text: /*
   * Copyright 2015 Google Inc.
   *
   * Use of this source code is governed by a BSD-style license that can be
   * found in the LICENSE file.
   */
  
  #include "GrTessellatingPathRenderer.h"
  
  #include "GrDefaultGeoProcFactory.h"
  #include "GrPathUtils.h"
  #include "SkChunkAlloc.h"
  #include "SkGeometry.h"
  
  #include <stdio.h>
  
  /*
   * This path renderer tessellates the path into triangles, uploads the triangles to a
   * vertex buffer, and renders them with a single draw call. It does not currently do
   * antialiasing, so it must be used in conjunction with multisampling.
   *
   * There are six stages to the algorithm:
   *
   * 1) Linearize the path contours into piecewise linear segments (path_to_contours()).
   * 2) Build a mesh of edges connecting the vertices (build_edges()).
   * 3) Sort the vertices in Y (and secondarily in X) (merge_sort()).
   * 4) Simplify the mesh by inserting new vertices at intersecting edges (simplify()).
   * 5) Tessellate the simplified mesh into monotone polygons (tessellate()).
   * 6) Triangulate the monotone polygons directly into a vertex buffer (polys_to_triangles()).
   *
   * The vertex sorting in step (3) is a merge sort, since it plays well with the linked list
   * of vertices (and the necessity of inserting new vertices on intersection).
   *
   * Stages (4) and (5) use an active edge list, which a list of all edges for which the
   * sweep line has crossed the top vertex, but not the bottom vertex.  It's sorted
   * left-to-right based on the point where both edges are active (when both top vertices
   * have been seen, so the "lower" top vertex of the two). If the top vertices are equal
   * (shared), it's sorted based on the last point where both edges are active, so the
   * "upper" bottom vertex.
   *
   * The most complex step is the simplification (4). It's based on the Bentley-Ottman
   * line-sweep algorithm, but due to floating point inaccuracy, the intersection points are
   * not exact and may violate the mesh topology or active edge list ordering. We
   * accommodate this by adjusting the topology of the mesh and AEL to match the intersection
   * points. This occurs in three ways:
   *
   * A) Intersections may cause a shortened edge to no longer be ordered with respect to its
   *    neighbouring edges at the top or bottom vertex. This is handled by merging the
   *    edges (merge_collinear_edges()).
   * B) Intersections may cause an edge to violate the left-to-right ordering of the
   *    active edge list. This is handled by splitting the neighbour edge on the
   *    intersected vertex (cleanup_active_edges()).
   * C) Shortening an edge may cause an active edge to become inactive or an inactive edge
   *    to become active. This is handled by removing or inserting the edge in the active
   *    edge list (fix_active_state()).
   *
   * The tessellation steps (5) and (6) are based on "Triangulating Simple Polygons and
   * Equivalent Problems" (Fournier and Montuno); also a line-sweep algorithm. Note that it
   * currently uses a linked list for the active edge list, rather than a 2-3 tree as the
   * paper describes. The 2-3 tree gives O(lg N) lookups, but insertion and removal also
   * become O(lg N). In all the test cases, it was found that the cost of frequent O(lg N)
   * insertions and removals was greater than the cost of infrequent O(N) lookups with the
   * linked list implementation. With the latter, all removals are O(1), and most insertions
   * are O(1), since we know the adjacent edge in the active edge list based on the topology.
   * Only type 2 vertices (see paper) require the O(N) lookups, and these are much less
   * frequent. There may be other data structures worth investigating, however.
   *
   * Note that there is a compile-time flag (SWEEP_IN_X) which changes the orientation of the
   * line sweep algorithms. When SWEEP_IN_X is unset, we sort vertices based on increasing
   * Y coordinate, and secondarily by increasing X coordinate. When SWEEP_IN_X is set, we sort by
   * increasing X coordinate, but secondarily by *decreasing* Y coordinate. This is so that the
   * "left" and "right" orientation in the code remains correct (edges to the left are increasing
   * in Y; edges to the right are decreasing in Y). That is, the setting rotates 90 degrees
   * counterclockwise, rather that transposing.
   *
   * The choice is arbitrary, but most test cases are wider than they are tall, so the
   * default is to sweep in X. In the future, we may want to make this a runtime parameter
   * and base it on the aspect ratio of the clip bounds.
   */
  #define LOGGING_ENABLED 0
  #define WIREFRAME 0
  #define SWEEP_IN_X 1
  
  #if LOGGING_ENABLED
  #define LOG printf
  #else
  #define LOG(...)
  #endif
  
  #define ALLOC_NEW(Type, args, alloc) \
      SkNEW_PLACEMENT_ARGS(alloc.allocThrow(sizeof(Type)), Type, args)
  
  namespace {
  
  struct Vertex;
  struct Edge;
  struct Poly;
  
  template <class T, T* T::*Prev, T* T::*Next>
  void insert(T* t, T* prev, T* next, T** head, T** tail) {
      t->*Prev = prev;
      t->*Next = next;
      if (prev) {
          prev->*Next = t;
      } else if (head) {
          *head = t;
      }
      if (next) {
          next->*Prev = t;
      } else if (tail) {
          *tail = t;
      }
  }
  
  template <class T, T* T::*Prev, T* T::*Next>
  void remove(T* t, T** head, T** tail) {
      if (t->*Prev) {
          t->*Prev->*Next = t->*Next;
      } else if (head) {
          *head = t->*Next;
      }
      if (t->*Next) {
          t->*Next->*Prev = t->*Prev;
      } else if (tail) {
          *tail = t->*Prev;
      }
      t->*Prev = t->*Next = NULL;
  }
  
  /**
   * Vertices are used in three ways: first, the path contours are converted into a
   * circularly-linked list of Vertices for each contour. After edge construction, the same Vertices
   * are re-ordered by the merge sort according to the sweep_lt comparator (usually, increasing
   * in Y) using the same fPrev/fNext pointers that were used for the contours, to avoid
   * reallocation. Finally, MonotonePolys are built containing a circularly-linked list of
   * Vertices. (Currently, those Vertices are newly-allocated for the MonotonePolys, since
   * an individual Vertex from the path mesh may belong to multiple
   * MonotonePolys, so the original Vertices cannot be re-used.
   */
  
  struct Vertex {
    Vertex(const SkPoint& point)
      : fPoint(point), fPrev(NULL), fNext(NULL)
      , fFirstEdgeAbove(NULL), fLastEdgeAbove(NULL)
      , fFirstEdgeBelow(NULL), fLastEdgeBelow(NULL)
      , fProcessed(false)
  #if LOGGING_ENABLED
      , fID (-1.0f)
  #endif
      {}
      SkPoint fPoint;           // Vertex position
      Vertex* fPrev;            // Linked list of contours, then Y-sorted vertices.
      Vertex* fNext;            // "
      Edge*   fFirstEdgeAbove;  // Linked list of edges above this vertex.
      Edge*   fLastEdgeAbove;   // "
      Edge*   fFirstEdgeBelow;  // Linked list of edges below this vertex.
      Edge*   fLastEdgeBelow;   // "
      bool    fProcessed;       // Has this vertex been seen in simplify()?
  #if LOGGING_ENABLED
      float   fID;              // Identifier used for logging.
  #endif
  };
  
  /***************************************************************************************/
  
  bool sweep_lt(const SkPoint& a, const SkPoint& b) {
  #if SWEEP_IN_X
      return a.fX == b.fX ? a.fY > b.fY : a.fX < b.fX;
  #else
      return a.fY == b.fY ? a.fX < b.fX : a.fY < b.fY;
  #endif
  }
  
  bool sweep_gt(const SkPoint& a, const SkPoint& b) {
  #if SWEEP_IN_X
      return a.fX == b.fX ? a.fY < b.fY : a.fX > b.fX;
  #else
      return a.fY == b.fY ? a.fX > b.fX : a.fY > b.fY;
  #endif
  }
  
  inline void* emit_vertex(Vertex* v, void* data) {
      SkPoint* d = static_cast<SkPoint*>(data);
      *d++ = v->fPoint;
      return d;
  }
  
  void* emit_triangle(Vertex* v0, Vertex* v1, Vertex* v2, void* data) {
  #if WIREFRAME
      data = emit_vertex(v0, data);
      data = emit_vertex(v1, data);
      data = emit_vertex(v1, data);
      data = emit_vertex(v2, data);
      data = emit_vertex(v2, data);
      data = emit_vertex(v0, data);
  #else
      data = emit_vertex(v0, data);
      data = emit_vertex(v1, data);
      data = emit_vertex(v2, data);
  #endif
      return data;
  }
  
  /**
   * An Edge joins a top Vertex to a bottom Vertex. Edge ordering for the list of "edges above" and
   * "edge below" a vertex as well as for the active edge list is handled by isLeftOf()/isRightOf().
   * Note that an Edge will give occasionally dist() != 0 for its own endpoints (because floating
   * point). For speed, that case is only tested by the callers which require it (e.g.,
   * cleanup_active_edges()). Edges also handle checking for intersection with other edges.
   * Currently, this converts the edges to the parametric form, in order to avoid doing a division
   * until an intersection has been confirmed. This is slightly slower in the "found" case, but
   * a lot faster in the "not found" case.
   *
   * The coefficients of the line equation stored in double precision to avoid catastrphic
   * cancellation in the isLeftOf() and isRightOf() checks. Using doubles ensures that the result is
   * correct in float, since it's a polynomial of degree 2. The intersect() function, being
   * degree 5, is still subject to catastrophic cancellation. We deal with that by assuming its
   * output may be incorrect, and adjusting the mesh topology to match (see comment at the top of
   * this file).
   */
  
  struct Edge {
      Edge(Vertex* top, Vertex* bottom, int winding)
          : fWinding(winding)
          , fTop(top)
          , fBottom(bottom)
          , fLeft(NULL)
          , fRight(NULL)
          , fPrevEdgeAbove(NULL)
          , fNextEdgeAbove(NULL)
          , fPrevEdgeBelow(NULL)
          , fNextEdgeBelow(NULL)
          , fLeftPoly(NULL)
          , fRightPoly(NULL) {
              recompute();
          }
      int      fWinding;          // 1 == edge goes downward; -1 = edge goes upward.
      Vertex*  fTop;              // The top vertex in vertex-sort-order (sweep_lt).
      Vertex*  fBottom;           // The bottom vertex in vertex-sort-order.
      Edge*    fLeft;             // The linked list of edges in the active edge list.
      Edge*    fRight;            // "
      Edge*    fPrevEdgeAbove;    // The linked list of edges in the bottom Vertex's "edges above".
      Edge*    fNextEdgeAbove;    // "
      Edge*    fPrevEdgeBelow;    // The linked list of edges in the top Vertex's "edges below".
      Edge*    fNextEdgeBelow;    // "
      Poly*    fLeftPoly;         // The Poly to the left of this edge, if any.
      Poly*    fRightPoly;        // The Poly to the right of this edge, if any.
      double   fDX;               // The line equation for this edge, in implicit form.
      double   fDY;               // fDY * x + fDX * y + fC = 0, for point (x, y) on the line.
      double   fC;
      double dist(const SkPoint& p) const {
          return fDY * p.fX - fDX * p.fY + fC;
      }
      bool isRightOf(Vertex* v) const {
          return dist(v->fPoint) < 0.0;
      }
      bool isLeftOf(Vertex* v) const {
          return dist(v->fPoint) > 0.0;
      }
      void recompute() {
          fDX = static_cast<double>(fBottom->fPoint.fX) - fTop->fPoint.fX;
          fDY = static_cast<double>(fBottom->fPoint.fY) - fTop->fPoint.fY;
          fC = static_cast<double>(fTop->fPoint.fY) * fBottom->fPoint.fX -
               static_cast<double>(fTop->fPoint.fX) * fBottom->fPoint.fY;
      }
      bool intersect(const Edge& other, SkPoint* p) {
          LOG("intersecting %g -> %g with %g -> %g\n",
                 fTop->fID, fBottom->fID,
                 other.fTop->fID, other.fBottom->fID);
          if (fTop == other.fTop || fBottom == other.fBottom) {
              return false;
          }
          double denom = fDX * other.fDY - fDY * other.fDX;
          if (denom == 0.0) {
              return false;
          }
          double dx = static_cast<double>(fTop->fPoint.fX) - other.fTop->fPoint.fX;
          double dy = static_cast<double>(fTop->fPoint.fY) - other.fTop->fPoint.fY;
          double sNumer = dy * other.fDX - dx * other.fDY;
          double tNumer = dy * fDX - dx * fDY;
          // If (sNumer / denom) or (tNumer / denom) is not in [0..1], exit early.
          // This saves us doing the divide below unless absolutely necessary.
          if (denom > 0.0 ? (sNumer < 0.0 || sNumer > denom || tNumer < 0.0 || tNumer > denom)
                          : (sNumer > 0.0 || sNumer < denom || tNumer > 0.0 || tNumer < denom)) {
              return false;
          }
          double s = sNumer / denom;
          SkASSERT(s >= 0.0 && s <= 1.0);
          p->fX = SkDoubleToScalar(fTop->fPoint.fX + s * fDX);
          p->fY = SkDoubleToScalar(fTop->fPoint.fY + s * fDY);
          return true;
      }
      bool isActive(Edge** activeEdges) const {
          return activeEdges && (fLeft || fRight || *activeEdges == this);
      }
  };
  
  /***************************************************************************************/
  
  struct Poly {
      Poly(int winding)
          : fWinding(winding)
          , fHead(NULL)
          , fTail(NULL)
          , fActive(NULL)
          , fNext(NULL)
          , fPartner(NULL)
          , fCount(0)
      {
  #if LOGGING_ENABLED
          static int gID = 0;
          fID = gID++;
          LOG("*** created Poly %d\n", fID);
  #endif
      }
      typedef enum { kNeither_Side, kLeft_Side, kRight_Side } Side;
      struct MonotonePoly {
          MonotonePoly()
              : fSide(kNeither_Side)
              , fHead(NULL)
              , fTail(NULL)
              , fPrev(NULL)
              , fNext(NULL) {}
          Side          fSide;
          Vertex*       fHead;
          Vertex*       fTail;
          MonotonePoly* fPrev;
          MonotonePoly* fNext;
          bool addVertex(Vertex* v, Side side, SkChunkAlloc& alloc) {
              Vertex* newV = ALLOC_NEW(Vertex, (v->fPoint), alloc);
              bool done = false;
              if (fSide == kNeither_Side) {
                  fSide = side;
              } else {
                  done = side != fSide;
              }
              if (fHead == NULL) {
                  fHead = fTail = newV;
              } else if (fSide == kRight_Side) {
                  newV->fPrev = fTail;
                  fTail->fNext = newV;
                  fTail = newV;
              } else {
                  newV->fNext = fHead;
                  fHead->fPrev = newV;
                  fHead = newV;
              }
              return done;
          }
  
          void* emit(void* data) {
              Vertex* first = fHead;
              Vertex* v = first->fNext;
              while (v != fTail) {
                  SkASSERT(v && v->fPrev && v->fNext);
  #ifdef SK_DEBUG
                  validate();
  #endif
                  Vertex* prev = v->fPrev;
                  Vertex* curr = v;
                  Vertex* next = v->fNext;
                  double ax = static_cast<double>(curr->fPoint.fX) - prev->fPoint.fX;
                  double ay = static_cast<double>(curr->fPoint.fY) - prev->fPoint.fY;
                  double bx = static_cast<double>(next->fPoint.fX) - curr->fPoint.fX;
                  double by = static_cast<double>(next->fPoint.fY) - curr->fPoint.fY;
                  if (ax * by - ay * bx >= 0.0) {
                      data = emit_triangle(prev, curr, next, data);
                      v->fPrev->fNext = v->fNext;
                      v->fNext->fPrev = v->fPrev;
                      if (v->fPrev == first) {
                          v = v->fNext;
                      } else {
                          v = v->fPrev;
                      }
                  } else {
                      v = v->fNext;
                      SkASSERT(v != fTail);
                  }
              }
              return data;
          }
  
  #ifdef SK_DEBUG
          void validate() {
              int winding = sweep_lt(fHead->fPoint, fTail->fPoint) ? 1 : -1;
              Vertex* top = winding < 0 ? fTail : fHead;
              Vertex* bottom = winding < 0 ? fHead : fTail;
              Edge e(top, bottom, winding);
              for (Vertex* v = fHead->fNext; v != fTail; v = v->fNext) {
                  if (fSide == kRight_Side) {
                      SkASSERT(!e.isRightOf(v));
                  } else if (fSide == Poly::kLeft_Side) {
                      SkASSERT(!e.isLeftOf(v));
                  }
              }
          }
  #endif
      };
      Poly* addVertex(Vertex* v, Side side, SkChunkAlloc& alloc) {
          LOG("addVertex() to %d at %g (%g, %g), %s side\n", fID, v->fID, v->fPoint.fX, v->fPoint.fY,
                 side == kLeft_Side ? "left" : side == kRight_Side ? "right" : "neither");
          Poly* partner = fPartner;
          Poly* poly = this;
          if (partner) {
              fPartner = partner->fPartner = NULL;
          }
          if (!fActive) {
              fActive = ALLOC_NEW(MonotonePoly, (), alloc);
          }
          if (fActive->addVertex(v, side, alloc)) {
  #ifdef SK_DEBUG
              fActive->validate();
  #endif
              if (fTail) {
                  fActive->fPrev = fTail;
                  fTail->fNext = fActive;
                  fTail = fActive;
              } else {
                  fHead = fTail = fActive;
              }
              if (partner) {
                  partner->addVertex(v, side, alloc);
                  poly = partner;
              } else {
                  Vertex* prev = fActive->fSide == Poly::kLeft_Side ?
                                 fActive->fHead->fNext : fActive->fTail->fPrev;
                  fActive = ALLOC_NEW(MonotonePoly, , alloc);
                  fActive->addVertex(prev, Poly::kNeither_Side, alloc);
                  fActive->addVertex(v, side, alloc);
              }
          }
          fCount++;
          return poly;
      }
      void end(Vertex* v, SkChunkAlloc& alloc) {
          LOG("end() %d at %g, %g\n", fID, v->fPoint.fX, v->fPoint.fY);
          if (fPartner) {
              fPartner = fPartner->fPartner = NULL;
          }
          addVertex(v, fActive->fSide == kLeft_Side ? kRight_Side : kLeft_Side, alloc);
      }
      void* emit(void *data) {
          if (fCount < 3) {
              return data;
          }
          LOG("emit() %d, size %d\n", fID, fCount);
          for (MonotonePoly* m = fHead; m != NULL; m = m->fNext) {
              data = m->emit(data);
          }
          return data;
      }
      int fWinding;
      MonotonePoly* fHead;
      MonotonePoly* fTail;
      MonotonePoly* fActive;
      Poly* fNext;
      Poly* fPartner;
      int fCount;
  #if LOGGING_ENABLED
      int fID;
  #endif
  };
  
  /***************************************************************************************/
  
  bool coincident(const SkPoint& a, const SkPoint& b) {
      return a == b;
  }
  
  Poly* new_poly(Poly** head, Vertex* v, int winding, SkChunkAlloc& alloc) {
      Poly* poly = ALLOC_NEW(Poly, (winding), alloc);
      poly->addVertex(v, Poly::kNeither_Side, alloc);
      poly->fNext = *head;
      *head = poly;
      return poly;
  }
  
  #ifdef SK_DEBUG
  void validate_edges(Edge* head) {
      for (Edge* e = head; e != NULL; e = e->fRight) {
          SkASSERT(e->fTop != e->fBottom);
          if (e->fLeft) {
              SkASSERT(e->fLeft->fRight == e);
              if (sweep_gt(e->fTop->fPoint, e->fLeft->fTop->fPoint)) {
                  SkASSERT(e->fLeft->isLeftOf(e->fTop));
              }
              if (sweep_lt(e->fBottom->fPoint, e->fLeft->fBottom->fPoint)) {
                  SkASSERT(e->fLeft->isLeftOf(e->fBottom));
              }
          } else {
              SkASSERT(e == head);
          }
          if (e->fRight) {
              SkASSERT(e->fRight->fLeft == e);
              if (sweep_gt(e->fTop->fPoint, e->fRight->fTop->fPoint)) {
                  SkASSERT(e->fRight->isRightOf(e->fTop));
              }
              if (sweep_lt(e->fBottom->fPoint, e->fRight->fBottom->fPoint)) {
                  SkASSERT(e->fRight->isRightOf(e->fBottom));
              }
          }
      }
  }
  
  void validate_connectivity(Vertex* v) {
      for (Edge* e = v->fFirstEdgeAbove; e != NULL; e = e->fNextEdgeAbove) {
          SkASSERT(e->fBottom == v);
          if (e->fPrevEdgeAbove) {
              SkASSERT(e->fPrevEdgeAbove->fNextEdgeAbove == e);
              SkASSERT(e->fPrevEdgeAbove->isLeftOf(e->fTop));
          } else {
              SkASSERT(e == v->fFirstEdgeAbove);
          }
          if (e->fNextEdgeAbove) {
              SkASSERT(e->fNextEdgeAbove->fPrevEdgeAbove == e);
              SkASSERT(e->fNextEdgeAbove->isRightOf(e->fTop));
          } else {
              SkASSERT(e == v->fLastEdgeAbove);
          }
      }
      for (Edge* e = v->fFirstEdgeBelow; e != NULL; e = e->fNextEdgeBelow) {
          SkASSERT(e->fTop == v);
          if (e->fPrevEdgeBelow) {
              SkASSERT(e->fPrevEdgeBelow->fNextEdgeBelow == e);
              SkASSERT(e->fPrevEdgeBelow->isLeftOf(e->fBottom));
          } else {
              SkASSERT(e == v->fFirstEdgeBelow);
          }
          if (e->fNextEdgeBelow) {
              SkASSERT(e->fNextEdgeBelow->fPrevEdgeBelow == e);
              SkASSERT(e->fNextEdgeBelow->isRightOf(e->fBottom));
          } else {
              SkASSERT(e == v->fLastEdgeBelow);
          }
      }
  }
  #endif
  
  Vertex* append_point_to_contour(const SkPoint& p, Vertex* prev, Vertex** head,
                                  SkChunkAlloc& alloc) {
      Vertex* v = ALLOC_NEW(Vertex, (p), alloc);
  #if LOGGING_ENABLED
      static float gID = 0.0f;
      v->fID = gID++;
  #endif
      if (prev) {
          prev->fNext = v;
          v->fPrev = prev;
      } else {
          *head = v;
      }
      return v;
  }
  
  Vertex* generate_quadratic_points(const SkPoint& p0,
                                    const SkPoint& p1,
                                    const SkPoint& p2,
                                    SkScalar tolSqd,
                                    Vertex* prev,
                                    Vertex** head,
                                    int pointsLeft,
                                    SkChunkAlloc& alloc) {
      SkScalar d = p1.distanceToLineSegmentBetweenSqd(p0, p2);
      if (pointsLeft < 2 || d < tolSqd || !SkScalarIsFinite(d)) {
          return append_point_to_contour(p2, prev, head, alloc);
      }
  
      const SkPoint q[] = {
          { SkScalarAve(p0.fX, p1.fX), SkScalarAve(p0.fY, p1.fY) },
          { SkScalarAve(p1.fX, p2.fX), SkScalarAve(p1.fY, p2.fY) },
      };
      const SkPoint r = { SkScalarAve(q[0].fX, q[1].fX), SkScalarAve(q[0].fY, q[1].fY) };
  
      pointsLeft >>= 1;
      prev = generate_quadratic_points(p0, q[0], r, tolSqd, prev, head, pointsLeft, alloc);
      prev = generate_quadratic_points(r, q[1], p2, tolSqd, prev, head, pointsLeft, alloc);
      return prev;
  }
  
  Vertex* generate_cubic_points(const SkPoint& p0,
                                const SkPoint& p1,
                                const SkPoint& p2,
                                const SkPoint& p3,
                                SkScalar tolSqd,
                                Vertex* prev,
                                Vertex** head,
                                int pointsLeft,
                                SkChunkAlloc& alloc) {
      SkScalar d1 = p1.distanceToLineSegmentBetweenSqd(p0, p3);
      SkScalar d2 = p2.distanceToLineSegmentBetweenSqd(p0, p3);
      if (pointsLeft < 2 || (d1 < tolSqd && d2 < tolSqd) ||
          !SkScalarIsFinite(d1) || !SkScalarIsFinite(d2)) {
          return append_point_to_contour(p3, prev, head, alloc);
      }
      const SkPoint q[] = {
          { SkScalarAve(p0.fX, p1.fX), SkScalarAve(p0.fY, p1.fY) },
          { SkScalarAve(p1.fX, p2.fX), SkScalarAve(p1.fY, p2.fY) },
          { SkScalarAve(p2.fX, p3.fX), SkScalarAve(p2.fY, p3.fY) }
      };
      const SkPoint r[] = {
          { SkScalarAve(q[0].fX, q[1].fX), SkScalarAve(q[0].fY, q[1].fY) },
          { SkScalarAve(q[1].fX, q[2].fX), SkScalarAve(q[1].fY, q[2].fY) }
      };
      const SkPoint s = { SkScalarAve(r[0].fX, r[1].fX), SkScalarAve(r[0].fY, r[1].fY) };
      pointsLeft >>= 1;
      prev = generate_cubic_points(p0, q[0], r[0], s, tolSqd, prev, head, pointsLeft, alloc);
      prev = generate_cubic_points(s, r[1], q[2], p3, tolSqd, prev, head, pointsLeft, alloc);
      return prev;
  }
  
  // Stage 1: convert the input path to a set of linear contours (linked list of Vertices).
  
  void path_to_contours(const SkPath& path, SkScalar tolerance, const SkRect& clipBounds,
                        Vertex** contours, SkChunkAlloc& alloc) {
  
      SkScalar toleranceSqd = tolerance * tolerance;
  
      SkPoint pts[4];
      bool done = false;
      SkPath::Iter iter(path, false);
      Vertex* prev = NULL;
      Vertex* head = NULL;
      if (path.isInverseFillType()) {
          SkPoint quad[4];
          clipBounds.toQuad(quad);
          for (int i = 3; i >= 0; i--) {
              prev = append_point_to_contour(quad[i], prev, &head, alloc);
          }
          head->fPrev = prev;
          prev->fNext = head;
          *contours++ = head;
          head = prev = NULL;
      }
      SkAutoConicToQuads converter;
      while (!done) {
          SkPath::Verb verb = iter.next(pts);
          switch (verb) {
              case SkPath::kConic_Verb: {
                  SkScalar weight = iter.conicWeight();
                  const SkPoint* quadPts = converter.computeQuads(pts, weight, toleranceSqd);
                  for (int i = 0; i < converter.countQuads(); ++i) {
                      int pointsLeft = GrPathUtils::quadraticPointCount(quadPts, toleranceSqd);
                      prev = generate_quadratic_points(quadPts[0], quadPts[1], quadPts[2],
                                                       toleranceSqd, prev, &head, pointsLeft, alloc);
                      quadPts += 2;
                  }
                  break;
              }
              case SkPath::kMove_Verb:
                  if (head) {
                      head->fPrev = prev;
                      prev->fNext = head;
                      *contours++ = head;
                  }
                  head = prev = NULL;
                  prev = append_point_to_contour(pts[0], prev, &head, alloc);
                  break;
              case SkPath::kLine_Verb: {
                  prev = append_point_to_contour(pts[1], prev, &head, alloc);
                  break;
              }
              case SkPath::kQuad_Verb: {
                  int pointsLeft = GrPathUtils::quadraticPointCount(pts, toleranceSqd);
                  prev = generate_quadratic_points(pts[0], pts[1], pts[2], toleranceSqd, prev,
                                                   &head, pointsLeft, alloc);
                  break;
              }
              case SkPath::kCubic_Verb: {
                  int pointsLeft = GrPathUtils::cubicPointCount(pts, toleranceSqd);
                  prev = generate_cubic_points(pts[0], pts[1], pts[2], pts[3],
                                  toleranceSqd, prev, &head, pointsLeft, alloc);
                  break;
              }
              case SkPath::kClose_Verb:
                  if (head) {
                      head->fPrev = prev;
                      prev->fNext = head;
                      *contours++ = head;
                  }
                  head = prev = NULL;
                  break;
              case SkPath::kDone_Verb:
                  if (head) {
                      head->fPrev = prev;
                      prev->fNext = head;
                      *contours++ = head;
                  }
                  done = true;
                  break;
          }
      }
  }
  
  inline bool apply_fill_type(SkPath::FillType fillType, int winding) {
      switch (fillType) {
          case SkPath::kWinding_FillType:
              return winding != 0;
          case SkPath::kEvenOdd_FillType:
              return (winding & 1) != 0;
          case SkPath::kInverseWinding_FillType:
              return winding == 1;
          case SkPath::kInverseEvenOdd_FillType:
              return (winding & 1) == 1;
          default:
              SkASSERT(false);
              return false;
      }
  }
  
  Edge* new_edge(Vertex* prev, Vertex* next, SkChunkAlloc& alloc) {
      int winding = sweep_lt(prev->fPoint, next->fPoint) ? 1 : -1;
      Vertex* top = winding < 0 ? next : prev;
      Vertex* bottom = winding < 0 ? prev : next;
      return ALLOC_NEW(Edge, (top, bottom, winding), alloc);
  }
  
  void remove_edge(Edge* edge, Edge** head) {
      LOG("removing edge %g -> %g\n", edge->fTop->fID, edge->fBottom->fID);
      SkASSERT(edge->isActive(head));
      remove<Edge, &Edge::fLeft, &Edge::fRight>(edge, head, NULL);
  }
  
  void insert_edge(Edge* edge, Edge* prev, Edge** head) {
      LOG("inserting edge %g -> %g\n", edge->fTop->fID, edge->fBottom->fID);
      SkASSERT(!edge->isActive(head));
      Edge* next = prev ? prev->fRight : *head;
      insert<Edge, &Edge::fLeft, &Edge::fRight>(edge, prev, next, head, NULL);
  }
  
  void find_enclosing_edges(Vertex* v, Edge* head, Edge** left, Edge** right) {
      if (v->fFirstEdgeAbove) {
          *left = v->fFirstEdgeAbove->fLeft;
          *right = v->fLastEdgeAbove->fRight;
          return;
      }
      Edge* prev = NULL;
      Edge* next;
      for (next = head; next != NULL; next = next->fRight) {
          if (next->isRightOf(v)) {
              break;
          }
          prev = next;
      }
      *left = prev;
      *right = next;
      return;
  }
  
  void find_enclosing_edges(Edge* edge, Edge* head, Edge** left, Edge** right) {
      Edge* prev = NULL;
      Edge* next;
      for (next = head; next != NULL; next = next->fRight) {
          if ((sweep_gt(edge->fTop->fPoint, next->fTop->fPoint) && next->isRightOf(edge->fTop)) ||
              (sweep_gt(next->fTop->fPoint, edge->fTop->fPoint) && edge->isLeftOf(next->fTop)) ||
              (sweep_lt(edge->fBottom->fPoint, next->fBottom->fPoint) &&
               next->isRightOf(edge->fBottom)) ||
              (sweep_lt(next->fBottom->fPoint, edge->fBottom->fPoint) &&
               edge->isLeftOf(next->fBottom))) {
              break;
          }
          prev = next;
      }
      *left = prev;
      *right = next;
      return;
  }
  
  void fix_active_state(Edge* edge, Edge** activeEdges) {
      if (edge->isActive(activeEdges)) {
          if (edge->fBottom->fProcessed || !edge->fTop->fProcessed) {
              remove_edge(edge, activeEdges);
          }
      } else if (edge->fTop->fProcessed && !edge->fBottom->fProcessed) {
          Edge* left;
          Edge* right;
          find_enclosing_edges(edge, *activeEdges, &left, &right);
          insert_edge(edge, left, activeEdges);
      }
  }
  
  void insert_edge_above(Edge* edge, Vertex* v) {
      if (edge->fTop->fPoint == edge->fBottom->fPoint ||
          sweep_gt(edge->fTop->fPoint, edge->fBottom->fPoint)) {
          SkASSERT(false);
          return;
      }
      LOG("insert edge (%g -> %g) above vertex %g\n", edge->fTop->fID, edge->fBottom->fID, v->fID);
      Edge* prev = NULL;
      Edge* next;
      for (next = v->fFirstEdgeAbove; next; next = next->fNextEdgeAbove) {
          if (next->isRightOf(edge->fTop)) {
              break;
          }
          prev = next;
      }
      insert<Edge, &Edge::fPrevEdgeAbove, &Edge::fNextEdgeAbove>(
          edge, prev, next, &v->fFirstEdgeAbove, &v->fLastEdgeAbove);
  }
  
  void insert_edge_below(Edge* edge, Vertex* v) {
      if (edge->fTop->fPoint == edge->fBottom->fPoint ||
          sweep_gt(edge->fTop->fPoint, edge->fBottom->fPoint)) {
          SkASSERT(false);
          return;
      }
      LOG("insert edge (%g -> %g) below vertex %g\n", edge->fTop->fID, edge->fBottom->fID, v->fID);
      Edge* prev = NULL;
      Edge* next;
      for (next = v->fFirstEdgeBelow; next; next = next->fNextEdgeBelow) {
          if (next->isRightOf(edge->fBottom)) {
              break;
          }
          prev = next;
      }
      insert<Edge, &Edge::fPrevEdgeBelow, &Edge::fNextEdgeBelow>(
          edge, prev, next, &v->fFirstEdgeBelow, &v->fLastEdgeBelow);
  }
  
  void remove_edge_above(Edge* edge) {
      LOG("removing edge (%g -> %g) above vertex %g\n", edge->fTop->fID, edge->fBottom->fID,
          edge->fBottom->fID);
      remove<Edge, &Edge::fPrevEdgeAbove, &Edge::fNextEdgeAbove>(
          edge, &edge->fBottom->fFirstEdgeAbove, &edge->fBottom->fLastEdgeAbove);
  }
  
  void remove_edge_below(Edge* edge) {
      LOG("removing edge (%g -> %g) below vertex %g\n", edge->fTop->fID, edge->fBottom->fID,
          edge->fTop->fID);
      remove<Edge, &Edge::fPrevEdgeBelow, &Edge::fNextEdgeBelow>(
          edge, &edge->fTop->fFirstEdgeBelow, &edge->fTop->fLastEdgeBelow);
  }
  
  void erase_edge_if_zero_winding(Edge* edge, Edge** head) {
      if (edge->fWinding != 0) {
          return;
      }
      LOG("erasing edge (%g -> %g)\n", edge->fTop->fID, edge->fBottom->fID);
      remove_edge_above(edge);
      remove_edge_below(edge);
      if (edge->isActive(head)) {
          remove_edge(edge, head);
      }
  }
  
  void merge_collinear_edges(Edge* edge, Edge** activeEdges);
  
  void set_top(Edge* edge, Vertex* v, Edge** activeEdges) {
      remove_edge_below(edge);
      edge->fTop = v;
      edge->recompute();
      insert_edge_below(edge, v);
      fix_active_state(edge, activeEdges);
      merge_collinear_edges(edge, activeEdges);
  }
  
  void set_bottom(Edge* edge, Vertex* v, Edge** activeEdges) {
      remove_edge_above(edge);
      edge->fBottom = v;
      edge->recompute();
      insert_edge_above(edge, v);
      fix_active_state(edge, activeEdges);
      merge_collinear_edges(edge, activeEdges);
  }
  
  void merge_edges_above(Edge* edge, Edge* other, Edge** activeEdges) {
      if (coincident(edge->fTop->fPoint, other->fTop->fPoint)) {
          LOG("merging coincident above edges (%g, %g) -> (%g, %g)\n",
              edge->fTop->fPoint.fX, edge->fTop->fPoint.fY,
              edge->fBottom->fPoint.fX, edge->fBottom->fPoint.fY);
          other->fWinding += edge->fWinding;
          erase_edge_if_zero_winding(other, activeEdges);
          edge->fWinding = 0;
          erase_edge_if_zero_winding(edge, activeEdges);
      } else if (sweep_lt(edge->fTop->fPoint, other->fTop->fPoint)) {
          other->fWinding += edge->fWinding;
          erase_edge_if_zero_winding(other, activeEdges);
          set_bottom(edge, other->fTop, activeEdges);
      } else {
          edge->fWinding += other->fWinding;
          erase_edge_if_zero_winding(edge, activeEdges);
          set_bottom(other, edge->fTop, activeEdges);
      }
  }
  
  void merge_edges_below(Edge* edge, Edge* other, Edge** activeEdges) {
      if (coincident(edge->fBottom->fPoint, other->fBottom->fPoint)) {
          LOG("merging coincident below edges (%g, %g) -> (%g, %g)\n",
              edge->fTop->fPoint.fX, edge->fTop->fPoint.fY,
              edge->fBottom->fPoint.fX, edge->fBottom->fPoint.fY);
          other->fWinding += edge->fWinding;
          erase_edge_if_zero_winding(other, activeEdges);
          edge->fWinding = 0;
          erase_edge_if_zero_winding(edge, activeEdges);
      } else if (sweep_lt(edge->fBottom->fPoint, other->fBottom->fPoint)) {
          edge->fWinding += other->fWinding;
          erase_edge_if_zero_winding(edge, activeEdges);
          set_top(other, edge->fBottom, activeEdges);
      } else {
          other->fWinding += edge->fWinding;
          erase_edge_if_zero_winding(other, activeEdges);
          set_top(edge, other->fBottom, activeEdges);
      }
  }
  
  void merge_collinear_edges(Edge* edge, Edge** activeEdges) {
      if (edge->fPrevEdgeAbove && (edge->fTop == edge->fPrevEdgeAbove->fTop ||
                                   !edge->fPrevEdgeAbove->isLeftOf(edge->fTop))) {
          merge_edges_above(edge, edge->fPrevEdgeAbove, activeEdges);
      } else if (edge->fNextEdgeAbove && (edge->fTop == edge->fNextEdgeAbove->fTop ||
                                          !edge->isLeftOf(edge->fNextEdgeAbove->fTop))) {
          merge_edges_above(edge, edge->fNextEdgeAbove, activeEdges);
      }
      if (edge->fPrevEdgeBelow && (edge->fBottom == edge->fPrevEdgeBelow->fBottom ||
                                   !edge->fPrevEdgeBelow->isLeftOf(edge->fBottom))) {
          merge_edges_below(edge, edge->fPrevEdgeBelow, activeEdges);
      } else if (edge->fNextEdgeBelow && (edge->fBottom == edge->fNextEdgeBelow->fBottom ||
                                          !edge->isLeftOf(edge->fNextEdgeBelow->fBottom))) {
          merge_edges_below(edge, edge->fNextEdgeBelow, activeEdges);
      }
  }
  
  void split_edge(Edge* edge, Vertex* v, Edge** activeEdges, SkChunkAlloc& alloc);
  
  void cleanup_active_edges(Edge* edge, Edge** activeEdges, SkChunkAlloc& alloc) {
      Vertex* top = edge->fTop;
      Vertex* bottom = edge->fBottom;
      if (edge->fLeft) {
          Vertex* leftTop = edge->fLeft->fTop;
          Vertex* leftBottom = edge->fLeft->fBottom;
          if (sweep_gt(top->fPoint, leftTop->fPoint) && !edge->fLeft->isLeftOf(top)) {
              split_edge(edge->fLeft, edge->fTop, activeEdges, alloc);
          } else if (sweep_gt(leftTop->fPoint, top->fPoint) && !edge->isRightOf(leftTop)) {
              split_edge(edge, leftTop, activeEdges, alloc);
          } else if (sweep_lt(bottom->fPoint, leftBottom->fPoint) && !edge->fLeft->isLeftOf(bottom)) {
              split_edge(edge->fLeft, bottom, activeEdges, alloc);
          } else if (sweep_lt(leftBottom->fPoint, bottom->fPoint) && !edge->isRightOf(leftBottom)) {
              split_edge(edge, leftBottom, activeEdges, alloc);
          }
      }
      if (edge->fRight) {
          Vertex* rightTop = edge->fRight->fTop;
          Vertex* rightBottom = edge->fRight->fBottom;
          if (sweep_gt(top->fPoint, rightTop->fPoint) && !edge->fRight->isRightOf(top)) {
              split_edge(edge->fRight, top, activeEdges, alloc);
          } else if (sweep_gt(rightTop->fPoint, top->fPoint) && !edge->isLeftOf(rightTop)) {
              split_edge(edge, rightTop, activeEdges, alloc);
          } else if (sweep_lt(bottom->fPoint, rightBottom->fPoint) &&
                     !edge->fRight->isRightOf(bottom)) {
              split_edge(edge->fRight, bottom, activeEdges, alloc);
          } else if (sweep_lt(rightBottom->fPoint, bottom->fPoint) &&
                     !edge->isLeftOf(rightBottom)) {
              split_edge(edge, rightBottom, activeEdges, alloc);
          }
      }
  }
  
  void split_edge(Edge* edge, Vertex* v, Edge** activeEdges, SkChunkAlloc& alloc) {
      LOG("splitting edge (%g -> %g) at vertex %g (%g, %g)\n",
          edge->fTop->fID, edge->fBottom->fID,
          v->fID, v->fPoint.fX, v->fPoint.fY);
      Edge* newEdge = ALLOC_NEW(Edge, (v, edge->fBottom, edge->fWinding), alloc);
      insert_edge_below(newEdge, v);
      insert_edge_above(newEdge, edge->fBottom);
      set_bottom(edge, v, activeEdges);
      cleanup_active_edges(edge, activeEdges, alloc);
      fix_active_state(newEdge, activeEdges);
      merge_collinear_edges(newEdge, activeEdges);
  }
  
  void merge_vertices(Vertex* src, Vertex* dst, Vertex** head, SkChunkAlloc& alloc) {
      LOG("found coincident verts at %g, %g; merging %g into %g\n", src->fPoint.fX, src->fPoint.fY,
          src->fID, dst->fID);
      for (Edge* edge = src->fFirstEdgeAbove; edge;) {
          Edge* next = edge->fNextEdgeAbove;
          set_bottom(edge, dst, NULL);
          edge = next;
      }
      for (Edge* edge = src->fFirstEdgeBelow; edge;) {
          Edge* next = edge->fNextEdgeBelow;
          set_top(edge, dst, NULL);
          edge = next;
      }
      remove<Vertex, &Vertex::fPrev, &Vertex::fNext>(src, head, NULL);
  }
  
  Vertex* check_for_intersection(Edge* edge, Edge* other, Edge** activeEdges, SkChunkAlloc& alloc) {
      SkPoint p;
      if (!edge || !other) {
          return NULL;
      }
      if (edge->intersect(*other, &p)) {
          Vertex* v;
          LOG("found intersection, pt is %g, %g\n", p.fX, p.fY);
          if (p == edge->fTop->fPoint || sweep_lt(p, edge->fTop->fPoint)) {
              split_edge(other, edge->fTop, activeEdges, alloc);
              v = edge->fTop;
          } else if (p == edge->fBottom->fPoint || sweep_gt(p, edge->fBottom->fPoint)) {
              split_edge(other, edge->fBottom, activeEdges, alloc);
              v = edge->fBottom;
          } else if (p == other->fTop->fPoint || sweep_lt(p, other->fTop->fPoint)) {
              split_edge(edge, other->fTop, activeEdges, alloc);
              v = other->fTop;
          } else if (p == other->fBottom->fPoint || sweep_gt(p, other->fBottom->fPoint)) {
              split_edge(edge, other->fBottom, activeEdges, alloc);
              v = other->fBottom;
          } else {
              Vertex* nextV = edge->fTop;
              while (sweep_lt(p, nextV->fPoint)) {
                  nextV = nextV->fPrev;
              }
              while (sweep_lt(nextV->fPoint, p)) {
                  nextV = nextV->fNext;
              }
              Vertex* prevV = nextV->fPrev;
              if (coincident(prevV->fPoint, p)) {
                  v = prevV;
              } else if (coincident(nextV->fPoint, p)) {
                  v = nextV;
              } else {
                  v = ALLOC_NEW(Vertex, (p), alloc);
                  LOG("inserting between %g (%g, %g) and %g (%g, %g)\n",
                      prevV->fID, prevV->fPoint.fX, prevV->fPoint.fY,
                      nextV->fID, nextV->fPoint.fX, nextV->fPoint.fY);
  #if LOGGING_ENABLED
                  v->fID = (nextV->fID + prevV->fID) * 0.5f;
  #endif
                  v->fPrev = prevV;
                  v->fNext = nextV;
                  prevV->fNext = v;
                  nextV->fPrev = v;
              }
              split_edge(edge, v, activeEdges, alloc);
              split_edge(other, v, activeEdges, alloc);
          }
  #ifdef SK_DEBUG
          validate_connectivity(v);
  #endif
          return v;
      }
      return NULL;
  }
  
  void sanitize_contours(Vertex** contours, int contourCnt) {
      for (int i = 0; i < contourCnt; ++i) {
          SkASSERT(contours[i]);
          for (Vertex* v = contours[i];;) {
              if (coincident(v->fPrev->fPoint, v->fPoint)) {
                  LOG("vertex %g,%g coincident; removing\n", v->fPoint.fX, v->fPoint.fY);
                  if (v->fPrev == v) {
                      contours[i] = NULL;
                      break;
                  }
                  v->fPrev->fNext = v->fNext;
                  v->fNext->fPrev = v->fPrev;
                  if (contours[i] == v) {
                      contours[i] = v->fNext;
                  }
                  v = v->fPrev;
              } else {
                  v = v->fNext;
                  if (v == contours[i]) break;
              }
          }
      }
  }
  
  void merge_coincident_vertices(Vertex** vertices, SkChunkAlloc& alloc) {
      for (Vertex* v = (*vertices)->fNext; v != NULL; v = v->fNext) {
          if (sweep_lt(v->fPoint, v->fPrev->fPoint)) {
              v->fPoint = v->fPrev->fPoint;
          }
          if (coincident(v->fPrev->fPoint, v->fPoint)) {
              merge_vertices(v->fPrev, v, vertices, alloc);
          }
      }
  }
  
  // Stage 2: convert the contours to a mesh of edges connecting the vertices.
  
  Vertex* build_edges(Vertex** contours, int contourCnt, SkChunkAlloc& alloc) {
      Vertex* vertices = NULL;
      Vertex* prev = NULL;
      for (int i = 0; i < contourCnt; ++i) {
          for (Vertex* v = contours[i]; v != NULL;) {
              Vertex* vNext = v->fNext;
              Edge* edge = new_edge(v->fPrev, v, alloc);
              if (edge->fWinding > 0) {
                  insert_edge_below(edge, v->fPrev);
                  insert_edge_above(edge, v);
              } else {
                  insert_edge_below(edge, v);
                  insert_edge_above(edge, v->fPrev);
              }
              merge_collinear_edges(edge, NULL);
              if (prev) {
                  prev->fNext = v;
                  v->fPrev = prev;
              } else {
                  vertices = v;
              }
              prev = v;
              v = vNext;
              if (v == contours[i]) break;
          }
      }
      if (prev) {
          prev->fNext = vertices->fPrev = NULL;
      }
      return vertices;
  }
  
  // Stage 3: sort the vertices by increasing Y (or X if SWEEP_IN_X is on).
  
  Vertex* sorted_merge(Vertex* a, Vertex* b);
  
  void front_back_split(Vertex* v, Vertex** pFront, Vertex** pBack) {
      Vertex* fast;
      Vertex* slow;
      if (!v || !v->fNext) {
          *pFront = v;
          *pBack = NULL;
      } else {
          slow = v;
          fast = v->fNext;
  
          while (fast != NULL) {
              fast = fast->fNext;
              if (fast != NULL) {
                  slow = slow->fNext;
                  fast = fast->fNext;
              }
          }
  
          *pFront = v;
          *pBack = slow->fNext;
          slow->fNext->fPrev = NULL;
          slow->fNext = NULL;
      }
  }
  
  void merge_sort(Vertex** head) {
      if (!*head || !(*head)->fNext) {
          return;
      }
  
      Vertex* a;
      Vertex* b;
      front_back_split(*head, &a, &b);
  
      merge_sort(&a);
      merge_sort(&b);
  
      *head = sorted_merge(a, b);
  }
  
  Vertex* sorted_merge(Vertex* a, Vertex* b) {
      if (!a) {
          return b;
      } else if (!b) {
          return a;
      }
  
      Vertex* result = NULL;
  
      if (sweep_lt(a->fPoint, b->fPoint)) {
          result = a;
          result->fNext = sorted_merge(a->fNext, b);
      } else {
          result = b;
          result->fNext = sorted_merge(a, b->fNext);
      }
      result->fNext->fPrev = result;
      return result;
  }
  
  // Stage 4: Simplify the mesh by inserting new vertices at intersecting edges.
  
  void simplify(Vertex* vertices, SkChunkAlloc& alloc) {
      LOG("simplifying complex polygons\n");
      Edge* activeEdges = NULL;
      for (Vertex* v = vertices; v != NULL; v = v->fNext) {
          if (!v->fFirstEdgeAbove && !v->fFirstEdgeBelow) {
              continue;
          }
  #if LOGGING_ENABLED
          LOG("\nvertex %g: (%g,%g)\n", v->fID, v->fPoint.fX, v->fPoint.fY);
  #endif
  #ifdef SK_DEBUG
          validate_connectivity(v);
  #endif
          Edge* leftEnclosingEdge = NULL;
          Edge* rightEnclosingEdge = NULL;
          bool restartChecks;
          do {
              restartChecks = false;
              find_enclosing_edges(v, activeEdges, &leftEnclosingEdge, &rightEnclosingEdge);
              if (v->fFirstEdgeBelow) {
                  for (Edge* edge = v->fFirstEdgeBelow; edge != NULL; edge = edge->fNextEdgeBelow) {
                      if (check_for_intersection(edge, leftEnclosingEdge, &activeEdges, alloc)) {
                          restartChecks = true;
                          break;
                      }
                      if (check_for_intersection(edge, rightEnclosingEdge, &activeEdges, alloc)) {
                          restartChecks = true;
                          break;
                      }
                  }
              } else {
                  if (Vertex* pv = check_for_intersection(leftEnclosingEdge, rightEnclosingEdge,
                                                          &activeEdges, alloc)) {
                      if (sweep_lt(pv->fPoint, v->fPoint)) {
                          v = pv;
                      }
                      restartChecks = true;
                  }
  
              }
          } while (restartChecks);
          SkASSERT(!leftEnclosingEdge || leftEnclosingEdge->isLeftOf(v));
          SkASSERT(!rightEnclosingEdge || rightEnclosingEdge->isRightOf(v));
  #ifdef SK_DEBUG
          validate_edges(activeEdges);
  #endif
          for (Edge* e = v->fFirstEdgeAbove; e; e = e->fNextEdgeAbove) {
              remove_edge(e, &activeEdges);
          }
          Edge* leftEdge = leftEnclosingEdge;
          for (Edge* e = v->fFirstEdgeBelow; e; e = e->fNextEdgeBelow) {
              insert_edge(e, leftEdge, &activeEdges);
              leftEdge = e;
          }
          v->fProcessed = true;
      }
  }
  
  // Stage 5: Tessellate the simplified mesh into monotone polygons.
  
  Poly* tessellate(Vertex* vertices, SkChunkAlloc& alloc) {
      LOG("tessellating simple polygons\n");
      Edge* activeEdges = NULL;
      Poly* polys = NULL;
      for (Vertex* v = vertices; v != NULL; v = v->fNext) {
          if (!v->fFirstEdgeAbove && !v->fFirstEdgeBelow) {
              continue;
          }
  #if LOGGING_ENABLED
          LOG("\nvertex %g: (%g,%g)\n", v->fID, v->fPoint.fX, v->fPoint.fY);
  #endif
  #ifdef SK_DEBUG
          validate_connectivity(v);
  #endif
          Edge* leftEnclosingEdge = NULL;
          Edge* rightEnclosingEdge = NULL;
          find_enclosing_edges(v, activeEdges, &leftEnclosingEdge, &rightEnclosingEdge);
          SkASSERT(!leftEnclosingEdge || leftEnclosingEdge->isLeftOf(v));
          SkASSERT(!rightEnclosingEdge || rightEnclosingEdge->isRightOf(v));
  #ifdef SK_DEBUG
          validate_edges(activeEdges);
  #endif
          Poly* leftPoly = NULL;
          Poly* rightPoly = NULL;
          if (v->fFirstEdgeAbove) {
              leftPoly = v->fFirstEdgeAbove->fLeftPoly;
              rightPoly = v->fLastEdgeAbove->fRightPoly;
          } else {
              leftPoly = leftEnclosingEdge ? leftEnclosingEdge->fRightPoly : NULL;
              rightPoly = rightEnclosingEdge ? rightEnclosingEdge->fLeftPoly : NULL;
          }
  #if LOGGING_ENABLED
          LOG("edges above:\n");
          for (Edge* e = v->fFirstEdgeAbove; e; e = e->fNextEdgeAbove) {
              LOG("%g -> %g, lpoly %d, rpoly %d\n", e->fTop->fID, e->fBottom->fID,
                  e->fLeftPoly ? e->fLeftPoly->fID : -1, e->fRightPoly ? e->fRightPoly->fID : -1);
          }
          LOG("edges below:\n");
          for (Edge* e = v->fFirstEdgeBelow; e; e = e->fNextEdgeBelow) {
              LOG("%g -> %g, lpoly %d, rpoly %d\n", e->fTop->fID, e->fBottom->fID,
                  e->fLeftPoly ? e->fLeftPoly->fID : -1, e->fRightPoly ? e->fRightPoly->fID : -1);
          }
  #endif
          if (v->fFirstEdgeAbove) {
              if (leftPoly) {
                  leftPoly = leftPoly->addVertex(v, Poly::kRight_Side, alloc);
              }
              if (rightPoly) {
                  rightPoly = rightPoly->addVertex(v, Poly::kLeft_Side, alloc);
              }
              for (Edge* e = v->fFirstEdgeAbove; e != v->fLastEdgeAbove; e = e->fNextEdgeAbove) {
                  Edge* leftEdge = e;
                  Edge* rightEdge = e->fNextEdgeAbove;
                  SkASSERT(rightEdge->isRightOf(leftEdge->fTop));
                  remove_edge(leftEdge, &activeEdges);
                  if (leftEdge->fRightPoly) {
                      leftEdge->fRightPoly->end(v, alloc);
                  }
                  if (rightEdge->fLeftPoly && rightEdge->fLeftPoly != leftEdge->fRightPoly) {
                      rightEdge->fLeftPoly->end(v, alloc);
                  }
              }
              remove_edge(v->fLastEdgeAbove, &activeEdges);
              if (!v->fFirstEdgeBelow) {
                  if (leftPoly && rightPoly && leftPoly != rightPoly) {
                      SkASSERT(leftPoly->fPartner == NULL && rightPoly->fPartner == NULL);
                      rightPoly->fPartner = leftPoly;
                      leftPoly->fPartner = rightPoly;
                  }
              }
          }
          if (v->fFirstEdgeBelow) {
              if (!v->fFirstEdgeAbove) {
                  if (leftPoly && leftPoly == rightPoly) {
                      // Split the poly.
                      if (leftPoly->fActive->fSide == Poly::kLeft_Side) {
                          leftPoly = new_poly(&polys, leftEnclosingEdge->fTop, leftPoly->fWinding,
                                              alloc);
                          leftPoly->addVertex(v, Poly::kRight_Side, alloc);
                          rightPoly->addVertex(v, Poly::kLeft_Side, alloc);
                          leftEnclosingEdge->fRightPoly = leftPoly;
                      } else {
                          rightPoly = new_poly(&polys, rightEnclosingEdge->fTop, rightPoly->fWinding,
                                               alloc);
                          rightPoly->addVertex(v, Poly::kLeft_Side, alloc);
                          leftPoly->addVertex(v, Poly::kRight_Side, alloc);
                          rightEnclosingEdge->fLeftPoly = rightPoly;
                      }
                  } else {
                      if (leftPoly) {
                          leftPoly = leftPoly->addVertex(v, Poly::kRight_Side, alloc);
                      }
                      if (rightPoly) {
                          rightPoly = rightPoly->addVertex(v, Poly::kLeft_Side, alloc);
                      }
                  }
              }
              Edge* leftEdge = v->fFirstEdgeBelow;
              leftEdge->fLeftPoly = leftPoly;
              insert_edge(leftEdge, leftEnclosingEdge, &activeEdges);
              for (Edge* rightEdge = leftEdge->fNextEdgeBelow; rightEdge;
                   rightEdge = rightEdge->fNextEdgeBelow) {
                  insert_edge(rightEdge, leftEdge, &activeEdges);
                  int winding = leftEdge->fLeftPoly ? leftEdge->fLeftPoly->fWinding : 0;
                  winding += leftEdge->fWinding;
                  if (winding != 0) {
                      Poly* poly = new_poly(&polys, v, winding, alloc);
                      leftEdge->fRightPoly = rightEdge->fLeftPoly = poly;
                  }
                  leftEdge = rightEdge;
              }
              v->fLastEdgeBelow->fRightPoly = rightPoly;
          }
  #ifdef SK_DEBUG
          validate_edges(activeEdges);
  #endif
  #if LOGGING_ENABLED
          LOG("\nactive edges:\n");
          for (Edge* e = activeEdges; e != NULL; e = e->fRight) {
              LOG("%g -> %g, lpoly %d, rpoly %d\n", e->fTop->fID, e->fBottom->fID,
                  e->fLeftPoly ? e->fLeftPoly->fID : -1, e->fRightPoly ? e->fRightPoly->fID : -1);
          }
  #endif
      }
      return polys;
  }
  
  // This is a driver function which calls stages 2-5 in turn.
  
  Poly* contours_to_polys(Vertex** contours, int contourCnt, SkChunkAlloc& alloc) {
  #if LOGGING_ENABLED
      for (int i = 0; i < contourCnt; ++i) {
          Vertex* v = contours[i];
          SkASSERT(v);
          LOG("path.moveTo(%20.20g, %20.20g);\n", v->fPoint.fX, v->fPoint.fY);
          for (v = v->fNext; v != contours[i]; v = v->fNext) {
              LOG("path.lineTo(%20.20g, %20.20g);\n", v->fPoint.fX, v->fPoint.fY);
          }
      }
  #endif
      sanitize_contours(contours, contourCnt);
      Vertex* vertices = build_edges(contours, contourCnt, alloc);
      if (!vertices) {
          return NULL;
      }
  
      // Sort vertices in Y (secondarily in X).
      merge_sort(&vertices);
      merge_coincident_vertices(&vertices, alloc);
  #if LOGGING_ENABLED
      for (Vertex* v = vertices; v != NULL; v = v->fNext) {
          static float gID = 0.0f;
          v->fID = gID++;
      }
  #endif
      simplify(vertices, alloc);
      return tessellate(vertices, alloc);
  }
  
  // Stage 6: Triangulate the monotone polygons into a vertex buffer.
  
  void* polys_to_triangles(Poly* polys, SkPath::FillType fillType, void* data) {
      void* d = data;
      for (Poly* poly = polys; poly; poly = poly->fNext) {
          if (apply_fill_type(fillType, poly->fWinding)) {
              d = poly->emit(d);
          }
      }
      return d;
  }
  
  };
  
  GrTessellatingPathRenderer::GrTessellatingPathRenderer() {
  }
  
  GrPathRenderer::StencilSupport GrTessellatingPathRenderer::onGetStencilSupport(
                                                              const GrDrawTarget*,
                                                              const GrPipelineBuilder*,
                                                              const SkPath&,
                                                              const SkStrokeRec&) const {
      return GrPathRenderer::kNoSupport_StencilSupport;
  }
  
  bool GrTessellatingPathRenderer::canDrawPath(const GrDrawTarget* target,
                                               const GrPipelineBuilder* pipelineBuilder,
                                               const SkMatrix& viewMatrix,
                                               const SkPath& path,
                                               const SkStrokeRec& stroke,
                                               bool antiAlias) const {
      // This path renderer can draw all fill styles, but does not do antialiasing. It can do convex
      // and concave paths, but we'll leave the convex ones to simpler algorithms.
      return stroke.isFillStyle() && !antiAlias && !path.isConvex();
  }
  
  bool GrTessellatingPathRenderer::onDrawPath(GrDrawTarget* target,
                                              GrPipelineBuilder* pipelineBuilder,
                                              GrColor color,
                                              const SkMatrix& viewM,
                                              const SkPath& path,
                                              const SkStrokeRec& stroke,
                                              bool antiAlias) {
      SkASSERT(!antiAlias);
      const GrRenderTarget* rt = pipelineBuilder->getRenderTarget();
      if (NULL == rt) {
          return false;
      }
  
      SkScalar tol = GrPathUtils::scaleToleranceToSrc(SK_Scalar1, viewM, path.getBounds());
  
      int contourCnt;
      int maxPts = GrPathUtils::worstCasePointCount(path, &contourCnt, tol);
      if (maxPts <= 0) {
          return false;
      }
      if (maxPts > ((int)SK_MaxU16 + 1)) {
          SkDebugf("Path not rendered, too many verts (%d)\n", maxPts);
          return false;
      }
      SkPath::FillType fillType = path.getFillType();
      if (SkPath::IsInverseFillType(fillType)) {
          contourCnt++;
      }
  
      LOG("got %d pts, %d contours\n", maxPts, contourCnt);
  
      SkAutoTDeleteArray<Vertex*> contours(SkNEW_ARRAY(Vertex *, contourCnt));
  
      // For the initial size of the chunk allocator, estimate based on the point count:
      // one vertex per point for the initial passes, plus two for the vertices in the
      // resulting Polys, since the same point may end up in two Polys.  Assume minimal
      // connectivity of one Edge per Vertex (will grow for intersections).
      SkChunkAlloc alloc(maxPts * (3 * sizeof(Vertex) + sizeof(Edge)));
      SkIRect clipBoundsI;
      pipelineBuilder->clip().getConservativeBounds(rt, &clipBoundsI);
      SkRect clipBounds = SkRect::Make(clipBoundsI);
      SkMatrix vmi;
      if (!viewM.invert(&vmi)) {
          return false;
      }
      vmi.mapRect(&clipBounds);
      path_to_contours(path, tol, clipBounds, contours.get(), alloc);
      Poly* polys;
      uint32_t flags = GrDefaultGeoProcFactory::kPosition_GPType;
      polys = contours_to_polys(contours.get(), contourCnt, alloc);
      SkAutoTUnref<const GrGeometryProcessor> gp(
          GrDefaultGeoProcFactory::Create(flags, color, viewM, SkMatrix::I()));
      int count = 0;
      for (Poly* poly = polys; poly; poly = poly->fNext) {
          if (apply_fill_type(fillType, poly->fWinding) && poly->fCount >= 3) {
              count += (poly->fCount - 2) * (WIREFRAME ? 6 : 3);
          }
      }
  
      size_t stride = gp->getVertexStride();
      GrDrawTarget::AutoReleaseGeometry arg;
      if (!arg.set(target, count, stride, 0)) {
          return false;
      }
      LOG("emitting %d verts\n", count);
      void* end = polys_to_triangles(polys, fillType, arg.vertices());
      int actualCount = static_cast<int>((static_cast<char*>(end) - static_cast<char*>(arg.vertices())) / stride);
      LOG("actual count: %d\n", actualCount);
      SkASSERT(actualCount <= count);
  
      GrPrimitiveType primitiveType = WIREFRAME ? kLines_GrPrimitiveType
                                                : kTriangles_GrPrimitiveType;
      target->drawNonIndexed(pipelineBuilder, gp, primitiveType, 0, actualCount);
  
      return true;
  }
  