Add implementation of path ops

src/pathops/SkOpSegment.cpp

Index: src/pathops/SkOpSegment.cpp
===================================================================
--- src/pathops/SkOpSegment.cpp	(revision 0)
+++ src/pathops/SkOpSegment.cpp	(revision 0)
@@ -0,0 +1,2939 @@
+/*
+ * Copyright 2012 Google Inc.
+ *
+ * Use of this source code is governed by a BSD-style license that can be
+ * found in the LICENSE file.
+ */
+#include "SkIntersections.h"
+#include "SkOpSegment.h"
+#include "SkPathWriter.h"
+#include "TSearch.h"
+
+#define F (false)      // discard the edge
+#define T (true)       // keep the edge
+
+static const bool gUnaryActiveEdge[2][2] = {
+//  from=0  from=1
+//  to=0,1  to=0,1
+    {F, T}, {T, F},
+};
+
+// FIXME: add support for kReverseDifference_Op
+static const bool gActiveEdge[kXOR_PathOp + 1][2][2][2][2] = {
+//                 miFrom=0                              miFrom=1
+//         miTo=0            miTo=1              miTo=0             miTo=1
+//    suFrom=0    1     suFrom=0     1      suFrom=0    1      suFrom=0    1
+//   suTo=0,1 suTo=0,1  suTo=0,1 suTo=0,1  suTo=0,1 suTo=0,1  suTo=0,1 suTo=0,1
+    {{{{F, F}, {F, F}}, {{T, F}, {T, F}}}, {{{T, T}, {F, F}}, {{F, T}, {T, F}}}}, // mi - su
+    {{{{F, F}, {F, F}}, {{F, T}, {F, T}}}, {{{F, F}, {T, T}}, {{F, T}, {T, F}}}}, // mi & su
+    {{{{F, T}, {T, F}}, {{T, T}, {F, F}}}, {{{T, F}, {T, F}}, {{F, F}, {F, F}}}}, // mi | su
+    {{{{F, T}, {T, F}}, {{T, F}, {F, T}}}, {{{T, F}, {F, T}}, {{F, T}, {T, F}}}}, // mi ^ su
+};
+
+#undef F
+#undef T
+
+// OPTIMIZATION: does the following also work, and is it any faster?
+// return outerWinding * innerWinding > 0
+//      || ((outerWinding + innerWinding < 0) ^ ((outerWinding - innerWinding) < 0)))
+bool SkOpSegment::UseInnerWinding(int outerWinding, int innerWinding) {
+    SkASSERT(outerWinding != SK_MaxS32);
+    SkASSERT(innerWinding != SK_MaxS32);
+    int absOut = abs(outerWinding);
+    int absIn = abs(innerWinding);
+    bool result = absOut == absIn ? outerWinding < 0 : absOut < absIn;
+    return result;
+}
+
+bool SkOpSegment::activeAngle(int index, int& done, SkTDArray<SkOpAngle>& angles) {
+    if (activeAngleInner(index, done, angles)) {
+        return true;
+    }
+    int lesser = index;
+    while (--lesser >= 0 && equalPoints(index, lesser)) {
+        if (activeAngleOther(lesser, done, angles)) {
+            return true;
+        }
+    }
+    lesser = index;
+    do {
+        if (activeAngleOther(index, done, angles)) {
+            return true;
+        }
+    } while (++index < fTs.count() && equalPoints(index, lesser));
+    return false;
+}
+
+bool SkOpSegment::activeAngleOther(int index, int& done, SkTDArray<SkOpAngle>& angles) {
+    SkOpSpan* span = &fTs[index];
+    SkOpSegment* other = span->fOther;
+    int oIndex = span->fOtherIndex;
+    return other->activeAngleInner(oIndex, done, angles);
+}
+
+bool SkOpSegment::activeAngleInner(int index, int& done, SkTDArray<SkOpAngle>& angles) {
+    int next = nextExactSpan(index, 1);
+    if (next > 0) {
+        SkOpSpan& upSpan = fTs[index];
+        if (upSpan.fWindValue || upSpan.fOppValue) {
+            addAngle(angles, index, next);
+            if (upSpan.fDone || upSpan.fUnsortableEnd) {
+                done++;
+            } else if (upSpan.fWindSum != SK_MinS32) {
+                return true;
+            }
+        } else if (!upSpan.fDone) {
+            upSpan.fDone = true;
+            fDoneSpans++;
+        }
+    }
+    int prev = nextExactSpan(index, -1);
+    // edge leading into junction
+    if (prev >= 0) {
+        SkOpSpan& downSpan = fTs[prev];
+        if (downSpan.fWindValue || downSpan.fOppValue) {
+            addAngle(angles, index, prev);
+            if (downSpan.fDone) {
+                done++;
+             } else if (downSpan.fWindSum != SK_MinS32) {
+                return true;
+            }
+        } else if (!downSpan.fDone) {
+            downSpan.fDone = true;
+            fDoneSpans++;
+        }
+    }
+    return false;
+}
+
+SkPoint SkOpSegment::activeLeftTop(bool onlySortable, int* firstT) const {
+    SkASSERT(!done());
+    SkPoint topPt = {SK_ScalarMax, SK_ScalarMax};
+    int count = fTs.count();
+    // see if either end is not done since we want smaller Y of the pair
+    bool lastDone = true;
+    bool lastUnsortable = false;
+    double lastT = -1;
+    for (int index = 0; index < count; ++index) {
+        const SkOpSpan& span = fTs[index];
+        if (onlySortable && (span.fUnsortableStart || lastUnsortable)) {
+            goto next;
+        }
+        if (span.fDone && lastDone) {
+            goto next;
+        }
+        if (approximately_negative(span.fT - lastT)) {
+            goto next;
+        }
+        {
+            const SkPoint& xy = xyAtT(&span);
+            if (topPt.fY > xy.fY || (topPt.fY == xy.fY && topPt.fX > xy.fX)) {
+                topPt = xy;
+                if (firstT) {
+                    *firstT = index;
+                }
+            }
+            if (fVerb != SkPath::kLine_Verb && !lastDone) {
+                SkPoint curveTop = (*CurveTop[fVerb])(fPts, lastT, span.fT);
+                if (topPt.fY > curveTop.fY || (topPt.fY == curveTop.fY
+                        && topPt.fX > curveTop.fX)) {
+                    topPt = curveTop;
+                    if (firstT) {
+                        *firstT = index;
+                    }
+                }
+            }
+            lastT = span.fT;
+        }
+next:
+        lastDone = span.fDone;
+        lastUnsortable = span.fUnsortableEnd;
+    }
+    return topPt;
+}
+
+bool SkOpSegment::activeOp(int index, int endIndex, int xorMiMask, int xorSuMask, SkPathOp op) {
+    int sumMiWinding = updateWinding(endIndex, index);
+    int sumSuWinding = updateOppWinding(endIndex, index);
+    if (fOperand) {
+        SkTSwap<int>(sumMiWinding, sumSuWinding);
+    }
+    int maxWinding, sumWinding, oppMaxWinding, oppSumWinding;
+    return activeOp(xorMiMask, xorSuMask, index, endIndex, op, sumMiWinding, sumSuWinding,
+            maxWinding, sumWinding, oppMaxWinding, oppSumWinding);
+}
+
+bool SkOpSegment::activeOp(int xorMiMask, int xorSuMask, int index, int endIndex, SkPathOp op,
+        int& sumMiWinding, int& sumSuWinding,
+        int& maxWinding, int& sumWinding, int& oppMaxWinding, int& oppSumWinding) {
+    setUpWindings(index, endIndex, sumMiWinding, sumSuWinding,
+            maxWinding, sumWinding, oppMaxWinding, oppSumWinding);
+    bool miFrom;
+    bool miTo;
+    bool suFrom;
+    bool suTo;
+    if (operand()) {
+        miFrom = (oppMaxWinding & xorMiMask) != 0;
+        miTo = (oppSumWinding & xorMiMask) != 0;
+        suFrom = (maxWinding & xorSuMask) != 0;
+        suTo = (sumWinding & xorSuMask) != 0;
+    } else {
+        miFrom = (maxWinding & xorMiMask) != 0;
+        miTo = (sumWinding & xorMiMask) != 0;
+        suFrom = (oppMaxWinding & xorSuMask) != 0;
+        suTo = (oppSumWinding & xorSuMask) != 0;
+    }
+    bool result = gActiveEdge[op][miFrom][miTo][suFrom][suTo];
+#if DEBUG_ACTIVE_OP
+    SkDebugf("%s op=%s miFrom=%d miTo=%d suFrom=%d suTo=%d result=%d\n", __FUNCTION__,
+            kPathOpStr[op], miFrom, miTo, suFrom, suTo, result);
+#endif
+    SkASSERT(result != -1);
+    return result;
+}
+
+bool SkOpSegment::activeWinding(int index, int endIndex) {
+    int sumWinding = updateWinding(endIndex, index);
+    int maxWinding;
+    return activeWinding(index, endIndex, maxWinding, sumWinding);
+}
+
+bool SkOpSegment::activeWinding(int index, int endIndex, int& maxWinding, int& sumWinding) {
+    setUpWinding(index, endIndex, maxWinding, sumWinding);
+    bool from = maxWinding != 0;
+    bool to = sumWinding  != 0;
+    bool result = gUnaryActiveEdge[from][to];
+    SkASSERT(result != -1);
+    return result;
+}
+
+void SkOpSegment::addAngle(SkTDArray<SkOpAngle>& angles, int start, int end) const {
+    SkASSERT(start != end);
+    SkOpAngle* angle = angles.append();
+#if DEBUG_ANGLE
+    if (angles.count() > 1 && !fTs[start].fTiny) {
+        SkPoint angle0Pt = (*CurvePointAtT[angles[0].verb()])(angles[0].pts(),
+                (*angles[0].spans())[angles[0].start()].fT);
+        SkPoint newPt = (*CurvePointAtT[fVerb])(fPts, fTs[start].fT);
+        SkASSERT(AlmostEqualUlps(angle0Pt.fX, newPt.fX));
+        SkASSERT(AlmostEqualUlps(angle0Pt.fY, newPt.fY));
+    }
+#endif
+    angle->set(fPts, fVerb, this, start, end, fTs);
+}
+
+void SkOpSegment::addCancelOutsides(double tStart, double oStart, SkOpSegment& other, double oEnd) {
+    int tIndex = -1;
+    int tCount = fTs.count();
+    int oIndex = -1;
+    int oCount = other.fTs.count();
+    do {
+        ++tIndex;
+    } while (!approximately_negative(tStart - fTs[tIndex].fT) && tIndex < tCount);
+    int tIndexStart = tIndex;
+    do {
+        ++oIndex;
+    } while (!approximately_negative(oStart - other.fTs[oIndex].fT) && oIndex < oCount);
+    int oIndexStart = oIndex;
+    double nextT;
+    do {
+        nextT = fTs[++tIndex].fT;
+    } while (nextT < 1 && approximately_negative(nextT - tStart));
+    double oNextT;
+    do {
+        oNextT = other.fTs[++oIndex].fT;
+    } while (oNextT < 1 && approximately_negative(oNextT - oStart));
+    // at this point, spans before and after are at:
+    //  fTs[tIndexStart - 1], fTs[tIndexStart], fTs[tIndex]
+    // if tIndexStart == 0, no prior span
+    // if nextT == 1, no following span
+
+    // advance the span with zero winding
+    // if the following span exists (not past the end, non-zero winding)
+    // connect the two edges
+    if (!fTs[tIndexStart].fWindValue) {
+        if (tIndexStart > 0 && fTs[tIndexStart - 1].fWindValue) {
+#if DEBUG_CONCIDENT
+            SkDebugf("%s 1 this=%d other=%d t [%d] %1.9g (%1.9g,%1.9g)\n",
+                    __FUNCTION__, fID, other.fID, tIndexStart - 1,
+                    fTs[tIndexStart].fT, xyAtT(tIndexStart).fX,
+                    xyAtT(tIndexStart).fY);
+#endif
+            addTPair(fTs[tIndexStart].fT, other, other.fTs[oIndex].fT, false,
+                    fTs[tIndexStart].fPt);
+        }
+        if (nextT < 1 && fTs[tIndex].fWindValue) {
+#if DEBUG_CONCIDENT
+            SkDebugf("%s 2 this=%d other=%d t [%d] %1.9g (%1.9g,%1.9g)\n",
+                    __FUNCTION__, fID, other.fID, tIndex,
+                    fTs[tIndex].fT, xyAtT(tIndex).fX,
+                    xyAtT(tIndex).fY);
+#endif
+            addTPair(fTs[tIndex].fT, other, other.fTs[oIndexStart].fT, false, fTs[tIndex].fPt);
+        }
+    } else {
+        SkASSERT(!other.fTs[oIndexStart].fWindValue);
+        if (oIndexStart > 0 && other.fTs[oIndexStart - 1].fWindValue) {
+#if DEBUG_CONCIDENT
+            SkDebugf("%s 3 this=%d other=%d t [%d] %1.9g (%1.9g,%1.9g)\n",
+                    __FUNCTION__, fID, other.fID, oIndexStart - 1,
+                    other.fTs[oIndexStart].fT, other.xyAtT(oIndexStart).fX,
+                    other.xyAtT(oIndexStart).fY);
+            other.debugAddTPair(other.fTs[oIndexStart].fT, *this, fTs[tIndex].fT);
+#endif
+        }
+        if (oNextT < 1 && other.fTs[oIndex].fWindValue) {
+#if DEBUG_CONCIDENT
+            SkDebugf("%s 4 this=%d other=%d t [%d] %1.9g (%1.9g,%1.9g)\n",
+                    __FUNCTION__, fID, other.fID, oIndex,
+                    other.fTs[oIndex].fT, other.xyAtT(oIndex).fX,
+                    other.xyAtT(oIndex).fY);
+            other.debugAddTPair(other.fTs[oIndex].fT, *this, fTs[tIndexStart].fT);
+#endif
+        }
+    }
+}
+
+void SkOpSegment::addCoinOutsides(const SkTDArray<double>& outsideTs, SkOpSegment& other,
+                                  double oEnd) {
+    // walk this to outsideTs[0]
+    // walk other to outsideTs[1]
+    // if either is > 0, add a pointer to the other, copying adjacent winding
+    int tIndex = -1;
+    int oIndex = -1;
+    double tStart = outsideTs[0];
+    double oStart = outsideTs[1];
+    do {
+        ++tIndex;
+    } while (!approximately_negative(tStart - fTs[tIndex].fT));
+    SkPoint ptStart = fTs[tIndex].fPt;
+    do {
+        ++oIndex;
+    } while (!approximately_negative(oStart - other.fTs[oIndex].fT));
+    if (tIndex > 0 || oIndex > 0 || fOperand != other.fOperand) {
+        addTPair(tStart, other, oStart, false, ptStart);
+    }
+    tStart = fTs[tIndex].fT;
+    oStart = other.fTs[oIndex].fT;
+    do {
+        double nextT;
+        do {
+            nextT = fTs[++tIndex].fT;
+        } while (approximately_negative(nextT - tStart));
+        tStart = nextT;
+        ptStart = fTs[tIndex].fPt;
+        do {
+            nextT = other.fTs[++oIndex].fT;
+        } while (approximately_negative(nextT - oStart));
+        oStart = nextT;
+        if (tStart == 1 && oStart == 1 && fOperand == other.fOperand) {
+            break;
+        }
+        addTPair(tStart, other, oStart, false, ptStart);
+    } while (tStart < 1 && oStart < 1 && !approximately_negative(oEnd - oStart));
+}
+
+void SkOpSegment::addCubic(const SkPoint pts[4], bool operand, bool evenOdd) {
+    init(pts, SkPath::kCubic_Verb, operand, evenOdd);
+    fBounds.setCubicBounds(pts);
+}
+
+void SkOpSegment::addCurveTo(int start, int end, SkPathWriter& path, bool active) const {
+    SkPoint edge[4];
+    const SkPoint* ePtr;
+    int lastT = fTs.count() - 1;
+    if (lastT < 0 || (start == 0 && end == lastT) || (start == lastT && end == 0)) {
+        ePtr = fPts;
+    } else {
+    // OPTIMIZE? if not active, skip remainder and return xyAtT(end)
+        subDivide(start, end, edge);
+        ePtr = edge;
+    }
+    if (active) {
+        bool reverse = ePtr == fPts && start != 0;
+        if (reverse) {
+            path.deferredMoveLine(ePtr[fVerb]);
+            switch (fVerb) {
+                case SkPath::kLine_Verb:
+                    path.deferredLine(ePtr[0]);
+                    break;
+                case SkPath::kQuad_Verb:
+                    path.quadTo(ePtr[1], ePtr[0]);
+                    break;
+                case SkPath::kCubic_Verb:
+                    path.cubicTo(ePtr[2], ePtr[1], ePtr[0]);
+                    break;
+                default:
+                    SkASSERT(0);
+            }
+   //         return ePtr[0];
+       } else {
+            path.deferredMoveLine(ePtr[0]);
+            switch (fVerb) {
+                case SkPath::kLine_Verb:
+                    path.deferredLine(ePtr[1]);
+                    break;
+                case SkPath::kQuad_Verb:
+                    path.quadTo(ePtr[1], ePtr[2]);
+                    break;
+                case SkPath::kCubic_Verb:
+                    path.cubicTo(ePtr[1], ePtr[2], ePtr[3]);
+                    break;
+                default:
+                    SkASSERT(0);
+            }
+        }
+    }
+  //  return ePtr[fVerb];
+}
+
+void SkOpSegment::addLine(const SkPoint pts[2], bool operand, bool evenOdd) {
+    init(pts, SkPath::kLine_Verb, operand, evenOdd);
+    fBounds.set(pts, 2);
+}
+
+// add 2 to edge or out of range values to get T extremes
+void SkOpSegment::addOtherT(int index, double otherT, int otherIndex) {
+    SkOpSpan& span = fTs[index];
+#if PIN_ADD_T
+    if (precisely_less_than_zero(otherT)) {
+        otherT = 0;
+    } else if (precisely_greater_than_one(otherT)) {
+        otherT = 1;
+    }
+#endif
+    span.fOtherT = otherT;
+    span.fOtherIndex = otherIndex;
+}
+
+void SkOpSegment::addQuad(const SkPoint pts[3], bool operand, bool evenOdd) {
+    init(pts, SkPath::kQuad_Verb, operand, evenOdd);
+    fBounds.setQuadBounds(pts);
+}
+
+    // Defer all coincident edge processing until
+    // after normal intersections have been computed
+
+// no need to be tricky; insert in normal T order
+// resolve overlapping ts when considering coincidence later
+
+    // add non-coincident intersection. Resulting edges are sorted in T.
+int SkOpSegment::addT(SkOpSegment* other, const SkPoint& pt, double newT) {
+    // FIXME: in the pathological case where there is a ton of intercepts,
+    //  binary search?
+    int insertedAt = -1;
+    size_t tCount = fTs.count();
+    for (size_t index = 0; index < tCount; ++index) {
+        // OPTIMIZATION: if there are three or more identical Ts, then
+        // the fourth and following could be further insertion-sorted so
+        // that all the edges are clockwise or counterclockwise.
+        // This could later limit segment tests to the two adjacent
+        // neighbors, although it doesn't help with determining which
+        // circular direction to go in.
+        if (newT < fTs[index].fT) {
+            insertedAt = index;
+            break;
+        }
+    }
+    SkOpSpan* span;
+    if (insertedAt >= 0) {
+        span = fTs.insert(insertedAt);
+    } else {
+        insertedAt = tCount;
+        span = fTs.append();
+    }
+    span->fT = newT;
+    span->fOther = other;
+    span->fPt = pt;
+    span->fWindSum = SK_MinS32;
+    span->fOppSum = SK_MinS32;
+    span->fWindValue = 1;
+    span->fOppValue = 0;
+    span->fTiny = false;
+    span->fLoop = false;
+    if ((span->fDone = newT == 1)) {
+        ++fDoneSpans;
+    }
+    span->fUnsortableStart = false;
+    span->fUnsortableEnd = false;
+    int less = -1;
+    while (&span[less + 1] - fTs.begin() > 0 && xyAtT(&span[less]) == xyAtT(span)) {
+        if (span[less].fDone) {
+            break;
+        }
+        double tInterval = newT - span[less].fT;
+        if (precisely_negative(tInterval)) {
+            break;
+        }
+        if (fVerb == SkPath::kCubic_Verb) {
+            double tMid = newT - tInterval / 2;
+            SkDPoint midPt = dcubic_xy_at_t(fPts, tMid);
+            if (!midPt.approximatelyEqual(xyAtT(span))) {
+                break;
+            }
+        }
+        span[less].fTiny = true;
+        span[less].fDone = true;
+        if (approximately_negative(newT - span[less].fT)) {
+            if (approximately_greater_than_one(newT)) {
+                span[less].fUnsortableStart = true;
+                span[less - 1].fUnsortableEnd = true;
+            }
+            if (approximately_less_than_zero(span[less].fT)) {
+                span[less + 1].fUnsortableStart = true;
+                span[less].fUnsortableEnd = true;
+            }
+        }
+        ++fDoneSpans;
+        --less;
+    }
+    int more = 1;
+    while (fTs.end() - &span[more - 1] > 1 && xyAtT(&span[more]) == xyAtT(span)) {
+        if (span[more - 1].fDone) {
+            break;
+        }
+        double tEndInterval = span[more].fT - newT;
+        if (precisely_negative(tEndInterval)) {
+            break;
+        }
+        if (fVerb == SkPath::kCubic_Verb) {
+            double tMid = newT - tEndInterval / 2;
+            SkDPoint midEndPt = dcubic_xy_at_t(fPts, tMid);
+            if (!midEndPt.approximatelyEqual(xyAtT(span))) {
+                break;
+            }
+        }
+        span[more - 1].fTiny = true;
+        span[more - 1].fDone = true;
+        if (approximately_negative(span[more].fT - newT)) {
+            if (approximately_greater_than_one(span[more].fT)) {
+                span[more + 1].fUnsortableStart = true;
+                span[more].fUnsortableEnd = true;
+            }
+            if (approximately_less_than_zero(newT)) {
+                span[more].fUnsortableStart = true;
+                span[more - 1].fUnsortableEnd = true;
+            }
+        }
+        ++fDoneSpans;
+        ++more;
+    }
+    return insertedAt;
+}
+
+// set spans from start to end to decrement by one
+// note this walks other backwards
+// FIMXE: there's probably an edge case that can be constructed where
+// two span in one segment are separated by float epsilon on one span but
+// not the other, if one segment is very small. For this
+// case the counts asserted below may or may not be enough to separate the
+// spans. Even if the counts work out, what if the spans aren't correctly
+// sorted? It feels better in such a case to match the span's other span
+// pointer since both coincident segments must contain the same spans.
+void SkOpSegment::addTCancel(double startT, double endT, SkOpSegment& other,
+        double oStartT, double oEndT) {
+    SkASSERT(!approximately_negative(endT - startT));
+    SkASSERT(!approximately_negative(oEndT - oStartT));
+    bool binary = fOperand != other.fOperand;
+    int index = 0;
+    while (!approximately_negative(startT - fTs[index].fT)) {
+        ++index;
+    }
+    int oIndex = other.fTs.count();
+    while (approximately_positive(other.fTs[--oIndex].fT - oEndT))
+        ;
+    double tRatio = (oEndT - oStartT) / (endT - startT);
+    SkOpSpan* test = &fTs[index];
+    SkOpSpan* oTest = &other.fTs[oIndex];
+    SkTDArray<double> outsideTs;
+    SkTDArray<double> oOutsideTs;
+    do {
+        bool decrement = test->fWindValue && oTest->fWindValue && !binary;
+        bool track = test->fWindValue || oTest->fWindValue;
+        double testT = test->fT;
+        double oTestT = oTest->fT;
+        SkOpSpan* span = test;
+        do {
+            if (decrement) {
+                decrementSpan(span);
+            } else if (track && span->fT < 1 && oTestT < 1) {
+                TrackOutside(outsideTs, span->fT, oTestT);
+            }
+            span = &fTs[++index];
+        } while (approximately_negative(span->fT - testT));
+        SkOpSpan* oSpan = oTest;
+        double otherTMatchStart = oEndT - (span->fT - startT) * tRatio;
+        double otherTMatchEnd = oEndT - (test->fT - startT) * tRatio;
+        SkDEBUGCODE(int originalWindValue = oSpan->fWindValue);
+        while (approximately_negative(otherTMatchStart - oSpan->fT)
+                && !approximately_negative(otherTMatchEnd - oSpan->fT)) {
+    #ifdef SK_DEBUG
+            SkASSERT(originalWindValue == oSpan->fWindValue);
+    #endif
+            if (decrement) {
+                other.decrementSpan(oSpan);
+            } else if (track && oSpan->fT < 1 && testT < 1) {
+                TrackOutside(oOutsideTs, oSpan->fT, testT);
+            }
+            if (!oIndex) {
+                break;
+            }
+            oSpan = &other.fTs[--oIndex];
+        }
+        test = span;
+        oTest = oSpan;
+    } while (!approximately_negative(endT - test->fT));
+    SkASSERT(!oIndex || approximately_negative(oTest->fT - oStartT));
+    // FIXME: determine if canceled edges need outside ts added
+    if (!done() && outsideTs.count()) {
+        double tStart = outsideTs[0];
+        double oStart = outsideTs[1];
+        addCancelOutsides(tStart, oStart, other, oEndT);
+        int count = outsideTs.count();
+        if (count > 2) {
+            double tStart = outsideTs[count - 2];
+            double oStart = outsideTs[count - 1];
+            addCancelOutsides(tStart, oStart, other, oEndT);
+        }
+    }
+    if (!other.done() && oOutsideTs.count()) {
+        double tStart = oOutsideTs[0];
+        double oStart = oOutsideTs[1];
+        other.addCancelOutsides(tStart, oStart, *this, endT);
+    }
+}
+
+int SkOpSegment::addSelfT(SkOpSegment* other, const SkPoint& pt, double newT) {
+    int result = addT(other, pt, newT);
+    SkOpSpan* span = &fTs[result];
+    span->fLoop = true;
+    return result;
+}
+
+int SkOpSegment::addUnsortableT(SkOpSegment* other, bool start, const SkPoint& pt, double newT) {
+    int result = addT(other, pt, newT);
+    SkOpSpan* span = &fTs[result];
+    if (start) {
+        if (result > 0) {
+            span[result - 1].fUnsortableEnd = true;
+        }
+        span[result].fUnsortableStart = true;
+    } else {
+        span[result].fUnsortableEnd = true;
+        if (result + 1 < fTs.count()) {
+            span[result + 1].fUnsortableStart = true;
+        }
+    }
+    return result;
+}
+
+int SkOpSegment::bumpCoincidentThis(const SkOpSpan* oTest, bool opp, int index,
+        SkTDArray<double>& outsideTs) {
+    int oWindValue = oTest->fWindValue;
+    int oOppValue = oTest->fOppValue;
+    if (opp) {
+        SkTSwap<int>(oWindValue, oOppValue);
+    }
+    SkOpSpan* const test = &fTs[index];
+    SkOpSpan* end = test;
+    const double oStartT = oTest->fT;
+    do {
+        if (bumpSpan(end, oWindValue, oOppValue)) {
+            TrackOutside(outsideTs, end->fT, oStartT);
+        }
+        end = &fTs[++index];
+    } while (approximately_negative(end->fT - test->fT));
+    return index;
+}
+
+// because of the order in which coincidences are resolved, this and other
+// may not have the same intermediate points. Compute the corresponding
+// intermediate T values (using this as the master, other as the follower)
+// and walk other conditionally -- hoping that it catches up in the end
+int SkOpSegment::bumpCoincidentOther(const SkOpSpan* test, double oEndT, int& oIndex,
+        SkTDArray<double>& oOutsideTs) {
+    SkOpSpan* const oTest = &fTs[oIndex];
+    SkOpSpan* oEnd = oTest;
+    const double startT = test->fT;
+    const double oStartT = oTest->fT;
+    while (!approximately_negative(oEndT - oEnd->fT)
+            && approximately_negative(oEnd->fT - oStartT)) {
+        zeroSpan(oEnd);
+        TrackOutside(oOutsideTs, oEnd->fT, startT);
+        oEnd = &fTs[++oIndex];
+    }
+    return oIndex;
+}
+
+// FIXME: need to test this case:
+// contourA has two segments that are coincident
+// contourB has two segments that are coincident in the same place
+// each ends up with +2/0 pairs for winding count
+// since logic below doesn't transfer count (only increments/decrements) can this be
+// resolved to +4/0 ?
+
+// set spans from start to end to increment the greater by one and decrement
+// the lesser
+void SkOpSegment::addTCoincident(double startT, double endT, SkOpSegment& other, double oStartT,
+                                 double oEndT) {
+    SkASSERT(!approximately_negative(endT - startT));
+    SkASSERT(!approximately_negative(oEndT - oStartT));
+    bool opp = fOperand ^ other.fOperand;
+    int index = 0;
+    while (!approximately_negative(startT - fTs[index].fT)) {
+        ++index;
+    }
+    int oIndex = 0;
+    while (!approximately_negative(oStartT - other.fTs[oIndex].fT)) {
+        ++oIndex;
+    }
+    SkOpSpan* test = &fTs[index];
+    SkOpSpan* oTest = &other.fTs[oIndex];
+    SkTDArray<double> outsideTs;
+    SkTDArray<double> oOutsideTs;
+    do {
+        // if either span has an opposite value and the operands don't match, resolve first
+ //       SkASSERT(!test->fDone || !oTest->fDone);
+        if (test->fDone || oTest->fDone) {
+            index = advanceCoincidentThis(oTest, opp, index);
+            oIndex = other.advanceCoincidentOther(test, oEndT, oIndex);
+        } else {
+            index = bumpCoincidentThis(oTest, opp, index, outsideTs);
+            oIndex = other.bumpCoincidentOther(test, oEndT, oIndex, oOutsideTs);
+        }
+        test = &fTs[index];
+        oTest = &other.fTs[oIndex];
+    } while (!approximately_negative(endT - test->fT));
+    SkASSERT(approximately_negative(oTest->fT - oEndT));
+    SkASSERT(approximately_negative(oEndT - oTest->fT));
+    if (!done() && outsideTs.count()) {
+        addCoinOutsides(outsideTs, other, oEndT);
+    }
+    if (!other.done() && oOutsideTs.count()) {
+        other.addCoinOutsides(oOutsideTs, *this, endT);
+    }
+}
+
+// FIXME: this doesn't prevent the same span from being added twice
+// fix in caller, SkASSERT here?
+void SkOpSegment::addTPair(double t, SkOpSegment& other, double otherT, bool borrowWind,
+                           const SkPoint& pt) {
+    int tCount = fTs.count();
+    for (int tIndex = 0; tIndex < tCount; ++tIndex) {
+        const SkOpSpan& span = fTs[tIndex];
+        if (!approximately_negative(span.fT - t)) {
+            break;
+        }
+        if (approximately_negative(span.fT - t) && span.fOther == &other
+                && approximately_equal(span.fOtherT, otherT)) {
+#if DEBUG_ADD_T_PAIR
+            SkDebugf("%s addTPair duplicate this=%d %1.9g other=%d %1.9g\n",
+                    __FUNCTION__, fID, t, other.fID, otherT);
+#endif
+            return;
+        }
+    }
+#if DEBUG_ADD_T_PAIR
+    SkDebugf("%s addTPair this=%d %1.9g other=%d %1.9g\n",
+            __FUNCTION__, fID, t, other.fID, otherT);
+#endif
+    int insertedAt = addT(&other, pt, t);
+    int otherInsertedAt = other.addT(this, pt, otherT);
+    addOtherT(insertedAt, otherT, otherInsertedAt);
+    other.addOtherT(otherInsertedAt, t, insertedAt);
+    matchWindingValue(insertedAt, t, borrowWind);
+    other.matchWindingValue(otherInsertedAt, otherT, borrowWind);
+}
+
+void SkOpSegment::addTwoAngles(int start, int end, SkTDArray<SkOpAngle>& angles) const {
+    // add edge leading into junction
+    int min = SkMin32(end, start);
+    if (fTs[min].fWindValue > 0 || fTs[min].fOppValue > 0) {
+        addAngle(angles, end, start);
+    }
+    // add edge leading away from junction
+    int step = SkSign32(end - start);
+    int tIndex = nextExactSpan(end, step);
+    min = SkMin32(end, tIndex);
+    if (tIndex >= 0 && (fTs[min].fWindValue > 0 || fTs[min].fOppValue > 0)) {
+        addAngle(angles, end, tIndex);
+    }
+}
+
+int SkOpSegment::advanceCoincidentThis(const SkOpSpan* oTest, bool opp, int index) {
+    SkOpSpan* const test = &fTs[index];
+    SkOpSpan* end;
+    do {
+        end = &fTs[++index];
+    } while (approximately_negative(end->fT - test->fT));
+    return index;
+}
+
+int SkOpSegment::advanceCoincidentOther(const SkOpSpan* test, double oEndT, int& oIndex) {
+    SkOpSpan* const oTest = &fTs[oIndex];
+    SkOpSpan* oEnd = oTest;
+    const double oStartT = oTest->fT;
+    while (!approximately_negative(oEndT - oEnd->fT)
+            && approximately_negative(oEnd->fT - oStartT)) {
+        oEnd = &fTs[++oIndex];
+    }
+    return oIndex;
+}
+
+bool SkOpSegment::betweenTs(int lesser, double testT, int greater) {
+    if (lesser > greater) {
+        SkTSwap<int>(lesser, greater);
+    }
+    return approximately_between(fTs[lesser].fT, testT, fTs[greater].fT);
+}
+
+void SkOpSegment::buildAngles(int index, SkTDArray<SkOpAngle>& angles, bool includeOpp) const {
+    double referenceT = fTs[index].fT;
+    int lesser = index;
+    while (--lesser >= 0 && (includeOpp || fTs[lesser].fOther->fOperand == fOperand)
+            && precisely_negative(referenceT - fTs[lesser].fT)) {
+        buildAnglesInner(lesser, angles);
+    }
+    do {
+        buildAnglesInner(index, angles);
+    } while (++index < fTs.count() && (includeOpp || fTs[index].fOther->fOperand == fOperand)
+            && precisely_negative(fTs[index].fT - referenceT));
+}
+
+void SkOpSegment::buildAnglesInner(int index, SkTDArray<SkOpAngle>& angles) const {
+    const SkOpSpan* span = &fTs[index];
+    SkOpSegment* other = span->fOther;
+// if there is only one live crossing, and no coincidence, continue
+// in the same direction
+// if there is coincidence, the only choice may be to reverse direction
+    // find edge on either side of intersection
+    int oIndex = span->fOtherIndex;
+    // if done == -1, prior span has already been processed
+    int step = 1;
+    int next = other->nextExactSpan(oIndex, step);
+   if (next < 0) {
+        step = -step;
+        next = other->nextExactSpan(oIndex, step);
+    }
+    // add candidate into and away from junction
+    other->addTwoAngles(next, oIndex, angles);
+}
+
+int SkOpSegment::computeSum(int startIndex, int endIndex, bool binary) {
+    SkTDArray<SkOpAngle> angles;
+    addTwoAngles(startIndex, endIndex, angles);
+    buildAngles(endIndex, angles, false);
+    // OPTIMIZATION: check all angles to see if any have computed wind sum
+    // before sorting (early exit if none)
+    SkTDArray<SkOpAngle*> sorted;
+    bool sortable = SortAngles(angles, sorted);
+#if DEBUG_SORT
+    sorted[0]->segment()->debugShowSort(__FUNCTION__, sorted, 0, 0, 0);
+#endif
+    if (!sortable) {
+        return SK_MinS32;
+    }
+    int angleCount = angles.count();
+    const SkOpAngle* angle;
+    const SkOpSegment* base;
+    int winding;
+    int oWinding;
+    int firstIndex = 0;
+    do {
+        angle = sorted[firstIndex];
+        base = angle->segment();
+        winding = base->windSum(angle);
+        if (winding != SK_MinS32) {
+            oWinding = base->oppSum(angle);
+            break;
+        }
+        if (++firstIndex == angleCount) {
+            return SK_MinS32;
+        }
+    } while (true);
+    // turn winding into contourWinding
+    int spanWinding = base->spanSign(angle);
+    bool inner = UseInnerWinding(winding + spanWinding, winding);
+#if DEBUG_WINDING
+    SkDebugf("%s spanWinding=%d winding=%d sign=%d inner=%d result=%d\n", __FUNCTION__,
+        spanWinding, winding, angle->sign(), inner,
+        inner ? winding + spanWinding : winding);
+#endif
+    if (inner) {
+        winding += spanWinding;
+    }
+#if DEBUG_SORT
+    base->debugShowSort(__FUNCTION__, sorted, firstIndex, winding, oWinding);
+#endif
+    int nextIndex = firstIndex + 1;
+    int lastIndex = firstIndex != 0 ? firstIndex : angleCount;
+    winding -= base->spanSign(angle);
+    oWinding -= base->oppSign(angle);
+    do {
+        if (nextIndex == angleCount) {
+            nextIndex = 0;
+        }
+        angle = sorted[nextIndex];
+        SkOpSegment* segment = angle->segment();
+        bool opp = base->fOperand ^ segment->fOperand;
+        int maxWinding, oMaxWinding;
+        int spanSign = segment->spanSign(angle);
+        int oppoSign = segment->oppSign(angle);
+        if (opp) {
+            oMaxWinding = oWinding;
+            oWinding -= spanSign;
+            maxWinding = winding;
+            if (oppoSign) {
+                winding -= oppoSign;
+            }
+        } else {
+            maxWinding = winding;
+            winding -= spanSign;
+            oMaxWinding = oWinding;
+            if (oppoSign) {
+                oWinding -= oppoSign;
+            }
+        }
+        if (segment->windSum(angle) == SK_MinS32) {
+            if (opp) {
+                if (UseInnerWinding(oMaxWinding, oWinding)) {
+                    oMaxWinding = oWinding;
+                }
+                if (oppoSign && UseInnerWinding(maxWinding, winding)) {
+                    maxWinding = winding;
+                }
+                (void) segment->markAndChaseWinding(angle, oMaxWinding, maxWinding);
+            } else {
+                if (UseInnerWinding(maxWinding, winding)) {
+                    maxWinding = winding;
+                }
+                if (oppoSign && UseInnerWinding(oMaxWinding, oWinding)) {
+                    oMaxWinding = oWinding;
+                }
+                (void) segment->markAndChaseWinding(angle, maxWinding,
+                        binary ? oMaxWinding : 0);
+            }
+        }
+    } while (++nextIndex != lastIndex);
+    int minIndex = SkMin32(startIndex, endIndex);
+    return windSum(minIndex);
+}
+
+int SkOpSegment::crossedSpanY(const SkPoint& basePt, SkScalar& bestY, double& hitT,
+                              bool& hitSomething, double mid, bool opp, bool current) const {
+    SkScalar bottom = fBounds.fBottom;
+    int bestTIndex = -1;
+    if (bottom <= bestY) {
+        return bestTIndex;
+    }
+    SkScalar top = fBounds.fTop;
+    if (top >= basePt.fY) {
+        return bestTIndex;
+    }
+    if (fBounds.fLeft > basePt.fX) {
+        return bestTIndex;
+    }
+    if (fBounds.fRight < basePt.fX) {
+        return bestTIndex;
+    }
+    if (fBounds.fLeft == fBounds.fRight) {
+        // if vertical, and directly above test point, wait for another one
+        return AlmostEqualUlps(basePt.fX, fBounds.fLeft) ? SK_MinS32 : bestTIndex;
+    }
+    // intersect ray starting at basePt with edge
+    SkIntersections intersections;
+    // OPTIMIZE: use specialty function that intersects ray with curve,
+    // returning t values only for curve (we don't care about t on ray)
+    int pts = (intersections.*CurveVertical[fVerb])(fPts, top, bottom, basePt.fX, false);
+    if (pts == 0 || (current && pts == 1)) {
+        return bestTIndex;
+    }
+    if (current) {
+        SkASSERT(pts > 1);
+        int closestIdx = 0;
+        double closest = fabs(intersections[0][0] - mid);
+        for (int idx = 1; idx < pts; ++idx) {
+            double test = fabs(intersections[0][idx] - mid);
+            if (closest > test) {
+                closestIdx = idx;
+                closest = test;
+            }
+        }
+        intersections.quickRemoveOne(closestIdx, --pts);
+    }
+    double bestT = -1;
+    for (int index = 0; index < pts; ++index) {
+        double foundT = intersections[0][index];
+        if (approximately_less_than_zero(foundT)
+                    || approximately_greater_than_one(foundT)) {
+            continue;
+        }
+        SkScalar testY = (*CurvePointAtT[fVerb])(fPts, foundT).fY;
+        if (approximately_negative(testY - bestY)
+                || approximately_negative(basePt.fY - testY)) {
+            continue;
+        }
+        if (pts > 1 && fVerb == SkPath::kLine_Verb) {
+            return SK_MinS32; // if the intersection is edge on, wait for another one
+        }
+        if (fVerb > SkPath::kLine_Verb) {
+            SkScalar dx = (*CurveSlopeAtT[fVerb])(fPts, foundT).fX;
+            if (approximately_zero(dx)) {
+                return SK_MinS32; // hit vertical, wait for another one
+            }
+        }
+        bestY = testY;
+        bestT = foundT;
+    }
+    if (bestT < 0) {
+        return bestTIndex;
+    }
+    SkASSERT(bestT >= 0);
+    SkASSERT(bestT <= 1);
+    int start;
+    int end = 0;
+    do {
+        start = end;
+        end = nextSpan(start, 1);
+    } while (fTs[end].fT < bestT);
+    // FIXME: see next candidate for a better pattern to find the next start/end pair
+    while (start + 1 < end && fTs[start].fDone) {
+        ++start;
+    }
+    if (!isCanceled(start)) {
+        hitT = bestT;
+        bestTIndex = start;
+        hitSomething = true;
+    }
+    return bestTIndex;
+}
+
+void SkOpSegment::decrementSpan(SkOpSpan* span) {
+    SkASSERT(span->fWindValue > 0);
+    if (--(span->fWindValue) == 0) {
+        if (!span->fOppValue && !span->fDone) {
+            span->fDone = true;
+            ++fDoneSpans;
+        }
+    }
+}
+
+bool SkOpSegment::bumpSpan(SkOpSpan* span, int windDelta, int oppDelta) {
+    SkASSERT(!span->fDone);
+    span->fWindValue += windDelta;
+    SkASSERT(span->fWindValue >= 0);
+    span->fOppValue += oppDelta;
+    SkASSERT(span->fOppValue >= 0);
+    if (fXor) {
+        span->fWindValue &= 1;
+    }
+    if (fOppXor) {
+        span->fOppValue &= 1;
+    }
+    if (!span->fWindValue && !span->fOppValue) {
+        span->fDone = true;
+        ++fDoneSpans;
+        return true;
+    }
+    return false;
+}
+
+bool SkOpSegment::equalPoints(int greaterTIndex, int lesserTIndex) {
+    SkASSERT(greaterTIndex >= lesserTIndex);
+    double greaterT = fTs[greaterTIndex].fT;
+    double lesserT = fTs[lesserTIndex].fT;
+    if (greaterT == lesserT) {
+        return true;
+    }
+    if (!approximately_negative(greaterT - lesserT)) {
+        return false;
+    }
+    return xyAtT(greaterTIndex) == xyAtT(lesserTIndex);
+}
+
+/*
+ The M and S variable name parts stand for the operators.
+   Mi stands for Minuend (see wiki subtraction, analogous to difference)
+   Su stands for Subtrahend
+ The Opp variable name part designates that the value is for the Opposite operator.
+ Opposite values result from combining coincident spans.
+ */
+SkOpSegment* SkOpSegment::findNextOp(SkTDArray<SkOpSpan*>& chase, int& nextStart, int& nextEnd,
+                                     bool& unsortable, SkPathOp op, const int xorMiMask,
+                                     const int xorSuMask) {
+    const int startIndex = nextStart;
+    const int endIndex = nextEnd;
+    SkASSERT(startIndex != endIndex);
+    SkDEBUGCODE(const int count = fTs.count());
+    SkASSERT(startIndex < endIndex ? startIndex < count - 1 : startIndex > 0);
+    const int step = SkSign32(endIndex - startIndex);
+    const int end = nextExactSpan(startIndex, step);
+    SkASSERT(end >= 0);
+    SkOpSpan* endSpan = &fTs[end];
+    SkOpSegment* other;
+    if (isSimple(end)) {
+    // mark the smaller of startIndex, endIndex done, and all adjacent
+    // spans with the same T value (but not 'other' spans)
+#if DEBUG_WINDING
+        SkDebugf("%s simple\n", __FUNCTION__);
+#endif
+        int min = SkMin32(startIndex, endIndex);
+        if (fTs[min].fDone) {
+            return NULL;
+        }
+        markDoneBinary(min);
+        other = endSpan->fOther;
+        nextStart = endSpan->fOtherIndex;
+        double startT = other->fTs[nextStart].fT;
+        nextEnd = nextStart;
+        do {
+            nextEnd += step;
+        }
+        while (precisely_zero(startT - other->fTs[nextEnd].fT));
+        SkASSERT(step < 0 ? nextEnd >= 0 : nextEnd < other->fTs.count());
+        return other;
+    }
+    // more than one viable candidate -- measure angles to find best
+    SkTDArray<SkOpAngle> angles;
+    SkASSERT(startIndex - endIndex != 0);
+    SkASSERT((startIndex - endIndex < 0) ^ (step < 0));
+    addTwoAngles(startIndex, end, angles);
+    buildAngles(end, angles, true);
+    SkTDArray<SkOpAngle*> sorted;
+    bool sortable = SortAngles(angles, sorted);
+    int angleCount = angles.count();
+    int firstIndex = findStartingEdge(sorted, startIndex, end);
+    SkASSERT(firstIndex >= 0);
+#if DEBUG_SORT
+    debugShowSort(__FUNCTION__, sorted, firstIndex);
+#endif
+    if (!sortable) {
+        unsortable = true;
+        return NULL;
+    }
+    SkASSERT(sorted[firstIndex]->segment() == this);
+#if DEBUG_WINDING
+    SkDebugf("%s firstIndex=[%d] sign=%d\n", __FUNCTION__, firstIndex,
+            sorted[firstIndex]->sign());
+#endif
+    int sumMiWinding = updateWinding(endIndex, startIndex);
+    int sumSuWinding = updateOppWinding(endIndex, startIndex);
+    if (operand()) {
+        SkTSwap<int>(sumMiWinding, sumSuWinding);
+    }
+    int nextIndex = firstIndex + 1;
+    int lastIndex = firstIndex != 0 ? firstIndex : angleCount;
+    const SkOpAngle* foundAngle = NULL;
+    bool foundDone = false;
+    // iterate through the angle, and compute everyone's winding
+    SkOpSegment* nextSegment;
+    int activeCount = 0;
+    do {
+        SkASSERT(nextIndex != firstIndex);
+        if (nextIndex == angleCount) {
+            nextIndex = 0;
+        }
+        const SkOpAngle* nextAngle = sorted[nextIndex];
+        nextSegment = nextAngle->segment();
+        int maxWinding, sumWinding, oppMaxWinding, oppSumWinding;
+        bool activeAngle = nextSegment->activeOp(xorMiMask, xorSuMask, nextAngle->start(),
+                nextAngle->end(), op, sumMiWinding, sumSuWinding,
+                maxWinding, sumWinding, oppMaxWinding, oppSumWinding);
+        if (activeAngle) {
+            ++activeCount;
+            if (!foundAngle || (foundDone && activeCount & 1)) {
+                if (nextSegment->tiny(nextAngle)) {
+                    unsortable = true;
+                    return NULL;
+                }
+                foundAngle = nextAngle;
+                foundDone = nextSegment->done(nextAngle) && !nextSegment->tiny(nextAngle);
+            }
+        }
+        if (nextSegment->done()) {
+            continue;
+        }
+        if (nextSegment->windSum(nextAngle) != SK_MinS32) {
+            continue;
+        }
+        SkOpSpan* last = nextSegment->markAngle(maxWinding, sumWinding, oppMaxWinding,
+                oppSumWinding, activeAngle, nextAngle);
+        if (last) {
+            *chase.append() = last;
+#if DEBUG_WINDING
+            SkDebugf("%s chase.append id=%d\n", __FUNCTION__,
+                    last->fOther->fTs[last->fOtherIndex].fOther->debugID());
+#endif
+        }
+    } while (++nextIndex != lastIndex);
+    markDoneBinary(SkMin32(startIndex, endIndex));
+    if (!foundAngle) {
+        return NULL;
+    }
+    nextStart = foundAngle->start();
+    nextEnd = foundAngle->end();
+    nextSegment = foundAngle->segment();
+
+#if DEBUG_WINDING
+    SkDebugf("%s from:[%d] to:[%d] start=%d end=%d\n",
+            __FUNCTION__, debugID(), nextSegment->debugID(), nextStart, nextEnd);
+ #endif
+    return nextSegment;
+}
+
+SkOpSegment* SkOpSegment::findNextWinding(SkTDArray<SkOpSpan*>& chase, int& nextStart,
+                                          int& nextEnd, bool& unsortable) {
+    const int startIndex = nextStart;
+    const int endIndex = nextEnd;
+    SkASSERT(startIndex != endIndex);
+    SkDEBUGCODE(const int count = fTs.count());
+    SkASSERT(startIndex < endIndex ? startIndex < count - 1 : startIndex > 0);
+    const int step = SkSign32(endIndex - startIndex);
+    const int end = nextExactSpan(startIndex, step);
+    SkASSERT(end >= 0);
+    SkOpSpan* endSpan = &fTs[end];
+    SkOpSegment* other;
+    if (isSimple(end)) {
+    // mark the smaller of startIndex, endIndex done, and all adjacent
+    // spans with the same T value (but not 'other' spans)
+#if DEBUG_WINDING
+        SkDebugf("%s simple\n", __FUNCTION__);
+#endif
+        int min = SkMin32(startIndex, endIndex);
+        if (fTs[min].fDone) {
+            return NULL;
+        }
+        markDoneUnary(min);
+        other = endSpan->fOther;
+        nextStart = endSpan->fOtherIndex;
+        double startT = other->fTs[nextStart].fT;
+        nextEnd = nextStart;
+        do {
+            nextEnd += step;
+        }
+        while (precisely_zero(startT - other->fTs[nextEnd].fT));
+        SkASSERT(step < 0 ? nextEnd >= 0 : nextEnd < other->fTs.count());
+        return other;
+    }
+    // more than one viable candidate -- measure angles to find best
+    SkTDArray<SkOpAngle> angles;
+    SkASSERT(startIndex - endIndex != 0);
+    SkASSERT((startIndex - endIndex < 0) ^ (step < 0));
+    addTwoAngles(startIndex, end, angles);
+    buildAngles(end, angles, true);
+    SkTDArray<SkOpAngle*> sorted;
+    bool sortable = SortAngles(angles, sorted);
+    int angleCount = angles.count();
+    int firstIndex = findStartingEdge(sorted, startIndex, end);
+    SkASSERT(firstIndex >= 0);
+#if DEBUG_SORT
+    debugShowSort(__FUNCTION__, sorted, firstIndex);
+#endif
+    if (!sortable) {
+        unsortable = true;
+        return NULL;
+    }
+    SkASSERT(sorted[firstIndex]->segment() == this);
+#if DEBUG_WINDING
+    SkDebugf("%s firstIndex=[%d] sign=%d\n", __FUNCTION__, firstIndex,
+            sorted[firstIndex]->sign());
+#endif
+    int sumWinding = updateWinding(endIndex, startIndex);
+    int nextIndex = firstIndex + 1;
+    int lastIndex = firstIndex != 0 ? firstIndex : angleCount;
+    const SkOpAngle* foundAngle = NULL;
+    bool foundDone = false;
+    // iterate through the angle, and compute everyone's winding
+    SkOpSegment* nextSegment;
+    int activeCount = 0;
+    do {
+        SkASSERT(nextIndex != firstIndex);
+        if (nextIndex == angleCount) {
+            nextIndex = 0;
+        }
+        const SkOpAngle* nextAngle = sorted[nextIndex];
+        nextSegment = nextAngle->segment();
+        int maxWinding;
+        bool activeAngle = nextSegment->activeWinding(nextAngle->start(), nextAngle->end(),
+                maxWinding, sumWinding);
+        if (activeAngle) {
+            ++activeCount;
+            if (!foundAngle || (foundDone && activeCount & 1)) {
+                if (nextSegment->tiny(nextAngle)) {
+                    unsortable = true;
+                    return NULL;
+                }
+                foundAngle = nextAngle;
+                foundDone = nextSegment->done(nextAngle);
+            }
+        }
+        if (nextSegment->done()) {
+            continue;
+        }
+        if (nextSegment->windSum(nextAngle) != SK_MinS32) {
+            continue;
+        }
+        SkOpSpan* last = nextSegment->markAngle(maxWinding, sumWinding, activeAngle, nextAngle);
+        if (last) {
+            *chase.append() = last;
+#if DEBUG_WINDING
+            SkDebugf("%s chase.append id=%d\n", __FUNCTION__,
+                    last->fOther->fTs[last->fOtherIndex].fOther->debugID());
+#endif
+        }
+    } while (++nextIndex != lastIndex);
+    markDoneUnary(SkMin32(startIndex, endIndex));
+    if (!foundAngle) {
+        return NULL;
+    }
+    nextStart = foundAngle->start();
+    nextEnd = foundAngle->end();
+    nextSegment = foundAngle->segment();
+#if DEBUG_WINDING
+    SkDebugf("%s from:[%d] to:[%d] start=%d end=%d\n",
+            __FUNCTION__, debugID(), nextSegment->debugID(), nextStart, nextEnd);
+ #endif
+    return nextSegment;
+}
+
+SkOpSegment* SkOpSegment::findNextXor(int& nextStart, int& nextEnd, bool& unsortable) {
+    const int startIndex = nextStart;
+    const int endIndex = nextEnd;
+    SkASSERT(startIndex != endIndex);
+    SkDEBUGCODE(int count = fTs.count());
+    SkASSERT(startIndex < endIndex ? startIndex < count - 1
+            : startIndex > 0);
+    int step = SkSign32(endIndex - startIndex);
+    int end = nextExactSpan(startIndex, step);
+    SkASSERT(end >= 0);
+    SkOpSpan* endSpan = &fTs[end];
+    SkOpSegment* other;
+    if (isSimple(end)) {
+#if DEBUG_WINDING
+        SkDebugf("%s simple\n", __FUNCTION__);
+#endif
+        int min = SkMin32(startIndex, endIndex);
+        if (fTs[min].fDone) {
+            return NULL;
+        }
+        markDone(min, 1);
+        other = endSpan->fOther;
+        nextStart = endSpan->fOtherIndex;
+        double startT = other->fTs[nextStart].fT;
+        // FIXME: I don't know why the logic here is difference from the winding case
+        SkDEBUGCODE(bool firstLoop = true;)
+        if ((approximately_less_than_zero(startT) && step < 0)
+                || (approximately_greater_than_one(startT) && step > 0)) {
+            step = -step;
+            SkDEBUGCODE(firstLoop = false;)
+        }
+        do {
+            nextEnd = nextStart;
+            do {
+                nextEnd += step;
+            }
+             while (precisely_zero(startT - other->fTs[nextEnd].fT));
+            if (other->fTs[SkMin32(nextStart, nextEnd)].fWindValue) {
+                break;
+            }
+#ifdef SK_DEBUG
+            SkASSERT(firstLoop);
+#endif
+            SkDEBUGCODE(firstLoop = false;)
+            step = -step;
+        } while (true);
+        SkASSERT(step < 0 ? nextEnd >= 0 : nextEnd < other->fTs.count());
+        return other;
+    }
+    SkTDArray<SkOpAngle> angles;
+    SkASSERT(startIndex - endIndex != 0);
+    SkASSERT((startIndex - endIndex < 0) ^ (step < 0));
+    addTwoAngles(startIndex, end, angles);
+    buildAngles(end, angles, false);
+    SkTDArray<SkOpAngle*> sorted;
+    bool sortable = SortAngles(angles, sorted);
+    if (!sortable) {
+        unsortable = true;
+#if DEBUG_SORT
+        debugShowSort(__FUNCTION__, sorted, findStartingEdge(sorted, startIndex, end), 0, 0);
+#endif
+        return NULL;
+    }
+    int angleCount = angles.count();
+    int firstIndex = findStartingEdge(sorted, startIndex, end);
+    SkASSERT(firstIndex >= 0);
+#if DEBUG_SORT
+    debugShowSort(__FUNCTION__, sorted, firstIndex, 0, 0);
+#endif
+    SkASSERT(sorted[firstIndex]->segment() == this);
+    int nextIndex = firstIndex + 1;
+    int lastIndex = firstIndex != 0 ? firstIndex : angleCount;
+    const SkOpAngle* foundAngle = NULL;
+    bool foundDone = false;
+    SkOpSegment* nextSegment;
+    int activeCount = 0;
+    do {
+        SkASSERT(nextIndex != firstIndex);
+        if (nextIndex == angleCount) {
+            nextIndex = 0;
+        }
+        const SkOpAngle* nextAngle = sorted[nextIndex];
+        nextSegment = nextAngle->segment();
+        ++activeCount;
+        if (!foundAngle || (foundDone && activeCount & 1)) {
+            if (nextSegment->tiny(nextAngle)) {
+                unsortable = true;
+                return NULL;
+            }
+            foundAngle = nextAngle;
+            foundDone = nextSegment->done(nextAngle);
+        }
+        if (nextSegment->done()) {
+            continue;
+        }
+    } while (++nextIndex != lastIndex);
+    markDone(SkMin32(startIndex, endIndex), 1);
+    if (!foundAngle) {
+        return NULL;
+    }
+    nextStart = foundAngle->start();
+    nextEnd = foundAngle->end();
+    nextSegment = foundAngle->segment();
+#if DEBUG_WINDING
+    SkDebugf("%s from:[%d] to:[%d] start=%d end=%d\n",
+            __FUNCTION__, debugID(), nextSegment->debugID(), nextStart, nextEnd);
+ #endif
+    return nextSegment;
+}
+
+int SkOpSegment::findStartingEdge(SkTDArray<SkOpAngle*>& sorted, int start, int end) {
+    int angleCount = sorted.count();
+    int firstIndex = -1;
+    for (int angleIndex = 0; angleIndex < angleCount; ++angleIndex) {
+        const SkOpAngle* angle = sorted[angleIndex];
+        if (angle->segment() == this && angle->start() == end &&
+                angle->end() == start) {
+            firstIndex = angleIndex;
+            break;
+        }
+    }
+    return firstIndex;
+}
+
+// FIXME: this is tricky code; needs its own unit test
+// note that fOtherIndex isn't computed yet, so it can't be used here
+void SkOpSegment::findTooCloseToCall() {
+    int count = fTs.count();
+    if (count < 3) { // require t=0, x, 1 at minimum
+        return;
+    }
+    int matchIndex = 0;
+    int moCount;
+    SkOpSpan* match;
+    SkOpSegment* mOther;
+    do {
+        match = &fTs[matchIndex];
+        mOther = match->fOther;
+        // FIXME: allow quads, cubics to be near coincident?
+        if (mOther->fVerb == SkPath::kLine_Verb) {
+            moCount = mOther->fTs.count();
+            if (moCount >= 3) {
+                break;
+            }
+        }
+        if (++matchIndex >= count) {
+            return;
+        }
+    } while (true); // require t=0, x, 1 at minimum
+    // OPTIMIZATION: defer matchPt until qualifying toCount is found?
+    const SkPoint* matchPt = &xyAtT(match);
+    // look for a pair of nearby T values that map to the same (x,y) value
+    // if found, see if the pair of other segments share a common point. If
+    // so, the span from here to there is coincident.
+    for (int index = matchIndex + 1; index < count; ++index) {
+        SkOpSpan* test = &fTs[index];
+        if (test->fDone) {
+            continue;
+        }
+        SkOpSegment* tOther = test->fOther;
+        if (tOther->fVerb != SkPath::kLine_Verb) {
+            continue; // FIXME: allow quads, cubics to be near coincident?
+        }
+        int toCount = tOther->fTs.count();
+        if (toCount < 3) { // require t=0, x, 1 at minimum
+            continue;
+        }
+        const SkPoint* testPt = &xyAtT(test);
+        if (*matchPt != *testPt) {
+            matchIndex = index;
+            moCount = toCount;
+            match = test;
+            mOther = tOther;
+            matchPt = testPt;
+            continue;
+        }
+        int moStart = -1;
+        int moEnd = -1;
+        double moStartT, moEndT;
+        for (int moIndex = 0; moIndex < moCount; ++moIndex) {
+            SkOpSpan& moSpan = mOther->fTs[moIndex];
+            if (moSpan.fDone) {
+                continue;
+            }
+            if (moSpan.fOther == this) {
+                if (moSpan.fOtherT == match->fT) {
+                    moStart = moIndex;
+                    moStartT = moSpan.fT;
+                }
+                continue;
+            }
+            if (moSpan.fOther == tOther) {
+                if (tOther->windValueAt(moSpan.fOtherT) == 0) {
+                    moStart = -1;
+                    break;
+                }
+                SkASSERT(moEnd == -1);
+                moEnd = moIndex;
+                moEndT = moSpan.fT;
+            }
+        }
+        if (moStart < 0 || moEnd < 0) {
+            continue;
+        }
+        // FIXME: if moStartT, moEndT are initialized to NaN, can skip this test
+        if (approximately_equal(moStartT, moEndT)) {
+            continue;
+        }
+        int toStart = -1;
+        int toEnd = -1;
+        double toStartT, toEndT;
+        for (int toIndex = 0; toIndex < toCount; ++toIndex) {
+            SkOpSpan& toSpan = tOther->fTs[toIndex];
+            if (toSpan.fDone) {
+                continue;
+            }
+            if (toSpan.fOther == this) {
+                if (toSpan.fOtherT == test->fT) {
+                    toStart = toIndex;
+                    toStartT = toSpan.fT;
+                }
+                continue;
+            }
+            if (toSpan.fOther == mOther && toSpan.fOtherT == moEndT) {
+                if (mOther->windValueAt(toSpan.fOtherT) == 0) {
+                    moStart = -1;
+                    break;
+                }
+                SkASSERT(toEnd == -1);
+                toEnd = toIndex;
+                toEndT = toSpan.fT;
+            }
+        }
+        // FIXME: if toStartT, toEndT are initialized to NaN, can skip this test
+        if (toStart <= 0 || toEnd <= 0) {
+            continue;
+        }
+        if (approximately_equal(toStartT, toEndT)) {
+            continue;
+        }
+        // test to see if the segment between there and here is linear
+        if (!mOther->isLinear(moStart, moEnd)
+                || !tOther->isLinear(toStart, toEnd)) {
+            continue;
+        }
+        bool flipped = (moStart - moEnd) * (toStart - toEnd) < 1;
+        if (flipped) {
+            mOther->addTCancel(moStartT, moEndT, *tOther, toEndT, toStartT);
+        } else {
+            mOther->addTCoincident(moStartT, moEndT, *tOther, toStartT, toEndT);
+        }
+    }
+}
+
+// FIXME: either:
+// a) mark spans with either end unsortable as done, or
+// b) rewrite findTop / findTopSegment / findTopContour to iterate further
+//    when encountering an unsortable span
+
+// OPTIMIZATION : for a pair of lines, can we compute points at T (cached)
+// and use more concise logic like the old edge walker code?
+// FIXME: this needs to deal with coincident edges
+SkOpSegment* SkOpSegment::findTop(int& tIndex, int& endIndex, bool& unsortable, bool onlySortable) {
+    // iterate through T intersections and return topmost
+    // topmost tangent from y-min to first pt is closer to horizontal
+    SkASSERT(!done());
+    int firstT = -1;
+    /* SkPoint topPt = */ activeLeftTop(onlySortable, &firstT);
+    if (firstT < 0) {
+        unsortable = true;
+        firstT = 0;
+        while (fTs[firstT].fDone) {
+            SkASSERT(firstT < fTs.count());
+            ++firstT;
+        }
+        tIndex = firstT;
+        endIndex = nextExactSpan(firstT, 1);
+        return this;
+    }
+    // sort the edges to find the leftmost
+    int step = 1;
+    int end = nextSpan(firstT, step);
+    if (end == -1) {
+        step = -1;
+        end = nextSpan(firstT, step);
+        SkASSERT(end != -1);
+    }
+    // if the topmost T is not on end, or is three-way or more, find left
+    // look for left-ness from tLeft to firstT (matching y of other)
+    SkTDArray<SkOpAngle> angles;
+    SkASSERT(firstT - end != 0);
+    addTwoAngles(end, firstT, angles);
+    buildAngles(firstT, angles, true);
+    SkTDArray<SkOpAngle*> sorted;
+    bool sortable = SortAngles(angles, sorted);
+    int first = SK_MaxS32;
+    SkScalar top = SK_ScalarMax;
+    int count = sorted.count();
+    for (int index = 0; index < count; ++index) {
+        const SkOpAngle* angle = sorted[index];
+        SkOpSegment* next = angle->segment();
+        SkPathOpsBounds bounds;
+        next->subDivideBounds(angle->end(), angle->start(), bounds);
+        if (approximately_greater(top, bounds.fTop)) {
+            top = bounds.fTop;
+            first = index;
+        }
+    }
+    SkASSERT(first < SK_MaxS32);
+#if DEBUG_SORT // || DEBUG_SWAP_TOP
+    sorted[first]->segment()->debugShowSort(__FUNCTION__, sorted, first, 0, 0);
+#endif
+    if (onlySortable && !sortable) {
+        unsortable = true;
+        return NULL;
+    }
+    // skip edges that have already been processed
+    firstT = first - 1;
+    SkOpSegment* leftSegment;
+    do {
+        if (++firstT == count) {
+            firstT = 0;
+        }
+        const SkOpAngle* angle = sorted[firstT];
+        SkASSERT(!onlySortable || !angle->unsortable());
+        leftSegment = angle->segment();
+        tIndex = angle->end();
+        endIndex = angle->start();
+    } while (leftSegment->fTs[SkMin32(tIndex, endIndex)].fDone);
+    if (leftSegment->verb() >= SkPath::kQuad_Verb) {
+        if (!leftSegment->clockwise(tIndex, endIndex)) {
+            bool swap = !leftSegment->monotonicInY(tIndex, endIndex)
+                    && !leftSegment->serpentine(tIndex, endIndex);
+    #if DEBUG_SWAP_TOP
+            SkDebugf("%s swap=%d serpentine=%d containedByEnds=%d monotonic=%d\n", __FUNCTION__,
+                    swap,
+                    leftSegment->serpentine(tIndex, endIndex),
+                    leftSegment->controlsContainedByEnds(tIndex, endIndex),
+                    leftSegment->monotonicInY(tIndex, endIndex));
+    #endif
+            if (swap) {
+    // FIXME: I doubt it makes sense to (necessarily) swap if the edge was not the first
+    // sorted but merely the first not already processed (i.e., not done)
+                SkTSwap(tIndex, endIndex);
+            }
+        }
+    }
+    SkASSERT(!leftSegment->fTs[SkMin32(tIndex, endIndex)].fTiny);
+    return leftSegment;
+}
+
+// FIXME: not crazy about this
+// when the intersections are performed, the other index is into an
+// incomplete array. As the array grows, the indices become incorrect
+// while the following fixes the indices up again, it isn't smart about
+// skipping segments whose indices are already correct
+// assuming we leave the code that wrote the index in the first place
+void SkOpSegment::fixOtherTIndex() {
+    int iCount = fTs.count();
+    for (int i = 0; i < iCount; ++i) {
+        SkOpSpan& iSpan = fTs[i];
+        double oT = iSpan.fOtherT;
+        SkOpSegment* other = iSpan.fOther;
+        int oCount = other->fTs.count();
+        for (int o = 0; o < oCount; ++o) {
+            SkOpSpan& oSpan = other->fTs[o];
+            if (oT == oSpan.fT && this == oSpan.fOther && oSpan.fOtherT == iSpan.fT) {
+                iSpan.fOtherIndex = o;
+                break;
+            }
+        }
+    }
+}
+
+void SkOpSegment::init(const SkPoint pts[], SkPath::Verb verb, bool operand, bool evenOdd) {
+    fDoneSpans = 0;
+    fOperand = operand;
+    fXor = evenOdd;
+    fPts = pts;
+    fVerb = verb;
+}
+
+void SkOpSegment::initWinding(int start, int end) {
+    int local = spanSign(start, end);
+    int oppLocal = oppSign(start, end);
+    (void) markAndChaseWinding(start, end, local, oppLocal);
+    // OPTIMIZATION: the reverse mark and chase could skip the first marking
+    (void) markAndChaseWinding(end, start, local, oppLocal);
+}
+
+void SkOpSegment::initWinding(int start, int end, int winding, int oppWinding) {
+    int local = spanSign(start, end);
+    if (local * winding >= 0) {
+        winding += local;
+    }
+    int oppLocal = oppSign(start, end);
+    if (oppLocal * oppWinding >= 0) {
+        oppWinding += oppLocal;
+    }
+    (void) markAndChaseWinding(start, end, winding, oppWinding);
+}
+
+/*
+when we start with a vertical intersect, we try to use the dx to determine if the edge is to
+the left or the right of vertical. This determines if we need to add the span's
+sign or not. However, this isn't enough.
+If the supplied sign (winding) is zero, then we didn't hit another vertical span, so dx is needed.
+If there was a winding, then it may or may not need adjusting. If the span the winding was borrowed
+from has the same x direction as this span, the winding should change. If the dx is opposite, then
+the same winding is shared by both.
+*/
+void SkOpSegment::initWinding(int start, int end, double tHit, int winding, SkScalar hitDx,
+                              int oppWind, SkScalar hitOppDx) {
+    SkASSERT(hitDx || !winding);
+    SkScalar dx = (*CurveSlopeAtT[fVerb])(fPts, tHit).fX;
+    SkASSERT(dx);
+    int windVal = windValue(SkMin32(start, end));
+#if DEBUG_WINDING_AT_T
+    SkDebugf("%s oldWinding=%d hitDx=%c dx=%c windVal=%d", __FUNCTION__, winding,
+            hitDx ? hitDx > 0 ? '+' : '-' : '0', dx > 0 ? '+' : '-', windVal);
+#endif
+    if (!winding) {
+        winding = dx < 0 ? windVal : -windVal;
+    } else if (winding * dx < 0) {
+        int sideWind = winding + (dx < 0 ? windVal : -windVal);
+        if (abs(winding) < abs(sideWind)) {
+            winding = sideWind;
+        }
+    }
+#if DEBUG_WINDING_AT_T
+    SkDebugf(" winding=%d\n", winding);
+#endif
+    SkDEBUGCODE(int oppLocal = oppSign(start, end));
+    SkASSERT(hitOppDx || !oppWind || !oppLocal);
+    int oppWindVal = oppValue(SkMin32(start, end));
+    if (!oppWind) {
+        oppWind = dx < 0 ? oppWindVal : -oppWindVal;
+    } else if (hitOppDx * dx >= 0) {
+        int oppSideWind = oppWind + (dx < 0 ? oppWindVal : -oppWindVal);
+        if (abs(oppWind) < abs(oppSideWind)) {
+            oppWind = oppSideWind;
+        }
+    }
+    (void) markAndChaseWinding(start, end, winding, oppWind);
+}
+
+bool SkOpSegment::isLinear(int start, int end) const {
+    if (fVerb == SkPath::kLine_Verb) {
+        return true;
+    }
+    if (fVerb == SkPath::kQuad_Verb) {
+        SkDQuad qPart = SkDQuad::SubDivide(fPts, fTs[start].fT, fTs[end].fT);
+        return qPart.isLinear(0, 2);
+    } else {
+        SkASSERT(fVerb == SkPath::kCubic_Verb);
+        SkDCubic cPart = SkDCubic::SubDivide(fPts, fTs[start].fT, fTs[end].fT);
+        return cPart.isLinear(0, 3);
+    }
+}
+
+// OPTIMIZE: successive calls could start were the last leaves off
+// or calls could specialize to walk forwards or backwards
+bool SkOpSegment::isMissing(double startT) const {
+    size_t tCount = fTs.count();
+    for (size_t index = 0; index < tCount; ++index) {
+        if (approximately_zero(startT - fTs[index].fT)) {
+            return false;
+        }
+    }
+    return true;
+}
+
+bool SkOpSegment::isSimple(int end) const {
+    int count = fTs.count();
+    if (count == 2) {
+        return true;
+    }
+    double t = fTs[end].fT;
+    if (approximately_less_than_zero(t)) {
+        return !approximately_less_than_zero(fTs[1].fT);
+    }
+    if (approximately_greater_than_one(t)) {
+        return !approximately_greater_than_one(fTs[count - 2].fT);
+    }
+    return false;
+}
+
+// this span is excluded by the winding rule -- chase the ends
+// as long as they are unambiguous to mark connections as done
+// and give them the same winding value
+SkOpSpan* SkOpSegment::markAndChaseDone(const SkOpAngle* angle, int winding) {
+    int index = angle->start();
+    int endIndex = angle->end();
+    return markAndChaseDone(index, endIndex, winding);
+}
+
+SkOpSpan* SkOpSegment::markAndChaseDone(int index, int endIndex, int winding) {
+    int step = SkSign32(endIndex - index);
+    int min = SkMin32(index, endIndex);
+    markDone(min, winding);
+    SkOpSpan* last;
+    SkOpSegment* other = this;
+    while ((other = other->nextChase(index, step, min, last))) {
+        other->markDone(min, winding);
+    }
+    return last;
+}
+
+SkOpSpan* SkOpSegment::markAndChaseDoneBinary(const SkOpAngle* angle, int winding, int oppWinding) {
+    int index = angle->start();
+    int endIndex = angle->end();
+    int step = SkSign32(endIndex - index);
+    int min = SkMin32(index, endIndex);
+    markDoneBinary(min, winding, oppWinding);
+    SkOpSpan* last;
+    SkOpSegment* other = this;
+    while ((other = other->nextChase(index, step, min, last))) {
+        other->markDoneBinary(min, winding, oppWinding);
+    }
+    return last;
+}
+
+SkOpSpan* SkOpSegment::markAndChaseDoneBinary(int index, int endIndex) {
+    int step = SkSign32(endIndex - index);
+    int min = SkMin32(index, endIndex);
+    markDoneBinary(min);
+    SkOpSpan* last;
+    SkOpSegment* other = this;
+    while ((other = other->nextChase(index, step, min, last))) {
+        if (other->done()) {
+            return NULL;
+        }
+        other->markDoneBinary(min);
+    }
+    return last;
+}
+
+SkOpSpan* SkOpSegment::markAndChaseDoneUnary(int index, int endIndex) {
+    int step = SkSign32(endIndex - index);
+    int min = SkMin32(index, endIndex);
+    markDoneUnary(min);
+    SkOpSpan* last;
+    SkOpSegment* other = this;
+    while ((other = other->nextChase(index, step, min, last))) {
+        if (other->done()) {
+            return NULL;
+        }
+        other->markDoneUnary(min);
+    }
+    return last;
+}
+
+SkOpSpan* SkOpSegment::markAndChaseDoneUnary(const SkOpAngle* angle, int winding) {
+    int index = angle->start();
+    int endIndex = angle->end();
+    return markAndChaseDone(index, endIndex, winding);
+}
+
+SkOpSpan* SkOpSegment::markAndChaseWinding(const SkOpAngle* angle, const int winding) {
+    int index = angle->start();
+    int endIndex = angle->end();
+    int step = SkSign32(endIndex - index);
+    int min = SkMin32(index, endIndex);
+    markWinding(min, winding);
+    SkOpSpan* last;
+    SkOpSegment* other = this;
+    while ((other = other->nextChase(index, step, min, last))) {
+        if (other->fTs[min].fWindSum != SK_MinS32) {
+            SkASSERT(other->fTs[min].fWindSum == winding);
+            return NULL;
+        }
+        other->markWinding(min, winding);
+    }
+    return last;
+}
+
+SkOpSpan* SkOpSegment::markAndChaseWinding(int index, int endIndex, int winding, int oppWinding) {
+    int min = SkMin32(index, endIndex);
+    int step = SkSign32(endIndex - index);
+    markWinding(min, winding, oppWinding);
+    SkOpSpan* last;
+    SkOpSegment* other = this;
+    while ((other = other->nextChase(index, step, min, last))) {
+        if (other->fTs[min].fWindSum != SK_MinS32) {
+            SkASSERT(other->fTs[min].fWindSum == winding || other->fTs[min].fLoop);
+            return NULL;
+        }
+        other->markWinding(min, winding, oppWinding);
+    }
+    return last;
+}
+
+SkOpSpan* SkOpSegment::markAndChaseWinding(const SkOpAngle* angle, int winding, int oppWinding) {
+    int start = angle->start();
+    int end = angle->end();
+    return markAndChaseWinding(start, end, winding, oppWinding);
+}
+
+SkOpSpan* SkOpSegment::markAngle(int maxWinding, int sumWinding, bool activeAngle,
+                                const SkOpAngle* angle) {
+    SkASSERT(angle->segment() == this);
+    if (UseInnerWinding(maxWinding, sumWinding)) {
+        maxWinding = sumWinding;
+    }
+    SkOpSpan* last;
+    if (activeAngle) {
+        last = markAndChaseWinding(angle, maxWinding);
+    } else {
+        last = markAndChaseDoneUnary(angle, maxWinding);
+    }
+    return last;
+}
+
+SkOpSpan* SkOpSegment::markAngle(int maxWinding, int sumWinding, int oppMaxWinding,
+                                 int oppSumWinding, bool activeAngle, const SkOpAngle* angle) {
+    SkASSERT(angle->segment() == this);
+    if (UseInnerWinding(maxWinding, sumWinding)) {
+        maxWinding = sumWinding;
+    }
+    if (oppMaxWinding != oppSumWinding && UseInnerWinding(oppMaxWinding, oppSumWinding)) {
+        oppMaxWinding = oppSumWinding;
+    }
+    SkOpSpan* last;
+    if (activeAngle) {
+        last = markAndChaseWinding(angle, maxWinding, oppMaxWinding);
+    } else {
+        last = markAndChaseDoneBinary(angle, maxWinding, oppMaxWinding);
+    }
+    return last;
+}
+
+// FIXME: this should also mark spans with equal (x,y)
+// This may be called when the segment is already marked done. While this
+// wastes time, it shouldn't do any more than spin through the T spans.
+// OPTIMIZATION: abort on first done found (assuming that this code is
+// always called to mark segments done).
+void SkOpSegment::markDone(int index, int winding) {
+  //  SkASSERT(!done());
+    SkASSERT(winding);
+    double referenceT = fTs[index].fT;
+    int lesser = index;
+    while (--lesser >= 0 && precisely_negative(referenceT - fTs[lesser].fT)) {
+        markOneDone(__FUNCTION__, lesser, winding);
+    }
+    do {
+        markOneDone(__FUNCTION__, index, winding);
+    } while (++index < fTs.count() && precisely_negative(fTs[index].fT - referenceT));
+}
+
+void SkOpSegment::markDoneBinary(int index, int winding, int oppWinding) {
+  //  SkASSERT(!done());
+    SkASSERT(winding || oppWinding);
+    double referenceT = fTs[index].fT;
+    int lesser = index;
+    while (--lesser >= 0 && precisely_negative(referenceT - fTs[lesser].fT)) {
+        markOneDoneBinary(__FUNCTION__, lesser, winding, oppWinding);
+    }
+    do {
+        markOneDoneBinary(__FUNCTION__, index, winding, oppWinding);
+    } while (++index < fTs.count() && precisely_negative(fTs[index].fT - referenceT));
+}
+
+void SkOpSegment::markDoneBinary(int index) {
+    double referenceT = fTs[index].fT;
+    int lesser = index;
+    while (--lesser >= 0 && precisely_negative(referenceT - fTs[lesser].fT)) {
+        markOneDoneBinary(__FUNCTION__, lesser);
+    }
+    do {
+        markOneDoneBinary(__FUNCTION__, index);
+    } while (++index < fTs.count() && precisely_negative(fTs[index].fT - referenceT));
+}
+
+void SkOpSegment::markDoneUnary(int index, int winding) {
+  //  SkASSERT(!done());
+    SkASSERT(winding);
+    double referenceT = fTs[index].fT;
+    int lesser = index;
+    while (--lesser >= 0 && precisely_negative(referenceT - fTs[lesser].fT)) {
+        markOneDoneUnary(__FUNCTION__, lesser, winding);
+    }
+    do {
+        markOneDoneUnary(__FUNCTION__, index, winding);
+    } while (++index < fTs.count() && precisely_negative(fTs[index].fT - referenceT));
+}
+
+void SkOpSegment::markDoneUnary(int index) {
+    double referenceT = fTs[index].fT;
+    int lesser = index;
+    while (--lesser >= 0 && precisely_negative(referenceT - fTs[lesser].fT)) {
+        markOneDoneUnary(__FUNCTION__, lesser);
+    }
+    do {
+        markOneDoneUnary(__FUNCTION__, index);
+    } while (++index < fTs.count() && precisely_negative(fTs[index].fT - referenceT));
+}
+
+void SkOpSegment::markOneDone(const char* funName, int tIndex, int winding) {
+    SkOpSpan* span = markOneWinding(funName, tIndex, winding);
+    if (!span) {
+        return;
+    }
+    span->fDone = true;
+    fDoneSpans++;
+}
+
+void SkOpSegment::markOneDoneBinary(const char* funName, int tIndex) {
+    SkOpSpan* span = verifyOneWinding(funName, tIndex);
+    if (!span) {
+        return;
+    }
+    span->fDone = true;
+    fDoneSpans++;
+}
+
+void SkOpSegment::markOneDoneBinary(const char* funName, int tIndex, int winding, int oppWinding) {
+    SkOpSpan* span = markOneWinding(funName, tIndex, winding, oppWinding);
+    if (!span) {
+        return;
+    }
+    span->fDone = true;
+    fDoneSpans++;
+}
+
+void SkOpSegment::markOneDoneUnary(const char* funName, int tIndex) {
+    SkOpSpan* span = verifyOneWindingU(funName, tIndex);
+    if (!span) {
+        return;
+    }
+    span->fDone = true;
+    fDoneSpans++;
+}
+
+void SkOpSegment::markOneDoneUnary(const char* funName, int tIndex, int winding) {
+    SkOpSpan* span = markOneWinding(funName, tIndex, winding);
+    if (!span) {
+        return;
+    }
+    span->fDone = true;
+    fDoneSpans++;
+}
+
+SkOpSpan* SkOpSegment::markOneWinding(const char* funName, int tIndex, int winding) {
+    SkOpSpan& span = fTs[tIndex];
+    if (span.fDone) {
+        return NULL;
+    }
+#if DEBUG_MARK_DONE
+    debugShowNewWinding(funName, span, winding);
+#endif
+    SkASSERT(span.fWindSum == SK_MinS32 || span.fWindSum == winding);
+#ifdef SK_DEBUG
+    SkASSERT(abs(winding) <= gDebugMaxWindSum);
+#endif
+    span.fWindSum = winding;
+    return &span;
+}
+
+SkOpSpan* SkOpSegment::markOneWinding(const char* funName, int tIndex, int winding,
+                                      int oppWinding) {
+    SkOpSpan& span = fTs[tIndex];
+    if (span.fDone) {
+        return NULL;
+    }
+#if DEBUG_MARK_DONE
+    debugShowNewWinding(funName, span, winding, oppWinding);
+#endif
+    SkASSERT(span.fWindSum == SK_MinS32 || span.fWindSum == winding);
+#ifdef SK_DEBUG
+    SkASSERT(abs(winding) <= gDebugMaxWindSum);
+#endif
+    span.fWindSum = winding;
+    SkASSERT(span.fOppSum == SK_MinS32 || span.fOppSum == oppWinding);
+#ifdef SK_DEBUG
+    SkASSERT(abs(oppWinding) <= gDebugMaxWindSum);
+#endif
+    span.fOppSum = oppWinding;
+    return &span;
+}
+
+// from http://stackoverflow.com/questions/1165647/how-to-determine-if-a-list-of-polygon-points-are-in-clockwise-order
+bool SkOpSegment::clockwise(int tStart, int tEnd) const {
+    SkASSERT(fVerb != SkPath::kLine_Verb);
+    SkPoint edge[4];
+    subDivide(tStart, tEnd, edge);
+    double sum = (edge[0].fX - edge[fVerb].fX) * (edge[0].fY + edge[fVerb].fY);
+    if (fVerb == SkPath::kCubic_Verb) {
+        SkScalar lesser = SkTMin(edge[0].fY, edge[3].fY);
+        if (edge[1].fY < lesser && edge[2].fY < lesser) {
+            SkDLine tangent1 = {{ {edge[0].fX, edge[0].fY}, {edge[1].fX, edge[1].fY} }};
+            SkDLine tangent2 = {{ {edge[2].fX, edge[2].fY}, {edge[3].fX, edge[3].fY} }};
+            if (SkIntersections::Test(tangent1, tangent2)) {
+                SkPoint topPt = cubic_top(fPts, fTs[tStart].fT, fTs[tEnd].fT);
+                sum += (topPt.fX - edge[0].fX) * (topPt.fY + edge[0].fY);
+                sum += (edge[3].fX - topPt.fX) * (edge[3].fY + topPt.fY);
+                return sum <= 0;
+            }
+        }
+    }
+    for (int idx = 0; idx < fVerb; ++idx){
+        sum += (edge[idx + 1].fX - edge[idx].fX) * (edge[idx + 1].fY + edge[idx].fY);
+    }
+    return sum <= 0;
+}
+
+bool SkOpSegment::monotonicInY(int tStart, int tEnd) const {
+    if (fVerb == SkPath::kLine_Verb) {
+        return false;
+    }
+    if (fVerb == SkPath::kQuad_Verb) {
+        SkDQuad dst = SkDQuad::SubDivide(fPts, fTs[tStart].fT, fTs[tEnd].fT);
+        return dst.monotonicInY();
+    }
+    SkASSERT(fVerb == SkPath::kCubic_Verb);
+    SkDCubic dst = SkDCubic::SubDivide(fPts, fTs[tStart].fT, fTs[tEnd].fT);
+    return dst.monotonicInY();
+}
+
+bool SkOpSegment::serpentine(int tStart, int tEnd) const {
+    if (fVerb != SkPath::kCubic_Verb) {
+        return false;
+    }
+    SkDCubic dst = SkDCubic::SubDivide(fPts, fTs[tStart].fT, fTs[tEnd].fT);
+    return dst.serpentine();
+}
+
+SkOpSpan* SkOpSegment::verifyOneWinding(const char* funName, int tIndex) {
+    SkOpSpan& span = fTs[tIndex];
+    if (span.fDone) {
+        return NULL;
+    }
+#if DEBUG_MARK_DONE
+    debugShowNewWinding(funName, span, span.fWindSum, span.fOppSum);
+#endif
+    SkASSERT(span.fWindSum != SK_MinS32);
+    SkASSERT(span.fOppSum != SK_MinS32);
+    return &span;
+}
+
+SkOpSpan* SkOpSegment::verifyOneWindingU(const char* funName, int tIndex) {
+    SkOpSpan& span = fTs[tIndex];
+    if (span.fDone) {
+        return NULL;
+    }
+#if DEBUG_MARK_DONE
+    debugShowNewWinding(funName, span, span.fWindSum);
+#endif
+    SkASSERT(span.fWindSum != SK_MinS32);
+    return &span;
+}
+
+// note that just because a span has one end that is unsortable, that's
+// not enough to mark it done. The other end may be sortable, allowing the
+// span to be added.
+// FIXME: if abs(start - end) > 1, mark intermediates as unsortable on both ends
+void SkOpSegment::markUnsortable(int start, int end) {
+    SkOpSpan* span = &fTs[start];
+    if (start < end) {
+#if DEBUG_UNSORTABLE
+        debugShowNewWinding(__FUNCTION__, *span, 0);
+#endif
+        span->fUnsortableStart = true;
+    } else {
+        --span;
+#if DEBUG_UNSORTABLE
+        debugShowNewWinding(__FUNCTION__, *span, 0);
+#endif
+        span->fUnsortableEnd = true;
+    }
+    if (!span->fUnsortableStart || !span->fUnsortableEnd || span->fDone) {
+        return;
+    }
+    span->fDone = true;
+    fDoneSpans++;
+}
+
+void SkOpSegment::markWinding(int index, int winding) {
+//    SkASSERT(!done());
+    SkASSERT(winding);
+    double referenceT = fTs[index].fT;
+    int lesser = index;
+    while (--lesser >= 0 && precisely_negative(referenceT - fTs[lesser].fT)) {
+        markOneWinding(__FUNCTION__, lesser, winding);
+    }
+    do {
+        markOneWinding(__FUNCTION__, index, winding);
+   } while (++index < fTs.count() && precisely_negative(fTs[index].fT - referenceT));
+}
+
+void SkOpSegment::markWinding(int index, int winding, int oppWinding) {
+//    SkASSERT(!done());
+    SkASSERT(winding || oppWinding);
+    double referenceT = fTs[index].fT;
+    int lesser = index;
+    while (--lesser >= 0 && precisely_negative(referenceT - fTs[lesser].fT)) {
+        markOneWinding(__FUNCTION__, lesser, winding, oppWinding);
+    }
+    do {
+        markOneWinding(__FUNCTION__, index, winding, oppWinding);
+   } while (++index < fTs.count() && precisely_negative(fTs[index].fT - referenceT));
+}
+
+void SkOpSegment::matchWindingValue(int tIndex, double t, bool borrowWind) {
+    int nextDoorWind = SK_MaxS32;
+    int nextOppWind = SK_MaxS32;
+    if (tIndex > 0) {
+        const SkOpSpan& below = fTs[tIndex - 1];
+        if (approximately_negative(t - below.fT)) {
+            nextDoorWind = below.fWindValue;
+            nextOppWind = below.fOppValue;
+        }
+    }
+    if (nextDoorWind == SK_MaxS32 && tIndex + 1 < fTs.count()) {
+        const SkOpSpan& above = fTs[tIndex + 1];
+        if (approximately_negative(above.fT - t)) {
+            nextDoorWind = above.fWindValue;
+            nextOppWind = above.fOppValue;
+        }
+    }
+    if (nextDoorWind == SK_MaxS32 && borrowWind && tIndex > 0 && t < 1) {
+        const SkOpSpan& below = fTs[tIndex - 1];
+        nextDoorWind = below.fWindValue;
+        nextOppWind = below.fOppValue;
+    }
+    if (nextDoorWind != SK_MaxS32) {
+        SkOpSpan& newSpan = fTs[tIndex];
+        newSpan.fWindValue = nextDoorWind;
+        newSpan.fOppValue = nextOppWind;
+        if (!nextDoorWind && !nextOppWind && !newSpan.fDone) {
+            newSpan.fDone = true;
+            ++fDoneSpans;
+        }
+    }
+}
+
+bool SkOpSegment::moreHorizontal(int index, int endIndex, bool& unsortable) const {
+    // find bounds
+    SkPathOpsBounds bounds;
+    bounds.setPointBounds(xyAtT(index));
+    bounds.add(xyAtT(endIndex));
+    SkScalar width = bounds.width();
+    SkScalar height = bounds.height();
+    if (width > height) {
+        if (approximately_negative(width)) {
+            unsortable = true; // edge is too small to resolve meaningfully
+        }
+        return false;
+    } else {
+        if (approximately_negative(height)) {
+            unsortable = true; // edge is too small to resolve meaningfully
+        }
+        return true;
+    }
+}
+
+// return span if when chasing, two or more radiating spans are not done
+// OPTIMIZATION: ? multiple spans is detected when there is only one valid
+// candidate and the remaining spans have windValue == 0 (canceled by
+// coincidence). The coincident edges could either be removed altogether,
+// or this code could be more complicated in detecting this case. Worth it?
+bool SkOpSegment::multipleSpans(int end) const {
+    return end > 0 && end < fTs.count() - 1;
+}
+
+bool SkOpSegment::nextCandidate(int& start, int& end) const {
+    while (fTs[end].fDone) {
+        if (fTs[end].fT == 1) {
+            return false;
+        }
+        ++end;
+    }
+    start = end;
+    end = nextExactSpan(start, 1);
+    return true;
+}
+
+SkOpSegment* SkOpSegment::nextChase(int& index, const int step, int& min, SkOpSpan*& last) {
+    int end = nextExactSpan(index, step);
+    SkASSERT(end >= 0);
+    if (multipleSpans(end)) {
+        last = &fTs[end];
+        return NULL;
+    }
+    const SkOpSpan& endSpan = fTs[end];
+    SkOpSegment* other = endSpan.fOther;
+    index = endSpan.fOtherIndex;
+    SkASSERT(index >= 0);
+    int otherEnd = other->nextExactSpan(index, step);
+    SkASSERT(otherEnd >= 0);
+    min = SkMin32(index, otherEnd);
+    return other;
+}
+
+// This has callers for two different situations: one establishes the end
+// of the current span, and one establishes the beginning of the next span
+// (thus the name). When this is looking for the end of the current span,
+// coincidence is found when the beginning Ts contain -step and the end
+// contains step. When it is looking for the beginning of the next, the
+// first Ts found can be ignored and the last Ts should contain -step.
+// OPTIMIZATION: probably should split into two functions
+int SkOpSegment::nextSpan(int from, int step) const {
+    const SkOpSpan& fromSpan = fTs[from];
+    int count = fTs.count();
+    int to = from;
+    while (step > 0 ? ++to < count : --to >= 0) {
+        const SkOpSpan& span = fTs[to];
+        if (approximately_zero(span.fT - fromSpan.fT)) {
+            continue;
+        }
+        return to;
+    }
+    return -1;
+}
+
+// FIXME
+// this returns at any difference in T, vs. a preset minimum. It may be
+// that all callers to nextSpan should use this instead.
+// OPTIMIZATION splitting this into separate loops for up/down steps
+// would allow using precisely_negative instead of precisely_zero
+int SkOpSegment::nextExactSpan(int from, int step) const {
+    const SkOpSpan& fromSpan = fTs[from];
+    int count = fTs.count();
+    int to = from;
+    while (step > 0 ? ++to < count : --to >= 0) {
+        const SkOpSpan& span = fTs[to];
+        if (precisely_zero(span.fT - fromSpan.fT)) {
+            continue;
+        }
+        return to;
+    }
+    return -1;
+}
+
+void SkOpSegment::setUpWindings(int index, int endIndex, int& sumMiWinding, int& sumSuWinding,
+        int& maxWinding, int& sumWinding, int& oppMaxWinding, int& oppSumWinding) {
+    int deltaSum = spanSign(index, endIndex);
+    int oppDeltaSum = oppSign(index, endIndex);
+    if (operand()) {
+        maxWinding = sumSuWinding;
+        sumWinding = sumSuWinding -= deltaSum;
+        oppMaxWinding = sumMiWinding;
+        oppSumWinding = sumMiWinding -= oppDeltaSum;
+    } else {
+        maxWinding = sumMiWinding;
+        sumWinding = sumMiWinding -= deltaSum;
+        oppMaxWinding = sumSuWinding;
+        oppSumWinding = sumSuWinding -= oppDeltaSum;
+    }
+}
+
+// This marks all spans unsortable so that this info is available for early
+// exclusion in find top and others. This could be optimized to only mark
+// adjacent spans that unsortable. However, this makes it difficult to later
+// determine starting points for edge detection in find top and the like.
+bool SkOpSegment::SortAngles(SkTDArray<SkOpAngle>& angles, SkTDArray<SkOpAngle*>& angleList) {
+    bool sortable = true;
+    int angleCount = angles.count();
+    int angleIndex;
+    angleList.setReserve(angleCount);
+    for (angleIndex = 0; angleIndex < angleCount; ++angleIndex) {
+        SkOpAngle& angle = angles[angleIndex];
+        *angleList.append() = &angle;
+        sortable &= !angle.unsortable();
+    }
+    if (sortable) {
+        QSort<SkOpAngle>(angleList.begin(), angleList.end() - 1);
+        for (angleIndex = 0; angleIndex < angleCount; ++angleIndex) {
+            if (angles[angleIndex].unsortable()) {
+                sortable = false;
+                break;
+            }
+        }
+    }
+    if (!sortable) {
+        for (angleIndex = 0; angleIndex < angleCount; ++angleIndex) {
+            SkOpAngle& angle = angles[angleIndex];
+            angle.segment()->markUnsortable(angle.start(), angle.end());
+        }
+    }
+    return sortable;
+}
+
+void SkOpSegment::subDivide(int start, int end, SkPoint edge[4]) const {
+    edge[0] = fTs[start].fPt;
+    edge[fVerb] = fTs[end].fPt;
+    if (fVerb == SkPath::kQuad_Verb || fVerb == SkPath::kCubic_Verb) {
+        SkDPoint sub[2] = {{ edge[0].fX, edge[0].fY}, {edge[fVerb].fX, edge[fVerb].fY }};
+        if (fVerb == SkPath::kQuad_Verb) {
+            edge[1] = SkDQuad::SubDivide(fPts, sub[0], sub[1], fTs[start].fT,
+                    fTs[end].fT).asSkPoint();
+        } else {
+            SkDCubic::SubDivide(fPts, sub[0], sub[1], fTs[start].fT, fTs[end].fT, sub);
+            edge[1] = sub[0].asSkPoint();
+            edge[2] = sub[1].asSkPoint();
+        }
+    }
+}
+
+void SkOpSegment::subDivideBounds(int start, int end, SkPathOpsBounds& bounds) const {
+    SkPoint edge[4];
+    subDivide(start, end, edge);
+    (bounds.*SetCurveBounds[fVerb])(edge);
+}
+
+bool SkOpSegment::tiny(const SkOpAngle* angle) const {
+    int start = angle->start();
+    int end = angle->end();
+    const SkOpSpan& mSpan = fTs[SkMin32(start, end)];
+    return mSpan.fTiny;
+}
+
+void SkOpSegment::TrackOutside(SkTDArray<double>& outsideTs, double end, double start) {
+    int outCount = outsideTs.count();
+    if (outCount == 0 || !approximately_negative(end - outsideTs[outCount - 2])) {
+        *outsideTs.append() = end;
+        *outsideTs.append() = start;
+    }
+}
+
+void SkOpSegment::undoneSpan(int& start, int& end) {
+    size_t tCount = fTs.count();
+    size_t index;
+    for (index = 0; index < tCount; ++index) {
+        if (!fTs[index].fDone) {
+            break;
+        }
+    }
+    SkASSERT(index < tCount - 1);
+    start = index;
+    double startT = fTs[index].fT;
+    while (approximately_negative(fTs[++index].fT - startT))
+        SkASSERT(index < tCount);
+    SkASSERT(index < tCount);
+    end = index;
+}
+
+int SkOpSegment::updateOppWinding(int index, int endIndex) const {
+    int lesser = SkMin32(index, endIndex);
+    int oppWinding = oppSum(lesser);
+    int oppSpanWinding = oppSign(index, endIndex);
+    if (oppSpanWinding && UseInnerWinding(oppWinding - oppSpanWinding, oppWinding)
+            && oppWinding != SK_MaxS32) {
+        oppWinding -= oppSpanWinding;
+    }
+    return oppWinding;
+}
+
+int SkOpSegment::updateOppWinding(const SkOpAngle* angle) const {
+    int startIndex = angle->start();
+    int endIndex = angle->end();
+    return updateOppWinding(endIndex, startIndex);
+}
+
+int SkOpSegment::updateOppWindingReverse(const SkOpAngle* angle) const {
+    int startIndex = angle->start();
+    int endIndex = angle->end();
+    return updateOppWinding(startIndex, endIndex);
+}
+
+int SkOpSegment::updateWinding(int index, int endIndex) const {
+    int lesser = SkMin32(index, endIndex);
+    int winding = windSum(lesser);
+    int spanWinding = spanSign(index, endIndex);
+    if (winding && UseInnerWinding(winding - spanWinding, winding) && winding != SK_MaxS32) {
+        winding -= spanWinding;
+    }
+    return winding;
+}
+
+int SkOpSegment::updateWinding(const SkOpAngle* angle) const {
+    int startIndex = angle->start();
+    int endIndex = angle->end();
+    return updateWinding(endIndex, startIndex);
+}
+
+int SkOpSegment::updateWindingReverse(const SkOpAngle* angle) const {
+    int startIndex = angle->start();
+    int endIndex = angle->end();
+    return updateWinding(startIndex, endIndex);
+}
+
+int SkOpSegment::windingAtT(double tHit, int tIndex, bool crossOpp, SkScalar& dx) const {
+    if (approximately_zero(tHit - t(tIndex))) { // if we hit the end of a span, disregard
+        return SK_MinS32;
+    }
+    int winding = crossOpp ? oppSum(tIndex) : windSum(tIndex);
+    SkASSERT(winding != SK_MinS32);
+    int windVal = crossOpp ? oppValue(tIndex) : windValue(tIndex);
+#if DEBUG_WINDING_AT_T
+    SkDebugf("%s oldWinding=%d windValue=%d", __FUNCTION__, winding, windVal);
+#endif
+    // see if a + change in T results in a +/- change in X (compute x'(T))
+    dx = (*CurveSlopeAtT[fVerb])(fPts, tHit).fX;
+    if (fVerb > SkPath::kLine_Verb && approximately_zero(dx)) {
+        dx = fPts[2].fX - fPts[1].fX - dx;
+    }
+    if (dx == 0) {
+#if DEBUG_WINDING_AT_T
+        SkDebugf(" dx=0 winding=SK_MinS32\n");
+#endif
+        return SK_MinS32;
+    }
+    if (winding * dx > 0) { // if same signs, result is negative
+        winding += dx > 0 ? -windVal : windVal;
+    }
+#if DEBUG_WINDING_AT_T
+    SkDebugf(" dx=%c winding=%d\n", dx > 0 ? '+' : '-', winding);
+#endif
+    return winding;
+}
+
+int SkOpSegment::windSum(const SkOpAngle* angle) const {
+    int start = angle->start();
+    int end = angle->end();
+    int index = SkMin32(start, end);
+    return windSum(index);
+}
+
+int SkOpSegment::windValue(const SkOpAngle* angle) const {
+    int start = angle->start();
+    int end = angle->end();
+    int index = SkMin32(start, end);
+    return windValue(index);
+}
+
+int SkOpSegment::windValueAt(double t) const {
+    int count = fTs.count();
+    for (int index = 0; index < count; ++index) {
+        if (fTs[index].fT == t) {
+            return fTs[index].fWindValue;
+        }
+    }
+    SkASSERT(0);
+    return 0;
+}
+
+void SkOpSegment::zeroCoincidentOpp(SkOpSpan* oTest, int index) {
+    SkOpSpan* const test = &fTs[index];
+    SkOpSpan* end = test;
+    do {
+        end->fOppValue = 0;
+        end = &fTs[++index];
+    } while (approximately_negative(end->fT - test->fT));
+}
+
+void SkOpSegment::zeroCoincidentOther(SkOpSpan* test, const double tRatio, const double oEndT,
+                                      int oIndex) {
+    SkOpSpan* const oTest = &fTs[oIndex];
+    SkOpSpan* oEnd = oTest;
+    const double startT = test->fT;
+    const double oStartT = oTest->fT;
+    double otherTMatch = (test->fT - startT) * tRatio + oStartT;
+    while (!approximately_negative(oEndT - oEnd->fT)
+            && approximately_negative(oEnd->fT - otherTMatch)) {
+        oEnd->fOppValue = 0;
+        oEnd = &fTs[++oIndex];
+    }
+}
+
+void SkOpSegment::zeroSpan(SkOpSpan* span) {
+    SkASSERT(span->fWindValue > 0 || span->fOppValue > 0);
+    span->fWindValue = 0;
+    span->fOppValue = 0;
+    SkASSERT(!span->fDone);
+    span->fDone = true;
+    ++fDoneSpans;
+}
+    
+#if DEBUG_SWAP_TOP
+bool SkOpSegment::controlsContainedByEnds(int tStart, int tEnd) const {
+    if (fVerb != SkPath::kCubic_Verb) {
+        return false;
+    }
+    SkDCubic dst = SkDCubic::SubDivide(fPts, fTs[tStart].fT, fTs[tEnd].fT);
+    return dst.controlsContainedByEnds();
+}
+#endif
+
+#if DEBUG_DUMP
+void SkOpSegment::dump() const {
+    const char className[] = "SkOpSegment";
+    const int tab = 4;
+    for (int i = 0; i < fTs.count(); ++i) {
+        SkPoint out = (*CurvePointAtT[fVerb])(fPts, t(i));
+        SkDebugf("%*s [%d] %s.fTs[%d]=%1.9g (%1.9g,%1.9g) other=%d"
+                " otherT=%1.9g windSum=%d\n",
+                tab + sizeof(className), className, fID,
+                kLVerbStr[fVerb], i, fTs[i].fT, out.fX, out.fY,
+                fTs[i].fOther->fID, fTs[i].fOtherT, fTs[i].fWindSum);
+    }
+    SkDebugf("%*s [%d] fBounds=(l:%1.9g, t:%1.9g r:%1.9g, b:%1.9g)",
+            tab + sizeof(className), className, fID,
+            fBounds.fLeft, fBounds.fTop, fBounds.fRight, fBounds.fBottom);
+}
+#endif
+
+#if DEBUG_CONCIDENT
+// SkASSERT if pair has not already been added
+ void SkOpSegment::debugAddTPair(double t, const SkOpSegment& other, double otherT) const {
+    for (int i = 0; i < fTs.count(); ++i) {
+        if (fTs[i].fT == t && fTs[i].fOther == &other && fTs[i].fOtherT == otherT) {
+            return;
+        }
+    }
+    SkASSERT(0);
+ }
+#endif
+
+#if DEBUG_WINDING
+void SkOpSegment::debugShowSums() const {
+    SkDebugf("%s id=%d (%1.9g,%1.9g %1.9g,%1.9g)", __FUNCTION__, fID,
+        fPts[0].fX, fPts[0].fY, fPts[fVerb].fX, fPts[fVerb].fY);
+    for (int i = 0; i < fTs.count(); ++i) {
+        const SkOpSpan& span = fTs[i];
+        SkDebugf(" [t=%1.3g %1.9g,%1.9g w=", span.fT, xAtT(&span), yAtT(&span));
+        if (span.fWindSum == SK_MinS32) {
+            SkDebugf("?");
+        } else {
+            SkDebugf("%d", span.fWindSum);
+        }
+        SkDebugf("]");
+    }
+    SkDebugf("\n");
+}
+#endif
+
+#if DEBUG_CONCIDENT
+void SkOpSegment::debugShowTs() const {
+    SkDebugf("%s id=%d", __FUNCTION__, fID);
+    int lastWind = -1;
+    int lastOpp = -1;
+    double lastT = -1;
+    int i;
+    for (i = 0; i < fTs.count(); ++i) {
+        bool change = lastT != fTs[i].fT || lastWind != fTs[i].fWindValue
+                || lastOpp != fTs[i].fOppValue;
+        if (change && lastWind >= 0) {
+            SkDebugf(" t=%1.3g %1.9g,%1.9g w=%d o=%d]",
+                    lastT, xyAtT(i - 1).fX, xyAtT(i - 1).fY, lastWind, lastOpp);
+        }
+        if (change) {
+            SkDebugf(" [o=%d", fTs[i].fOther->fID);
+            lastWind = fTs[i].fWindValue;
+            lastOpp = fTs[i].fOppValue;
+            lastT = fTs[i].fT;
+        } else {
+            SkDebugf(",%d", fTs[i].fOther->fID);
+        }
+    }
+    if (i <= 0) {
+        return;
+    }
+    SkDebugf(" t=%1.3g %1.9g,%1.9g w=%d o=%d]",
+            lastT, xyAtT(i - 1).fX, xyAtT(i - 1).fY, lastWind, lastOpp);
+    if (fOperand) {
+        SkDebugf(" operand");
+    }
+    if (done()) {
+        SkDebugf(" done");
+    }
+    SkDebugf("\n");
+}
+#endif
+
+#if DEBUG_ACTIVE_SPANS
+void SkOpSegment::debugShowActiveSpans() const {
+    if (done()) {
+        return;
+    }
+#if DEBUG_ACTIVE_SPANS_SHORT_FORM
+    int lastId = -1;
+    double lastT = -1;
+#endif
+    for (int i = 0; i < fTs.count(); ++i) {
+        SkASSERT(&fTs[i] == &fTs[i].fOther->fTs[fTs[i].fOtherIndex].fOther->
+                fTs[fTs[i].fOther->fTs[fTs[i].fOtherIndex].fOtherIndex]);
+        if (fTs[i].fDone) {
+            continue;
+        }
+#if DEBUG_ACTIVE_SPANS_SHORT_FORM
+        if (lastId == fID && lastT == fTs[i].fT) {
+            continue;
+        }
+        lastId = fID;
+        lastT = fTs[i].fT;
+#endif
+        SkDebugf("%s id=%d", __FUNCTION__, fID);
+        SkDebugf(" (%1.9g,%1.9g", fPts[0].fX, fPts[0].fY);
+        for (int vIndex = 1; vIndex <= fVerb; ++vIndex) {
+            SkDebugf(" %1.9g,%1.9g", fPts[vIndex].fX, fPts[vIndex].fY);
+        }
+        const SkOpSpan* span = &fTs[i];
+        SkDebugf(") t=%1.9g (%1.9g,%1.9g)", fTs[i].fT,
+                 xAtT(span), yAtT(span));
+        int iEnd = i + 1;
+        while (fTs[iEnd].fT < 1 && approximately_equal(fTs[i].fT, fTs[iEnd].fT)) {
+            ++iEnd;
+        }
+        SkDebugf(" tEnd=%1.9g", fTs[iEnd].fT);
+        const SkOpSegment* other = fTs[i].fOther;
+        SkDebugf(" other=%d otherT=%1.9g otherIndex=%d windSum=",
+                other->fID, fTs[i].fOtherT, fTs[i].fOtherIndex);
+        if (fTs[i].fWindSum == SK_MinS32) {
+            SkDebugf("?");
+        } else {
+            SkDebugf("%d", fTs[i].fWindSum);
+        }
+        SkDebugf(" windValue=%d oppValue=%d\n", fTs[i].fWindValue, fTs[i].fOppValue);
+    }
+}
+
+// This isn't useful yet -- but leaving it in for now in case i think of something
+// to use it for
+void SkOpSegment::validateActiveSpans() const {
+    if (done()) {
+        return;
+    }
+    int tCount = fTs.count();
+    for (int index = 0; index < tCount; ++index) {
+        if (fTs[index].fDone) {
+            continue;
+        }
+        // count number of connections which are not done
+        int first = index;
+        double baseT = fTs[index].fT;
+        while (first > 0 && approximately_equal(fTs[first - 1].fT, baseT)) {
+            --first;
+        }
+        int last = index;
+        while (last < tCount - 1 && approximately_equal(fTs[last + 1].fT, baseT)) {
+            ++last;
+        }
+        int connections = 0;
+        connections += first > 0 && !fTs[first - 1].fDone;
+        for (int test = first; test <= last; ++test) {
+            connections += !fTs[test].fDone;
+            const SkOpSegment* other = fTs[test].fOther;
+            int oIndex = fTs[test].fOtherIndex;
+            connections += !other->fTs[oIndex].fDone;
+            connections += oIndex > 0 && !other->fTs[oIndex - 1].fDone;
+        }
+  //      SkASSERT(!(connections & 1));
+    }
+}
+#endif
+
+
+#if DEBUG_MARK_DONE || DEBUG_UNSORTABLE
+void SkOpSegment::debugShowNewWinding(const char* fun, const SkOpSpan& span, int winding) {
+    const SkPoint& pt = xyAtT(&span);
+    SkDebugf("%s id=%d", fun, fID);
+    SkDebugf(" (%1.9g,%1.9g", fPts[0].fX, fPts[0].fY);
+    for (int vIndex = 1; vIndex <= fVerb; ++vIndex) {
+        SkDebugf(" %1.9g,%1.9g", fPts[vIndex].fX, fPts[vIndex].fY);
+    }
+    SkASSERT(&span == &span.fOther->fTs[span.fOtherIndex].fOther->
+            fTs[span.fOther->fTs[span.fOtherIndex].fOtherIndex]);
+    SkDebugf(") t=%1.9g [%d] (%1.9g,%1.9g) tEnd=%1.9g newWindSum=%d windSum=",
+            span.fT, span.fOther->fTs[span.fOtherIndex].fOtherIndex, pt.fX, pt.fY,
+            (&span)[1].fT, winding);
+    if (span.fWindSum == SK_MinS32) {
+        SkDebugf("?");
+    } else {
+        SkDebugf("%d", span.fWindSum);
+    }
+    SkDebugf(" windValue=%d\n", span.fWindValue);
+}
+
+void SkOpSegment::debugShowNewWinding(const char* fun, const SkOpSpan& span, int winding,
+                                      int oppWinding) {
+    const SkPoint& pt = xyAtT(&span);
+    SkDebugf("%s id=%d", fun, fID);
+    SkDebugf(" (%1.9g,%1.9g", fPts[0].fX, fPts[0].fY);
+    for (int vIndex = 1; vIndex <= fVerb; ++vIndex) {
+        SkDebugf(" %1.9g,%1.9g", fPts[vIndex].fX, fPts[vIndex].fY);
+    }
+    SkASSERT(&span == &span.fOther->fTs[span.fOtherIndex].fOther->
+            fTs[span.fOther->fTs[span.fOtherIndex].fOtherIndex]);
+    SkDebugf(") t=%1.9g [%d] (%1.9g,%1.9g) tEnd=%1.9g newWindSum=%d newOppSum=%d oppSum=",
+            span.fT, span.fOther->fTs[span.fOtherIndex].fOtherIndex, pt.fX, pt.fY,
+            (&span)[1].fT, winding, oppWinding);
+    if (span.fOppSum == SK_MinS32) {
+        SkDebugf("?");
+    } else {
+        SkDebugf("%d", span.fOppSum);
+    }
+    SkDebugf(" windSum=");
+    if (span.fWindSum == SK_MinS32) {
+        SkDebugf("?");
+    } else {
+        SkDebugf("%d", span.fWindSum);
+    }
+    SkDebugf(" windValue=%d\n", span.fWindValue);
+}
+#endif
+
+#if DEBUG_SORT || DEBUG_SWAP_TOP
+void SkOpSegment::debugShowSort(const char* fun, const SkTDArray<SkOpAngle*>& angles, int first,
+        const int contourWinding, const int oppContourWinding) const {
+    if (--gDebugSortCount < 0) {
+        return;
+    }
+    SkASSERT(angles[first]->segment() == this);
+    SkASSERT(angles.count() > 1);
+    int lastSum = contourWinding;
+    int oppLastSum = oppContourWinding;
+    const SkOpAngle* firstAngle = angles[first];
+    int windSum = lastSum - spanSign(firstAngle);
+    int oppoSign = oppSign(firstAngle);
+    int oppWindSum = oppLastSum - oppoSign;
+    #define WIND_AS_STRING(x) char x##Str[12]; if (!valid_wind(x)) strcpy(x##Str, "?"); \
+        else snprintf(x##Str, sizeof(x##Str), "%d", x)
+    WIND_AS_STRING(contourWinding);
+    WIND_AS_STRING(oppContourWinding);
+    SkDebugf("%s %s contourWinding=%s oppContourWinding=%s sign=%d\n", fun, __FUNCTION__,
+            contourWindingStr, oppContourWindingStr, spanSign(angles[first]));
+    int index = first;
+    bool firstTime = true;
+    do {
+        const SkOpAngle& angle = *angles[index];
+        const SkOpSegment& segment = *angle.segment();
+        int start = angle.start();
+        int end = angle.end();
+        const SkOpSpan& sSpan = segment.fTs[start];
+        const SkOpSpan& eSpan = segment.fTs[end];
+        const SkOpSpan& mSpan = segment.fTs[SkMin32(start, end)];
+        bool opp = segment.fOperand ^ fOperand;
+        if (!firstTime) {
+            oppoSign = segment.oppSign(&angle);
+            if (opp) {
+                oppLastSum = oppWindSum;
+                oppWindSum -= segment.spanSign(&angle);
+                if (oppoSign) {
+                    lastSum = windSum;
+                    windSum -= oppoSign;
+                }
+            } else {
+                lastSum = windSum;
+                windSum -= segment.spanSign(&angle);
+                if (oppoSign) {
+                    oppLastSum = oppWindSum;
+                    oppWindSum -= oppoSign;
+                }
+            }
+        }
+        SkDebugf("%s [%d] %s", __FUNCTION__, index,
+                angle.unsortable() ? "*** UNSORTABLE *** " : "");
+    #if COMPACT_DEBUG_SORT
+        SkDebugf("id=%d %s start=%d (%1.9g,%,1.9g) end=%d (%1.9g,%,1.9g)",
+                segment.fID, kLVerbStr[segment.fVerb],
+                start, segment.xAtT(&sSpan), segment.yAtT(&sSpan), end,
+                segment.xAtT(&eSpan), segment.yAtT(&eSpan));
+    #else
+        switch (segment.fVerb) {
+            case SkPath::kLine_Verb:
+                SkDebugf(LINE_DEBUG_STR, LINE_DEBUG_DATA(segment.fPts));
+                break;
+            case SkPath::kQuad_Verb:
+                SkDebugf(QUAD_DEBUG_STR, QUAD_DEBUG_DATA(segment.fPts));
+                break;
+            case SkPath::kCubic_Verb:
+                SkDebugf(CUBIC_DEBUG_STR, CUBIC_DEBUG_DATA(segment.fPts));
+                break;
+            default:
+                SkASSERT(0);
+        }
+        SkDebugf(" tStart=%1.9g tEnd=%1.9g", sSpan.fT, eSpan.fT);
+    #endif
+        SkDebugf(" sign=%d windValue=%d windSum=", angle.sign(), mSpan.fWindValue);
+        #ifdef SK_DEBUG
+            winding_printf(mSpan.fWindSum);
+        #endif
+        int last, wind;
+        if (opp) {
+            last = oppLastSum;
+            wind = oppWindSum;
+        } else {
+            last = lastSum;
+            wind = windSum;
+        }
+        bool useInner = valid_wind(last) && valid_wind(wind) && UseInnerWinding(last, wind);
+        WIND_AS_STRING(last);
+        WIND_AS_STRING(wind);
+        WIND_AS_STRING(lastSum);
+        WIND_AS_STRING(oppLastSum);
+        WIND_AS_STRING(windSum);
+        WIND_AS_STRING(oppWindSum);
+        #undef WIND_AS_STRING
+        if (!oppoSign) {
+            SkDebugf(" %s->%s (max=%s)", lastStr, windStr, useInner ? windStr : lastStr);
+        } else {
+            SkDebugf(" %s->%s (%s->%s)", lastStr, windStr, opp ? lastSumStr : oppLastSumStr,
+                    opp ? windSumStr : oppWindSumStr);
+        }
+        SkDebugf(" done=%d tiny=%d opp=%d\n", mSpan.fDone, mSpan.fTiny, opp);
+#if false && DEBUG_ANGLE
+        angle.debugShow(segment.xyAtT(&sSpan));
+#endif
+        ++index;
+        if (index == angles.count()) {
+            index = 0;
+        }
+        if (firstTime) {
+            firstTime = false;
+        }
+    } while (index != first);
+}
+
+void SkOpSegment::debugShowSort(const char* fun, const SkTDArray<SkOpAngle*>& angles, int first) {
+    const SkOpAngle* firstAngle = angles[first];
+    const SkOpSegment* segment = firstAngle->segment();
+    int winding = segment->updateWinding(firstAngle);
+    int oppWinding = segment->updateOppWinding(firstAngle);
+    debugShowSort(fun, angles, first, winding, oppWinding);
+}
+
+#endif
+
+#if DEBUG_SHOW_WINDING
+int SkOpSegment::debugShowWindingValues(int slotCount, int ofInterest) const {
+    if (!(1 << fID & ofInterest)) {
+        return 0;
+    }
+    int sum = 0;
+    SkTDArray<char> slots;
+    slots.setCount(slotCount * 2);
+    memset(slots.begin(), ' ', slotCount * 2);
+    for (int i = 0; i < fTs.count(); ++i) {
+   //     if (!(1 << fTs[i].fOther->fID & ofInterest)) {
+   //         continue;
+   //     }
+        sum += fTs[i].fWindValue;
+        slots[fTs[i].fOther->fID - 1] = as_digit(fTs[i].fWindValue);
+        sum += fTs[i].fOppValue;
+        slots[slotCount + fTs[i].fOther->fID - 1] = as_digit(fTs[i].fOppValue);
+    }
+    SkDebugf("%s id=%2d %.*s | %.*s\n", __FUNCTION__, fID, slotCount, slots.begin(), slotCount,
+            slots.begin() + slotCount);
+    return sum;
+}
+#endif