experimental/Intersection/CubicIntersection_Test.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 "CurveIntersection.h"
#include "CurveUtilities.h"
#include "CubicIntersection_TestData.h"
#include "Intersection_Tests.h"
#include "Intersections.h"
#include "TestUtilities.h"

const int firstCubicIntersectionTest = 9;

static void standardTestCases() {
    for (size_t index = firstCubicIntersectionTest; index < tests_count; ++index) {
        const Cubic& cubic1 = tests[index][0];
        const Cubic& cubic2 = tests[index][1];
        Cubic reduce1, reduce2;
        int order1 = reduceOrder(cubic1, reduce1, kReduceOrder_NoQuadraticsAllowed);
        int order2 = reduceOrder(cubic2, reduce2, kReduceOrder_NoQuadraticsAllowed);
        if (order1 < 4) {
            printf("%s [%d] cubic1 order=%d\n", __FUNCTION__, (int) index, order1);
            continue;
        }
        if (order2 < 4) {
            printf("%s [%d] cubic2 order=%d\n", __FUNCTION__, (int) index, order2);
            continue;
        }
        if (implicit_matches(reduce1, reduce2)) {
            printf("%s [%d] coincident\n", __FUNCTION__, (int) index);
            continue;
        }
        Intersections tIntersections;
        intersect(reduce1, reduce2, tIntersections);
        if (!tIntersections.intersected()) {
            printf("%s [%d] no intersection\n", __FUNCTION__, (int) index);
            continue;
        }
        for (int pt = 0; pt < tIntersections.used(); ++pt) {
            double tt1 = tIntersections.fT[0][pt];
            double tx1, ty1;
            xy_at_t(cubic1, tt1, tx1, ty1);
            double tt2 = tIntersections.fT[1][pt];
            double tx2, ty2;
            xy_at_t(cubic2, tt2, tx2, ty2);
            if (!AlmostEqualUlps(tx1, tx2)) {
                printf("%s [%d,%d] x!= t1=%g (%g,%g) t2=%g (%g,%g)\n",
                    __FUNCTION__, (int)index, pt, tt1, tx1, ty1, tt2, tx2, ty2);
            }
            if (!AlmostEqualUlps(ty1, ty2)) {
                printf("%s [%d,%d] y!= t1=%g (%g,%g) t2=%g (%g,%g)\n",
                    __FUNCTION__, (int)index, pt, tt1, tx1, ty1, tt2, tx2, ty2);
            }
        }
    }
}

static const Cubic testSet[] = {
{{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, 0}, {0, 1}, {1, 1}, {1, 0}},
{{1, 0}, {0, 0}, {0, 1}, {1, 1}},

{{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}},

{{67.4265481, 37.9937726}, {23.4836959, 90.4768632}, {35.5970651, 79.8724826}, {75.3863417, 18.24489}},
{{61.3365082, 82.6931328}, {44.6393809, 54.0748258}, {16.8156155, 20.0497047}, {41.866885, 56.7355037}},

{{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 void oneOff(const Cubic& cubic1, const Cubic& cubic2) {
    SkTDArray<Quadratic> quads1;
    cubic_to_quadratics(cubic1, calcPrecision(cubic1), quads1);
#if ONE_OFF_DEBUG
    for (int index = 0; index < quads1.count(); ++index) {
        const Quadratic& q = quads1[index];
        SkDebugf("  {{%1.9g,%1.9g}, {%1.9g,%1.9g}, {%1.9g,%1.9g}},\n", q[0].x, q[0].y,
                 q[1].x, q[1].y,  q[2].x, q[2].y);
    }
    SkDebugf("\n");
#endif
    SkTDArray<Quadratic> quads2;
    cubic_to_quadratics(cubic2, calcPrecision(cubic2), quads2);
#if ONE_OFF_DEBUG
    for (int index = 0; index < quads2.count(); ++index) {
        const Quadratic& q = quads2[index];
        SkDebugf("  {{%1.9g,%1.9g}, {%1.9g,%1.9g}, {%1.9g,%1.9g}},\n", q[0].x, q[0].y,
                 q[1].x, q[1].y,  q[2].x, q[2].y);
    }
    SkDebugf("\n");
#endif
    Intersections intersections2, intersections3;
    intersect2(cubic1, cubic2, intersections2);
    intersect3(cubic1, cubic2, intersections3);
    int pt1, pt2, pt3;
    bool found;
    double tt1, tt2, last = -1;
    _Point xy1, xy2;
    for (pt1 = 0; pt1 < intersections2.used(); ++pt1) {
        tt1 = intersections2.fT[0][pt1];
        SkASSERT(!approximately_equal(last, tt1));
        last = tt1;
        xy_at_t(cubic1, tt1, xy1.x, xy1.y);
        pt2 = intersections2.fFlip ? intersections2.used() - pt1 - 1 : pt1;
        tt2 = intersections2.fT[1][pt2];
        xy_at_t(cubic2, tt2, xy2.x, xy2.y);
#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.x, xy1.y, intersections2.fPt[pt1].x,
                intersections2.fPt[pt1].y, xy2.x, xy2.y, tt2);
#endif
        SkASSERT(xy1.approximatelyEqual(xy2));
#if SK_DEBUG
        found = false;
        for (pt3 = 0; pt3 < intersections3.used(); ++pt3) {
            if (roughly_equal(tt1, intersections3.fT[0][pt3])) {
                found = true;
                break;
            }
        }
        SkASSERT(found);
#endif
    }
    last = -1;
    for (pt3 = 0; pt3 < intersections3.used(); ++pt3) {
        found = false;
        double tt3 = intersections3.fT[0][pt3];
        SkASSERT(!approximately_equal(last, tt3));
        last = tt3;
        for (pt1 = 0; pt1 < intersections2.used(); ++pt1) {
            if (approximately_equal(tt3, intersections2.fT[0][pt1])) {
                found = true;
                break;
            }
        }
        if (!found) {
            tt1 = intersections3.fT[0][pt3];
            xy_at_t(cubic1, tt1, xy1.x, xy1.y);
            pt2 = intersections3.fFlip ? intersections3.used() - pt3 - 1 : pt3;
            tt2 = intersections3.fT[1][pt2];
            xy_at_t(cubic2, tt2, xy2.x, xy2.y);
    #if ONE_OFF_DEBUG
            SkDebugf("%s t3=%1.9g (%1.9g, %1.9g) (%1.9g, %1.9g) (%1.9g, %1.9g) t2=%1.9g\n",
                    __FUNCTION__, tt1, xy1.x, xy1.y, intersections3.fPt[pt1].x,
                    intersections3.fPt[pt1].y, xy2.x, xy2.y, tt2);
    #endif
            SkASSERT(xy1.approximatelyEqual(xy2));
            SkDebugf("%s missing in intersect2\n", __FUNCTION__);
        }
    }
}

static void oneOff3(const Cubic& cubic1, const Cubic& cubic2) {
    SkTDArray<Quadratic> quads1;
    cubic_to_quadratics(cubic1, calcPrecision(cubic1), quads1);
#if ONE_OFF_DEBUG
    for (int index = 0; index < quads1.count(); ++index) {
        const Quadratic& q = quads1[index];
        SkDebugf("  {{%1.9g,%1.9g}, {%1.9g,%1.9g}, {%1.9g,%1.9g}},\n", q[0].x, q[0].y,
                 q[1].x, q[1].y,  q[2].x, q[2].y);
    }
    SkDebugf("\n");
#endif
    SkTDArray<Quadratic> quads2;
    cubic_to_quadratics(cubic2, calcPrecision(cubic2), quads2);
#if ONE_OFF_DEBUG
    for (int index = 0; index < quads2.count(); ++index) {
        const Quadratic& q = quads2[index];
        SkDebugf("  {{%1.9g,%1.9g}, {%1.9g,%1.9g}, {%1.9g,%1.9g}},\n", q[0].x, q[0].y,
                 q[1].x, q[1].y,  q[2].x, q[2].y);
    }
    SkDebugf("\n");
#endif
    Intersections intersections3;
    intersect3(cubic1, cubic2, intersections3);
    int pt2, pt3;
    double tt1, tt2, last = -1;
    _Point xy1, xy2;
    for (pt3 = 0; pt3 < intersections3.used(); ++pt3) {
        double tt3 = intersections3.fT[0][pt3];
        SkASSERT(!approximately_equal(last, tt3));
        last = tt3;
        tt1 = intersections3.fT[0][pt3];
        xy_at_t(cubic1, tt1, xy1.x, xy1.y);
        pt2 = intersections3.fFlip ? intersections3.used() - pt3 - 1 : pt3;
        tt2 = intersections3.fT[1][pt2];
        xy_at_t(cubic2, tt2, xy2.x, xy2.y);
#if ONE_OFF_DEBUG
        SkDebugf("%s t3=%1.9g (%1.9g, %1.9g) (%1.9g, %1.9g) (%1.9g, %1.9g) t2=%1.9g\n",
                __FUNCTION__, tt1, xy1.x, xy1.y, intersections3.fPt[pt3].x,
                intersections3.fPt[pt3].y, xy2.x, xy2.y, tt2);
#endif
        SkASSERT(xy1.approximatelyEqual(xy2));
    }
}

static int fails[][2] = {   {0, 23}, // fails in intersect2 recursing
                            {2, 7},  // answers differ, but neither is correct ('3' is closer)
                            {3, 26}, // fails in intersect2 recursing
                            {4, 9},  // fails in intersect2 recursing
                            {4, 10}, // fails in intersect2 recursing
                            {10, 17}, // fails in intersect2 recursing
                            {12, 14}, // loops indefinitely
                            {12, 21}, // fails in intersect2 recursing
                            {13, 21}, // fails in intersect2 recursing
                            {14, 21}, // fails in intersect2 recursing
                            {17, 25}, // fails in intersect2 recursing
                            {23, 25}, // fails in intersect2 recursing
};

static int failCount = sizeof(fails) / sizeof(fails[0]);

static void oneOff(int outer, int inner) {
    const Cubic& cubic1 = testSet[outer];
    const Cubic& cubic2 = testSet[inner];
    bool failing = false;
    for (int i = 0; i < failCount; ++i) {
        if ((fails[i][0] == outer && fails[i][1] == inner)
                || (fails[i][1] == outer && fails[i][0] == inner)) {
            failing = true;
            break;
        }
    }
    if (!failing) {
        oneOff(cubic1, cubic2);
    } else {
        oneOff3(cubic1, cubic2);
    }
}

void CubicIntersection_OneOffTest() {
    oneOff(12, 14);
}

static void oneOffTests() {
    for (size_t outer = 0; outer < testSetCount - 1; ++outer) {
        for (size_t inner = outer + 1; inner < testSetCount; ++inner) {
            oneOff(outer, inner);
        }
    }
}

void CubicIntersection_OneOffTests() {
    oneOffTests();
}

#define DEBUG_CRASH 0

class CubicChopper {
public:

// only finds one intersection
CubicChopper(const Cubic& c1, const Cubic& c2)
    : cubic1(c1)
    , cubic2(c2)
    , depth(0) {
}

bool intersect(double minT1, double maxT1, double minT2, double maxT2) {
    Cubic sub1, sub2;
    // FIXME: carry last subdivide and reduceOrder result with cubic
    sub_divide(cubic1, minT1, maxT1, sub1);
    sub_divide(cubic2, minT2, maxT2, sub2);
    Intersections i;
    intersect2(sub1, sub2, i);
    if (i.used() == 0) {
        return false;
    }
    double x1, y1, x2, y2;
    t1 = minT1 + i.fT[0][0] * (maxT1 - minT1);
    t2 = minT2 + i.fT[1][0] * (maxT2 - minT2);
    xy_at_t(cubic1, t1, x1, y1);
    xy_at_t(cubic2, t2, x2, y2);
    if (AlmostEqualUlps(x1, x2) && AlmostEqualUlps(y1, y2)) {
        return true;
    }
    double half1 = (minT1 + maxT1) / 2;
    double half2 = (minT2 + maxT2) / 2;
    ++depth;
    bool result;
    if (depth & 1) {
        result = intersect(minT1, half1, minT2, maxT2) || intersect(half1, maxT1, minT2, maxT2)
            || intersect(minT1, maxT1, minT2, half2) || intersect(minT1, maxT1, half2, maxT2);
    } else {
        result = intersect(minT1, maxT1, minT2, half2) || intersect(minT1, maxT1, half2, maxT2)
            || intersect(minT1, half1, minT2, maxT2) || intersect(half1, maxT1, minT2, maxT2);
    }
    --depth;
    return result;
}

const Cubic& cubic1;
const Cubic& cubic2;
double t1;
double t2;
int depth;
};

#define TRY_OLD 0 // old way fails on test == 1

void CubicIntersection_RandTestOld() {
    srand(0);
    const int tests = 1000000; // 10000000;
    double largestFactor = DBL_MAX;
    for (int test = 0; test < tests; ++test) {
        Cubic cubic1, cubic2;
        for (int i = 0; i < 4; ++i) {
            cubic1[i].x = (double) rand() / RAND_MAX * 100;
            cubic1[i].y = (double) rand() / RAND_MAX * 100;
            cubic2[i].x = (double) rand() / RAND_MAX * 100;
            cubic2[i].y = (double) rand() / RAND_MAX * 100;
        }
        if (test == 2513) { // the pair crosses three times, but the quadratic approximation
            continue; // only sees one -- should be OK to ignore the other two?
        }
        if (test == 12932) { // this exposes a weakness when one cubic touches the other but
            continue; // does not touch the quad approximation. Captured in qc.htm as cubic15
        }
    #if DEBUG_CRASH
        char str[1024];
        sprintf(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].x, cubic1[0].y,  cubic1[1].x, cubic1[1].y, cubic1[2].x, cubic1[2].y,
                cubic1[3].x, cubic1[3].y,
                cubic2[0].x, cubic2[0].y,  cubic2[1].x, cubic2[1].y, cubic2[2].x, cubic2[2].y,
                cubic2[3].x, cubic2[3].y);
    #endif
        _Rect rect1, rect2;
        rect1.setBounds(cubic1);
        rect2.setBounds(cubic2);
        bool boundsIntersect = rect1.left <= rect2.right && rect2.left <= rect2.right
                && rect1.top <= rect2.bottom && rect2.top <= rect1.bottom;
        Intersections i1, i2;
    #if TRY_OLD
        bool oldIntersects = intersect(cubic1, cubic2, i1);
    #else
        bool oldIntersects = false;
    #endif
        if (test == -1) {
            SkDebugf("ready...\n");
        }
        bool newIntersects = intersect2(cubic1, cubic2, i2);
        if (!boundsIntersect && (oldIntersects || newIntersects)) {
    #if DEBUG_CRASH
            SkDebugf("%s %d unexpected intersection boundsIntersect=%d oldIntersects=%d"
                    " newIntersects=%d\n%s %s\n", __FUNCTION__, test, boundsIntersect,
                    oldIntersects, newIntersects, __FUNCTION__, str);
    #endif
            SkASSERT(0);
        }
        if (oldIntersects && !newIntersects) {
    #if DEBUG_CRASH
            SkDebugf("%s %d missing intersection oldIntersects=%d newIntersects=%d\n%s %s\n",
                    __FUNCTION__, test, oldIntersects, newIntersects, __FUNCTION__, str);
    #endif
            SkASSERT(0);
        }
        if (!oldIntersects && !newIntersects) {
            continue;
        }
        if (i2.used() > 1) {
            continue;
            // just look at single intercepts for simplicity
        }
        Intersections self1, self2; // self-intersect checks
        if (intersect(cubic1, self1)) {
            continue;
        }
        if (intersect(cubic2, self2)) {
            continue;
        }
        // binary search for range necessary to enclose real intersection
        CubicChopper c(cubic1, cubic2);
        bool result = c.intersect(0, 1, 0, 1);
        if (!result) {
            // FIXME: a failure here probably means that a core routine used by CubicChopper is failing
            continue;
        }
        double delta1 = fabs(c.t1 - i2.fT[0][0]);
        double delta2 = fabs(c.t2 - i2.fT[1][0]);
        double calc1 = calcPrecision(cubic1);
        double calc2 = calcPrecision(cubic2);
        double factor1 = calc1 / delta1;
        double factor2 = calc2 / delta2;
        SkDebugf("%s %d calc1=%1.9g delta1=%1.9g factor1=%1.9g calc2=%1.9g delta2=%1.9g"
                " factor2=%1.9g\n", __FUNCTION__, test,
                calc1, delta1, factor1, calc2, delta2, factor2);
        if (factor1 < largestFactor) {
            SkDebugf("WE HAVE A WINNER! %1.9g\n", factor1);
    #if DEBUG_CRASH
            SkDebugf("%s\n", str);
    #endif
            oneOff(cubic1, cubic2);
            largestFactor = factor1;
        }
        if (factor2 < largestFactor) {
            SkDebugf("WE HAVE A WINNER! %1.9g\n", factor2);
    #if DEBUG_CRASH
            SkDebugf("%s\n", str);
    #endif
            oneOff(cubic1, cubic2);
            largestFactor = factor2;
        }
    }
}

void CubicIntersection_RandTest() {
    srand(0);
    const int tests = 10000000;
    for (int test = 0; test < tests; ++test) {
        Cubic cubic1, cubic2;
        for (int i = 0; i < 4; ++i) {
            cubic1[i].x = (double) rand() / RAND_MAX * 100;
            cubic1[i].y = (double) rand() / RAND_MAX * 100;
            cubic2[i].x = (double) rand() / RAND_MAX * 100;
            cubic2[i].y = (double) rand() / RAND_MAX * 100;
        }
    #if DEBUG_CRASH
        char str[1024];
        sprintf(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].x, cubic1[0].y,  cubic1[1].x, cubic1[1].y, cubic1[2].x, cubic1[2].y,
                cubic1[3].x, cubic1[3].y,
                cubic2[0].x, cubic2[0].y,  cubic2[1].x, cubic2[1].y, cubic2[2].x, cubic2[2].y,
                cubic2[3].x, cubic2[3].y);
    #endif
        _Rect rect1, rect2;
        rect1.setBounds(cubic1);
        rect2.setBounds(cubic2);
        bool boundsIntersect = rect1.left <= rect2.right && rect2.left <= rect2.right
                && rect1.top <= rect2.bottom && rect2.top <= rect1.bottom;
        if (test == -1) {
            SkDebugf("ready...\n");
        }
        Intersections intersections2;
        bool newIntersects = intersect2(cubic1, cubic2, intersections2);
        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
            SkASSERT(0);
        }
        for (int pt = 0; pt < intersections2.used(); ++pt) {
            double tt1 = intersections2.fT[0][pt];
            _Point xy1, xy2;
            xy_at_t(cubic1, tt1, xy1.x, xy1.y);
            int pt2 = intersections2.fFlip ? intersections2.used() - pt - 1 : pt;
            double tt2 = intersections2.fT[1][pt2];
            xy_at_t(cubic2, tt2, xy2.x, xy2.y);
        #if 0
            SkDebugf("%s t1=%1.9g (%1.9g, %1.9g) (%1.9g, %1.9g) t2=%1.9g\n", __FUNCTION__,
                tt1, xy1.x, xy1.y, xy2.x, xy2.y, tt2);
        #endif
            SkASSERT(xy1.approximatelyEqual(xy2));
        }
    }
}

void CubicIntersection_IntersectionFinder() {
    const Cubic& cubic1 = testSet[2];
    const Cubic& cubic2 = testSet[7];

    double t1Seed = 0.254;
    double t2Seed = 0.245;
    double t1Step = 0.01;
    double t2Step = 0.01;
    _Point t1[3], t2[3];
    bool toggle = true;
    do {
        xy_at_t(cubic1, t1Seed - t1Step, t1[0].x, t1[0].y);
        xy_at_t(cubic1, t1Seed,          t1[1].x, t1[1].y);
        xy_at_t(cubic1, t1Seed + t1Step, t1[2].x, t1[2].y);
        xy_at_t(cubic2, t2Seed - t2Step, t2[0].x, t2[0].y);
        xy_at_t(cubic2, t2Seed,          t2[1].x, t2[1].y);
        xy_at_t(cubic2, t2Seed + t2Step, t2[2].x, t2[2].y);
        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;
    _Point test;
    while (!approximately_zero(t1Step)) {
        xy_at_t(cubic1, t10, test.x, test.y);
        t10 += t1[1].approximatelyEqual(test) ? -t1Step : t1Step;
        t1Step /= 2;
    }
    t1Step = 0.1;
    while (!approximately_zero(t1Step)) {
        xy_at_t(cubic1, t12, test.x, test.y);
        t12 -= t1[1].approximatelyEqual(test) ? -t1Step : t1Step;
        t1Step /= 2;
    }
    while (!approximately_zero(t2Step)) {
        xy_at_t(cubic2, t20, test.x, test.y);
        t20 += t2[1].approximatelyEqual(test) ? -t2Step : t2Step;
        t2Step /= 2;
    }
    t2Step = 0.1;
    while (!approximately_zero(t2Step)) {
        xy_at_t(cubic2, t22, test.x, test.y);
        t22 -= t2[1].approximatelyEqual(test) ? -t2Step : t2Step;
        t2Step /= 2;
    }
    SkDebugf("%s t1=(%1.9g<%1.9g<%1.9g) t2=(%1.9g<%1.9g<%1.9g)\n", __FUNCTION__,
        t10, t1Seed, t12, t20, t2Seed, t22);
    _Point p10 = xy_at_t(cubic1, t10);
    _Point p1Seed = xy_at_t(cubic1, t1Seed);
    _Point p12 = xy_at_t(cubic1, t12);
    SkDebugf("%s p1=(%1.9g,%1.9g)<(%1.9g,%1.9g)<(%1.9g,%1.9g)\n", __FUNCTION__,
        p10.x, p10.y, p1Seed.x, p1Seed.y, p12.x, p12.y);
    _Point p20 = xy_at_t(cubic2, t20);
    _Point p2Seed = xy_at_t(cubic2, t2Seed);
    _Point p22 = xy_at_t(cubic2, t22);
    SkDebugf("%s p2=(%1.9g,%1.9g)<(%1.9g,%1.9g)<(%1.9g,%1.9g)\n", __FUNCTION__,
        p20.x, p20.y, p2Seed.x, p2Seed.y, p22.x, p22.y);
}

static void coincidentTest() {
#if 0
    Cubic cubic1 = {{0, 1}, {0, 2}, {1, 0}, {1, 0}};
    Cubic cubic2 = {{0, 1}, {0, 2}, {1, 0}, {6, 1}};
#endif
}

void CubicIntersection_Test() {
    oneOffTests();
    coincidentTest();
    standardTestCases();
}