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Diffstat (limited to 'gfx/skia/skia/src/pathops/SkDQuadLineIntersection.cpp')
| -rw-r--r-- | gfx/skia/skia/src/pathops/SkDQuadLineIntersection.cpp | 478 |
1 files changed, 478 insertions, 0 deletions
diff --git a/gfx/skia/skia/src/pathops/SkDQuadLineIntersection.cpp b/gfx/skia/skia/src/pathops/SkDQuadLineIntersection.cpp new file mode 100644 index 0000000000..2caeaeb8d7 --- /dev/null +++ b/gfx/skia/skia/src/pathops/SkDQuadLineIntersection.cpp @@ -0,0 +1,478 @@ +/* + * 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 "include/core/SkPath.h" +#include "include/core/SkPoint.h" +#include "include/core/SkScalar.h" +#include "src/pathops/SkIntersections.h" +#include "src/pathops/SkPathOpsCurve.h" +#include "src/pathops/SkPathOpsDebug.h" +#include "src/pathops/SkPathOpsLine.h" +#include "src/pathops/SkPathOpsPoint.h" +#include "src/pathops/SkPathOpsQuad.h" +#include "src/pathops/SkPathOpsTypes.h" + +#include <cmath> + +/* +Find the intersection 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); +} |