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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);
- }
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