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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 "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(2);

     }

     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]);

             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(*lineT)) {

             return false;

         }

         if (!approximately_zero_or_more(*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 (gridPt == fLine[0].asSkPoint()) {

             *lineT = 0;

         } else if (gridPt == fLine[1].asSkPoint()) {

             *lineT = 1;

         }

         if (gridPt == fQuad[0].asSkPoint()) {

             *quadT = 0;

         } else if (gridPt == fQuad[2].asSkPoint()) {

             *quadT = 1;

         }

         return true;

     }

 private:

     const SkDQuad& fQuad;

     const SkDLine& fLine;

     SkIntersections* fIntersections;

     bool fAllowNear;

 };

 // utility for pairs of coincident quads

 static double horizontalIntersect(const SkDQuad& quad, const SkDPoint& pt) {

     LineQuadraticIntersections q(quad, *(static_cast<SkDLine*>(0)),

             static_cast<SkIntersections*>(0));

     double rootVals[2];

     int roots = q.horizontalIntersect(pt.fY, rootVals);

     for (int index = 0; index < roots; ++index) {

         double t = rootVals[index];

         SkDPoint qPt = quad.ptAtT(t);

         if (AlmostEqualUlps(qPt.fX, pt.fX)) {

             return t;

         }

     }

     return -1;

 }

 static double verticalIntersect(const SkDQuad& quad, const SkDPoint& pt) {

     LineQuadraticIntersections q(quad, *(static_cast<SkDLine*>(0)),

             static_cast<SkIntersections*>(0));

     double rootVals[2];

     int roots = q.verticalIntersect(pt.fX, rootVals);

     for (int index = 0; index < roots; ++index) {

         double t = rootVals[index];

         SkDPoint qPt = quad.ptAtT(t);

         if (AlmostEqualUlps(qPt.fY, pt.fY)) {

             return t;

         }

     }

     return -1;

 }

 double SkIntersections::Axial(const SkDQuad& q1, const SkDPoint& p, bool vertical) {

     if (vertical) {

         return verticalIntersect(q1, p);

     }

     return horizontalIntersect(q1, p);

 }

 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;

 }