Revert of pathops version two

src/pathops/SkDCubicIntersection.cpp

Index: src/pathops/SkDCubicIntersection.cpp
diff --git a/src/pathops/SkDCubicIntersection.cpp b/src/pathops/SkDCubicIntersection.cpp
new file mode 100644
index 0000000000000000000000000000000000000000..2fb35e182794dcec2bb86cb654a0d7d22a3b0af0
--- /dev/null
+++ b/src/pathops/SkDCubicIntersection.cpp
@@ -0,0 +1,704 @@
+/*
+ * 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 "SkPathOpsCubic.h"
+#include "SkPathOpsLine.h"
+#include "SkPathOpsPoint.h"
+#include "SkPathOpsQuad.h"
+#include "SkPathOpsRect.h"
+#include "SkReduceOrder.h"
+#include "SkTSort.h"
+
+#if ONE_OFF_DEBUG
+static const double tLimits1[2][2] = {{0.3, 0.4}, {0.8, 0.9}};
+static const double tLimits2[2][2] = {{-0.8, -0.9}, {-0.8, -0.9}};
+#endif
+
+#define DEBUG_QUAD_PART ONE_OFF_DEBUG && 1
+#define DEBUG_QUAD_PART_SHOW_SIMPLE DEBUG_QUAD_PART && 0
+#define SWAP_TOP_DEBUG 0
+
+static const int kCubicToQuadSubdivisionDepth = 8; // slots reserved for cubic to quads subdivision
+
+static int quadPart(const SkDCubic& cubic, double tStart, double tEnd, SkReduceOrder* reducer) {
+    SkDCubic part = cubic.subDivide(tStart, tEnd);
+    SkDQuad quad = part.toQuad();
+    // FIXME: should reduceOrder be looser in this use case if quartic is going to blow up on an
+    // extremely shallow quadratic?
+    int order = reducer->reduce(quad);
+#if DEBUG_QUAD_PART
+    SkDebugf("%s cubic=(%1.9g,%1.9g %1.9g,%1.9g %1.9g,%1.9g %1.9g,%1.9g)"
+            " t=(%1.9g,%1.9g)\n", __FUNCTION__, cubic[0].fX, cubic[0].fY,
+            cubic[1].fX, cubic[1].fY, cubic[2].fX, cubic[2].fY,
+            cubic[3].fX, cubic[3].fY, tStart, tEnd);
+    SkDebugf("  {{%1.9g,%1.9g}, {%1.9g,%1.9g}, {%1.9g,%1.9g}, {%1.9g,%1.9g}},\n"
+             "  {{%1.9g,%1.9g}, {%1.9g,%1.9g}, {%1.9g,%1.9g}},\n",
+            part[0].fX, part[0].fY, part[1].fX, part[1].fY, part[2].fX, part[2].fY,
+            part[3].fX, part[3].fY, quad[0].fX, quad[0].fY,
+            quad[1].fX, quad[1].fY, quad[2].fX, quad[2].fY);
+#if DEBUG_QUAD_PART_SHOW_SIMPLE
+    SkDebugf("%s simple=(%1.9g,%1.9g", __FUNCTION__, reducer->fQuad[0].fX, reducer->fQuad[0].fY);
+    if (order > 1) {
+        SkDebugf(" %1.9g,%1.9g", reducer->fQuad[1].fX, reducer->fQuad[1].fY);
+    }
+    if (order > 2) {
+        SkDebugf(" %1.9g,%1.9g", reducer->fQuad[2].fX, reducer->fQuad[2].fY);
+    }
+    SkDebugf(")\n");
+    SkASSERT(order < 4 && order > 0);
+#endif
+#endif
+    return order;
+}
+
+static void intersectWithOrder(const SkDQuad& simple1, int order1, const SkDQuad& simple2,
+        int order2, SkIntersections& i) {
+    if (order1 == 3 && order2 == 3) {
+        i.intersect(simple1, simple2);
+    } else if (order1 <= 2 && order2 <= 2) {
+        i.intersect((const SkDLine&) simple1, (const SkDLine&) simple2);
+    } else if (order1 == 3 && order2 <= 2) {
+        i.intersect(simple1, (const SkDLine&) simple2);
+    } else {
+        SkASSERT(order1 <= 2 && order2 == 3);
+        i.intersect(simple2, (const SkDLine&) simple1);
+        i.swapPts();
+    }
+}
+
+// this flavor centers potential intersections recursively. In contrast, '2' may inadvertently
+// chase intersections near quadratic ends, requiring odd hacks to find them.
+static void intersect(const SkDCubic& cubic1, double t1s, double t1e, const SkDCubic& cubic2,
+        double t2s, double t2e, double precisionScale, SkIntersections& i) {
+    i.upDepth();
+    SkDCubic c1 = cubic1.subDivide(t1s, t1e);
+    SkDCubic c2 = cubic2.subDivide(t2s, t2e);
+    SkSTArray<kCubicToQuadSubdivisionDepth, double, true> ts1;
+    // OPTIMIZE: if c1 == c2, call once (happens when detecting self-intersection)
+    c1.toQuadraticTs(c1.calcPrecision() * precisionScale, &ts1);
+    SkSTArray<kCubicToQuadSubdivisionDepth, double, true> ts2;
+    c2.toQuadraticTs(c2.calcPrecision() * precisionScale, &ts2);
+    double t1Start = t1s;
+    int ts1Count = ts1.count();
+    for (int i1 = 0; i1 <= ts1Count; ++i1) {
+        const double tEnd1 = i1 < ts1Count ? ts1[i1] : 1;
+        const double t1 = t1s + (t1e - t1s) * tEnd1;
+        SkReduceOrder s1;
+        int o1 = quadPart(cubic1, t1Start, t1, &s1);
+        double t2Start = t2s;
+        int ts2Count = ts2.count();
+        for (int i2 = 0; i2 <= ts2Count; ++i2) {
+            const double tEnd2 = i2 < ts2Count ? ts2[i2] : 1;
+            const double t2 = t2s + (t2e - t2s) * tEnd2;
+            if (&cubic1 == &cubic2 && t1Start >= t2Start) {
+                t2Start = t2;
+                continue;
+            }
+            SkReduceOrder s2;
+            int o2 = quadPart(cubic2, t2Start, t2, &s2);
+        #if ONE_OFF_DEBUG
+            char tab[] = "                  ";
+            if (tLimits1[0][0] >= t1Start && tLimits1[0][1] <= t1
+                    && tLimits1[1][0] >= t2Start && tLimits1[1][1] <= t2) {
+                SkDebugf("%.*s %s t1=(%1.9g,%1.9g) t2=(%1.9g,%1.9g)", i.depth()*2, tab,
+                        __FUNCTION__, t1Start, t1, t2Start, t2);
+                SkIntersections xlocals;
+                xlocals.allowNear(false);
+                xlocals.allowFlatMeasure(true);
+                intersectWithOrder(s1.fQuad, o1, s2.fQuad, o2, xlocals);
+                SkDebugf(" xlocals.fUsed=%d\n", xlocals.used());
+            }
+        #endif
+            SkIntersections locals;
+            locals.allowNear(false);
+            locals.allowFlatMeasure(true);
+            intersectWithOrder(s1.fQuad, o1, s2.fQuad, o2, locals);
+            int tCount = locals.used();
+            for (int tIdx = 0; tIdx < tCount; ++tIdx) {
+                double to1 = t1Start + (t1 - t1Start) * locals[0][tIdx];
+                double to2 = t2Start + (t2 - t2Start) * locals[1][tIdx];
+    // if the computed t is not sufficiently precise, iterate
+                SkDPoint p1 = cubic1.ptAtT(to1);
+                SkDPoint p2 = cubic2.ptAtT(to2);
+                if (p1.approximatelyEqual(p2)) {
+    // FIXME: local edge may be coincident -- experiment with not propagating coincidence to caller
+//                    SkASSERT(!locals.isCoincident(tIdx));
+                    if (&cubic1 != &cubic2 || !approximately_equal(to1, to2)) {
+                        if (i.swapped()) {  //  FIXME: insert should respect swap
+                            i.insert(to2, to1, p1);
+                        } else {
+                            i.insert(to1, to2, p1);
+                        }
+                    }
+                } else {
+/*for random cubics, 16 below catches 99.997% of the intersections. To test for the remaining 0.003%
+  look for nearly coincident curves. and check each 1/16th section.
+*/
+                    double offset = precisionScale / 16;  // FIXME: const is arbitrary: test, refine
+                    double c1Bottom = tIdx == 0 ? 0 :
+                            (t1Start + (t1 - t1Start) * locals[0][tIdx - 1] + to1) / 2;
+                    double c1Min = SkTMax(c1Bottom, to1 - offset);
+                    double c1Top = tIdx == tCount - 1 ? 1 :
+                            (t1Start + (t1 - t1Start) * locals[0][tIdx + 1] + to1) / 2;
+                    double c1Max = SkTMin(c1Top, to1 + offset);
+                    double c2Min = SkTMax(0., to2 - offset);
+                    double c2Max = SkTMin(1., to2 + offset);
+                #if ONE_OFF_DEBUG
+                    SkDebugf("%.*s %s 1 contains1=%d/%d contains2=%d/%d\n", i.depth()*2, tab,
+                            __FUNCTION__,
+                            c1Min <= tLimits1[0][1] && tLimits1[0][0] <= c1Max
+                         && c2Min <= tLimits1[1][1] && tLimits1[1][0] <= c2Max,
+                            to1 - offset <= tLimits1[0][1] && tLimits1[0][0] <= to1 + offset
+                         && to2 - offset <= tLimits1[1][1] && tLimits1[1][0] <= to2 + offset,
+                            c1Min <= tLimits2[0][1] && tLimits2[0][0] <= c1Max
+                         && c2Min <= tLimits2[1][1] && tLimits2[1][0] <= c2Max,
+                            to1 - offset <= tLimits2[0][1] && tLimits2[0][0] <= to1 + offset
+                         && to2 - offset <= tLimits2[1][1] && tLimits2[1][0] <= to2 + offset);
+                    SkDebugf("%.*s %s 1 c1Bottom=%1.9g c1Top=%1.9g c2Bottom=%1.9g c2Top=%1.9g"
+                            " 1-o=%1.9g 1+o=%1.9g 2-o=%1.9g 2+o=%1.9g offset=%1.9g\n",
+                            i.depth()*2, tab, __FUNCTION__, c1Bottom, c1Top, 0., 1.,
+                            to1 - offset, to1 + offset, to2 - offset, to2 + offset, offset);
+                    SkDebugf("%.*s %s 1 to1=%1.9g to2=%1.9g c1Min=%1.9g c1Max=%1.9g c2Min=%1.9g"
+                            " c2Max=%1.9g\n", i.depth()*2, tab, __FUNCTION__, to1, to2, c1Min,
+                            c1Max, c2Min, c2Max);
+                #endif
+                    intersect(cubic1, c1Min, c1Max, cubic2, c2Min, c2Max, offset, i);
+                #if ONE_OFF_DEBUG
+                    SkDebugf("%.*s %s 1 i.used=%d t=%1.9g\n", i.depth()*2, tab, __FUNCTION__,
+                            i.used(), i.used() > 0 ? i[0][i.used() - 1] : -1);
+                #endif
+                    if (tCount > 1) {
+                        c1Min = SkTMax(0., to1 - offset);
+                        c1Max = SkTMin(1., to1 + offset);
+                        double c2Bottom = tIdx == 0 ? to2 :
+                                (t2Start + (t2 - t2Start) * locals[1][tIdx - 1] + to2) / 2;
+                        double c2Top = tIdx == tCount - 1 ? to2 :
+                                (t2Start + (t2 - t2Start) * locals[1][tIdx + 1] + to2) / 2;
+                        if (c2Bottom > c2Top) {
+                            SkTSwap(c2Bottom, c2Top);
+                        }
+                        if (c2Bottom == to2) {
+                            c2Bottom = 0;
+                        }
+                        if (c2Top == to2) {
+                            c2Top = 1;
+                        }
+                        c2Min = SkTMax(c2Bottom, to2 - offset);
+                        c2Max = SkTMin(c2Top, to2 + offset);
+                    #if ONE_OFF_DEBUG
+                        SkDebugf("%.*s %s 2 contains1=%d/%d contains2=%d/%d\n", i.depth()*2, tab,
+                            __FUNCTION__,
+                            c1Min <= tLimits1[0][1] && tLimits1[0][0] <= c1Max
+                         && c2Min <= tLimits1[1][1] && tLimits1[1][0] <= c2Max,
+                            to1 - offset <= tLimits1[0][1] && tLimits1[0][0] <= to1 + offset
+                         && to2 - offset <= tLimits1[1][1] && tLimits1[1][0] <= to2 + offset,
+                            c1Min <= tLimits2[0][1] && tLimits2[0][0] <= c1Max
+                         && c2Min <= tLimits2[1][1] && tLimits2[1][0] <= c2Max,
+                            to1 - offset <= tLimits2[0][1] && tLimits2[0][0] <= to1 + offset
+                         && to2 - offset <= tLimits2[1][1] && tLimits2[1][0] <= to2 + offset);
+                        SkDebugf("%.*s %s 2 c1Bottom=%1.9g c1Top=%1.9g c2Bottom=%1.9g c2Top=%1.9g"
+                                " 1-o=%1.9g 1+o=%1.9g 2-o=%1.9g 2+o=%1.9g offset=%1.9g\n",
+                                i.depth()*2, tab, __FUNCTION__, 0., 1., c2Bottom, c2Top,
+                                to1 - offset, to1 + offset, to2 - offset, to2 + offset, offset);
+                        SkDebugf("%.*s %s 2 to1=%1.9g to2=%1.9g c1Min=%1.9g c1Max=%1.9g c2Min=%1.9g"
+                                " c2Max=%1.9g\n", i.depth()*2, tab, __FUNCTION__, to1, to2, c1Min,
+                                c1Max, c2Min, c2Max);
+                    #endif
+                        intersect(cubic1, c1Min, c1Max, cubic2, c2Min, c2Max, offset, i);
+                #if ONE_OFF_DEBUG
+                    SkDebugf("%.*s %s 2 i.used=%d t=%1.9g\n", i.depth()*2, tab, __FUNCTION__,
+                            i.used(), i.used() > 0 ? i[0][i.used() - 1] : -1);
+                #endif
+                        c1Min = SkTMax(c1Bottom, to1 - offset);
+                        c1Max = SkTMin(c1Top, to1 + offset);
+                    #if ONE_OFF_DEBUG
+                        SkDebugf("%.*s %s 3 contains1=%d/%d contains2=%d/%d\n", i.depth()*2, tab,
+                        __FUNCTION__,
+                            c1Min <= tLimits1[0][1] && tLimits1[0][0] <= c1Max
+                         && c2Min <= tLimits1[1][1] && tLimits1[1][0] <= c2Max,
+                            to1 - offset <= tLimits1[0][1] && tLimits1[0][0] <= to1 + offset
+                         && to2 - offset <= tLimits1[1][1] && tLimits1[1][0] <= to2 + offset,
+                            c1Min <= tLimits2[0][1] && tLimits2[0][0] <= c1Max
+                         && c2Min <= tLimits2[1][1] && tLimits2[1][0] <= c2Max,
+                            to1 - offset <= tLimits2[0][1] && tLimits2[0][0] <= to1 + offset
+                         && to2 - offset <= tLimits2[1][1] && tLimits2[1][0] <= to2 + offset);
+                        SkDebugf("%.*s %s 3 c1Bottom=%1.9g c1Top=%1.9g c2Bottom=%1.9g c2Top=%1.9g"
+                                " 1-o=%1.9g 1+o=%1.9g 2-o=%1.9g 2+o=%1.9g offset=%1.9g\n",
+                                i.depth()*2, tab, __FUNCTION__, 0., 1., c2Bottom, c2Top,
+                                to1 - offset, to1 + offset, to2 - offset, to2 + offset, offset);
+                        SkDebugf("%.*s %s 3 to1=%1.9g to2=%1.9g c1Min=%1.9g c1Max=%1.9g c2Min=%1.9g"
+                                " c2Max=%1.9g\n", i.depth()*2, tab, __FUNCTION__, to1, to2, c1Min,
+                                c1Max, c2Min, c2Max);
+                    #endif
+                        intersect(cubic1, c1Min, c1Max, cubic2, c2Min, c2Max, offset, i);
+                #if ONE_OFF_DEBUG
+                    SkDebugf("%.*s %s 3 i.used=%d t=%1.9g\n", i.depth()*2, tab, __FUNCTION__,
+                            i.used(), i.used() > 0 ? i[0][i.used() - 1] : -1);
+                #endif
+                    }
+          //          intersect(cubic1, c1Min, c1Max, cubic2, c2Min, c2Max, offset, i);
+                    // FIXME: if no intersection is found, either quadratics intersected where
+                    // cubics did not, or the intersection was missed. In the former case, expect
+                    // the quadratics to be nearly parallel at the point of intersection, and check
+                    // for that.
+                }
+            }
+            t2Start = t2;
+        }
+        t1Start = t1;
+    }
+    i.downDepth();
+}
+
+    // if two ends intersect, check middle for coincidence
+bool SkIntersections::cubicCheckCoincidence(const SkDCubic& c1, const SkDCubic& c2) {
+    if (fUsed < 2) {
+        return false;
+    }
+    int last = fUsed - 1;
+    double tRange1 = fT[0][last] - fT[0][0];
+    double tRange2 = fT[1][last] - fT[1][0];
+    for (int index = 1; index < 5; ++index) {
+        double testT1 = fT[0][0] + tRange1 * index / 5;
+        double testT2 = fT[1][0] + tRange2 * index / 5;
+        SkDPoint testPt1 = c1.ptAtT(testT1);
+        SkDPoint testPt2 = c2.ptAtT(testT2);
+        if (!testPt1.approximatelyEqual(testPt2)) {
+            return false;
+        }
+    }
+    if (fUsed > 2) {
+        fPt[1] = fPt[last];
+        fT[0][1] = fT[0][last];
+        fT[1][1] = fT[1][last];
+        fUsed = 2;
+    }
+    fIsCoincident[0] = fIsCoincident[1] = 0x03;
+    return true;
+}
+
+#define LINE_FRACTION 0.1
+
+// intersect the end of the cubic with the other. Try lines from the end to control and opposite
+// end to determine range of t on opposite cubic.
+bool SkIntersections::cubicExactEnd(const SkDCubic& cubic1, bool start, const SkDCubic& cubic2) {
+    int t1Index = start ? 0 : 3;
+    double testT = (double) !start;
+    bool swap = swapped();
+    // quad/quad at this point checks to see if exact matches have already been found
+    // cubic/cubic can't reject so easily since cubics can intersect same point more than once
+    SkDLine tmpLine;
+    tmpLine[0] = tmpLine[1] = cubic2[t1Index];
+    tmpLine[1].fX += cubic2[2 - start].fY - cubic2[t1Index].fY;
+    tmpLine[1].fY -= cubic2[2 - start].fX - cubic2[t1Index].fX;
+    SkIntersections impTs;
+    impTs.allowNear(false);
+    impTs.allowFlatMeasure(true);
+    impTs.intersectRay(cubic1, tmpLine);
+    for (int index = 0; index < impTs.used(); ++index) {
+        SkDPoint realPt = impTs.pt(index);
+        if (!tmpLine[0].approximatelyEqual(realPt)) {
+            continue;
+        }
+        if (swap) {
+            cubicInsert(testT, impTs[0][index], tmpLine[0], cubic2, cubic1);
+        } else {
+            cubicInsert(impTs[0][index], testT, tmpLine[0], cubic1, cubic2);
+        }
+        return true;
+    }
+    return false;
+}
+
+
+void SkIntersections::cubicInsert(double one, double two, const SkDPoint& pt,
+        const SkDCubic& cubic1, const SkDCubic& cubic2) {
+    for (int index = 0; index < fUsed; ++index) {
+        if (fT[0][index] == one) {
+            double oldTwo = fT[1][index];
+            if (oldTwo == two) {
+                return;
+            }
+            SkDPoint mid = cubic2.ptAtT((oldTwo + two) / 2);
+            if (mid.approximatelyEqual(fPt[index])) {
+                return;
+            }
+        }
+        if (fT[1][index] == two) {
+            SkDPoint mid = cubic1.ptAtT((fT[0][index] + two) / 2);
+            if (mid.approximatelyEqual(fPt[index])) {
+                return;
+            }
+        }
+    }
+    insert(one, two, pt);
+}
+
+void SkIntersections::cubicNearEnd(const SkDCubic& cubic1, bool start, const SkDCubic& cubic2,
+                         const SkDRect& bounds2) {
+    SkDLine line;
+    int t1Index = start ? 0 : 3;
+    double testT = (double) !start;
+   // don't bother if the two cubics are connnected
+    static const int kPointsInCubic = 4; // FIXME: move to DCubic, replace '4' with this
+    static const int kMaxLineCubicIntersections = 3;
+    SkSTArray<(kMaxLineCubicIntersections - 1) * kMaxLineCubicIntersections, double, true> tVals;
+    line[0] = cubic1[t1Index];
+    // this variant looks for intersections with the end point and lines parallel to other points
+    for (int index = 0; index < kPointsInCubic; ++index) {
+        if (index == t1Index) {
+            continue;
+        }
+        SkDVector dxy1 = cubic1[index] - line[0];
+        dxy1 /= SkDCubic::gPrecisionUnit;
+        line[1] = line[0] + dxy1;
+        SkDRect lineBounds;
+        lineBounds.setBounds(line);
+        if (!bounds2.intersects(&lineBounds)) {
+            continue;
+        }
+        SkIntersections local;
+        if (!local.intersect(cubic2, line)) {
+            continue;
+        }
+        for (int idx2 = 0; idx2 < local.used(); ++idx2) {
+            double foundT = local[0][idx2];
+            if (approximately_less_than_zero(foundT)
+                    || approximately_greater_than_one(foundT)) {
+                continue;
+            }
+            if (local.pt(idx2).approximatelyEqual(line[0])) {
+                if (swapped()) {  // FIXME: insert should respect swap
+                    insert(foundT, testT, line[0]);
+                } else {
+                    insert(testT, foundT, line[0]);
+                }
+            } else {
+                tVals.push_back(foundT);
+            }
+        }
+    }
+    if (tVals.count() == 0) {
+        return;
+    }
+    SkTQSort<double>(tVals.begin(), tVals.end() - 1);
+    double tMin1 = start ? 0 : 1 - LINE_FRACTION;
+    double tMax1 = start ? LINE_FRACTION : 1;
+    int tIdx = 0;
+    do {
+        int tLast = tIdx;
+        while (tLast + 1 < tVals.count() && roughly_equal(tVals[tLast + 1], tVals[tIdx])) {
+            ++tLast;
+        }
+        double tMin2 = SkTMax(tVals[tIdx] - LINE_FRACTION, 0.0);
+        double tMax2 = SkTMin(tVals[tLast] + LINE_FRACTION, 1.0);
+        int lastUsed = used();
+        if (start ? tMax1 < tMin2 : tMax2 < tMin1) {
+            ::intersect(cubic1, tMin1, tMax1, cubic2, tMin2, tMax2, 1, *this);
+        }
+        if (lastUsed == used()) {
+            tMin2 = SkTMax(tVals[tIdx] - (1.0 / SkDCubic::gPrecisionUnit), 0.0);
+            tMax2 = SkTMin(tVals[tLast] + (1.0 / SkDCubic::gPrecisionUnit), 1.0);
+            if (start ? tMax1 < tMin2 : tMax2 < tMin1) {
+                ::intersect(cubic1, tMin1, tMax1, cubic2, tMin2, tMax2, 1, *this);
+            }
+        }
+        tIdx = tLast + 1;
+    } while (tIdx < tVals.count());
+    return;
+}
+
+const double CLOSE_ENOUGH = 0.001;
+
+static bool closeStart(const SkDCubic& cubic, int cubicIndex, SkIntersections& i, SkDPoint& pt) {
+    if (i[cubicIndex][0] != 0 || i[cubicIndex][1] > CLOSE_ENOUGH) {
+        return false;
+    }
+    pt = cubic.ptAtT((i[cubicIndex][0] + i[cubicIndex][1]) / 2);
+    return true;
+}
+
+static bool closeEnd(const SkDCubic& cubic, int cubicIndex, SkIntersections& i, SkDPoint& pt) {
+    int last = i.used() - 1;
+    if (i[cubicIndex][last] != 1 || i[cubicIndex][last - 1] < 1 - CLOSE_ENOUGH) {
+        return false;
+    }
+    pt = cubic.ptAtT((i[cubicIndex][last] + i[cubicIndex][last - 1]) / 2);
+    return true;
+}
+
+static bool only_end_pts_in_common(const SkDCubic& c1, const SkDCubic& c2) {
+// the idea here is to see at minimum do a quick reject by rotating all points
+// to either side of the line formed by connecting the endpoints
+// if the opposite curves points are on the line or on the other side, the
+// curves at most intersect at the endpoints
+    for (int oddMan = 0; oddMan < 4; ++oddMan) {
+        const SkDPoint* endPt[3];
+        for (int opp = 1; opp < 4; ++opp) {
+            int end = oddMan ^ opp;  // choose a value not equal to oddMan
+            endPt[opp - 1] = &c1[end];
+        }
+        for (int triTest = 0; triTest < 3; ++triTest) {
+            double origX = endPt[triTest]->fX;
+            double origY = endPt[triTest]->fY;
+            int oppTest = triTest + 1;
+            if (3 == oppTest) {
+                oppTest = 0;
+            }
+            double adj = endPt[oppTest]->fX - origX;
+            double opp = endPt[oppTest]->fY - origY;
+            if (adj == 0 && opp == 0) {  // if the other point equals the test point, ignore it
+                continue;
+            }
+            double sign = (c1[oddMan].fY - origY) * adj - (c1[oddMan].fX - origX) * opp;
+            if (approximately_zero(sign)) {
+                goto tryNextHalfPlane;
+            }
+            for (int n = 0; n < 4; ++n) {
+                double test = (c2[n].fY - origY) * adj - (c2[n].fX - origX) * opp;
+                if (test * sign > 0 && !precisely_zero(test)) {
+                    goto tryNextHalfPlane;
+                }
+            }
+        }
+        return true;
+tryNextHalfPlane:
+        ;
+    }
+    return false;
+}
+
+int SkIntersections::intersect(const SkDCubic& c1, const SkDCubic& c2) {
+    if (fMax == 0) {
+        fMax = 9;
+    }
+    bool selfIntersect = &c1 == &c2;
+    if (selfIntersect) {
+        if (c1[0].approximatelyEqual(c1[3])) {
+            insert(0, 1, c1[0]);
+            return fUsed;
+        }
+    } else {
+        // OPTIMIZATION: set exact end bits here to avoid cubic exact end later
+        for (int i1 = 0; i1 < 4; i1 += 3) {
+            for (int i2 = 0; i2 < 4; i2 += 3) {
+                if (c1[i1].approximatelyEqual(c2[i2])) {
+                    insert(i1 >> 1, i2 >> 1, c1[i1]);
+                }
+            }
+        }
+    }
+    SkASSERT(fUsed < 4);
+    if (!selfIntersect) {
+        if (only_end_pts_in_common(c1, c2)) {
+            return fUsed;
+        }
+        if (only_end_pts_in_common(c2, c1)) {
+            return fUsed;
+        }
+    }
+    // quad/quad does linear test here -- cubic does not
+    // cubics which are really lines should have been detected in reduce step earlier
+    int exactEndBits = 0;
+    if (selfIntersect) {
+        if (fUsed) {
+            return fUsed;
+        }
+    } else {
+        exactEndBits |= cubicExactEnd(c1, false, c2) << 0;
+        exactEndBits |= cubicExactEnd(c1, true, c2) << 1;
+        swap();
+        exactEndBits |= cubicExactEnd(c2, false, c1) << 2;
+        exactEndBits |= cubicExactEnd(c2, true, c1) << 3;
+        swap();
+    }
+    if (cubicCheckCoincidence(c1, c2)) {
+        SkASSERT(!selfIntersect);
+        return fUsed;
+    }
+    // FIXME: pass in cached bounds from caller
+    SkDRect c2Bounds;
+    c2Bounds.setBounds(c2);
+    if (!(exactEndBits & 4)) {
+        cubicNearEnd(c1, false, c2, c2Bounds);
+    }
+    if (!(exactEndBits & 8)) {
+        if (selfIntersect && fUsed) {
+            return fUsed;
+        }
+        cubicNearEnd(c1, true, c2, c2Bounds);
+        if (selfIntersect && fUsed && ((approximately_less_than_zero(fT[0][0])
+                    && approximately_less_than_zero(fT[1][0]))
+                    || (approximately_greater_than_one(fT[0][0])
+                    && approximately_greater_than_one(fT[1][0])))) {
+            SkASSERT(fUsed == 1);
+            fUsed = 0;
+            return fUsed;
+        }
+    }
+    if (!selfIntersect) {
+        SkDRect c1Bounds;
+        c1Bounds.setBounds(c1);  // OPTIMIZE use setRawBounds ?
+        swap();
+        if (!(exactEndBits & 1)) {
+            cubicNearEnd(c2, false, c1, c1Bounds);
+        }
+        if (!(exactEndBits & 2)) {
+            cubicNearEnd(c2, true, c1, c1Bounds);
+        }
+        swap();
+    }
+    if (cubicCheckCoincidence(c1, c2)) {
+        SkASSERT(!selfIntersect);
+        return fUsed;
+    }
+    SkIntersections i;
+    i.fAllowNear = false;
+    i.fFlatMeasure = true;
+    i.fMax = 9;
+    ::intersect(c1, 0, 1, c2, 0, 1, 1, i);
+    int compCount = i.used();
+    if (compCount) {
+        int exactCount = used();
+        if (exactCount == 0) {
+            *this = i;
+        } else {
+            // at least one is exact or near, and at least one was computed. Eliminate duplicates
+            for (int exIdx = 0; exIdx < exactCount; ++exIdx) {
+                for (int cpIdx = 0; cpIdx < compCount; ) {
+                    if (fT[0][0] == i[0][0] && fT[1][0] == i[1][0]) {
+                        i.removeOne(cpIdx);
+                        --compCount;
+                        continue;
+                    }
+                    double tAvg = (fT[0][exIdx] + i[0][cpIdx]) / 2;
+                    SkDPoint pt = c1.ptAtT(tAvg);
+                    if (!pt.approximatelyEqual(fPt[exIdx])) {
+                        ++cpIdx;
+                        continue;
+                    }
+                    tAvg = (fT[1][exIdx] + i[1][cpIdx]) / 2;
+                    pt = c2.ptAtT(tAvg);
+                    if (!pt.approximatelyEqual(fPt[exIdx])) {
+                        ++cpIdx;
+                        continue;
+                    }
+                    i.removeOne(cpIdx);
+                    --compCount;
+                }
+            }
+            // if mid t evaluates to nearly the same point, skip the t
+            for (int cpIdx = 0; cpIdx < compCount - 1; ) {
+                double tAvg = (fT[0][cpIdx] + i[0][cpIdx + 1]) / 2;
+                SkDPoint pt = c1.ptAtT(tAvg);
+                if (!pt.approximatelyEqual(fPt[cpIdx])) {
+                    ++cpIdx;
+                    continue;
+                }
+                tAvg = (fT[1][cpIdx] + i[1][cpIdx + 1]) / 2;
+                pt = c2.ptAtT(tAvg);
+                if (!pt.approximatelyEqual(fPt[cpIdx])) {
+                    ++cpIdx;
+                    continue;
+                }
+                i.removeOne(cpIdx);
+                --compCount;
+            }
+            // in addition to adding below missing function, think about how to say
+            append(i);
+        }
+    }
+    // If an end point and a second point very close to the end is returned, the second
+    // point may have been detected because the approximate quads
+    // intersected at the end and close to it. Verify that the second point is valid.
+    if (fUsed <= 1) {
+        return fUsed;
+    }
+    SkDPoint pt[2];
+    if (closeStart(c1, 0, *this, pt[0]) && closeStart(c2, 1, *this, pt[1])
+            && pt[0].approximatelyEqual(pt[1])) {
+        removeOne(1);
+    }
+    if (closeEnd(c1, 0, *this, pt[0]) && closeEnd(c2, 1, *this, pt[1])
+            && pt[0].approximatelyEqual(pt[1])) {
+        removeOne(used() - 2);
+    }
+    // vet the pairs of t values to see if the mid value is also on the curve. If so, mark
+    // the span as coincident
+    if (fUsed >= 2 && !coincidentUsed()) {
+        int last = fUsed - 1;
+        int match = 0;
+        for (int index = 0; index < last; ++index) {
+            double mid1 = (fT[0][index] + fT[0][index + 1]) / 2;
+            double mid2 = (fT[1][index] + fT[1][index + 1]) / 2;
+            pt[0] = c1.ptAtT(mid1);
+            pt[1] = c2.ptAtT(mid2);
+            if (pt[0].approximatelyEqual(pt[1])) {
+                match |= 1 << index;
+            }
+        }
+        if (match) {
+#if DEBUG_CONCIDENT
+            if (((match + 1) & match) != 0) {
+                SkDebugf("%s coincident hole\n", __FUNCTION__);
+            }
+#endif
+            // for now, assume that everything from start to finish is coincident
+            if (fUsed > 2) {
+                  fPt[1] = fPt[last];
+                  fT[0][1] = fT[0][last];
+                  fT[1][1] = fT[1][last];
+                  fIsCoincident[0] = 0x03;
+                  fIsCoincident[1] = 0x03;
+                  fUsed = 2;
+            }
+        }
+    }
+    return fUsed;
+}
+
+// Up promote the quad to a cubic.
+// OPTIMIZATION If this is a common use case, optimize by duplicating
+// the intersect 3 loop to avoid the promotion  / demotion code
+int SkIntersections::intersect(const SkDCubic& cubic, const SkDQuad& quad) {
+    fMax = 7;
+    SkDCubic up = quad.toCubic();
+    (void) intersect(cubic, up);
+    return used();
+}
+
+/* http://www.ag.jku.at/compass/compasssample.pdf
+( Self-Intersection Problems and Approximate Implicitization by Jan B. Thomassen
+Centre of Mathematics for Applications, University of Oslo http://www.cma.uio.no janbth@math.uio.no
+SINTEF Applied Mathematics http://www.sintef.no )
+describes a method to find the self intersection of a cubic by taking the gradient of the implicit
+form dotted with the normal, and solving for the roots. My math foo is too poor to implement this.*/
+
+int SkIntersections::intersect(const SkDCubic& c) {
+    fMax = 1;
+    // check to see if x or y end points are the extrema. Are other quick rejects possible?
+    if (c.endsAreExtremaInXOrY()) {
+        return false;
+    }
+    // OPTIMIZATION: could quick reject if neither end point tangent ray intersected the line
+    // segment formed by the opposite end point to the control point
+    (void) intersect(c, c);
+    if (used() > 1) {
+        fUsed = 0;
+    } else if (used() > 0) {
+        if (approximately_equal_double(fT[0][0], fT[1][0])) {
+            fUsed = 0;
+        } else {
+            SkASSERT(used() == 1);
+            if (fT[0][0] > fT[1][0]) {
+                swapPts();
+            }
+        }
+    }
+    return used();
+}