| /* |
| * 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" |
| |
| static const double tClipLimit = 0.8; // http://cagd.cs.byu.edu/~tom/papers/bezclip.pdf see Multiple intersections |
| |
| class CubicIntersections : public Intersections { |
| public: |
| |
| CubicIntersections(const Cubic& c1, const Cubic& c2, Intersections& i) |
| : cubic1(c1) |
| , cubic2(c2) |
| , intersections(i) |
| , depth(0) |
| , splits(0) { |
| } |
| |
| bool intersect() { |
| double minT1, minT2, maxT1, maxT2; |
| if (!bezier_clip(cubic2, cubic1, minT1, maxT1)) { |
| return false; |
| } |
| if (!bezier_clip(cubic1, cubic2, minT2, maxT2)) { |
| return false; |
| } |
| int split; |
| if (maxT1 - minT1 < maxT2 - minT2) { |
| intersections.swap(); |
| minT2 = 0; |
| maxT2 = 1; |
| split = maxT1 - minT1 > tClipLimit; |
| } else { |
| minT1 = 0; |
| maxT1 = 1; |
| split = (maxT2 - minT2 > tClipLimit) << 1; |
| } |
| return chop(minT1, maxT1, minT2, maxT2, split); |
| } |
| |
| protected: |
| |
| bool intersect(double minT1, double maxT1, double minT2, double maxT2) { |
| Cubic smaller, larger; |
| // FIXME: carry last subdivide and reduceOrder result with cubic |
| sub_divide(cubic1, minT1, maxT1, intersections.swapped() ? larger : smaller); |
| sub_divide(cubic2, minT2, maxT2, intersections.swapped() ? smaller : larger); |
| Cubic smallResult; |
| if (reduceOrder(smaller, smallResult, |
| kReduceOrder_NoQuadraticsAllowed) <= 2) { |
| Cubic largeResult; |
| if (reduceOrder(larger, largeResult, |
| kReduceOrder_NoQuadraticsAllowed) <= 2) { |
| const _Line& smallLine = (const _Line&) smallResult; |
| const _Line& largeLine = (const _Line&) largeResult; |
| double smallT[2]; |
| double largeT[2]; |
| // FIXME: this doesn't detect or deal with coincident lines |
| if (!::intersect(smallLine, largeLine, smallT, largeT)) { |
| return false; |
| } |
| if (intersections.swapped()) { |
| smallT[0] = interp(minT2, maxT2, smallT[0]); |
| largeT[0] = interp(minT1, maxT1, largeT[0]); |
| } else { |
| smallT[0] = interp(minT1, maxT1, smallT[0]); |
| largeT[0] = interp(minT2, maxT2, largeT[0]); |
| } |
| intersections.add(smallT[0], largeT[0]); |
| return true; |
| } |
| } |
| double minT, maxT; |
| if (!bezier_clip(smaller, larger, minT, maxT)) { |
| if (minT == maxT) { |
| if (intersections.swapped()) { |
| minT1 = (minT1 + maxT1) / 2; |
| minT2 = interp(minT2, maxT2, minT); |
| } else { |
| minT1 = interp(minT1, maxT1, minT); |
| minT2 = (minT2 + maxT2) / 2; |
| } |
| intersections.add(minT1, minT2); |
| return true; |
| } |
| return false; |
| } |
| |
| int split; |
| if (intersections.swapped()) { |
| double newMinT1 = interp(minT1, maxT1, minT); |
| double newMaxT1 = interp(minT1, maxT1, maxT); |
| split = (newMaxT1 - newMinT1 > (maxT1 - minT1) * tClipLimit) << 1; |
| #define VERBOSE 0 |
| #if VERBOSE |
| printf("%s d=%d s=%d new1=(%g,%g) old1=(%g,%g) split=%d\n", |
| __FUNCTION__, depth, splits, newMinT1, newMaxT1, minT1, maxT1, |
| split); |
| #endif |
| minT1 = newMinT1; |
| maxT1 = newMaxT1; |
| } else { |
| double newMinT2 = interp(minT2, maxT2, minT); |
| double newMaxT2 = interp(minT2, maxT2, maxT); |
| split = newMaxT2 - newMinT2 > (maxT2 - minT2) * tClipLimit; |
| #if VERBOSE |
| printf("%s d=%d s=%d new2=(%g,%g) old2=(%g,%g) split=%d\n", |
| __FUNCTION__, depth, splits, newMinT2, newMaxT2, minT2, maxT2, |
| split); |
| #endif |
| minT2 = newMinT2; |
| maxT2 = newMaxT2; |
| } |
| return chop(minT1, maxT1, minT2, maxT2, split); |
| } |
| |
| bool chop(double minT1, double maxT1, double minT2, double maxT2, int split) { |
| ++depth; |
| intersections.swap(); |
| if (split) { |
| ++splits; |
| if (split & 2) { |
| double middle1 = (maxT1 + minT1) / 2; |
| intersect(minT1, middle1, minT2, maxT2); |
| intersect(middle1, maxT1, minT2, maxT2); |
| } else { |
| double middle2 = (maxT2 + minT2) / 2; |
| intersect(minT1, maxT1, minT2, middle2); |
| intersect(minT1, maxT1, middle2, maxT2); |
| } |
| --splits; |
| intersections.swap(); |
| --depth; |
| return intersections.intersected(); |
| } |
| bool result = intersect(minT1, maxT1, minT2, maxT2); |
| intersections.swap(); |
| --depth; |
| return result; |
| } |
| |
| private: |
| |
| const Cubic& cubic1; |
| const Cubic& cubic2; |
| Intersections& intersections; |
| int depth; |
| int splits; |
| }; |
| |
| bool intersect(const Cubic& c1, const Cubic& c2, Intersections& i) { |
| CubicIntersections c(c1, c2, i); |
| return c.intersect(); |
| } |
| |
| #include "CubicUtilities.h" |
| |
| static void cubicTangent(const Cubic& cubic, double t, _Line& tangent, _Point& pt, _Point& dxy) { |
| xy_at_t(cubic, t, tangent[0].x, tangent[0].y); |
| pt = tangent[1] = tangent[0]; |
| dxdy_at_t(cubic, t, dxy); |
| tangent[0] -= dxy; |
| tangent[1] += dxy; |
| } |
| |
| static double cubicDelta(const _Point& dxy, _Line& tangent, double scale) { |
| double tangentLen = dxy.length(); |
| tangent[0] -= tangent[1]; |
| double intersectLen = tangent[0].length(); |
| double result = intersectLen / tangentLen + scale; |
| return result; |
| } |
| |
| // FIXME: after testing, make this static |
| void computeDelta(const Cubic& c1, double t1, double scale1, const Cubic& c2, double t2, |
| double scale2, double& delta1, double& delta2) { |
| _Line tangent1, tangent2, line1, line2; |
| _Point dxy1, dxy2; |
| cubicTangent(c1, t1, line1, tangent1[0], dxy1); |
| cubicTangent(c2, t2, line2, tangent2[0], dxy2); |
| double range1[2], range2[2]; |
| int found = intersect(line1, line2, range1, range2); |
| if (found == 0) { |
| range1[0] = 0.5; |
| } else { |
| SkASSERT(found == 1); |
| } |
| xy_at_t(line1, range1[0], tangent1[1].x, tangent1[1].y); |
| #if SK_DEBUG |
| if (found == 1) { |
| xy_at_t(line2, range2[0], tangent2[1].x, tangent2[1].y); |
| SkASSERT(tangent2[1].approximatelyEqual(tangent1[1])); |
| } |
| #endif |
| tangent2[1] = tangent1[1]; |
| delta1 = cubicDelta(dxy1, tangent1, scale1 / precisionUnit); |
| delta2 = cubicDelta(dxy2, tangent2, scale2 / precisionUnit); |
| } |
| |
| #if SK_DEBUG |
| int debugDepth; |
| #endif |
| |
| // 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; |
| SkASSERT(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(); |
| } |
| |
| static bool intersectEnd(const Cubic& cubic1, bool start, const Cubic& cubic2, const _Rect& bounds2, |
| Intersections& i) { |
| _Line line1; |
| line1[0] = line1[1] = cubic1[start ? 0 : 3]; |
| _Point dxy1 = line1[0] - cubic1[start ? 1 : 2]; |
| dxy1 /= precisionUnit; |
| line1[1] += dxy1; |
| _Rect line1Bounds; |
| line1Bounds.setBounds(line1); |
| if (!bounds2.intersects(line1Bounds)) { |
| return false; |
| } |
| _Line line2; |
| line2[0] = line2[1] = line1[0]; |
| _Point dxy2 = line2[0] - cubic1[start ? 3 : 0]; |
| dxy2 /= precisionUnit; |
| line2[1] += dxy2; |
| #if 0 // this is so close to the first bounds test it isn't worth the short circuit test |
| _Rect line2Bounds; |
| line2Bounds.setBounds(line2); |
| if (!bounds2.intersects(line2Bounds)) { |
| return false; |
| } |
| #endif |
| Intersections local1; |
| if (!intersect(cubic2, line1, local1)) { |
| return false; |
| } |
| Intersections local2; |
| if (!intersect(cubic2, line2, local2)) { |
| return false; |
| } |
| double tMin, tMax; |
| tMin = tMax = local1.fT[0][0]; |
| for (int index = 1; index < local1.fUsed; ++index) { |
| tMin = SkTMin(tMin, local1.fT[0][index]); |
| tMax = SkTMax(tMax, local1.fT[0][index]); |
| } |
| for (int index = 1; index < local2.fUsed; ++index) { |
| tMin = SkTMin(tMin, local2.fT[0][index]); |
| tMax = SkTMax(tMax, local2.fT[0][index]); |
| } |
| #if SK_DEBUG |
| debugDepth = 0; |
| #endif |
| return intersect2(cubic1, start ? 0 : 1, start ? 1.0 / precisionUnit : 1 - 1.0 / precisionUnit, |
| cubic2, tMin, tMax, 1, i); |
| } |
| |
| // FIXME: add intersection of convex null 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) { |
| #if SK_DEBUG |
| debugDepth = 0; |
| #endif |
| 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; |
| } |
| |
| int intersect(const Cubic& cubic, const Quadratic& quad, Intersections& i) { |
| SkTDArray<double> ts; |
| double precision = calcPrecision(cubic); |
| cubic_to_quadratics(cubic, precision, ts); |
| double tStart = 0; |
| Cubic part; |
| int tsCount = ts.count(); |
| for (int idx = 0; idx <= tsCount; ++idx) { |
| double t = idx < tsCount ? ts[idx] : 1; |
| Quadratic q1; |
| sub_divide(cubic, tStart, t, part); |
| demote_cubic_to_quad(part, q1); |
| Intersections locals; |
| intersect2(q1, quad, locals); |
| for (int tIdx = 0; tIdx < locals.used(); ++tIdx) { |
| double globalT = tStart + (t - tStart) * locals.fT[0][tIdx]; |
| i.insertOne(globalT, 0); |
| globalT = locals.fT[1][tIdx]; |
| i.insertOne(globalT, 1); |
| } |
| tStart = t; |
| } |
| return i.used(); |
| } |
| |
| 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(); |
| } |
