Add intersections for path ops

src/pathops/SkDCubicIntersection.cpp

+/*
+ * 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 "SkTDArray.h"
+#include "TSearch.h"
+
+#if ONE_OFF_DEBUG
+static const double tLimits1[2][2] = {{0.36, 0.37}, {0.63, 0.64}};
+static const double tLimits2[2][2] = {{-0.865211397, -0.865215212}, {-0.865207696, -0.865208078}};
+#endif
+
+#define DEBUG_QUAD_PART 0
+#define SWAP_TOP_DEBUG 0
+
+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, SkReduceOrder::kFill_Style);
+#if DEBUG_QUAD_PART
+    SkDebugf("%s cubic=(%1.17g,%1.17g %1.17g,%1.17g %1.17g,%1.17g %1.17g,%1.17g)"
+            " t=(%1.17g,%1.17g)\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("%s part=(%1.17g,%1.17g %1.17g,%1.17g %1.17g,%1.17g %1.17g,%1.17g)"
+            " quad=(%1.17g,%1.17g %1.17g,%1.17g %1.17g,%1.17g)\n", __FUNCTION__,
+            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);
+    SkDebugf("%s simple=(%1.17g,%1.17g", __FUNCTION__, reducer->fQuad[0].fX, reducer->fQuad[0].fY);
+    if (order > 1) {
+        SkDebugf(" %1.17g,%1.17g", reducer->fQuad[1].fX, reducer->fQuad[1].fY);
+    }
+    if (order > 2) {
+        SkDebugf(" %1.17g,%1.17g", reducer->fQuad[2].fX, reducer->fQuad[2].fY);
+    }
+    SkDebugf(")\n");
+    SkASSERT(order < 4 && order > 0);
+#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);
+    SkTDArray<double> ts1;
+    // OPTIMIZE: if c1 == c2, call once (happens when detecting self-intersection)
+    c1.toQuadraticTs(c1.calcPrecision() * precisionScale, &ts1);
+    SkTDArray<double> 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) {
+                SkDCubic cSub1 = cubic1.subDivide(t1Start, t1);
+                SkDCubic cSub2 = cubic2.subDivide(t2Start, 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;
+                intersectWithOrder(s1.fQuad, o1, s2.fQuad, o2, xlocals);
+                SkDebugf(" xlocals.fUsed=%d\n", xlocals.used());
+            }
+        #endif
+            SkIntersections locals;
+            intersectWithOrder(s1.fQuad, o1, s2.fQuad, o2, locals);
+            double coStart[2] = { -1 };
+            SkDPoint coPoint;
+            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.xyAtT(to1);
+                SkDPoint p2 = cubic2.xyAtT(to2);
+                if (p1.approximatelyEqual(p2)) {
+                    if (locals.isCoincident(tIdx)) {
+                        if (coStart[0] < 0) {
+                            coStart[0] = to1;
+                            coStart[1] = to2;
+                            coPoint = p1;
+                        } else {
+                            i.insertCoincidentPair(coStart[0], to1, coStart[1], to2, coPoint, p1);
+                            coStart[0] = -1;
+                        }
+                    } else 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 {
+                    double offset = precisionScale / 16;  // FIME: const is arbitrary: test, refine
+#if 1
+                    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
+                    }
+#else
+                    double c1Bottom = tIdx == 0 ? 0 :
+                            (t1Start + (t1 - t1Start) * locals.fT[0][tIdx - 1] + to1) / 2;
+                    double c1Min = SkTMax(c1Bottom, to1 - offset);
+                    double c1Top = tIdx == tCount - 1 ? 1 :
+                            (t1Start + (t1 - t1Start) * locals.fT[0][tIdx + 1] + to1) / 2;
+                    double c1Max = SkTMin(c1Top, to1 + offset);
+                    double c2Bottom = tIdx == 0 ? to2 :
+                            (t2Start + (t2 - t2Start) * locals.fT[1][tIdx - 1] + to2) / 2;
+                    double c2Top = tIdx == tCount - 1 ? to2 :
+                            (t2Start + (t2 - t2Start) * locals.fT[1][tIdx + 1] + to2) / 2;
+                    if (c2Bottom > c2Top) {
+                        SkTSwap(c2Bottom, c2Top);
+                    }
+                    if (c2Bottom == to2) {
+                        c2Bottom = 0;
+                    }
+                    if (c2Top == to2) {
+                        c2Top = 1;
+                    }
+                    double c2Min = SkTMax(c2Bottom, to2 - offset);
+                    double c2Max = SkTMin(c2Top, to2 + offset);
+                #if ONE_OFF_DEBUG
+                    SkDebugf("%s contains1=%d/%d contains2=%d/%d\n", __FUNCTION__,
+                            c1Min <= 0.210357794 && 0.210357794 <= c1Max
+                         && c2Min <= 0.223476406 && 0.223476406 <= c2Max,
+                            to1 - offset <= 0.210357794 && 0.210357794 <= to1 + offset
+                         && to2 - offset <= 0.223476406 && 0.223476406 <= to2 + offset,
+                            c1Min <= 0.211324707 && 0.211324707 <= c1Max
+                         && c2Min <= 0.211327209 && 0.211327209 <= c2Max,
+                            to1 - offset <= 0.211324707 && 0.211324707 <= to1 + offset
+                         && to2 - offset <= 0.211327209 && 0.211327209 <= to2 + offset);
+                    SkDebugf("%s 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",
+                            __FUNCTION__, c1Bottom, c1Top, c2Bottom, c2Top,
+                            to1 - offset, to1 + offset, to2 - offset, to2 + offset, offset);
+                    SkDebugf("%s to1=%1.9g to2=%1.9g c1Min=%1.9g c1Max=%1.9g c2Min=%1.9g"
+                            " c2Max=%1.9g\n", __FUNCTION__, to1, to2, c1Min, c1Max, c2Min, c2Max);
+                #endif
+#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.
+                }
+            }
+            SkASSERT(coStart[0] == -1);
+            t2Start = t2;
+        }
+        t1Start = t1;
+    }
+    i.downDepth();
+}
+
+#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.
+static void intersectEnd(const SkDCubic& cubic1, bool start, const SkDCubic& cubic2,
+                         const SkDRect& bounds2, SkIntersections& i) {
+    SkDLine line;
+    int t1Index = start ? 0 : 3;
+    line[0] = cubic1[t1Index];
+    // don't bother if the two cubics are connnected
+    SkTDArray<double> tVals;  // OPTIMIZE: replace with hard-sized array
+    for (int index = 0; index < 4; ++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 (i.swapped()) {  // FIXME: insert should respect swap
+                    i.insert(foundT, start ? 0 : 1, line[0]);
+                } else {
+                    i.insert(start ? 0 : 1, foundT, line[0]);
+                }
+            } else {
+                *tVals.append() = local[0][idx2];
+            }
+        }
+    }
+    if (tVals.count() == 0) {
+        return;
+    }
+    QSort<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 = i.used();
+        intersect(cubic1, tMin1, tMax1, cubic2, tMin2, tMax2, 1, i);
+        if (lastUsed == i.used()) {
+            tMin2 = SkTMax(tVals[tIdx] - (1.0 / SkDCubic::gPrecisionUnit), 0.0);
+            tMax2 = SkTMin(tVals[tLast] + (1.0 / SkDCubic::gPrecisionUnit), 1.0);
+            intersect(cubic1, tMin1, tMax1, cubic2, tMin2, tMax2, 1, i);
+        }
+        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.xyAtT((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.xyAtT((i[cubicIndex][last] + i[cubicIndex][last - 1]) / 2);
+    return true;
+}
+
+int SkIntersections::intersect(const SkDCubic& c1, const SkDCubic& c2) {
+    ::intersect(c1, 0, 1, c2, 0, 1, 1, *this);
+    // FIXME: pass in cached bounds from caller
+    SkDRect c1Bounds, c2Bounds;
+    c1Bounds.setBounds(c1);  // OPTIMIZE use setRawBounds ?
+    c2Bounds.setBounds(c2);
+    intersectEnd(c1, false, c2, c2Bounds, *this);
+    intersectEnd(c1, true, c2, c2Bounds, *this);
+    bool selfIntersect = &c1 == &c2;
+    if (!selfIntersect) {
+        swap();
+        intersectEnd(c2, false, c1, c1Bounds, *this);
+        intersectEnd(c2, true, c1, c1Bounds, *this);
+        swap();
+    }
+    // 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 || coincidentUsed()) {
+        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);
+    }
+    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) {
+    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) {
+    // check to see if x or y end points are the extrema. Are other quick rejects possible?
+    if (c.endsAreExtremaInXOrY()) {
+        return false;
+    }
+    (void) intersect(c, c);
+    if (used() > 0) {
+        SkASSERT(used() == 1);
+        if (fT[0][0] > fT[1][0]) {
+            swapPts();
+        }
+    }
+    return used();
+}