Add intersections for path ops

This CL depends on 
https://codereview.chromium.org/12827020/
"Add base types for path ops"

The intersection of a line, quadratic, or cubic
with another curve (or with itself) is found by
solving the implicit equation for the curve pair.

The curves are first reduced to find the simplest
form that will describe the original, and to detect
degenerate or special-case data like horizontal and
vertical lines.

For cubic self-intersection, and for a pair of cubics,
the intersection is found by recursively
approximating the cubic with a series of quadratics.

The implicit solutions depend on the root finding
contained in the DCubic and DQuad structs, and
the quartic root finder included here.
Review URL: https://codereview.chromium.org/12880016

git-svn-id: http://skia.googlecode.com/svn/trunk@8552 2bbb7eff-a529-9590-31e7-b0007b416f81

1 files changed, 513 insertions, 0 deletions

diff --git a/tests/PathOpsCubicIntersectionTest.cpp b/tests/PathOpsCubicIntersectionTest.cpp
new file mode 100644
index 0000000000..5a6a336a3f
--- /dev/null
+++ b/tests/PathOpsCubicIntersectionTest.cpp
@@ -0,0 +1,513 @@
+/*
+ * 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 "PathOpsCubicIntersectionTestData.h"
+#include "PathOpsTestCommon.h"
+#include "SkIntersections.h"
+#include "SkPathOpsRect.h"
+#include "SkReduceOrder.h"
+#include "Test.h"
+
+const int firstCubicIntersectionTest = 9;
+
+static void standardTestCases(skiatest::Reporter* reporter) {
+    for (size_t index = firstCubicIntersectionTest; index < tests_count; ++index) {
+        int iIndex = static_cast<int>(index);
+        const SkDCubic& cubic1 = tests[index][0];
+        const SkDCubic& cubic2 = tests[index][1];
+        SkReduceOrder reduce1, reduce2;
+        int order1 = reduce1.reduce(cubic1, SkReduceOrder::kNo_Quadratics,
+            SkReduceOrder::kFill_Style);
+        int order2 = reduce2.reduce(cubic2, SkReduceOrder::kNo_Quadratics,
+            SkReduceOrder::kFill_Style);
+        const bool showSkipped = false;
+        if (order1 < 4) {
+            if (showSkipped) {
+                SkDebugf("%s [%d] cubic1 order=%d\n", __FUNCTION__, iIndex, order1);
+            }
+            continue;
+        }
+        if (order2 < 4) {
+            if (showSkipped) {
+                SkDebugf("%s [%d] cubic2 order=%d\n", __FUNCTION__, iIndex, order2);
+            }
+            continue;
+        }
+        SkIntersections tIntersections;
+        tIntersections.intersect(cubic1, cubic2);
+        if (!tIntersections.used()) {
+            if (showSkipped) {
+                SkDebugf("%s [%d] no intersection\n", __FUNCTION__, iIndex);
+            }
+            continue;
+        }
+        if (tIntersections.isCoincident(0)) {
+            if (showSkipped) {
+                SkDebugf("%s [%d] coincident\n", __FUNCTION__, iIndex);
+            }
+            continue;
+        }
+        for (int pt = 0; pt < tIntersections.used(); ++pt) {
+            double tt1 = tIntersections[0][pt];
+            SkDPoint xy1 = cubic1.xyAtT(tt1);
+            double tt2 = tIntersections[1][pt];
+            SkDPoint xy2 = cubic2.xyAtT(tt2);
+            if (!xy1.approximatelyEqual(xy2)) {
+                SkDebugf("%s [%d,%d] x!= t1=%g (%g,%g) t2=%g (%g,%g)\n",
+                    __FUNCTION__, (int)index, pt, tt1, xy1.fX, xy1.fY, tt2, xy2.fX, xy2.fY);
+            }
+            REPORTER_ASSERT(reporter, xy1.approximatelyEqual(xy2));
+        }
+    }
+}
+
+static const SkDCubic testSet[] = {
+// FIXME: uncommenting these two will cause this to fail
+// this results in two curves very nearly but not exactly coincident
+#if 0
+{{{67.426548091427676, 37.993772624988935}, {23.483695892376684, 90.476863174921306},
+      {35.597065061143162, 79.872482633158796}, {75.38634169631932, 18.244890038969412}}},
+{{{67.4265481, 37.9937726}, {23.4836959, 90.4768632}, {35.5970651, 79.8724826},
+      {75.3863417, 18.24489}}},
+#endif
+
+{{{0, 0}, {0, 1}, {1, 1}, {1, 0}}},
+{{{1, 0}, {0, 0}, {0, 1}, {1, 1}}},
+
+{{{0, 1}, {4, 5}, {1, 0}, {5, 3}}},
+{{{0, 1}, {3, 5}, {1, 0}, {5, 4}}},
+
+{{{0, 1}, {1, 6}, {1, 0}, {1, 0}}},
+{{{0, 1}, {0, 1}, {1, 0}, {6, 1}}},
+
+{{{0, 1}, {3, 4}, {1, 0}, {5, 1}}},
+{{{0, 1}, {1, 5}, {1, 0}, {4, 3}}},
+
+{{{0, 1}, {1, 2}, {1, 0}, {6, 1}}},
+{{{0, 1}, {1, 6}, {1, 0}, {2, 1}}},
+
+{{{0, 1}, {0, 5}, {1, 0}, {4, 0}}},
+{{{0, 1}, {0, 4}, {1, 0}, {5, 0}}},
+
+{{{0, 1}, {3, 4}, {1, 0}, {3, 0}}},
+{{{0, 1}, {0, 3}, {1, 0}, {4, 3}}},
+
+{{{0, 0}, {1, 2}, {3, 4}, {4, 4}}},
+{{{0, 0}, {1, 2}, {3, 4}, {4, 4}}},
+{{{4, 4}, {3, 4}, {1, 2}, {0, 0}}},
+
+{{{0, 1}, {2, 3}, {1, 0}, {1, 0}}},
+{{{0, 1}, {0, 1}, {1, 0}, {3, 2}}},
+
+{{{0, 2}, {0, 1}, {1, 0}, {1, 0}}},
+{{{0, 1}, {0, 1}, {2, 0}, {1, 0}}},
+
+{{{0, 1}, {0, 2}, {1, 0}, {1, 0}}},
+{{{0, 1}, {0, 1}, {1, 0}, {2, 0}}},
+
+{{{0, 1}, {1, 6}, {1, 0}, {2, 0}}},
+{{{0, 1}, {0, 2}, {1, 0}, {6, 1}}},
+
+{{{0, 1}, {5, 6}, {1, 0}, {1, 0}}},
+{{{0, 1}, {0, 1}, {1, 0}, {6, 5}}},
+
+{{{95.837747722788592, 45.025976907939643}, {16.564570095652982, 0.72959763963222402},
+        {63.209855865319199, 68.047528419665767}, {57.640240647662544, 59.524565264361243}}},
+{{{51.593891741518817, 38.53849970667553}, {62.34752929878772, 74.924924725166022},
+        {74.810149322641152, 34.17966562983564}, {29.368398119401373, 94.66719277886078}}},
+
+{{{39.765160968417838, 33.060396198677083}, {5.1922921581157908, 66.854301452103215},
+        {31.619281802149157, 25.269248720849514}, {81.541621071073038, 70.025341524754353}}},
+{{{46.078911165743556, 48.259962651999651}, {20.24450549867214, 49.403916182650214},
+        {0.26325131778756683, 24.46489805563581}, {15.915006546264051, 83.515023059917155}}},
+
+{{{65.454505973241524, 93.881892270353575}, {45.867360264932437, 92.723972719499827},
+        {2.1464054482739447, 74.636369140183717}, {33.774068594804994, 40.770872887582925}}},
+{{{72.963387832494163, 95.659300729473728}, {11.809496633619768, 82.209921247423594},
+        {13.456139067865974, 57.329313623406605}, {36.060621606214262, 70.867335643091849}}},
+
+{{{32.484981432782945, 75.082940782924624}, {42.467313093350882, 48.131159948246157},
+        {3.5963115764764657, 43.208665839959245}, {79.442476890721579, 89.709102357602262}}},
+{{{18.98573861410177, 93.308887208490106}, {40.405250173250792, 91.039661826118675},
+        {8.0467721950480584, 42.100282172719147}, {40.883324221187891, 26.030185504830527}}},
+
+{{{7.5374809128872498, 82.441702896003477}, {22.444346930107265, 22.138854312775123},
+        {66.76091829629658, 50.753805856571446}, {78.193478508942519, 97.7932997968948}}},
+{{{97.700573130371311, 53.53260215070685}, {87.72443481149358, 84.575876772671876},
+        {19.215031396232092, 47.032676472809484}, {11.989686410869325, 10.659507480757082}}},
+
+{{{26.192053931854691, 9.8504326817814416}, {10.174241480498686, 98.476562741434464},
+        {21.177712558385782, 33.814968789841501}, {75.329030899018534, 55.02231980442177}}},
+{{{56.222082700683771, 24.54395039218662}, {95.589995289030483, 81.050822735322086},
+        {28.180450866082897, 28.837706255185282}, {60.128952916771617, 87.311672180570511}}},
+
+{{{42.449716172390481, 52.379709366885805}, {27.896043159019225, 48.797373636065686},
+        {92.770268299044233, 89.899302036454571}, {12.102066544863426, 99.43241951960718}}},
+{{{45.77532924980639, 45.958701495993274}, {37.458701356062065, 68.393691335056758},
+        {37.569326692060258, 27.673713456687381}, {60.674866037757539, 62.47349659096146}}},
+
+{{{67.426548091427676, 37.993772624988935}, {23.483695892376684, 90.476863174921306},
+        {35.597065061143162, 79.872482633158796}, {75.38634169631932, 18.244890038969412}}},
+{{{61.336508189019057, 82.693132843213675}, {44.639380902349664, 54.074825790745592},
+        {16.815615499771951, 20.049704667203923}, {41.866884958868326, 56.735503699973002}}},
+
+{{{18.1312339, 31.6473732}, {95.5711034, 63.5350219}, {92.3283165, 62.0158945},
+        {18.5656052, 32.1268808}}},
+{{{97.402018, 35.7169972}, {33.1127443, 25.8935163}, {1.13970027, 54.9424981},
+        {56.4860195, 60.529264}}},
+};
+
+const size_t testSetCount = sizeof(testSet) / sizeof(testSet[0]);
+
+static const SkDCubic newTestSet[] = {
+{{{0, 1}, {1, 5}, {1, 0}, {1, 0}}},
+{{{0, 1}, {0, 1}, {1, 0}, {5, 1}}},
+
+{{{1, 3}, {5, 6}, {5, 3}, {5, 4}}},
+{{{3, 5}, {4, 5}, {3, 1}, {6, 5}}},
+
+{{{0, 5}, {0, 5}, {5, 4}, {6, 4}}},
+{{{4, 5}, {4, 6}, {5, 0}, {5, 0}}},
+
+{{{0, 4}, {1, 3}, {5, 4}, {4, 2}}},
+{{{4, 5}, {2, 4}, {4, 0}, {3, 1}}},
+
+{{{0, 2}, {1, 5}, {3, 2}, {4, 1}}},
+{{{2, 3}, {1, 4}, {2, 0}, {5, 1}}},
+
+{{{0, 2}, {2, 3}, {5, 1}, {3, 2}}},
+{{{1, 5}, {2, 3}, {2, 0}, {3, 2}}},
+
+{{{2, 6}, {4, 5}, {1, 0}, {6, 1}}},
+{{{0, 1}, {1, 6}, {6, 2}, {5, 4}}},
+
+{{{0, 1}, {1, 2}, {6, 5}, {5, 4}}},
+{{{5, 6}, {4, 5}, {1, 0}, {2, 1}}},
+
+{{{2.5119999999999996, 1.5710000000000002}, {2.6399999999999983, 1.6599999999999997},
+        {2.8000000000000007, 1.8000000000000003}, {3, 2}}},
+{{{2.4181876227114887, 1.9849772580462195}, {2.8269904869227211, 2.009330650246834},
+        {3.2004679292461624, 1.9942047174679169}, {3.4986199496818058, 2.0035994597094731}}},
+
+{{{2, 3}, {1, 4}, {1, 0}, {6, 0}}},
+{{{0, 1}, {0, 6}, {3, 2}, {4, 1}}},
+
+{{{0, 2}, {1, 5}, {1, 0}, {6, 1}}},
+{{{0, 1}, {1, 6}, {2, 0}, {5, 1}}},
+
+{{{0, 1}, {1, 5}, {2, 1}, {4, 0}}},
+{{{1, 2}, {0, 4}, {1, 0}, {5, 1}}},
+
+{{{0, 1}, {3, 5}, {2, 1}, {3, 1}}},
+{{{1, 2}, {1, 3}, {1, 0}, {5, 3}}},
+
+{{{0, 1}, {2, 5}, {6, 0}, {5, 3}}},
+{{{0, 6}, {3, 5}, {1, 0}, {5, 2}}},
+
+{{{0, 1}, {3, 6}, {1, 0}, {5, 2}}},
+{{{0, 1}, {2, 5}, {1, 0}, {6, 3}}},
+
+{{{1, 2}, {5, 6}, {1, 0}, {1, 0}}},
+{{{0, 1}, {0, 1}, {2, 1}, {6, 5}}},
+
+{{{0, 6}, {1, 2}, {1, 0}, {1, 0}}},
+{{{0, 1}, {0, 1}, {6, 0}, {2, 1}}},
+
+{{{0, 2}, {0, 1}, {3, 0}, {1, 0}}},
+{{{0, 3}, {0, 1}, {2, 0}, {1, 0}}},
+};
+
+const size_t newTestSetCount = sizeof(newTestSet) / sizeof(newTestSet[0]);
+
+static void oneOff(skiatest::Reporter* reporter, const SkDCubic& cubic1, const SkDCubic& cubic2) {
+#if ONE_OFF_DEBUG
+    SkDebugf("computed quadratics given\n");
+    SkDebugf("  {{{%1.9g,%1.9g}, {%1.9g,%1.9g}, {%1.9g,%1.9g}, {%1.9g,%1.9g}}},\n",
+        cubic1[0].fX, cubic1[0].fY, cubic1[1].fX, cubic1[1].fY,
+        cubic1[2].fX, cubic1[2].fY, cubic1[3].fX, cubic1[3].fY);
+    SkDebugf("  {{{%1.9g,%1.9g}, {%1.9g,%1.9g}, {%1.9g,%1.9g}, {%1.9g,%1.9g}}},\n",
+        cubic2[0].fX, cubic2[0].fY, cubic2[1].fX, cubic2[1].fY,
+        cubic2[2].fX, cubic2[2].fY, cubic2[3].fX, cubic2[3].fY);
+#endif
+    SkTDArray<SkDQuad> quads1;
+    CubicToQuads(cubic1, cubic1.calcPrecision(), quads1);
+#if ONE_OFF_DEBUG
+    SkDebugf("computed quadratics set 1\n");
+    for (int index = 0; index < quads1.count(); ++index) {
+        const SkDQuad& q = quads1[index];
+        SkDebugf("  {{{%1.9g,%1.9g}, {%1.9g,%1.9g}, {%1.9g,%1.9g}}},\n", q[0].fX, q[0].fY,
+                 q[1].fX, q[1].fY,  q[2].fX, q[2].fY);
+    }
+#endif
+    SkTDArray<SkDQuad> quads2;
+    CubicToQuads(cubic2, cubic2.calcPrecision(), quads2);
+#if ONE_OFF_DEBUG
+    SkDebugf("computed quadratics set 2\n");
+    for (int index = 0; index < quads2.count(); ++index) {
+        const SkDQuad& q = quads2[index];
+        SkDebugf("  {{{%1.9g,%1.9g}, {%1.9g,%1.9g}, {%1.9g,%1.9g}}},\n", q[0].fX, q[0].fY,
+                 q[1].fX, q[1].fY,  q[2].fX, q[2].fY);
+    }
+#endif
+    SkIntersections intersections;
+    intersections.intersect(cubic1, cubic2);
+    double tt1, tt2;
+    SkDPoint xy1, xy2;
+    for (int pt3 = 0; pt3 < intersections.used(); ++pt3) {
+        tt1 = intersections[0][pt3];
+        xy1 = cubic1.xyAtT(tt1);
+        tt2 = intersections[1][pt3];
+        xy2 = cubic2.xyAtT(tt2);
+#if ONE_OFF_DEBUG
+        SkDebugf("%s t1=%1.9g (%1.9g, %1.9g) (%1.9g, %1.9g) (%1.9g, %1.9g) t2=%1.9g\n",
+                __FUNCTION__, tt1, xy1.fX, xy1.fY, intersections.pt(pt3).fX,
+                intersections.pt(pt3).fY, xy2.fX, xy2.fY, tt2);
+#endif
+        REPORTER_ASSERT(reporter, xy1.approximatelyEqual(xy2));
+    }
+}
+
+static void oneOff(skiatest::Reporter* reporter, int outer, int inner) {
+    const SkDCubic& cubic1 = testSet[outer];
+    const SkDCubic& cubic2 = testSet[inner];
+    oneOff(reporter, cubic1, cubic2);
+}
+
+static void newOneOff(skiatest::Reporter* reporter, int outer, int inner) {
+    const SkDCubic& cubic1 = newTestSet[outer];
+    const SkDCubic& cubic2 = newTestSet[inner];
+    oneOff(reporter, cubic1, cubic2);
+}
+
+static void oneOffTest(skiatest::Reporter* reporter) {
+    newOneOff(reporter, 0, 1);
+    oneOff(reporter, 0, 1);
+}
+
+static void oneOffTests(skiatest::Reporter* reporter) {
+    for (size_t outer = 0; outer < testSetCount - 1; ++outer) {
+        for (size_t inner = outer + 1; inner < testSetCount; ++inner) {
+            oneOff(reporter, outer, inner);
+        }
+    }
+    for (size_t outer = 0; outer < newTestSetCount - 1; ++outer) {
+        for (size_t inner = outer + 1; inner < newTestSetCount; ++inner) {
+            oneOff(reporter, outer, inner);
+        }
+    }
+}
+
+#define DEBUG_CRASH 0
+
+static void CubicIntersection_RandTest(skiatest::Reporter* reporter) {
+    srand(0);
+    const int tests = 10000000;
+#ifndef SK_BUILD_FOR_WIN
+    unsigned seed = 0;
+#endif
+    for (int test = 0; test < tests; ++test) {
+        SkDCubic cubic1, cubic2;
+        for (int i = 0; i < 4; ++i) {
+            cubic1[i].fX = static_cast<double>(SK_RAND(seed)) / RAND_MAX * 100;
+            cubic1[i].fY = static_cast<double>(SK_RAND(seed)) / RAND_MAX * 100;
+            cubic2[i].fX = static_cast<double>(SK_RAND(seed)) / RAND_MAX * 100;
+            cubic2[i].fY = static_cast<double>(SK_RAND(seed)) / RAND_MAX * 100;
+        }
+    #if DEBUG_CRASH
+        char str[1024];
+        snprintf(str, sizeof(str),
+            "{{{%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}, {%1.9g, %1.9g}}},\n",
+                cubic1[0].fX, cubic1[0].fY,  cubic1[1].fX, cubic1[1].fY, cubic1[2].fX, cubic1[2].fY,
+                cubic1[3].fX, cubic1[3].fY,
+                cubic2[0].fX, cubic2[0].fY,  cubic2[1].fX, cubic2[1].fY, cubic2[2].fX, cubic2[2].fY,
+                cubic2[3].fX, cubic2[3].fY);
+    #endif
+        SkDRect rect1, rect2;
+        rect1.setBounds(cubic1);
+        rect2.setBounds(cubic2);
+        bool boundsIntersect = rect1.fLeft <= rect2.fRight && rect2.fLeft <= rect2.fRight
+                && rect1.fTop <= rect2.fBottom && rect2.fTop <= rect1.fBottom;
+        if (test == -1) {
+            SkDebugf("ready...\n");
+        }
+        SkIntersections intersections2;
+        int newIntersects = intersections2.intersect(cubic1, cubic2);
+        if (!boundsIntersect && newIntersects) {
+    #if DEBUG_CRASH
+            SkDebugf("%s %d unexpected intersection boundsIntersect=%d "
+                    " newIntersects=%d\n%s %s\n", __FUNCTION__, test, boundsIntersect,
+                    newIntersects, __FUNCTION__, str);
+    #endif
+            REPORTER_ASSERT(reporter, 0);
+        }
+        for (int pt = 0; pt < intersections2.used(); ++pt) {
+            double tt1 = intersections2[0][pt];
+            SkDPoint xy1 = cubic1.xyAtT(tt1);
+            double tt2 = intersections2[1][pt];
+            SkDPoint xy2 = cubic2.xyAtT(tt2);
+            REPORTER_ASSERT(reporter, xy1.approximatelyEqual(xy2));
+        }
+    }
+}
+
+static void intersectionFinder(int index0, int index1, double t1Seed, double t2Seed,
+        double t1Step, double t2Step) {
+    const SkDCubic& cubic1 = newTestSet[index0];
+    const SkDCubic& cubic2 = newTestSet[index1];
+    SkDPoint t1[3], t2[3];
+    bool toggle = true;
+    do {
+        t1[0] = cubic1.xyAtT(t1Seed - t1Step);
+        t1[1] = cubic1.xyAtT(t1Seed);
+        t1[2] = cubic1.xyAtT(t1Seed + t1Step);
+        t2[0] = cubic2.xyAtT(t2Seed - t2Step);
+        t2[1] = cubic2.xyAtT(t2Seed);
+        t2[2] = cubic2.xyAtT(t2Seed + t2Step);
+        double dist[3][3];
+        dist[1][1] = t1[1].distance(t2[1]);
+        int best_i = 1, best_j = 1;
+        for (int i = 0; i < 3; ++i) {
+            for (int j = 0; j < 3; ++j) {
+                if (i == 1 && j == 1) {
+                    continue;
+                }
+                dist[i][j] = t1[i].distance(t2[j]);
+                if (dist[best_i][best_j] > dist[i][j]) {
+                    best_i = i;
+                    best_j = j;
+                }
+            }
+        }
+        if (best_i == 0) {
+            t1Seed -= t1Step;
+        } else if (best_i == 2) {
+            t1Seed += t1Step;
+        }
+        if (best_j == 0) {
+            t2Seed -= t2Step;
+        } else if (best_j == 2) {
+            t2Seed += t2Step;
+        }
+        if (best_i == 1 && best_j == 1) {
+            if ((toggle ^= true)) {
+                t1Step /= 2;
+            } else {
+                t2Step /= 2;
+            }
+        }
+    } while (!t1[1].approximatelyEqual(t2[1]));
+    t1Step = t2Step = 0.1;
+    double t10 = t1Seed - t1Step * 2;
+    double t12 = t1Seed + t1Step * 2;
+    double t20 = t2Seed - t2Step * 2;
+    double t22 = t2Seed + t2Step * 2;
+    SkDPoint test;
+    while (!approximately_zero(t1Step)) {
+        test = cubic1.xyAtT(t10);
+        t10 += t1[1].approximatelyEqual(test) ? -t1Step : t1Step;
+        t1Step /= 2;
+    }
+    t1Step = 0.1;
+    while (!approximately_zero(t1Step)) {
+        test = cubic1.xyAtT(t12);
+        t12 -= t1[1].approximatelyEqual(test) ? -t1Step : t1Step;
+        t1Step /= 2;
+    }
+    while (!approximately_zero(t2Step)) {
+        test = cubic2.xyAtT(t20);
+        t20 += t2[1].approximatelyEqual(test) ? -t2Step : t2Step;
+        t2Step /= 2;
+    }
+    t2Step = 0.1;
+    while (!approximately_zero(t2Step)) {
+        test = cubic2.xyAtT(t22);
+        t22 -= t2[1].approximatelyEqual(test) ? -t2Step : t2Step;
+        t2Step /= 2;
+    }
+#if ONE_OFF_DEBUG
+    SkDebugf("%s t1=(%1.9g<%1.9g<%1.9g) t2=(%1.9g<%1.9g<%1.9g)\n", __FUNCTION__,
+        t10, t1Seed, t12, t20, t2Seed, t22);
+    SkDPoint p10 = cubic1.xyAtT(t10);
+    SkDPoint p1Seed = cubic1.xyAtT(t1Seed);
+    SkDPoint p12 = cubic1.xyAtT(t12);
+    SkDebugf("%s p1=(%1.9g,%1.9g)<(%1.9g,%1.9g)<(%1.9g,%1.9g)\n", __FUNCTION__,
+        p10.fX, p10.fY, p1Seed.fX, p1Seed.fY, p12.fX, p12.fY);
+    SkDPoint p20 = cubic2.xyAtT(t20);
+    SkDPoint p2Seed = cubic2.xyAtT(t2Seed);
+    SkDPoint p22 = cubic2.xyAtT(t22);
+    SkDebugf("%s p2=(%1.9g,%1.9g)<(%1.9g,%1.9g)<(%1.9g,%1.9g)\n", __FUNCTION__,
+        p20.fX, p20.fY, p2Seed.fX, p2Seed.fY, p22.fX, p22.fY);
+#endif
+}
+
+static void CubicIntersection_IntersectionFinder() {
+//   double t1Seed = 0.87;
+//   double t2Seed = 0.87;
+    double t1Step = 0.000001;
+    double t2Step = 0.000001;
+    intersectionFinder(0, 1, 0.855895664, 0.864850875, t1Step, t2Step);
+    intersectionFinder(0, 1, 0.865207906, 0.865207887, t1Step, t2Step);
+    intersectionFinder(0, 1, 0.865213351, 0.865208087, t1Step, t2Step);
+}
+
+static void cubicIntersectionSelfTest(skiatest::Reporter* reporter) {
+    const SkDCubic selfSet[] = {
+        {{{0, 2}, {2, 3}, {5, 1}, {3, 2}}},
+        {{{0, 2}, {3, 5}, {5, 0}, {4, 2}}},
+        {{{3.34, 8.98}, {1.95, 10.27}, {3.76, 7.65}, {4.96, 10.64}}},
+        {{{3.13, 2.74}, {1.08, 4.62}, {3.71, 0.94}, {2.01, 3.81}}},
+        {{{6.71, 3.14}, {7.99, 2.75}, {8.27, 1.96}, {6.35, 3.57}}},
+        {{{12.81, 7.27}, {7.22, 6.98}, {12.49, 8.97}, {11.42, 6.18}}},
+    };
+    size_t selfSetCount = sizeof(selfSet) / sizeof(selfSet[0]);
+    size_t firstFail = 1;
+    for (size_t index = firstFail; index < selfSetCount; ++index) {
+        const SkDCubic& cubic = selfSet[index];
+    #if ONE_OFF_DEBUG
+        int idx2;
+        double max[3];
+        int ts = cubic.findMaxCurvature(max);
+        for (idx2 = 0; idx2 < ts; ++idx2) {
+            SkDebugf("%s max[%d]=%1.9g (%1.9g, %1.9g)\n", __FUNCTION__, idx2,
+                    max[idx2], cubic.xyAtT(max[idx2]).fX, cubic.xyAtT(max[idx2]).fY);
+        }
+        SkTDArray<double> ts1;
+        SkTDArray<SkDQuad> quads1;
+        cubic.toQuadraticTs(cubic.calcPrecision(), &ts1);
+        for (idx2 = 0; idx2 < ts1.count(); ++idx2) {
+            SkDebugf("%s t[%d]=%1.9g\n", __FUNCTION__, idx2, ts1[idx2]);
+        }
+        CubicToQuads(cubic, cubic.calcPrecision(), quads1);
+        for (idx2 = 0; idx2 < quads1.count(); ++idx2) {
+            const SkDQuad& q = quads1[idx2];
+            SkDebugf("  {{{%1.9g,%1.9g}, {%1.9g,%1.9g}, {%1.9g,%1.9g}}},\n",
+                    q[0].fX, q[0].fY,  q[1].fX, q[1].fY,  q[2].fX, q[2].fY);
+        }
+        SkDebugf("\n");
+    #endif
+        SkIntersections i;
+        int result = i.intersect(cubic);
+        REPORTER_ASSERT(reporter, result == 1);
+        REPORTER_ASSERT(reporter, i.used() == 1);
+        REPORTER_ASSERT(reporter, !approximately_equal(i[0][0], i[1][0]));
+        SkDPoint pt1 = cubic.xyAtT(i[0][0]);
+        SkDPoint pt2 = cubic.xyAtT(i[1][0]);
+        REPORTER_ASSERT(reporter, pt1.approximatelyEqual(pt2));
+    }
+}
+
+static void TestCubicIntersection(skiatest::Reporter* reporter) {
+    oneOffTest(reporter);
+    oneOffTests(reporter);
+    cubicIntersectionSelfTest(reporter);
+    standardTestCases(reporter);
+    if (false) CubicIntersection_IntersectionFinder();
+    if (false) CubicIntersection_RandTest(reporter);
+}
+
+#include "TestClassDef.h"
+DEFINE_TESTCLASS("PathOpsCubicIntersectionTest", CubicIntersectionTestClass, TestCubicIntersection)