| // Another approach is to start with the implicit form of one curve and solve |
| // (seek implicit coefficients in QuadraticParameter.cpp |
| // by substituting in the parametric form of the other. |
| // The downside of this approach is that early rejects are difficult to come by. |
| // http://planetmath.org/encyclopedia/GaloisTheoreticDerivationOfTheQuarticFormula.html#step |
| |
| |
| #include "CubicUtilities.h" |
| #include "CurveIntersection.h" |
| #include "Intersections.h" |
| #include "QuadraticParameterization.h" |
| #include "QuarticRoot.h" |
| #include "QuadraticUtilities.h" |
| #include "TSearch.h" |
| |
| #if SK_DEBUG |
| #include "LineUtilities.h" |
| #endif |
| |
| /* given the implicit form 0 = Ax^2 + Bxy + Cy^2 + Dx + Ey + F |
| * and given x = at^2 + bt + c (the parameterized form) |
| * y = dt^2 + et + f |
| * then |
| * 0 = A(at^2+bt+c)(at^2+bt+c)+B(at^2+bt+c)(dt^2+et+f)+C(dt^2+et+f)(dt^2+et+f)+D(at^2+bt+c)+E(dt^2+et+f)+F |
| */ |
| |
| static int findRoots(const QuadImplicitForm& i, const Quadratic& q2, double roots[4], |
| bool oneHint, int firstCubicRoot) { |
| double a, b, c; |
| set_abc(&q2[0].x, a, b, c); |
| double d, e, f; |
| set_abc(&q2[0].y, d, e, f); |
| const double t4 = i.x2() * a * a |
| + i.xy() * a * d |
| + i.y2() * d * d; |
| const double t3 = 2 * i.x2() * a * b |
| + i.xy() * (a * e + b * d) |
| + 2 * i.y2() * d * e; |
| const double t2 = i.x2() * (b * b + 2 * a * c) |
| + i.xy() * (c * d + b * e + a * f) |
| + i.y2() * (e * e + 2 * d * f) |
| + i.x() * a |
| + i.y() * d; |
| const double t1 = 2 * i.x2() * b * c |
| + i.xy() * (c * e + b * f) |
| + 2 * i.y2() * e * f |
| + i.x() * b |
| + i.y() * e; |
| const double t0 = i.x2() * c * c |
| + i.xy() * c * f |
| + i.y2() * f * f |
| + i.x() * c |
| + i.y() * f |
| + i.c(); |
| int rootCount = reducedQuarticRoots(t4, t3, t2, t1, t0, oneHint, roots); |
| if (rootCount >= 0) { |
| return rootCount; |
| } |
| return quarticRootsReal(firstCubicRoot, t4, t3, t2, t1, t0, roots); |
| } |
| |
| static int addValidRoots(const double roots[4], const int count, double valid[4]) { |
| int result = 0; |
| int index; |
| for (index = 0; index < count; ++index) { |
| if (!approximately_zero_or_more(roots[index]) || !approximately_one_or_less(roots[index])) { |
| continue; |
| } |
| double t = 1 - roots[index]; |
| if (approximately_less_than_zero(t)) { |
| t = 0; |
| } else if (approximately_greater_than_one(t)) { |
| t = 1; |
| } |
| valid[result++] = t; |
| } |
| return result; |
| } |
| |
| static bool onlyEndPtsInCommon(const Quadratic& q1, const Quadratic& q2, Intersections& i) { |
| // the idea here is to see at minimum do a quick reject by rotating all points |
| // to either side of the line formed by connecting the endpoints |
| // if the opposite curves points are on the line or on the other side, the |
| // curves at most intersect at the endpoints |
| for (int oddMan = 0; oddMan < 3; ++oddMan) { |
| const _Point* endPt[2]; |
| for (int opp = 1; opp < 3; ++opp) { |
| int end = oddMan ^ opp; |
| if (end == 3) { |
| end = opp; |
| } |
| endPt[opp - 1] = &q1[end]; |
| } |
| double origX = endPt[0]->x; |
| double origY = endPt[0]->y; |
| double adj = endPt[1]->x - origX; |
| double opp = endPt[1]->y - origY; |
| double sign = (q1[oddMan].y - origY) * adj - (q1[oddMan].x - origX) * opp; |
| if (approximately_zero(sign)) { |
| goto tryNextHalfPlane; |
| } |
| for (int n = 0; n < 3; ++n) { |
| double test = (q2[n].y - origY) * adj - (q2[n].x - origX) * opp; |
| if (test * sign > 0) { |
| goto tryNextHalfPlane; |
| } |
| } |
| for (int i1 = 0; i1 < 3; i1 += 2) { |
| for (int i2 = 0; i2 < 3; i2 += 2) { |
| if (q1[i1] == q2[i2]) { |
| i.insert(i1 >> 1, i2 >> 1, q1[i1]); |
| } |
| } |
| } |
| SkASSERT(i.fUsed < 3); |
| return true; |
| tryNextHalfPlane: |
| ; |
| } |
| return false; |
| } |
| |
| // 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 addIntercept(const Quadratic& q1, const Quadratic& q2, double tMin, double tMax, |
| Intersections& i, bool* subDivide) { |
| double tMid = (tMin + tMax) / 2; |
| _Point mid; |
| xy_at_t(q2, tMid, mid.x, mid.y); |
| _Line line; |
| line[0] = line[1] = mid; |
| _Vector dxdy = dxdy_at_t(q2, tMid); |
| line[0] -= dxdy; |
| line[1] += dxdy; |
| Intersections rootTs; |
| int roots = intersect(q1, line, rootTs); |
| if (roots == 0) { |
| if (subDivide) { |
| *subDivide = true; |
| } |
| return true; |
| } |
| if (roots == 2) { |
| return false; |
| } |
| _Point pt2; |
| xy_at_t(q1, rootTs.fT[0][0], pt2.x, pt2.y); |
| if (!pt2.approximatelyEqualHalf(mid)) { |
| return false; |
| } |
| i.insertSwap(rootTs.fT[0][0], tMid, pt2); |
| return true; |
| } |
| |
| static bool isLinearInner(const Quadratic& q1, double t1s, double t1e, const Quadratic& q2, |
| double t2s, double t2e, Intersections& i, bool* subDivide) { |
| Quadratic hull; |
| sub_divide(q1, t1s, t1e, hull); |
| _Line line = {hull[2], hull[0]}; |
| const _Line* testLines[] = { &line, (const _Line*) &hull[0], (const _Line*) &hull[1] }; |
| size_t testCount = sizeof(testLines) / sizeof(testLines[0]); |
| SkTDArray<double> tsFound; |
| for (size_t index = 0; index < testCount; ++index) { |
| Intersections rootTs; |
| int roots = intersect(q2, *testLines[index], rootTs); |
| for (int idx2 = 0; idx2 < roots; ++idx2) { |
| double t = rootTs.fT[0][idx2]; |
| #if SK_DEBUG |
| _Point qPt, lPt; |
| xy_at_t(q2, t, qPt.x, qPt.y); |
| xy_at_t(*testLines[index], rootTs.fT[1][idx2], lPt.x, lPt.y); |
| SkASSERT(qPt.approximatelyEqual(lPt)); |
| #endif |
| if (approximately_negative(t - t2s) || approximately_positive(t - t2e)) { |
| continue; |
| } |
| *tsFound.append() = rootTs.fT[0][idx2]; |
| } |
| } |
| int tCount = tsFound.count(); |
| if (!tCount) { |
| return true; |
| } |
| double tMin, tMax; |
| if (tCount == 1) { |
| tMin = tMax = tsFound[0]; |
| } else if (tCount > 1) { |
| QSort<double>(tsFound.begin(), tsFound.end() - 1); |
| tMin = tsFound[0]; |
| tMax = tsFound[tsFound.count() - 1]; |
| } |
| _Point end; |
| xy_at_t(q2, t2s, end.x, end.y); |
| bool startInTriangle = point_in_hull(hull, end); |
| if (startInTriangle) { |
| tMin = t2s; |
| } |
| xy_at_t(q2, t2e, end.x, end.y); |
| bool endInTriangle = point_in_hull(hull, end); |
| if (endInTriangle) { |
| tMax = t2e; |
| } |
| int split = 0; |
| _Vector dxy1, dxy2; |
| if (tMin != tMax || tCount > 2) { |
| dxy2 = dxdy_at_t(q2, tMin); |
| for (int index = 1; index < tCount; ++index) { |
| dxy1 = dxy2; |
| dxy2 = dxdy_at_t(q2, tsFound[index]); |
| double dot = dxy1.dot(dxy2); |
| if (dot < 0) { |
| split = index - 1; |
| break; |
| } |
| } |
| |
| } |
| if (split == 0) { // there's one point |
| if (addIntercept(q1, q2, tMin, tMax, i, subDivide)) { |
| return true; |
| } |
| i.swap(); |
| return isLinearInner(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 (addIntercept(q1, q2, tMin, tsFound[split - 1], i, subDivide)) { |
| result = true; |
| } else { |
| i.swap(); |
| result = isLinearInner(q2, tMin, tsFound[split - 1], q1, t1s, t1e, i, subDivide); |
| } |
| if (addIntercept(q1, q2, tsFound[split], tMax, i, subDivide)) { |
| result = true; |
| } else { |
| i.swap(); |
| result |= isLinearInner(q2, tsFound[split], tMax, q1, t1s, t1e, i, subDivide); |
| } |
| return result; |
| } |
| |
| static double flatMeasure(const Quadratic& q) { |
| _Vector mid = q[1] - q[0]; |
| _Vector dxy = q[2] - q[0]; |
| double length = dxy.length(); // OPTIMIZE: get rid of sqrt |
| return fabs(mid.cross(dxy) / length); |
| } |
| |
| // FIXME ? should this measure both and then use the quad that is the flattest as the line? |
| static bool isLinear(const Quadratic& q1, const Quadratic& q2, Intersections& i) { |
| double measure = flatMeasure(q1); |
| // OPTIMIZE: (get rid of sqrt) use approximately_zero |
| if (!approximately_zero_sqrt(measure)) { |
| return false; |
| } |
| return isLinearInner(q1, 0, 1, q2, 0, 1, i, NULL); |
| } |
| |
| // FIXME: if flat measure is sufficiently large, then probably the quartic solution failed |
| static void relaxedIsLinear(const Quadratic& q1, const Quadratic& q2, Intersections& i) { |
| double m1 = flatMeasure(q1); |
| double m2 = flatMeasure(q2); |
| #if SK_DEBUG |
| double min = SkTMin(m1, m2); |
| if (min > 5) { |
| SkDebugf("%s maybe not flat enough.. %1.9g\n", __FUNCTION__, min); |
| } |
| #endif |
| i.reset(); |
| const Quadratic& rounder = m2 < m1 ? q1 : q2; |
| const Quadratic& flatter = m2 < m1 ? q2 : q1; |
| bool subDivide = false; |
| isLinearInner(flatter, 0, 1, rounder, 0, 1, i, &subDivide); |
| if (subDivide) { |
| QuadraticPair pair; |
| chop_at(flatter, pair, 0.5); |
| Intersections firstI, secondI; |
| relaxedIsLinear(pair.first(), rounder, firstI); |
| for (int index = 0; index < firstI.used(); ++index) { |
| i.insert(firstI.fT[0][index] * 0.5, firstI.fT[1][index], firstI.fPt[index]); |
| } |
| relaxedIsLinear(pair.second(), rounder, secondI); |
| for (int index = 0; index < secondI.used(); ++index) { |
| i.insert(0.5 + secondI.fT[0][index] * 0.5, secondI.fT[1][index], secondI.fPt[index]); |
| } |
| } |
| if (m2 < m1) { |
| i.swapPts(); |
| } |
| } |
| |
| #if 0 |
| static void unsortableExpanse(const Quadratic& q1, const Quadratic& q2, Intersections& i) { |
| const Quadratic* qs[2] = { &q1, &q2 }; |
| // need t values for start and end of unsortable expanse on both curves |
| // try projecting lines parallel to the end points |
| i.fT[0][0] = 0; |
| i.fT[0][1] = 1; |
| int flip = -1; // undecided |
| for (int qIdx = 0; qIdx < 2; qIdx++) { |
| for (int t = 0; t < 2; t++) { |
| _Point dxdy; |
| dxdy_at_t(*qs[qIdx], t, dxdy); |
| _Line perp; |
| perp[0] = perp[1] = (*qs[qIdx])[t == 0 ? 0 : 2]; |
| perp[0].x += dxdy.y; |
| perp[0].y -= dxdy.x; |
| perp[1].x -= dxdy.y; |
| perp[1].y += dxdy.x; |
| Intersections hitData; |
| int hits = intersectRay(*qs[qIdx ^ 1], perp, hitData); |
| SkASSERT(hits <= 1); |
| if (hits) { |
| if (flip < 0) { |
| _Point dxdy2; |
| dxdy_at_t(*qs[qIdx ^ 1], hitData.fT[0][0], dxdy2); |
| double dot = dxdy.dot(dxdy2); |
| flip = dot < 0; |
| i.fT[1][0] = flip; |
| i.fT[1][1] = !flip; |
| } |
| i.fT[qIdx ^ 1][t ^ flip] = hitData.fT[0][0]; |
| } |
| } |
| } |
| i.fUnsortable = true; // failed, probably coincident or near-coincident |
| i.fUsed = 2; |
| } |
| #endif |
| |
| // each time through the loop, this computes values it had from the last loop |
| // if i == j == 1, the center values are still good |
| // otherwise, for i != 1 or j != 1, four of the values are still good |
| // and if i == 1 ^ j == 1, an additional value is good |
| static bool binarySearch(const Quadratic& quad1, const Quadratic& quad2, double& t1Seed, |
| double& t2Seed, _Point& pt) { |
| double tStep = ROUGH_EPSILON; |
| _Point t1[3], t2[3]; |
| int calcMask = ~0; |
| do { |
| if (calcMask & (1 << 1)) t1[1] = xy_at_t(quad1, t1Seed); |
| if (calcMask & (1 << 4)) t2[1] = xy_at_t(quad2, t2Seed); |
| if (t1[1].approximatelyEqual(t2[1])) { |
| pt = t1[1]; |
| #if ONE_OFF_DEBUG |
| SkDebugf("%s t1=%1.9g t2=%1.9g (%1.9g,%1.9g) == (%1.9g,%1.9g)\n", __FUNCTION__, |
| t1Seed, t2Seed, t1[1].x, t1[1].y, t1[2].x, t1[2].y); |
| #endif |
| return true; |
| } |
| if (calcMask & (1 << 0)) t1[0] = xy_at_t(quad1, t1Seed - tStep); |
| if (calcMask & (1 << 2)) t1[2] = xy_at_t(quad1, t1Seed + tStep); |
| if (calcMask & (1 << 3)) t2[0] = xy_at_t(quad2, t2Seed - tStep); |
| if (calcMask & (1 << 5)) t2[2] = xy_at_t(quad2, t2Seed + tStep); |
| double dist[3][3]; |
| // OPTIMIZE: using calcMask value permits skipping some distance calcuations |
| // if prior loop's results are moved to correct slot for reuse |
| dist[1][1] = t1[1].distanceSquared(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].distanceSquared(t2[j]); |
| if (dist[best_i][best_j] > dist[i][j]) { |
| best_i = i; |
| best_j = j; |
| } |
| } |
| } |
| if (best_i == 1 && best_j == 1) { |
| tStep /= 2; |
| if (tStep < FLT_EPSILON_HALF) { |
| break; |
| } |
| calcMask = (1 << 0) | (1 << 2) | (1 << 3) | (1 << 5); |
| continue; |
| } |
| if (best_i == 0) { |
| t1Seed -= tStep; |
| t1[2] = t1[1]; |
| t1[1] = t1[0]; |
| calcMask = 1 << 0; |
| } else if (best_i == 2) { |
| t1Seed += tStep; |
| t1[0] = t1[1]; |
| t1[1] = t1[2]; |
| calcMask = 1 << 2; |
| } else { |
| calcMask = 0; |
| } |
| if (best_j == 0) { |
| t2Seed -= tStep; |
| t2[2] = t2[1]; |
| t2[1] = t2[0]; |
| calcMask |= 1 << 3; |
| } else if (best_j == 2) { |
| t2Seed += tStep; |
| t2[0] = t2[1]; |
| t2[1] = t2[2]; |
| calcMask |= 1 << 5; |
| } |
| } while (true); |
| #if ONE_OFF_DEBUG |
| SkDebugf("%s t1=%1.9g t2=%1.9g (%1.9g,%1.9g) != (%1.9g,%1.9g) %s\n", __FUNCTION__, |
| t1Seed, t2Seed, t1[1].x, t1[1].y, t1[2].x, t1[2].y); |
| #endif |
| return false; |
| } |
| |
| bool intersect2(const Quadratic& q1, const Quadratic& q2, Intersections& i) { |
| // if the quads share an end point, check to see if they overlap |
| |
| if (onlyEndPtsInCommon(q1, q2, i)) { |
| return i.intersected(); |
| } |
| if (onlyEndPtsInCommon(q2, q1, i)) { |
| i.swapPts(); |
| return i.intersected(); |
| } |
| // see if either quad is really a line |
| if (isLinear(q1, q2, i)) { |
| return i.intersected(); |
| } |
| if (isLinear(q2, q1, i)) { |
| i.swapPts(); |
| return i.intersected(); |
| } |
| QuadImplicitForm i1(q1); |
| QuadImplicitForm i2(q2); |
| if (i1.implicit_match(i2)) { |
| // FIXME: compute T values |
| // compute the intersections of the ends to find the coincident span |
| bool useVertical = fabs(q1[0].x - q1[2].x) < fabs(q1[0].y - q1[2].y); |
| double t; |
| if ((t = axialIntersect(q1, q2[0], useVertical)) >= 0) { |
| i.insertCoincident(t, 0, q2[0]); |
| } |
| if ((t = axialIntersect(q1, q2[2], useVertical)) >= 0) { |
| i.insertCoincident(t, 1, q2[2]); |
| } |
| useVertical = fabs(q2[0].x - q2[2].x) < fabs(q2[0].y - q2[2].y); |
| if ((t = axialIntersect(q2, q1[0], useVertical)) >= 0) { |
| i.insertCoincident(0, t, q1[0]); |
| } |
| if ((t = axialIntersect(q2, q1[2], useVertical)) >= 0) { |
| i.insertCoincident(1, t, q1[2]); |
| } |
| SkASSERT(i.coincidentUsed() <= 2); |
| return i.coincidentUsed() > 0; |
| } |
| int index; |
| bool useCubic = q1[0] == q2[0] || q1[0] == q2[2] || q1[2] == q2[0]; |
| double roots1[4]; |
| int rootCount = findRoots(i2, q1, roots1, useCubic, 0); |
| // OPTIMIZATION: could short circuit here if all roots are < 0 or > 1 |
| double roots1Copy[4]; |
| int r1Count = addValidRoots(roots1, rootCount, roots1Copy); |
| _Point pts1[4]; |
| for (index = 0; index < r1Count; ++index) { |
| xy_at_t(q1, roots1Copy[index], pts1[index].x, pts1[index].y); |
| } |
| double roots2[4]; |
| int rootCount2 = findRoots(i1, q2, roots2, useCubic, 0); |
| double roots2Copy[4]; |
| int r2Count = addValidRoots(roots2, rootCount2, roots2Copy); |
| _Point pts2[4]; |
| for (index = 0; index < r2Count; ++index) { |
| xy_at_t(q2, roots2Copy[index], pts2[index].x, pts2[index].y); |
| } |
| if (r1Count == r2Count && r1Count <= 1) { |
| if (r1Count == 1) { |
| if (pts1[0].approximatelyEqualHalf(pts2[0])) { |
| i.insert(roots1Copy[0], roots2Copy[0], pts1[0]); |
| } else if (pts1[0].moreRoughlyEqual(pts2[0])) { |
| // experiment: see if a different cubic solution provides the correct quartic answer |
| #if 0 |
| for (int cu1 = 0; cu1 < 3; ++cu1) { |
| rootCount = findRoots(i2, q1, roots1, useCubic, cu1); |
| r1Count = addValidRoots(roots1, rootCount, roots1Copy); |
| if (r1Count == 0) { |
| continue; |
| } |
| for (int cu2 = 0; cu2 < 3; ++cu2) { |
| if (cu1 == 0 && cu2 == 0) { |
| continue; |
| } |
| rootCount2 = findRoots(i1, q2, roots2, useCubic, cu2); |
| r2Count = addValidRoots(roots2, rootCount2, roots2Copy); |
| if (r2Count == 0) { |
| continue; |
| } |
| SkASSERT(r1Count == 1 && r2Count == 1); |
| SkDebugf("*** [%d,%d] (%1.9g,%1.9g) %s (%1.9g,%1.9g)\n", cu1, cu2, |
| pts1[0].x, pts1[0].y, pts1[0].approximatelyEqualHalf(pts2[0]) |
| ? "==" : "!=", pts2[0].x, pts2[0].y); |
| } |
| } |
| #endif |
| // experiment: try to find intersection by chasing t |
| rootCount = findRoots(i2, q1, roots1, useCubic, 0); |
| r1Count = addValidRoots(roots1, rootCount, roots1Copy); |
| rootCount2 = findRoots(i1, q2, roots2, useCubic, 0); |
| r2Count = addValidRoots(roots2, rootCount2, roots2Copy); |
| if (binarySearch(q1, q2, roots1Copy[0], roots2Copy[0], pts1[0])) { |
| i.insert(roots1Copy[0], roots2Copy[0], pts1[0]); |
| } |
| } |
| } |
| return i.intersected(); |
| } |
| int closest[4]; |
| double dist[4]; |
| bool foundSomething = false; |
| for (index = 0; index < r1Count; ++index) { |
| dist[index] = DBL_MAX; |
| closest[index] = -1; |
| for (int ndex2 = 0; ndex2 < r2Count; ++ndex2) { |
| if (!pts2[ndex2].approximatelyEqualHalf(pts1[index])) { |
| continue; |
| } |
| double dx = pts2[ndex2].x - pts1[index].x; |
| double dy = pts2[ndex2].y - pts1[index].y; |
| double distance = dx * dx + dy * dy; |
| if (dist[index] <= distance) { |
| continue; |
| } |
| for (int outer = 0; outer < index; ++outer) { |
| if (closest[outer] != ndex2) { |
| continue; |
| } |
| if (dist[outer] < distance) { |
| goto next; |
| } |
| closest[outer] = -1; |
| } |
| dist[index] = distance; |
| closest[index] = ndex2; |
| foundSomething = true; |
| next: |
| ; |
| } |
| } |
| if (r1Count && r2Count && !foundSomething) { |
| relaxedIsLinear(q1, q2, i); |
| return i.intersected(); |
| } |
| int used = 0; |
| do { |
| double lowest = DBL_MAX; |
| int lowestIndex = -1; |
| for (index = 0; index < r1Count; ++index) { |
| if (closest[index] < 0) { |
| continue; |
| } |
| if (roots1Copy[index] < lowest) { |
| lowestIndex = index; |
| lowest = roots1Copy[index]; |
| } |
| } |
| if (lowestIndex < 0) { |
| break; |
| } |
| i.insert(roots1Copy[lowestIndex], roots2Copy[closest[lowestIndex]], |
| pts1[lowestIndex]); |
| closest[lowestIndex] = -1; |
| } while (++used < r1Count); |
| i.fFlip = false; |
| return i.intersected(); |
| } |
