Example #1
0
/*
令Aij为矩阵的中间元素,按照这个元素将矩阵划为4份,如果num大于Aij,则num可能落在左下、右上、右下三个子矩阵,对它们分别求子问题。
小于时也是类似的讨论。每次都会求三次子问题,每个子问题元素个数是上次的1/4,所以复杂度是f(N)=3f(N/4)+c,其中N=m*n是元素个数,c是大于0的常量,
主方法可求复杂度为O(N^(log4为底3)),计算器大概算了一下log4为底3约等于0.8。所以复杂度为O((m*n)^0.8)
*/
bool quadPart(int mat[][N] , int key , int l , int u , int r , int d , int &row , int &col)
{
	if(l > r || u > d)
		return false;
	if(key < mat[u][l] || key > mat[d][r])
		return false;
	int mid_col = (l+r)>>1;
	int mid_row = (u+d)>>1;
	if(mat[mid_row][mid_col] == key)     //查找成功
	{
		row = mid_row;
		col = mid_col;
		return true;
	}
	else if(l == r && u == d)
		return false;

	if(mat[mid_row][mid_col] > key)
	{   // 分别在右上角、左上角、左下角中查找
		return quadPart(mat , key , mid_col+1 , u , r , mid_row , row , col) ||
		       quadPart(mat , key , l , mid_row+1 , mid_col , d , row , col) ||
			   quadPart(mat , key , l , u , mid_col , mid_row , row , col) ;
	}
	else
	{   // 分别在右上角、左下角、右下角中查找
		return quadPart(mat , key , mid_col+1 , u , r , mid_row , row , col) ||
		       quadPart(mat , key , l , mid_row+1 , mid_col , d , row , col) ||
			   quadPart(mat , key , mid_col+1 , mid_row+1 , r , d , row , col) ;
	}
}
Example #2
0
bool quadPart(int mat[][N] , int key , int &row , int &col)
{
	return quadPart(mat , key , 0 , 0 , N-1 , N-1 , row , col);
}
// this flavor centers potential intersections recursively. In contrast, '2' may inadvertently
// chase intersections near quadratic ends, requiring odd hacks to find them.
static bool intersect3(const Cubic& cubic1, double t1s, double t1e, const Cubic& cubic2,
        double t2s, double t2e, double precisionScale, Intersections& i) {
    i.upDepth();
    bool result = false;
    Cubic c1, c2;
    sub_divide(cubic1, t1s, t1e, c1);
    sub_divide(cubic2, t2s, t2e, c2);
    SkTDArray<double> ts1;
    // OPTIMIZE: if c1 == c2, call once (happens when detecting self-intersection)
    cubic_to_quadratics(c1, calcPrecision(c1) * precisionScale, ts1);
    SkTDArray<double> ts2;
    cubic_to_quadratics(c2, calcPrecision(c2) * 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;
        Quadratic 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;
            }
            Quadratic 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) {
                Cubic cSub1, cSub2;
                sub_divide(cubic1, t1Start, t1, cSub1);
                sub_divide(cubic2, t2Start, t2, cSub2);
                SkDebugf("%.*s %s t1=(%1.9g,%1.9g) t2=(%1.9g,%1.9g)", i.depth()*2, tab, __FUNCTION__,
                        t1Start, t1, t2Start, t2);
                Intersections xlocals;
                intersectWithOrder(s1, o1, s2, o2, xlocals);
                SkDebugf(" xlocals.fUsed=%d\n", xlocals.used());
            }
        #endif
            Intersections locals;
            intersectWithOrder(s1, o1, s2, o2, locals);
            double coStart[2] = { -1 };
            _Point coPoint;
            int tCount = locals.used();
            for (int tIdx = 0; tIdx < tCount; ++tIdx) {
                double to1 = t1Start + (t1 - t1Start) * locals.fT[0][tIdx];
                double to2 = t2Start + (t2 - t2Start) * locals.fT[1][tIdx];
    // if the computed t is not sufficiently precise, iterate
                _Point p1 = xy_at_t(cubic1, to1);
                _Point p2 = xy_at_t(cubic2, to2);
                if (p1.approximatelyEqual(p2)) {
                    if (locals.fIsCoincident[0] & 1 << 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;
                        }
                        result = true;
                    } 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);
                        }
                        result = true;
                    }
                } else {
                    double offset = precisionScale / 16; // FIME: const is arbitrary -- test & refine
#if 1
                    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 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
                    intersect3(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.fT[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.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;
                        }
                        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
                        intersect3(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.fT[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
                        intersect3(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.fT[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
                    intersect3(cubic1, c1Min, c1Max, cubic2, c2Min, c2Max, offset, i);
                    // TODO: 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();
    return result;
}
// 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);
                intersectWithOrder(s1.fQuad, o1, s2.fQuad, o2, xlocals);
                SkDebugf(" xlocals.fUsed=%d\n", xlocals.used());
            }
        #endif
            SkIntersections locals;
            locals.allowNear(false);
            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();
}