Tessellating GPU path renderer.

src/gpu/GrTessellatingPathRenderer.cpp

+/*
+ * 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);
+        }
+    }
+
+    int 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<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;
+}