is horizontal, then finding the roots set up matrix to rotate quad to x-axis |cos(a) -sin(a)| |sin(a) cos(a)| note that cos(a) = A(djacent) / Hypoteneuse sin(a) = O(pposite) / Hypoteneuse since we are computing Ts, we can ignore hypoteneuse, the scale factor: | A -O | | O A | A = line[1].fX - line[0].fX (adjacent side of the right triangle) O = line[1].fY - line[0].fY (opposite side of the right triangle) for each of the three points (e.g. n = 0 to 2) quad[n].fY' = (quad[n].fY - line[0].fY) * A - (quad[n].fX - line[0].fX) * O */ double adj = (*fLine)[1].fX - (*fLine)[0].fX; double opp = (*fLine)[1].fY - (*fLine)[0].fY; double r[3]; for (int n = 0; n < 3; ++n) { r[n] = (fQuad[n].fY - (*fLine)[0].fY) * adj - (fQuad[n].fX - (*fLine)[0].fX) * opp; } double A = r[2]; double B = r[1]; double C = r[0]; A += C - 2 * B; // A = a - 2*b + c B -= C; // B = -(b - c) return SkDQuad::RootsValidT(A, 2 * B, C, roots); } int intersect() { addExactEndPoints(); if (fAllowNear) { addNearEndPoints(); } double rootVals[2]; int roots = intersectRay(rootVals); for (int index = 0; index < roots; ++index) { double quadT = rootVals[index]; double lineT = findLineT(quadT); SkDPoint pt; if (pinTs(&quadT, &lineT, &pt, kPointUninitialized) && uniqueAnswer(quadT, pt)) { fIntersections->insert(quadT, lineT, pt); } } checkCoincident(); return fIntersections->used(); } int horizontalIntersect(double axisIntercept, double roots[2]) { double D = fQuad[2].fY; // f double E = fQuad[1].fY; // e double F = fQuad[0].fY; // d D += F - 2 * E; // D = d - 2*e + f E -= F; // E = -(d - e) F -= axisIntercept; return SkDQuad::RootsValidT(D, 2 * E, F, roots); } int horizontalIntersect(double axisIntercept, double left, double right, bool flipped) { addExactHorizontalEndPoints(left, right, axisIntercept); if (fAllowNear) { addNearHorizontalEndPoints(left, right, axisIntercept); } double rootVals[2]; int roots = horizontalIntersect(axisIntercept, rootVals); for (int index = 0; index < roots; ++index) { double quadT = rootVals[index]; SkDPoint pt = fQuad.ptAtT(quadT); double lineT = (pt.fX - left) / (right - left); if (pinTs(&quadT, &lineT, &pt, kPointInitialized) && uniqueAnswer(quadT, pt)) { fIntersections->insert(quadT, lineT, pt); } } if (flipped) { fIntersections->flip(); } checkCoincident(); return fIntersections->used(); } bool uniqueAnswer(double quadT, const SkDPoint& pt) { for (int inner = 0; inner < fIntersections->used(); ++inner) { if (fIntersections->pt(inner) != pt) { continue; } double existingQuadT = (*fIntersections)[0][inner]; if (quadT == existingQuadT) { return false; } // check if midway on quad is also same point. If so, discard this double quadMidT = (existingQuadT + quadT) / 2; SkDPoint quadMidPt = fQuad.ptAtT(quadMidT); if (quadMidPt.approximatelyEqual(pt)) { return false; } } #if ONE_OFF_DEBUG SkDPoint qPt = fQuad.ptAtT(quadT); SkDebugf("%s pt=(%1.9g,%1.9g) cPt=(%1.9g,%1.9g)\n", __FUNCTION__, pt.fX, pt.fY, qPt.fX, qPt.fY); #endif return true; } int verticalIntersect(double axisIntercept, double roots[2]) { double D = fQuad[2].fX; // f double E = fQuad[1].fX; // e double F = fQuad[0].fX; // d D += F - 2 * E; // D = d - 2*e + f E -= F; // E = -(d - e) F -= axisIntercept; return SkDQuad::RootsValidT(D, 2 * E, F, roots); } int verticalIntersect(double axisIntercept, double top, double bottom, bool flipped) { addExactVerticalEndPoints(top, bottom, axisIntercept); if (fAllowNear) { addNearVerticalEndPoints(top, bottom, axisIntercept); } double rootVals[2]; int roots = verticalIntersect(axisIntercept, rootVals); for (int index = 0; index < roots; ++index) { double quadT = rootVals[index]; SkDPoint pt = fQuad.ptAtT(quadT); double lineT = (pt.fY - top) / (bottom - top); if (pinTs(&quadT, &lineT, &pt, kPointInitialized) && uniqueAnswer(quadT, pt)) { fIntersections->insert(quadT, lineT, pt); } } if (flipped) { fIntersections->flip(); } checkCoincident(); return fIntersections->used(); } protected: // add endpoints first to get zero and one t values exactly void addExactEndPoints() { for (int qIndex = 0; qIndex < 3; qIndex += 2) { double lineT = fLine->exactPoint(fQuad[qIndex]); if (lineT < 0) { continue; } double quadT = (double) (qIndex >> 1); fIntersections->insert(quadT, lineT, fQuad[qIndex]); } } void addNearEndPoints() { for (int qIndex = 0; qIndex < 3; qIndex += 2) { double quadT = (double) (qIndex >> 1); if (fIntersections->hasT(quadT)) { continue; } double lineT = fLine->nearPoint(fQuad[qIndex], nullptr); if (lineT < 0) { continue; } fIntersections->insert(quadT, lineT, fQuad[qIndex]); } this->addLineNearEndPoints(); } void addLineNearEndPoints() { for (int lIndex = 0; lIndex < 2; ++lIndex) { double lineT = (double) lIndex; if (fIntersections->hasOppT(lineT)) { continue; } double quadT = ((SkDCurve*) &fQuad)->nearPoint(SkPath::kQuad_Verb, (*fLine)[lIndex], (*fLine)[!lIndex]); if (quadT < 0) { continue; } fIntersections->insert(quadT, lineT, (*fLine)[lIndex]); } } void addExactHorizontalEndPoints(double left, double right, double y) { for (int qIndex = 0; qIndex < 3; qIndex += 2) { double lineT = SkDLine::ExactPointH(fQuad[qIndex], left, right, y); if (lineT < 0) { continue; } double quadT = (double) (qIndex >> 1); fIntersections->insert(quadT, lineT, fQuad[qIndex]); } } void addNearHorizontalEndPoints(double left, double right, double y) { for (int qIndex = 0; qIndex < 3; qIndex += 2) { double quadT = (double) (qIndex >> 1); if (fIntersections->hasT(quadT)) { continue; } double lineT = SkDLine::NearPointH(fQuad[qIndex], left, right, y); if (lineT < 0) { continue; } fIntersections->insert(quadT, lineT, fQuad[qIndex]); } this->addLineNearEndPoints(); } void addExactVerticalEndPoints(double top, double bottom, double x) { for (int qIndex = 0; qIndex < 3; qIndex += 2) { double lineT = SkDLine::ExactPointV(fQuad[qIndex], top, bottom, x); if (lineT < 0) { continue; } double quadT = (double) (qIndex >> 1); fIntersections->insert(quadT, lineT, fQuad[qIndex]); } } void addNearVerticalEndPoints(double top, double bottom, double x) { for (int qIndex = 0; qIndex < 3; qIndex += 2) { double quadT = (double) (qIndex >> 1); if (fIntersections->hasT(quadT)) { continue; } double lineT = SkDLine::NearPointV(fQuad[qIndex], top, bottom, x); if (lineT < 0) { continue; } fIntersections->insert(quadT, lineT, fQuad[qIndex]); } this->addLineNearEndPoints(); } double findLineT(double t) { SkDPoint xy = fQuad.ptAtT(t); double dx = (*fLine)[1].fX - (*fLine)[0].fX; double dy = (*fLine)[1].fY - (*fLine)[0].fY; if (fabs(dx) > fabs(dy)) { return (xy.fX - (*fLine)[0].fX) / dx; } return (xy.fY - (*fLine)[0].fY) / dy; } bool pinTs(double* quadT, double* lineT, SkDPoint* pt, PinTPoint ptSet) { if (!approximately_one_or_less_double(*lineT)) { return false; } if (!approximately_zero_or_more_double(*lineT)) { return false; } double qT = *quadT = SkPinT(*quadT); double lT = *lineT = SkPinT(*lineT); if (lT == 0 || lT == 1 || (ptSet == kPointUninitialized && qT != 0 && qT != 1)) { *pt = (*fLine).ptAtT(lT); } else if (ptSet == kPointUninitialized) { *pt = fQuad.ptAtT(qT); } SkPoint gridPt = pt->asSkPoint(); if (SkDPoint::ApproximatelyEqual(gridPt, (*fLine)[0].asSkPoint())) { *pt = (*fLine)[0]; *lineT = 0; } else if (SkDPoint::ApproximatelyEqual(gridPt, (*fLine)[1].asSkPoint())) { *pt = (*fLine)[1]; *lineT = 1; } if (fIntersections->used() > 0 && approximately_equal((*fIntersections)[1][0], *lineT)) { return false; } if (gridPt == fQuad[0].asSkPoint()) { *pt = fQuad[0]; *quadT = 0; } else if (gridPt == fQuad[2].asSkPoint()) { *pt = fQuad[2]; *quadT = 1; } return true; } private: const SkDQuad& fQuad; const SkDLine* fLine; SkIntersections* fIntersections; bool fAllowNear; }; int SkIntersections::horizontal(const SkDQuad& quad, double left, double right, double y, bool flipped) { SkDLine line = {{{ left, y }, { right, y }}}; LineQuadraticIntersections q(quad, line, this); return q.horizontalIntersect(y, left, right, flipped); } int SkIntersections::vertical(const SkDQuad& quad, double top, double bottom, double x, bool flipped) { SkDLine line = {{{ x, top }, { x, bottom }}}; LineQuadraticIntersections q(quad, line, this); return q.verticalIntersect(x, top, bottom, flipped); } int SkIntersections::intersect(const SkDQuad& quad, const SkDLine& line) { LineQuadraticIntersections q(quad, line, this); q.allowNear(fAllowNear); return q.intersect(); } int SkIntersections::intersectRay(const SkDQuad& quad, const SkDLine& line) { LineQuadraticIntersections q(quad, line, this); fUsed = q.intersectRay(fT[0]); for (int index = 0; index < fUsed; ++index) { fPt[index] = quad.ptAtT(fT[0][index]); } return fUsed; } int SkIntersections::HorizontalIntercept(const SkDQuad& quad, SkScalar y, double* roots) { LineQuadraticIntersections q(quad); return q.horizontalIntersect(y, roots); } int SkIntersections::VerticalIntercept(const SkDQuad& quad, SkScalar x, double* roots) { LineQuadraticIntersections q(quad); return q.verticalIntersect(x, roots); } // SkDQuad accessors to Intersection utilities int SkDQuad::horizontalIntersect(double yIntercept, double roots[2]) const { return SkIntersections::HorizontalIntercept(*this, yIntercept, roots); } int SkDQuad::verticalIntersect(double xIntercept, double roots[2]) const { return SkIntersections::VerticalIntercept(*this, xIntercept, roots); }