| /* |
| * 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 "SkIntersections.h" |
| #include "SkPathOpsLine.h" |
| #include "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(3); // allow short partial coincidence plus discrete intersection |
| } |
| |
| void allowNear(bool allow) { |
| fAllowNear = allow; |
| } |
| |
| 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(); |
| } |
| if (fIntersections->used() == 2) { |
| // FIXME : need sharable code that turns spans into coincident if middle point is on |
| } else { |
| 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)) { |
| fIntersections->insert(quadT, lineT, pt); |
| } |
| } |
| } |
| 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)) { |
| fIntersections->insert(quadT, lineT, pt); |
| } |
| } |
| if (flipped) { |
| fIntersections->flip(); |
| } |
| return fIntersections->used(); |
| } |
| |
| 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)) { |
| fIntersections->insert(quadT, lineT, pt); |
| } |
| } |
| if (flipped) { |
| fIntersections->flip(); |
| } |
| 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], NULL); |
| if (lineT < 0) { |
| continue; |
| } |
| fIntersections->insert(quadT, lineT, fQuad[qIndex]); |
| } |
| // FIXME: see if line end is nearly on quad |
| } |
| |
| 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]); |
| } |
| // FIXME: see if line end is nearly on quad |
| } |
| |
| 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]); |
| } |
| // FIXME: see if line end is nearly on quad |
| } |
| |
| 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; |
| } |
