| /* |
| * 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 "CubicUtilities.h" |
| #include "CurveIntersection.h" |
| #include "Intersections.h" |
| #include "IntersectionUtilities.h" |
| #include "LineIntersection.h" |
| #include "LineUtilities.h" |
| #include "QuadraticUtilities.h" |
| |
| #if ONE_OFF_DEBUG |
| static const double tLimits[2][2] = {{0.772784538, 0.77278492}, {0.999111748, 0.999112129}}; |
| #endif |
| |
| #define DEBUG_QUAD_PART 0 |
| #define SWAP_TOP_DEBUG 0 |
| |
| static int quadPart(const Cubic& cubic, double tStart, double tEnd, Quadratic& simple) { |
| Cubic part; |
| sub_divide(cubic, tStart, tEnd, part); |
| Quadratic quad; |
| demote_cubic_to_quad(part, quad); |
| // FIXME: should reduceOrder be looser in this use case if quartic is going to blow up on an |
| // extremely shallow quadratic? |
| int order = reduceOrder(quad, simple, kReduceOrder_TreatAsFill); |
| #if DEBUG_QUAD_PART |
| SkDebugf("%s cubic=(%1.17g,%1.17g %1.17g,%1.17g %1.17g,%1.17g %1.17g,%1.17g) t=(%1.17g,%1.17g)\n", |
| __FUNCTION__, cubic[0].x, cubic[0].y, cubic[1].x, cubic[1].y, cubic[2].x, cubic[2].y, |
| cubic[3].x, cubic[3].y, tStart, tEnd); |
| SkDebugf("%s part=(%1.17g,%1.17g %1.17g,%1.17g %1.17g,%1.17g %1.17g,%1.17g)" |
| " quad=(%1.17g,%1.17g %1.17g,%1.17g %1.17g,%1.17g)\n", __FUNCTION__, part[0].x, part[0].y, |
| part[1].x, part[1].y, part[2].x, part[2].y, part[3].x, part[3].y, quad[0].x, quad[0].y, |
| quad[1].x, quad[1].y, quad[2].x, quad[2].y); |
| SkDebugf("%s simple=(%1.17g,%1.17g", __FUNCTION__, simple[0].x, simple[0].y); |
| if (order > 1) { |
| SkDebugf(" %1.17g,%1.17g", simple[1].x, simple[1].y); |
| } |
| if (order > 2) { |
| SkDebugf(" %1.17g,%1.17g", simple[2].x, simple[2].y); |
| } |
| SkDebugf(")\n"); |
| SkASSERT(order < 4 && order > 0); |
| #endif |
| return order; |
| } |
| |
| static void intersectWithOrder(const Quadratic& simple1, int order1, const Quadratic& simple2, |
| int order2, Intersections& i) { |
| if (order1 == 3 && order2 == 3) { |
| intersect2(simple1, simple2, i); |
| } else if (order1 <= 2 && order2 <= 2) { |
| intersect((const _Line&) simple1, (const _Line&) simple2, i); |
| } else if (order1 == 3 && order2 <= 2) { |
| intersect(simple1, (const _Line&) simple2, i); |
| } else { |
| SkASSERT(order1 <= 2 && order2 == 3); |
| intersect(simple2, (const _Line&) simple1, i); |
| for (int s = 0; s < i.fUsed; ++s) { |
| SkTSwap(i.fT[0][s], i.fT[1][s]); |
| } |
| } |
| } |
| |
| static double distanceFromEnd(double t) { |
| return t > 0.5 ? 1 - t : t; |
| } |
| |
| // OPTIMIZATION: this used to try to guess the value for delta, and that may still be worthwhile |
| static void bumpForRetry(double t1, double t2, double& s1, double& e1, double& s2, double& e2) { |
| double dt1 = distanceFromEnd(t1); |
| double dt2 = distanceFromEnd(t2); |
| double delta = 1.0 / gPrecisionUnit; |
| if (dt1 < dt2) { |
| if (t1 == dt1) { |
| s1 = SkTMax(s1 - delta, 0.); |
| } else { |
| e1 = SkTMin(e1 + delta, 1.); |
| } |
| } else { |
| if (t2 == dt2) { |
| s2 = SkTMax(s2 - delta, 0.); |
| } else { |
| e2 = SkTMin(e2 + delta, 1.); |
| } |
| } |
| } |
| |
| static bool doIntersect(const Cubic& cubic1, double t1s, double t1m, double t1e, |
| const Cubic& cubic2, double t2s, double t2m, double t2e, Intersections& i) { |
| bool result = false; |
| i.upDepth(); |
| // divide the quadratics at the new t value and try again |
| double p1s = t1s; |
| double p1e = t1m; |
| for (int p1 = 0; p1 < 2; ++p1) { |
| Quadratic s1a; |
| int o1a = quadPart(cubic1, p1s, p1e, s1a); |
| double p2s = t2s; |
| double p2e = t2m; |
| for (int p2 = 0; p2 < 2; ++p2) { |
| Quadratic s2a; |
| int o2a = quadPart(cubic2, p2s, p2e, s2a); |
| Intersections locals; |
| #if ONE_OFF_DEBUG |
| if (tLimits[0][0] >= p1s && tLimits[0][1] <= p1e |
| && tLimits[1][0] >= p2s && tLimits[1][1] <= p2e) { |
| SkDebugf("t1=(%1.9g,%1.9g) o1=%d t2=(%1.9g,%1.9g) o2=%d\n", |
| p1s, p1e, o1a, p2s, p2e, o2a); |
| if (o1a == 2) { |
| SkDebugf("{{%1.9g,%1.9g}, {%1.9g,%1.9g}},\n", |
| s1a[0].x, s1a[0].y, s1a[1].x, s1a[1].y); |
| } else { |
| SkDebugf("{{%1.9g,%1.9g}, {%1.9g,%1.9g}, {%1.9g,%1.9g}},\n", |
| s1a[0].x, s1a[0].y, s1a[1].x, s1a[1].y, s1a[2].x, s1a[2].y); |
| } |
| if (o2a == 2) { |
| SkDebugf("{{%1.9g,%1.9g}, {%1.9g,%1.9g}},\n", |
| s2a[0].x, s2a[0].y, s2a[1].x, s2a[1].y); |
| } else { |
| SkDebugf("{{%1.9g,%1.9g}, {%1.9g,%1.9g}, {%1.9g,%1.9g}},\n", |
| s2a[0].x, s2a[0].y, s2a[1].x, s2a[1].y, s2a[2].x, s2a[2].y); |
| } |
| Intersections xlocals; |
| intersectWithOrder(s1a, o1a, s2a, o2a, xlocals); |
| SkDebugf("xlocals.fUsed=%d depth=%d\n", xlocals.used(), i.depth()); |
| } |
| #endif |
| intersectWithOrder(s1a, o1a, s2a, o2a, locals); |
| for (int tIdx = 0; tIdx < locals.used(); ++tIdx) { |
| double to1 = p1s + (p1e - p1s) * locals.fT[0][tIdx]; |
| double to2 = p2s + (p2e - p2s) * 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 ONE_OFF_DEBUG |
| SkDebugf("to1=%1.9g p1=(%1.9g,%1.9g) to2=%1.9g p2=(%1.9g,%1.9g) d=%1.9g\n", |
| to1, p1.x, p1.y, to2, p2.x, p2.y, p1.distance(p2)); |
| |
| #endif |
| if (p1.approximatelyEqualHalf(p2)) { |
| i.insertSwap(to1, to2, p1); |
| result = true; |
| } else { |
| result = doIntersect(cubic1, p1s, to1, p1e, cubic2, p2s, to2, p2e, i); |
| if (!result && p1.approximatelyEqual(p2)) { |
| i.insertSwap(to1, to2, p1); |
| #if SWAP_TOP_DEBUG |
| SkDebugf("!!!\n"); |
| #endif |
| result = true; |
| } else |
| // if both cubics curve in the same direction, the quadratic intersection |
| // may mark a range that does not contain the cubic intersection. If no |
| // intersection is found, look again including the t distance of the |
| // of the quadratic intersection nearest a quadratic end (which in turn is |
| // nearest the actual cubic) |
| if (!result) { |
| double b1s = p1s; |
| double b1e = p1e; |
| double b2s = p2s; |
| double b2e = p2e; |
| bumpForRetry(locals.fT[0][tIdx], locals.fT[1][tIdx], b1s, b1e, b2s, b2e); |
| result = doIntersect(cubic1, b1s, to1, b1e, cubic2, b2s, to2, b2e, i); |
| } |
| } |
| } |
| p2s = p2e; |
| p2e = t2e; |
| } |
| p1s = p1e; |
| p1e = t1e; |
| } |
| i.downDepth(); |
| return result; |
| } |
| |
| // 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) |
| 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; |
| 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; |
| Quadratic s2; |
| int o2 = quadPart(cubic2, t2Start, t2, s2); |
| #if ONE_OFF_DEBUG |
| if (tLimits[0][0] >= t1Start && tLimits[0][1] <= t1 |
| && tLimits[1][0] >= t2Start && tLimits[1][1] <= t2) { |
| Cubic cSub1, cSub2; |
| sub_divide(cubic1, t1Start, tEnd1, cSub1); |
| sub_divide(cubic2, t2Start, tEnd2, cSub2); |
| SkDebugf("t1=(%1.9g,%1.9g) t2=(%1.9g,%1.9g)\n", |
| 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); |
| |
| 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(to1, to2, p1); |
| } else { |
| #if ONE_OFF_DEBUG |
| if (tLimits[0][0] >= t1Start && tLimits[0][1] <= t1 |
| && tLimits[1][0] >= t2Start && tLimits[1][1] <= t2) { |
| SkDebugf("t1=(%1.9g,%1.9g) t2=(%1.9g,%1.9g)\n", |
| t1Start, t1, t2Start, t2); |
| } |
| #endif |
| bool found = doIntersect(cubic1, t1Start, to1, t1, cubic2, t2Start, to2, t2, i); |
| if (!found) { |
| double b1s = t1Start; |
| double b1e = t1; |
| double b2s = t2Start; |
| double b2e = t2; |
| bumpForRetry(locals.fT[0][tIdx], locals.fT[1][tIdx], b1s, b1e, b2s, b2e); |
| doIntersect(cubic1, b1s, to1, b1e, cubic2, b2s, to2, b2e, i); |
| } |
| } |
| } |
| int coincidentCount = locals.coincidentUsed(); |
| if (coincidentCount) { |
| // FIXME: one day, we'll probably need to allow coincident + non-coincident pts |
| SkASSERT(coincidentCount == locals.used()); |
| SkASSERT(coincidentCount == 2); |
| double coTs[2][2]; |
| for (int tIdx = 0; tIdx < coincidentCount; ++tIdx) { |
| if (locals.fIsCoincident[0] & (1 << tIdx)) { |
| coTs[0][tIdx] = t1Start + (t1 - t1Start) * locals.fT[0][tIdx]; |
| } |
| if (locals.fIsCoincident[1] & (1 << tIdx)) { |
| coTs[1][tIdx] = t2Start + (t2 - t2Start) * locals.fT[1][tIdx]; |
| } |
| } |
| i.insertCoincidentPair(coTs[0][0], coTs[0][1], coTs[1][0], coTs[1][1], |
| locals.fPt[0], locals.fPt[1]); |
| } |
| t2Start = t2; |
| } |
| t1Start = t1; |
| } |
| return i.intersected(); |
| } |
| |
| // 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; |
| 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 |
| if (tLimits[0][0] >= t1Start && tLimits[0][1] <= t1 |
| && tLimits[1][0] >= t2Start && tLimits[1][1] <= t2) { |
| Cubic cSub1, cSub2; |
| sub_divide(cubic1, t1Start, tEnd1, cSub1); |
| sub_divide(cubic2, t2Start, tEnd2, cSub2); |
| SkDebugf("t1=(%1.9g,%1.9g) t2=(%1.9g,%1.9g)\n", |
| 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; |
| 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)) { |
| 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 |
| double c1Min = SkTMax(0., to1 - offset); |
| double c1Max = SkTMin(1., to1 + offset); |
| double c2Min = SkTMax(0., to2 - offset); |
| double c2Max = SkTMin(1., to2 + offset); |
| bool found = intersect3(cubic1, c1Min, c1Max, cubic2, c2Min, c2Max, offset, i); |
| if (false && !found) { |
| // either offset was overagressive or cubics didn't really intersect |
| // if they didn't intersect, then quad tangents ought to be nearly parallel |
| offset = precisionScale / 2; // try much less agressive offset |
| c1Min = SkTMax(0., to1 - offset); |
| c1Max = SkTMin(1., to1 + offset); |
| c2Min = SkTMax(0., to2 - offset); |
| c2Max = SkTMin(1., to2 + offset); |
| found = intersect3(cubic1, c1Min, c1Max, cubic2, c2Min, c2Max, offset, i); |
| if (found) { |
| SkDebugf("%s *** over-aggressive? offset=%1.9g depth=%d\n", __FUNCTION__, |
| offset, i.depth()); |
| } |
| // try parallel measure |
| _Vector d1 = dxdy_at_t(cubic1, to1); |
| _Vector d2 = dxdy_at_t(cubic2, to2); |
| double shallow = d1.cross(d2); |
| #if 1 || ONE_OFF_DEBUG // not sure this is worth debugging |
| if (!approximately_zero(shallow)) { |
| SkDebugf("%s *** near-miss? shallow=%1.9g depth=%d\n", __FUNCTION__, |
| offset, i.depth()); |
| } |
| #endif |
| if (i.depth() == 1 && shallow < 0.6) { |
| SkDebugf("%s !!! near-miss? shallow=%1.9g depth=%d\n", __FUNCTION__, |
| offset, i.depth()); |
| } |
| } |
| } |
| } |
| SkASSERT(coStart[0] == -1); |
| t2Start = t2; |
| } |
| t1Start = t1; |
| } |
| i.downDepth(); |
| return result; |
| } |
| |
| // 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. |
| static bool intersectEnd(const Cubic& cubic1, bool start, const Cubic& cubic2, const _Rect& bounds2, |
| Intersections& i) { |
| _Line line; |
| int t1Index = start ? 0 : 3; |
| line[0] = cubic1[t1Index]; |
| // don't bother if the two cubics are connnected |
| if (line[0].approximatelyEqual(cubic2[0]) || line[0].approximatelyEqual(cubic2[3])) { |
| return false; |
| } |
| double tMin = 1, tMax = 0; |
| for (int index = 0; index < 4; ++index) { |
| if (index == t1Index) { |
| continue; |
| } |
| _Vector dxy1 = cubic1[index] - line[0]; |
| dxy1 /= gPrecisionUnit; |
| line[1] = line[0] + dxy1; |
| _Rect lineBounds; |
| lineBounds.setBounds(line); |
| if (!bounds2.intersects(lineBounds)) { |
| continue; |
| } |
| Intersections local; |
| if (!intersect(cubic2, line, local)) { |
| continue; |
| } |
| for (int index = 0; index < local.fUsed; ++index) { |
| tMin = SkTMin(tMin, local.fT[0][index]); |
| tMax = SkTMax(tMax, local.fT[0][index]); |
| } |
| } |
| if (tMin > tMax) { |
| return false; |
| } |
| double tMin1 = start ? 0 : 1 - 1.0 / gPrecisionUnit; |
| double tMax1 = start ? 1.0 / gPrecisionUnit : 1; |
| double tMin2 = SkTMax(tMin - 1.0 / gPrecisionUnit, 0.0); |
| double tMax2 = SkTMin(tMax + 1.0 / gPrecisionUnit, 1.0); |
| return intersect3(cubic1, tMin1, tMax1, cubic2, tMin2, tMax2, 1, i); |
| } |
| |
| // FIXME: add intersection of convex hull on cubics' ends with the opposite cubic. The hull line |
| // segments can be constructed to be only as long as the calculated precision suggests. If the hull |
| // line segments intersect the cubic, then use the intersections to construct a subdivision for |
| // quadratic curve fitting. |
| bool intersect2(const Cubic& c1, const Cubic& c2, Intersections& i) { |
| bool result = intersect2(c1, 0, 1, c2, 0, 1, 1, i); |
| // FIXME: pass in cached bounds from caller |
| _Rect c1Bounds, c2Bounds; |
| c1Bounds.setBounds(c1); // OPTIMIZE use setRawBounds ? |
| c2Bounds.setBounds(c2); |
| result |= intersectEnd(c1, false, c2, c2Bounds, i); |
| result |= intersectEnd(c1, true, c2, c2Bounds, i); |
| i.swap(); |
| result |= intersectEnd(c2, false, c1, c1Bounds, i); |
| result |= intersectEnd(c2, true, c1, c1Bounds, i); |
| i.swap(); |
| return result; |
| } |
| |
| const double CLOSE_ENOUGH = 0.001; |
| |
| static bool closeStart(const Cubic& cubic, int cubicIndex, Intersections& i, _Point& pt) { |
| if (i.fT[cubicIndex][0] != 0 || i.fT[cubicIndex][1] > CLOSE_ENOUGH) { |
| return false; |
| } |
| pt = xy_at_t(cubic, (i.fT[cubicIndex][0] + i.fT[cubicIndex][1]) / 2); |
| return true; |
| } |
| |
| static bool closeEnd(const Cubic& cubic, int cubicIndex, Intersections& i, _Point& pt) { |
| int last = i.used() - 1; |
| if (i.fT[cubicIndex][last] != 1 || i.fT[cubicIndex][last - 1] < 1 - CLOSE_ENOUGH) { |
| return false; |
| } |
| pt = xy_at_t(cubic, (i.fT[cubicIndex][last] + i.fT[cubicIndex][last - 1]) / 2); |
| return true; |
| } |
| |
| bool intersect3(const Cubic& c1, const Cubic& c2, Intersections& i) { |
| bool result = intersect3(c1, 0, 1, c2, 0, 1, 1, i); |
| // FIXME: pass in cached bounds from caller |
| _Rect c1Bounds, c2Bounds; |
| c1Bounds.setBounds(c1); // OPTIMIZE use setRawBounds ? |
| c2Bounds.setBounds(c2); |
| result |= intersectEnd(c1, false, c2, c2Bounds, i); |
| result |= intersectEnd(c1, true, c2, c2Bounds, i); |
| i.swap(); |
| result |= intersectEnd(c2, false, c1, c1Bounds, i); |
| result |= intersectEnd(c2, true, c1, c1Bounds, i); |
| i.swap(); |
| // If an end point and a second point very close to the end is returned, the second |
| // point may have been detected because the approximate quads |
| // intersected at the end and close to it. Verify that the second point is valid. |
| if (i.used() <= 1 || i.coincidentUsed()) { |
| return result; |
| } |
| _Point pt[2]; |
| if (closeStart(c1, 0, i, pt[0]) && closeStart(c2, 1, i, pt[1]) |
| && pt[0].approximatelyEqual(pt[1])) { |
| i.removeOne(1); |
| } |
| if (closeEnd(c1, 0, i, pt[0]) && closeEnd(c2, 1, i, pt[1]) |
| && pt[0].approximatelyEqual(pt[1])) { |
| i.removeOne(i.used() - 2); |
| } |
| return result; |
| } |
| |
| // Up promote the quad to a cubic. |
| // OPTIMIZATION If this is a common use case, optimize by duplicating |
| // the intersect 3 loop to avoid the promotion / demotion code |
| int intersect(const Cubic& cubic, const Quadratic& quad, Intersections& i) { |
| Cubic up; |
| toCubic(quad, up); |
| (void) intersect3(cubic, up, i); |
| return i.used(); |
| } |
| |
| /* http://www.ag.jku.at/compass/compasssample.pdf |
| ( Self-Intersection Problems and Approximate Implicitization by Jan B. Thomassen |
| Centre of Mathematics for Applications, University of Oslo http://www.cma.uio.no janbth@math.uio.no |
| SINTEF Applied Mathematics http://www.sintef.no ) |
| describes a method to find the self intersection of a cubic by taking the gradient of the implicit |
| form dotted with the normal, and solving for the roots. My math foo is too poor to implement this.*/ |
| |
| int intersect(const Cubic& c, Intersections& i) { |
| // check to see if x or y end points are the extrema. Are other quick rejects possible? |
| if ((between(c[0].x, c[1].x, c[3].x) && between(c[0].x, c[2].x, c[3].x)) |
| || (between(c[0].y, c[1].y, c[3].y) && between(c[0].y, c[2].y, c[3].y))) { |
| return false; |
| } |
| (void) intersect3(c, c, i); |
| return i.used(); |
| } |