pathops version two

src/pathops/SkDCubicIntersection.cpp

Index: src/pathops/SkDCubicIntersection.cpp
diff --git a/src/pathops/SkDCubicIntersection.cpp b/src/pathops/SkDCubicIntersection.cpp
deleted file mode 100644
index 2fb35e182794dcec2bb86cb654a0d7d22a3b0af0..0000000000000000000000000000000000000000
--- a/src/pathops/SkDCubicIntersection.cpp
+++ /dev/null
@@ -1,704 +0,0 @@
-/*
- * 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();
-}