// returns false if there's more than one intercept or the intercept doesn't match the point
// returns true if the intercept was successfully added or if the
// original quads need to be subdivided
static bool add_intercept(const SkDQuad& q1, const SkDQuad& q2, double tMin, double tMax,
SkIntersections* i, bool* subDivide) {
double tMid = (tMin + tMax) / 2;
SkDPoint mid = q2.ptAtT(tMid);
SkDLine line;
line[0] = line[1] = mid;
SkDVector dxdy = q2.dxdyAtT(tMid);
line[0] -= dxdy;
line[1] += dxdy;
SkIntersections rootTs;
rootTs.allowNear(false);
int roots = rootTs.intersect(q1, line);
if (roots == 0) {
if (subDivide) {
*subDivide = true;
}
return true;
}
if (roots == 2) {
return false;
}
SkDPoint pt2 = q1.ptAtT(rootTs[0][0]);
if (!pt2.approximatelyEqualHalf(mid)) {
return false;
}
i->insertSwap(rootTs[0][0], tMid, pt2);
return true;
}
// intersect the end of the cubic with the other. Try lines from the end to control and opposite
// end to determine range of t on opposite cubic.
bool SkIntersections::cubicExactEnd(const SkDCubic& cubic1, bool start, const SkDCubic& cubic2) {
int t1Index = start ? 0 : 3;
double testT = (double) !start;
bool swap = swapped();
// quad/quad at this point checks to see if exact matches have already been found
// cubic/cubic can't reject so easily since cubics can intersect same point more than once
SkDLine tmpLine;
tmpLine[0] = tmpLine[1] = cubic2[t1Index];
tmpLine[1].fX += cubic2[2 - start].fY - cubic2[t1Index].fY;
tmpLine[1].fY -= cubic2[2 - start].fX - cubic2[t1Index].fX;
SkIntersections impTs;
impTs.allowNear(false);
impTs.intersectRay(cubic1, tmpLine);
for (int index = 0; index < impTs.used(); ++index) {
SkDPoint realPt = impTs.pt(index);
if (!tmpLine[0].approximatelyEqual(realPt)) {
continue;
}
if (swap) {
cubicInsert(testT, impTs[0][index], tmpLine[0], cubic2, cubic1);
} else {
cubicInsert(impTs[0][index], testT, tmpLine[0], cubic1, cubic2);
}
return true;
}
return false;
}
// determine that slop required after quad/quad finds a candidate intersection
// use the cross of the tangents plus the distance from 1 or 0 as knobs
DEF_TEST(PathOpsCubicQuadSlop, reporter) {
// create a random non-selfintersecting cubic
// break it into quadratics
// offset the quadratic, measuring the slop required to find the intersection
if (!gPathOpCubicQuadSlopVerbose) { // takes a while to run -- so exclude it by default
return;
}
int results[101];
sk_bzero(results, sizeof(results));
double minCross[101];
sk_bzero(minCross, sizeof(minCross));
double maxCross[101];
sk_bzero(maxCross, sizeof(maxCross));
double sumCross[101];
sk_bzero(sumCross, sizeof(sumCross));
int foundOne = 0;
int slopCount = 1;
SkRandom ran;
for (int index = 0; index < 10000000; ++index) {
if (index % 1000 == 999) SkDebugf(".");
SkDCubic cubic = {{
{ran.nextRangeF(-1000, 1000), ran.nextRangeF(-1000, 1000)},
{ran.nextRangeF(-1000, 1000), ran.nextRangeF(-1000, 1000)},
{ran.nextRangeF(-1000, 1000), ran.nextRangeF(-1000, 1000)},
{ran.nextRangeF(-1000, 1000), ran.nextRangeF(-1000, 1000)}
}};
SkIntersections i;
if (i.intersect(cubic)) {
continue;
}
SkSTArray<kCubicToQuadSubdivisionDepth, double, true> ts;
cubic.toQuadraticTs(cubic.calcPrecision(), &ts);
double tStart = 0;
int tsCount = ts.count();
for (int i1 = 0; i1 <= tsCount; ++i1) {
const double tEnd = i1 < tsCount ? ts[i1] : 1;
SkDCubic part = cubic.subDivide(tStart, tEnd);
SkDQuad quad = part.toQuad();
SkReduceOrder reducer;
int order = reducer.reduce(quad);
if (order != 3) {
continue;
}
for (int i2 = 0; i2 < 100; ++i2) {
SkDPoint endDisplacement = {ran.nextRangeF(-100, 100), ran.nextRangeF(-100, 100)};
SkDQuad nearby = {{
{quad[0].fX + endDisplacement.fX, quad[0].fY + endDisplacement.fY},
{quad[1].fX + ran.nextRangeF(-100, 100), quad[1].fY + ran.nextRangeF(-100, 100)},
{quad[2].fX - endDisplacement.fX, quad[2].fY - endDisplacement.fY}
}};
order = reducer.reduce(nearby);
if (order != 3) {
continue;
}
SkIntersections locals;
locals.allowNear(false);
locals.intersect(quad, nearby);
if (locals.used() != 1) {
continue;
}
// brute force find actual intersection
SkDLine cubicLine = {{ {0, 0}, {cubic[0].fX, cubic[0].fY } }};
SkIntersections liner;
int i3;
int found = -1;
int foundErr = true;
for (i3 = 1; i3 <= 1000; ++i3) {
cubicLine[0] = cubicLine[1];
cubicLine[1] = cubic.ptAtT(i3 / 1000.);
liner.reset();
liner.allowNear(false);
liner.intersect(nearby, cubicLine);
if (liner.used() == 0) {
continue;
}
if (liner.used() > 1) {
foundErr = true;
break;
}
if (found > 0) {
foundErr = true;
break;
}
foundErr = false;
found = i3;
}
if (foundErr) {
continue;
}
SkDVector dist = liner.pt(0) - locals.pt(0);
SkDVector qV = nearby.dxdyAtT(locals[0][0]);
double cubicT = (found - 1 + liner[1][0]) / 1000.;
SkDVector cV = cubic.dxdyAtT(cubicT);
double qxc = qV.crossCheck(cV);
double qvLen = qV.length();
double cvLen = cV.length();
double maxLen = SkTMax(qvLen, cvLen);
qxc /= maxLen;
double quadT = tStart + (tEnd - tStart) * locals[0][0];
double diffT = fabs(cubicT - quadT);
int diffIdx = (int) (diffT * 100);
results[diffIdx]++;
double absQxc = fabs(qxc);
if (sumCross[diffIdx] == 0) {
minCross[diffIdx] = maxCross[diffIdx] = sumCross[diffIdx] = absQxc;
} else {
minCross[diffIdx] = SkTMin(minCross[diffIdx], absQxc);
maxCross[diffIdx] = SkTMax(maxCross[diffIdx], absQxc);
sumCross[diffIdx] += absQxc;
}
if (diffIdx >= 20) {
#if 01
SkDebugf("cubic={{{%1.9g,%1.9g}, {%1.9g,%1.9g}, {%1.9g,%1.9g}, {%1.9g,%1.9g}}}"
" quad={{{%1.9g,%1.9g}, {%1.9g,%1.9g}, {%1.9g,%1.9g}}}"
" {{{%1.9g,%1.9g}, {%1.9g,%1.9g}}}"
" qT=%1.9g cT=%1.9g dist=%1.9g cross=%1.9g\n",
cubic[0].fX, cubic[0].fY, cubic[1].fX, cubic[1].fY,
cubic[2].fX, cubic[2].fY, cubic[3].fX, cubic[3].fY,
nearby[0].fX, nearby[0].fY, nearby[1].fX, nearby[1].fY,
nearby[2].fX, nearby[2].fY,
liner.pt(0).fX, liner.pt(0).fY,
locals.pt(0).fX, locals.pt(0).fY, quadT, cubicT, dist.length(), qxc);
#else
SkDebugf("qT=%1.9g cT=%1.9g dist=%1.9g cross=%1.9g\n",
quadT, cubicT, dist.length(), qxc);
SkDebugf("<div id=\"slop%d\">\n", ++slopCount);
SkDebugf("{{{%1.9g,%1.9g}, {%1.9g,%1.9g}, {%1.9g,%1.9g}, {%1.9g,%1.9g}}}\n"
"{{{%1.9g,%1.9g}, {%1.9g,%1.9g}, {%1.9g,%1.9g}}}\n"
"{{{%1.9g,%1.9g}, {%1.9g,%1.9g}}}\n",
cubic[0].fX, cubic[0].fY, cubic[1].fX, cubic[1].fY,
cubic[2].fX, cubic[2].fY, cubic[3].fX, cubic[3].fY,
nearby[0].fX, nearby[0].fY, nearby[1].fX, nearby[1].fY,
nearby[2].fX, nearby[2].fY,
liner.pt(0).fX, liner.pt(0).fY,
locals.pt(0).fX, locals.pt(0).fY);
SkDebugf("</div>\n\n");
#endif
}
++foundOne;
}
tStart = tEnd;
}
if (++foundOne >= 100000) {
break;
}
}
#if 01
SkDebugf("slopCount=%d\n", slopCount);
int max = 100;
while (results[max] == 0) {
--max;
}
for (int i = 0; i <= max; ++i) {
if (i > 0 && i % 10 == 0) {
SkDebugf("\n");
}
SkDebugf("%d ", results[i]);
}
SkDebugf("min\n");
for (int i = 0; i <= max; ++i) {
if (i > 0 && i % 10 == 0) {
SkDebugf("\n");
}
SkDebugf("%1.9g ", minCross[i]);
}
SkDebugf("max\n");
for (int i = 0; i <= max; ++i) {
if (i > 0 && i % 10 == 0) {
SkDebugf("\n");
}
SkDebugf("%1.9g ", maxCross[i]);
}
SkDebugf("avg\n");
for (int i = 0; i <= max; ++i) {
if (i > 0 && i % 10 == 0) {
SkDebugf("\n");
}
SkDebugf("%1.9g ", sumCross[i] / results[i]);
}
#else
for (int i = 1; i < slopCount; ++i) {
SkDebugf(" slop%d,\n", i);
}
#endif
SkDebugf("\n");
}
static bool is_linear_inner(const SkDQuad& q1, double t1s, double t1e, const SkDQuad& q2,
double t2s, double t2e, SkIntersections* i, bool* subDivide) {
SkDQuad hull = q1.subDivide(t1s, t1e);
SkDLine line = {{hull[2], hull[0]}};
const SkDLine* testLines[] = { &line, (const SkDLine*) &hull[0], (const SkDLine*) &hull[1] };
const size_t kTestCount = SK_ARRAY_COUNT(testLines);
SkSTArray<kTestCount * 2, double, true> tsFound;
for (size_t index = 0; index < kTestCount; ++index) {
SkIntersections rootTs;
rootTs.allowNear(false);
int roots = rootTs.intersect(q2, *testLines[index]);
for (int idx2 = 0; idx2 < roots; ++idx2) {
double t = rootTs[0][idx2];
#ifdef SK_DEBUG
SkDPoint qPt = q2.ptAtT(t);
SkDPoint lPt = testLines[index]->ptAtT(rootTs[1][idx2]);
SkASSERT(qPt.approximatelyEqual(lPt));
#endif
if (approximately_negative(t - t2s) || approximately_positive(t - t2e)) {
continue;
}
tsFound.push_back(rootTs[0][idx2]);
}
}
int tCount = tsFound.count();
if (tCount <= 0) {
return true;
}
double tMin, tMax;
if (tCount == 1) {
tMin = tMax = tsFound[0];
} else {
SkASSERT(tCount > 1);
SkTQSort<double>(tsFound.begin(), tsFound.end() - 1);
tMin = tsFound[0];
tMax = tsFound[tsFound.count() - 1];
}
SkDPoint end = q2.ptAtT(t2s);
bool startInTriangle = hull.pointInHull(end);
if (startInTriangle) {
tMin = t2s;
}
end = q2.ptAtT(t2e);
bool endInTriangle = hull.pointInHull(end);
if (endInTriangle) {
tMax = t2e;
}
int split = 0;
SkDVector dxy1, dxy2;
if (tMin != tMax || tCount > 2) {
dxy2 = q2.dxdyAtT(tMin);
for (int index = 1; index < tCount; ++index) {
dxy1 = dxy2;
dxy2 = q2.dxdyAtT(tsFound[index]);
double dot = dxy1.dot(dxy2);
if (dot < 0) {
split = index - 1;
break;
}
}
}
if (split == 0) { // there's one point
if (add_intercept(q1, q2, tMin, tMax, i, subDivide)) {
return true;
}
i->swap();
return is_linear_inner(q2, tMin, tMax, q1, t1s, t1e, i, subDivide);
}
// At this point, we have two ranges of t values -- treat each separately at the split
bool result;
if (add_intercept(q1, q2, tMin, tsFound[split - 1], i, subDivide)) {
result = true;
} else {
i->swap();
result = is_linear_inner(q2, tMin, tsFound[split - 1], q1, t1s, t1e, i, subDivide);
}
if (add_intercept(q1, q2, tsFound[split], tMax, i, subDivide)) {
result = true;
} else {
i->swap();
result |= is_linear_inner(q2, tsFound[split], tMax, q1, t1s, t1e, i, subDivide);
}
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();
}
