void CubicIntersection_Test() {
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);
}
}
}
}
bool intersect(const Cubic& cubic, Intersections& i) {
SkTDArray<double> ts;
double precision = calcPrecision(cubic);
cubic_to_quadratics(cubic, precision, ts);
int tsCount = ts.count();
if (tsCount == 1) {
return false;
}
double t1Start = 0;
Cubic part;
for (int idx = 0; idx < tsCount; ++idx) {
double t1 = ts[idx];
Quadratic q1;
sub_divide(cubic, t1Start, t1, part);
demote_cubic_to_quad(part, q1);
double t2Start = t1;
for (int i2 = idx + 1; i2 <= tsCount; ++i2) {
const double t2 = i2 < tsCount ? ts[i2] : 1;
Quadratic q2;
sub_divide(cubic, t2Start, t2, part);
demote_cubic_to_quad(part, q2);
Intersections locals;
intersect2(q1, q2, locals);
for (int tIdx = 0; tIdx < locals.used(); ++tIdx) {
// discard intersections at cusp? (maximum curvature)
double t1sect = locals.fT[0][tIdx];
double t2sect = locals.fT[1][tIdx];
if (idx + 1 == i2 && t1sect == 1 && t2sect == 0) {
continue;
}
double to1 = t1Start + (t1 - t1Start) * t1sect;
double to2 = t2Start + (t2 - t2Start) * t2sect;
i.insert(to1, to2);
}
t2Start = t2;
}
t1Start = t1;
}
return i.intersected();
}
static void standardTestCases() {
for (size_t index = firstQuadIntersectionTest; index < quadraticTests_count; ++index) {
const Quadratic& quad1 = quadraticTests[index][0];
const Quadratic& quad2 = quadraticTests[index][1];
Quadratic reduce1, reduce2;
int order1 = reduceOrder(quad1, reduce1, kReduceOrder_TreatAsFill);
int order2 = reduceOrder(quad2, reduce2, kReduceOrder_TreatAsFill);
if (order1 < 3) {
printf("[%d] quad1 order=%d\n", (int) index, order1);
}
if (order2 < 3) {
printf("[%d] quad2 order=%d\n", (int) index, order2);
}
if (order1 == 3 && order2 == 3) {
Intersections intersections;
intersect2(reduce1, reduce2, intersections);
if (intersections.intersected()) {
for (int pt = 0; pt < intersections.used(); ++pt) {
double tt1 = intersections.fT[0][pt];
double tx1, ty1;
xy_at_t(quad1, tt1, tx1, ty1);
double tt2 = intersections.fT[1][pt];
double tx2, ty2;
xy_at_t(quad2, tt2, tx2, ty2);
if (!approximately_equal(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 (!approximately_equal(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);
}
}
}
}
}
}
// this flavor approximates the cubics with quads to find the intersecting ts
// OPTIMIZE: if this strategy proves successful, the quad approximations, or the ts used
// to create the approximations, could be stored in the cubic segment
// FIXME: this strategy needs to intersect the convex hull on either end with the opposite to
// account for inset quadratics that cause the endpoint intersection to avoid detection
// the segments can be very short -- the length of the maximum quadratic error (precision)
// FIXME: this needs to recurse on itself, taking a range of T values and computing the new
// t range ala is linear inner. The range can be figured by taking the dx/dy and determining
// the fraction that matches the precision. That fraction is the change in t for the smaller cubic.
static bool intersect2(const Cubic& cubic1, double t1s, double t1e, const Cubic& cubic2,
double t2s, double t2e, double precisionScale, Intersections& i) {
Cubic c1, c2;
sub_divide(cubic1, t1s, t1e, c1);
sub_divide(cubic2, t2s, t2e, c2);
SkTDArray<double> ts1;
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;
Cubic part1;
sub_divide(cubic1, t1Start, t1, part1);
Quadratic q1;
demote_cubic_to_quad(part1, q1);
// start here;
// should reduceOrder be looser in this use case if quartic is going to blow up on an
// extremely shallow quadratic?
Quadratic s1;
int o1 = reduceOrder(q1, 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;
Cubic part2;
sub_divide(cubic2, t2Start, t2, part2);
Quadratic q2;
demote_cubic_to_quad(part2, q2);
Quadratic s2;
double o2 = reduceOrder(q2, s2);
Intersections locals;
if (o1 == 3 && o2 == 3) {
intersect2(q1, q2, locals);
} else if (o1 <= 2 && o2 <= 2) {
locals.fUsed = intersect((const _Line&) s1, (const _Line&) s2, locals.fT[0],
locals.fT[1]);
} else if (o1 == 3 && o2 <= 2) {
intersect(q1, (const _Line&) s2, locals);
} else {
SkASSERT(o1 <= 2 && o2 == 3);
intersect(q2, (const _Line&) s1, locals);
for (int s = 0; s < locals.fUsed; ++s) {
SkTSwap(locals.fT[0][s], locals.fT[1][s]);
}
}
for (int tIdx = 0; tIdx < locals.used(); ++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, p2;
xy_at_t(cubic1, to1, p1.x, p1.y);
xy_at_t(cubic2, to2, p2.x, p2.y);
if (p1.approximatelyEqual(p2)) {
i.insert(i.swapped() ? to2 : to1, i.swapped() ? to1 : to2);
} else {
double dt1, dt2;
computeDelta(cubic1, to1, (t1e - t1s), cubic2, to2, (t2e - t2s), dt1, dt2);
double scale = precisionScale;
if (dt1 > 0.125 || dt2 > 0.125) {
scale /= 2;
SkDebugf("%s scale=%1.9g\n", __FUNCTION__, scale);
}
#if SK_DEBUG
++debugDepth;
assert(debugDepth < 10);
#endif
i.swap();
intersect2(cubic2, SkTMax(to2 - dt2, 0.), SkTMin(to2 + dt2, 1.),
cubic1, SkTMax(to1 - dt1, 0.), SkTMin(to1 + dt1, 1.), scale, i);
i.swap();
#if SK_DEBUG
--debugDepth;
#endif
}
}
t2Start = t2;
}
t1Start = t1;
}
return i.intersected();
}
