michael@0: /*
michael@0:  * Copyright 2012 Google Inc.
michael@0:  *
michael@0:  * Use of this source code is governed by a BSD-style license that can be
michael@0:  * found in the LICENSE file.
michael@0:  */
michael@0: #include "SkIntersections.h"
michael@0: #include "SkPathOpsLine.h"
michael@0: #include "SkPathOpsQuad.h"
michael@0: 
michael@0: /*
michael@0: Find the interection of a line and quadratic by solving for valid t values.
michael@0: 
michael@0: From http://stackoverflow.com/questions/1853637/how-to-find-the-mathematical-function-defining-a-bezier-curve
michael@0: 
michael@0: "A Bezier curve is a parametric function. A quadratic Bezier curve (i.e. three
michael@0: control points) can be expressed as: F(t) = A(1 - t)^2 + B(1 - t)t + Ct^2 where
michael@0: A, B and C are points and t goes from zero to one.
michael@0: 
michael@0: This will give you two equations:
michael@0: 
michael@0:   x = a(1 - t)^2 + b(1 - t)t + ct^2
michael@0:   y = d(1 - t)^2 + e(1 - t)t + ft^2
michael@0: 
michael@0: If you add for instance the line equation (y = kx + m) to that, you'll end up
michael@0: with three equations and three unknowns (x, y and t)."
michael@0: 
michael@0: Similar to above, the quadratic is represented as
michael@0:   x = a(1-t)^2 + 2b(1-t)t + ct^2
michael@0:   y = d(1-t)^2 + 2e(1-t)t + ft^2
michael@0: and the line as
michael@0:   y = g*x + h
michael@0: 
michael@0: Using Mathematica, solve for the values of t where the quadratic intersects the
michael@0: line:
michael@0: 
michael@0:   (in)  t1 = Resultant[a*(1 - t)^2 + 2*b*(1 - t)*t + c*t^2 - x,
michael@0:                        d*(1 - t)^2 + 2*e*(1 - t)*t  + f*t^2 - g*x - h, x]
michael@0:   (out) -d + h + 2 d t - 2 e t - d t^2 + 2 e t^2 - f t^2 +
michael@0:          g  (a - 2 a t + 2 b t + a t^2 - 2 b t^2 + c t^2)
michael@0:   (in)  Solve[t1 == 0, t]
michael@0:   (out) {
michael@0:     {t -> (-2 d + 2 e +   2 a g - 2 b g    -
michael@0:       Sqrt[(2 d - 2 e -   2 a g + 2 b g)^2 -
michael@0:           4 (-d + 2 e - f + a g - 2 b g    + c g) (-d + a g + h)]) /
michael@0:          (2 (-d + 2 e - f + a g - 2 b g    + c g))
michael@0:          },
michael@0:     {t -> (-2 d + 2 e +   2 a g - 2 b g    +
michael@0:       Sqrt[(2 d - 2 e -   2 a g + 2 b g)^2 -
michael@0:           4 (-d + 2 e - f + a g - 2 b g    + c g) (-d + a g + h)]) /
michael@0:          (2 (-d + 2 e - f + a g - 2 b g    + c g))
michael@0:          }
michael@0:         }
michael@0: 
michael@0: Using the results above (when the line tends towards horizontal)
michael@0:        A =   (-(d - 2*e + f) + g*(a - 2*b + c)     )
michael@0:        B = 2*( (d -   e    ) - g*(a -   b    )     )
michael@0:        C =   (-(d          ) + g*(a          ) + h )
michael@0: 
michael@0: If g goes to infinity, we can rewrite the line in terms of x.
michael@0:   x = g'*y + h'
michael@0: 
michael@0: And solve accordingly in Mathematica:
michael@0: 
michael@0:   (in)  t2 = Resultant[a*(1 - t)^2 + 2*b*(1 - t)*t + c*t^2 - g'*y - h',
michael@0:                        d*(1 - t)^2 + 2*e*(1 - t)*t  + f*t^2 - y, y]
michael@0:   (out)  a - h' - 2 a t + 2 b t + a t^2 - 2 b t^2 + c t^2 -
michael@0:          g'  (d - 2 d t + 2 e t + d t^2 - 2 e t^2 + f t^2)
michael@0:   (in)  Solve[t2 == 0, t]
michael@0:   (out) {
michael@0:     {t -> (2 a - 2 b -   2 d g' + 2 e g'    -
michael@0:     Sqrt[(-2 a + 2 b +   2 d g' - 2 e g')^2 -
michael@0:           4 (a - 2 b + c - d g' + 2 e g' - f g') (a - d g' - h')]) /
michael@0:          (2 (a - 2 b + c - d g' + 2 e g' - f g'))
michael@0:          },
michael@0:     {t -> (2 a - 2 b -   2 d g' + 2 e g'    +
michael@0:     Sqrt[(-2 a + 2 b +   2 d g' - 2 e g')^2 -
michael@0:           4 (a - 2 b + c - d g' + 2 e g' - f g') (a - d g' - h')])/
michael@0:          (2 (a - 2 b + c - d g' + 2 e g' - f g'))
michael@0:          }
michael@0:         }
michael@0: 
michael@0: Thus, if the slope of the line tends towards vertical, we use:
michael@0:        A =   ( (a - 2*b + c) - g'*(d  - 2*e + f)      )
michael@0:        B = 2*(-(a -   b    ) + g'*(d  -   e    )      )
michael@0:        C =   ( (a          ) - g'*(d           ) - h' )
michael@0:  */
michael@0: 
michael@0: 
michael@0: class LineQuadraticIntersections {
michael@0: public:
michael@0:     enum PinTPoint {
michael@0:         kPointUninitialized,
michael@0:         kPointInitialized
michael@0:     };
michael@0: 
michael@0:     LineQuadraticIntersections(const SkDQuad& q, const SkDLine& l, SkIntersections* i)
michael@0:         : fQuad(q)
michael@0:         , fLine(l)
michael@0:         , fIntersections(i)
michael@0:         , fAllowNear(true) {
michael@0:         i->setMax(2);
michael@0:     }
michael@0: 
michael@0:     void allowNear(bool allow) {
michael@0:         fAllowNear = allow;
michael@0:     }
michael@0: 
michael@0:     int intersectRay(double roots[2]) {
michael@0:     /*
michael@0:         solve by rotating line+quad so line is horizontal, then finding the roots
michael@0:         set up matrix to rotate quad to x-axis
michael@0:         |cos(a) -sin(a)|
michael@0:         |sin(a)  cos(a)|
michael@0:         note that cos(a) = A(djacent) / Hypoteneuse
michael@0:                   sin(a) = O(pposite) / Hypoteneuse
michael@0:         since we are computing Ts, we can ignore hypoteneuse, the scale factor:
michael@0:         |  A     -O    |
michael@0:         |  O      A    |
michael@0:         A = line[1].fX - line[0].fX (adjacent side of the right triangle)
michael@0:         O = line[1].fY - line[0].fY (opposite side of the right triangle)
michael@0:         for each of the three points (e.g. n = 0 to 2)
michael@0:         quad[n].fY' = (quad[n].fY - line[0].fY) * A - (quad[n].fX - line[0].fX) * O
michael@0:     */
michael@0:         double adj = fLine[1].fX - fLine[0].fX;
michael@0:         double opp = fLine[1].fY - fLine[0].fY;
michael@0:         double r[3];
michael@0:         for (int n = 0; n < 3; ++n) {
michael@0:             r[n] = (fQuad[n].fY - fLine[0].fY) * adj - (fQuad[n].fX - fLine[0].fX) * opp;
michael@0:         }
michael@0:         double A = r[2];
michael@0:         double B = r[1];
michael@0:         double C = r[0];
michael@0:         A += C - 2 * B;  // A = a - 2*b + c
michael@0:         B -= C;  // B = -(b - c)
michael@0:         return SkDQuad::RootsValidT(A, 2 * B, C, roots);
michael@0:     }
michael@0: 
michael@0:     int intersect() {
michael@0:         addExactEndPoints();
michael@0:         if (fAllowNear) {
michael@0:             addNearEndPoints();
michael@0:         }
michael@0:         if (fIntersections->used() == 2) {
michael@0:             // FIXME : need sharable code that turns spans into coincident if middle point is on
michael@0:         } else {
michael@0:             double rootVals[2];
michael@0:             int roots = intersectRay(rootVals);
michael@0:             for (int index = 0; index < roots; ++index) {
michael@0:                 double quadT = rootVals[index];
michael@0:                 double lineT = findLineT(quadT);
michael@0:                 SkDPoint pt;
michael@0:                 if (pinTs(&quadT, &lineT, &pt, kPointUninitialized)) {
michael@0:                     fIntersections->insert(quadT, lineT, pt);
michael@0:                 }
michael@0:             }
michael@0:         }
michael@0:         return fIntersections->used();
michael@0:     }
michael@0: 
michael@0:     int horizontalIntersect(double axisIntercept, double roots[2]) {
michael@0:         double D = fQuad[2].fY;  // f
michael@0:         double E = fQuad[1].fY;  // e
michael@0:         double F = fQuad[0].fY;  // d
michael@0:         D += F - 2 * E;         // D = d - 2*e + f
michael@0:         E -= F;                 // E = -(d - e)
michael@0:         F -= axisIntercept;
michael@0:         return SkDQuad::RootsValidT(D, 2 * E, F, roots);
michael@0:     }
michael@0: 
michael@0:     int horizontalIntersect(double axisIntercept, double left, double right, bool flipped) {
michael@0:         addExactHorizontalEndPoints(left, right, axisIntercept);
michael@0:         if (fAllowNear) {
michael@0:             addNearHorizontalEndPoints(left, right, axisIntercept);
michael@0:         }
michael@0:         double rootVals[2];
michael@0:         int roots = horizontalIntersect(axisIntercept, rootVals);
michael@0:         for (int index = 0; index < roots; ++index) {
michael@0:             double quadT = rootVals[index];
michael@0:             SkDPoint pt = fQuad.ptAtT(quadT);
michael@0:             double lineT = (pt.fX - left) / (right - left);
michael@0:             if (pinTs(&quadT, &lineT, &pt, kPointInitialized)) {
michael@0:                 fIntersections->insert(quadT, lineT, pt);
michael@0:             }
michael@0:         }
michael@0:         if (flipped) {
michael@0:             fIntersections->flip();
michael@0:         }
michael@0:         return fIntersections->used();
michael@0:     }
michael@0: 
michael@0:     int verticalIntersect(double axisIntercept, double roots[2]) {
michael@0:         double D = fQuad[2].fX;  // f
michael@0:         double E = fQuad[1].fX;  // e
michael@0:         double F = fQuad[0].fX;  // d
michael@0:         D += F - 2 * E;         // D = d - 2*e + f
michael@0:         E -= F;                 // E = -(d - e)
michael@0:         F -= axisIntercept;
michael@0:         return SkDQuad::RootsValidT(D, 2 * E, F, roots);
michael@0:     }
michael@0: 
michael@0:     int verticalIntersect(double axisIntercept, double top, double bottom, bool flipped) {
michael@0:         addExactVerticalEndPoints(top, bottom, axisIntercept);
michael@0:         if (fAllowNear) {
michael@0:             addNearVerticalEndPoints(top, bottom, axisIntercept);
michael@0:         }
michael@0:         double rootVals[2];
michael@0:         int roots = verticalIntersect(axisIntercept, rootVals);
michael@0:         for (int index = 0; index < roots; ++index) {
michael@0:             double quadT = rootVals[index];
michael@0:             SkDPoint pt = fQuad.ptAtT(quadT);
michael@0:             double lineT = (pt.fY - top) / (bottom - top);
michael@0:             if (pinTs(&quadT, &lineT, &pt, kPointInitialized)) {
michael@0:                 fIntersections->insert(quadT, lineT, pt);
michael@0:             }
michael@0:         }
michael@0:         if (flipped) {
michael@0:             fIntersections->flip();
michael@0:         }
michael@0:         return fIntersections->used();
michael@0:     }
michael@0: 
michael@0: protected:
michael@0:     // add endpoints first to get zero and one t values exactly
michael@0:     void addExactEndPoints() {
michael@0:         for (int qIndex = 0; qIndex < 3; qIndex += 2) {
michael@0:             double lineT = fLine.exactPoint(fQuad[qIndex]);
michael@0:             if (lineT < 0) {
michael@0:                 continue;
michael@0:             }
michael@0:             double quadT = (double) (qIndex >> 1);
michael@0:             fIntersections->insert(quadT, lineT, fQuad[qIndex]);
michael@0:         }
michael@0:     }
michael@0: 
michael@0:     void addNearEndPoints() {
michael@0:         for (int qIndex = 0; qIndex < 3; qIndex += 2) {
michael@0:             double quadT = (double) (qIndex >> 1);
michael@0:             if (fIntersections->hasT(quadT)) {
michael@0:                 continue;
michael@0:             }
michael@0:             double lineT = fLine.nearPoint(fQuad[qIndex]);
michael@0:             if (lineT < 0) {
michael@0:                 continue;
michael@0:             }
michael@0:             fIntersections->insert(quadT, lineT, fQuad[qIndex]);
michael@0:         }
michael@0:         // FIXME: see if line end is nearly on quad
michael@0:     }
michael@0: 
michael@0:     void addExactHorizontalEndPoints(double left, double right, double y) {
michael@0:         for (int qIndex = 0; qIndex < 3; qIndex += 2) {
michael@0:             double lineT = SkDLine::ExactPointH(fQuad[qIndex], left, right, y);
michael@0:             if (lineT < 0) {
michael@0:                 continue;
michael@0:             }
michael@0:             double quadT = (double) (qIndex >> 1);
michael@0:             fIntersections->insert(quadT, lineT, fQuad[qIndex]);
michael@0:         }
michael@0:     }
michael@0: 
michael@0:     void addNearHorizontalEndPoints(double left, double right, double y) {
michael@0:         for (int qIndex = 0; qIndex < 3; qIndex += 2) {
michael@0:             double quadT = (double) (qIndex >> 1);
michael@0:             if (fIntersections->hasT(quadT)) {
michael@0:                 continue;
michael@0:             }
michael@0:             double lineT = SkDLine::NearPointH(fQuad[qIndex], left, right, y);
michael@0:             if (lineT < 0) {
michael@0:                 continue;
michael@0:             }
michael@0:             fIntersections->insert(quadT, lineT, fQuad[qIndex]);
michael@0:         }
michael@0:         // FIXME: see if line end is nearly on quad
michael@0:     }
michael@0: 
michael@0:     void addExactVerticalEndPoints(double top, double bottom, double x) {
michael@0:         for (int qIndex = 0; qIndex < 3; qIndex += 2) {
michael@0:             double lineT = SkDLine::ExactPointV(fQuad[qIndex], top, bottom, x);
michael@0:             if (lineT < 0) {
michael@0:                 continue;
michael@0:             }
michael@0:             double quadT = (double) (qIndex >> 1);
michael@0:             fIntersections->insert(quadT, lineT, fQuad[qIndex]);
michael@0:         }
michael@0:     }
michael@0: 
michael@0:     void addNearVerticalEndPoints(double top, double bottom, double x) {
michael@0:         for (int qIndex = 0; qIndex < 3; qIndex += 2) {
michael@0:             double quadT = (double) (qIndex >> 1);
michael@0:             if (fIntersections->hasT(quadT)) {
michael@0:                 continue;
michael@0:             }
michael@0:             double lineT = SkDLine::NearPointV(fQuad[qIndex], top, bottom, x);
michael@0:             if (lineT < 0) {
michael@0:                 continue;
michael@0:             }
michael@0:             fIntersections->insert(quadT, lineT, fQuad[qIndex]);
michael@0:         }
michael@0:         // FIXME: see if line end is nearly on quad
michael@0:     }
michael@0: 
michael@0:     double findLineT(double t) {
michael@0:         SkDPoint xy = fQuad.ptAtT(t);
michael@0:         double dx = fLine[1].fX - fLine[0].fX;
michael@0:         double dy = fLine[1].fY - fLine[0].fY;
michael@0:         if (fabs(dx) > fabs(dy)) {
michael@0:             return (xy.fX - fLine[0].fX) / dx;
michael@0:         }
michael@0:         return (xy.fY - fLine[0].fY) / dy;
michael@0:     }
michael@0: 
michael@0:     bool pinTs(double* quadT, double* lineT, SkDPoint* pt, PinTPoint ptSet) {
michael@0:         if (!approximately_one_or_less(*lineT)) {
michael@0:             return false;
michael@0:         }
michael@0:         if (!approximately_zero_or_more(*lineT)) {
michael@0:             return false;
michael@0:         }
michael@0:         double qT = *quadT = SkPinT(*quadT);
michael@0:         double lT = *lineT = SkPinT(*lineT);
michael@0:         if (lT == 0 || lT == 1 || (ptSet == kPointUninitialized && qT != 0 && qT != 1)) {
michael@0:             *pt = fLine.ptAtT(lT);
michael@0:         } else if (ptSet == kPointUninitialized) {
michael@0:             *pt = fQuad.ptAtT(qT);
michael@0:         }
michael@0:         SkPoint gridPt = pt->asSkPoint();
michael@0:         if (gridPt == fLine[0].asSkPoint()) {
michael@0:             *lineT = 0;
michael@0:         } else if (gridPt == fLine[1].asSkPoint()) {
michael@0:             *lineT = 1;
michael@0:         }
michael@0:         if (gridPt == fQuad[0].asSkPoint()) {
michael@0:             *quadT = 0;
michael@0:         } else if (gridPt == fQuad[2].asSkPoint()) {
michael@0:             *quadT = 1;
michael@0:         }
michael@0:         return true;
michael@0:     }
michael@0: 
michael@0: private:
michael@0:     const SkDQuad& fQuad;
michael@0:     const SkDLine& fLine;
michael@0:     SkIntersections* fIntersections;
michael@0:     bool fAllowNear;
michael@0: };
michael@0: 
michael@0: // utility for pairs of coincident quads
michael@0: static double horizontalIntersect(const SkDQuad& quad, const SkDPoint& pt) {
michael@0:     LineQuadraticIntersections q(quad, *(static_cast<SkDLine*>(0)),
michael@0:             static_cast<SkIntersections*>(0));
michael@0:     double rootVals[2];
michael@0:     int roots = q.horizontalIntersect(pt.fY, rootVals);
michael@0:     for (int index = 0; index < roots; ++index) {
michael@0:         double t = rootVals[index];
michael@0:         SkDPoint qPt = quad.ptAtT(t);
michael@0:         if (AlmostEqualUlps(qPt.fX, pt.fX)) {
michael@0:             return t;
michael@0:         }
michael@0:     }
michael@0:     return -1;
michael@0: }
michael@0: 
michael@0: static double verticalIntersect(const SkDQuad& quad, const SkDPoint& pt) {
michael@0:     LineQuadraticIntersections q(quad, *(static_cast<SkDLine*>(0)),
michael@0:             static_cast<SkIntersections*>(0));
michael@0:     double rootVals[2];
michael@0:     int roots = q.verticalIntersect(pt.fX, rootVals);
michael@0:     for (int index = 0; index < roots; ++index) {
michael@0:         double t = rootVals[index];
michael@0:         SkDPoint qPt = quad.ptAtT(t);
michael@0:         if (AlmostEqualUlps(qPt.fY, pt.fY)) {
michael@0:             return t;
michael@0:         }
michael@0:     }
michael@0:     return -1;
michael@0: }
michael@0: 
michael@0: double SkIntersections::Axial(const SkDQuad& q1, const SkDPoint& p, bool vertical) {
michael@0:     if (vertical) {
michael@0:         return verticalIntersect(q1, p);
michael@0:     }
michael@0:     return horizontalIntersect(q1, p);
michael@0: }
michael@0: 
michael@0: int SkIntersections::horizontal(const SkDQuad& quad, double left, double right, double y,
michael@0:                                 bool flipped) {
michael@0:     SkDLine line = {{{ left, y }, { right, y }}};
michael@0:     LineQuadraticIntersections q(quad, line, this);
michael@0:     return q.horizontalIntersect(y, left, right, flipped);
michael@0: }
michael@0: 
michael@0: int SkIntersections::vertical(const SkDQuad& quad, double top, double bottom, double x,
michael@0:                               bool flipped) {
michael@0:     SkDLine line = {{{ x, top }, { x, bottom }}};
michael@0:     LineQuadraticIntersections q(quad, line, this);
michael@0:     return q.verticalIntersect(x, top, bottom, flipped);
michael@0: }
michael@0: 
michael@0: int SkIntersections::intersect(const SkDQuad& quad, const SkDLine& line) {
michael@0:     LineQuadraticIntersections q(quad, line, this);
michael@0:     q.allowNear(fAllowNear);
michael@0:     return q.intersect();
michael@0: }
michael@0: 
michael@0: int SkIntersections::intersectRay(const SkDQuad& quad, const SkDLine& line) {
michael@0:     LineQuadraticIntersections q(quad, line, this);
michael@0:     fUsed = q.intersectRay(fT[0]);
michael@0:     for (int index = 0; index < fUsed; ++index) {
michael@0:         fPt[index] = quad.ptAtT(fT[0][index]);
michael@0:     }
michael@0:     return fUsed;
michael@0: }