| /* |
| * 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 "CurveIntersection.h" |
| #include "Intersections.h" |
| #include "IntersectionUtilities.h" |
| #include "LineIntersection.h" |
| #include "LineUtilities.h" |
| #include "QuadraticLineSegments.h" |
| #include "QuadraticUtilities.h" |
| #include <algorithm> // for swap |
| |
| class QuadraticIntersections : public Intersections { |
| public: |
| |
| QuadraticIntersections(const Quadratic& q1, const Quadratic& q2, Intersections& i) |
| : quad1(q1) |
| , quad2(q2) |
| , intersections(i) |
| , depth(0) |
| , splits(0) { |
| } |
| |
| bool intersect() { |
| double minT1, minT2, maxT1, maxT2; |
| if (!bezier_clip(quad2, quad1, minT1, maxT1)) { |
| return false; |
| } |
| if (!bezier_clip(quad1, quad2, minT2, maxT2)) { |
| return false; |
| } |
| quad1Divisions = 1 / subDivisions(quad1); |
| quad2Divisions = 1 / subDivisions(quad2); |
| 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) { |
| bool t1IsLine = maxT1 - minT1 <= quad1Divisions; |
| bool t2IsLine = maxT2 - minT2 <= quad2Divisions; |
| if (t1IsLine | t2IsLine) { |
| return intersectAsLine(minT1, maxT1, minT2, maxT2, t1IsLine, t2IsLine); |
| } |
| Quadratic smaller, larger; |
| // FIXME: carry last subdivide and reduceOrder result with quad |
| sub_divide(quad1, minT1, maxT1, intersections.swapped() ? larger : smaller); |
| sub_divide(quad2, minT2, maxT2, intersections.swapped() ? smaller : larger); |
| double minT, maxT; |
| if (!bezier_clip(smaller, larger, minT, maxT)) { |
| if (approximately_equal(minT, maxT)) { |
| double smallT, largeT; |
| _Point q2pt, q1pt; |
| if (intersections.swapped()) { |
| largeT = interp(minT2, maxT2, minT); |
| xy_at_t(quad2, largeT, q2pt.x, q2pt.y); |
| xy_at_t(quad1, minT1, q1pt.x, q1pt.y); |
| if (approximately_equal(q2pt.x, q1pt.x) && approximately_equal(q2pt.y, q1pt.y)) { |
| smallT = minT1; |
| } else { |
| xy_at_t(quad1, maxT1, q1pt.x, q1pt.y); // FIXME: debug code |
| assert(approximately_equal(q2pt.x, q1pt.x) && approximately_equal(q2pt.y, q1pt.y)); |
| smallT = maxT1; |
| } |
| } else { |
| smallT = interp(minT1, maxT1, minT); |
| xy_at_t(quad1, smallT, q1pt.x, q1pt.y); |
| xy_at_t(quad2, minT2, q2pt.x, q2pt.y); |
| if (approximately_equal(q2pt.x, q1pt.x) && approximately_equal(q2pt.y, q1pt.y)) { |
| largeT = minT2; |
| } else { |
| xy_at_t(quad2, maxT2, q2pt.x, q2pt.y); // FIXME: debug code |
| assert(approximately_equal(q2pt.x, q1pt.x) && approximately_equal(q2pt.y, q1pt.y)); |
| largeT = maxT2; |
| } |
| } |
| intersections.add(smallT, largeT); |
| 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 intersectAsLine(double minT1, double maxT1, double minT2, double maxT2, |
| bool treat1AsLine, bool treat2AsLine) |
| { |
| _Line line1, line2; |
| if (intersections.swapped()) { |
| std::swap(treat1AsLine, treat2AsLine); |
| std::swap(minT1, minT2); |
| std::swap(maxT1, maxT2); |
| } |
| // do line/quadratic or even line/line intersection instead |
| if (treat1AsLine) { |
| xy_at_t(quad1, minT1, line1[0].x, line1[0].y); |
| xy_at_t(quad1, maxT1, line1[1].x, line1[1].y); |
| } |
| if (treat2AsLine) { |
| xy_at_t(quad2, minT2, line2[0].x, line2[0].y); |
| xy_at_t(quad2, maxT2, line2[1].x, line2[1].y); |
| } |
| int pts; |
| double smallT, largeT; |
| if (treat1AsLine & treat2AsLine) { |
| double t1[2], t2[2]; |
| pts = ::intersect(line1, line2, t1, t2); |
| for (int index = 0; index < pts; ++index) { |
| smallT = interp(minT1, maxT1, t1[index]); |
| largeT = interp(minT2, maxT2, t2[index]); |
| if (pts == 2) { |
| intersections.addCoincident(smallT, largeT, true); |
| } else { |
| intersections.add(smallT, largeT); |
| } |
| } |
| } else { |
| Intersections lq; |
| pts = ::intersect(treat1AsLine ? quad2 : quad1, |
| treat1AsLine ? line1 : line2, lq); |
| bool coincident = false; |
| if (pts == 2) { // if the line and edge are coincident treat differently |
| _Point midQuad, midLine; |
| double midQuadT = (lq.fT[0][0] + lq.fT[0][1]) / 2; |
| xy_at_t(treat1AsLine ? quad2 : quad1, midQuadT, midQuad.x, midQuad.y); |
| double lineT = t_at(treat1AsLine ? line1 : line2, midQuad); |
| xy_at_t(treat1AsLine ? line1 : line2, lineT, midLine.x, midLine.y); |
| coincident = approximately_equal(midQuad.x, midLine.x) |
| && approximately_equal(midQuad.y, midLine.y); |
| } |
| for (int index = 0; index < pts; ++index) { |
| smallT = lq.fT[0][index]; |
| largeT = lq.fT[1][index]; |
| if (treat1AsLine) { |
| smallT = interp(minT1, maxT1, smallT); |
| } else { |
| largeT = interp(minT2, maxT2, largeT); |
| } |
| if (coincident) { |
| intersections.addCoincident(smallT, largeT, true); |
| } else { |
| intersections.add(smallT, largeT); |
| } |
| } |
| } |
| return pts > 0; |
| } |
| |
| 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: |
| |
| static const double tClipLimit = 0.8; // http://cagd.cs.byu.edu/~tom/papers/bezclip.pdf see Multiple intersections |
| const Quadratic& quad1; |
| const Quadratic& quad2; |
| Intersections& intersections; |
| int depth; |
| int splits; |
| double quad1Divisions; // line segments to approximate original within error |
| double quad2Divisions; |
| }; |
| |
| bool intersect(const Quadratic& q1, const Quadratic& q2, Intersections& i) { |
| if (implicit_matches(q1, q2)) { |
| // 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.addCoincident(t, 0, false); |
| } |
| if ((t = axialIntersect(q1, q2[2], useVertical)) >= 0) { |
| i.addCoincident(t, 1, false); |
| } |
| useVertical = fabs(q2[0].x - q2[2].x) < fabs(q2[0].y - q2[2].y); |
| if ((t = axialIntersect(q2, q1[0], useVertical)) >= 0) { |
| i.addCoincident(0, t, false); |
| } |
| if ((t = axialIntersect(q2, q1[2], useVertical)) >= 0) { |
| i.addCoincident(1, t, false); |
| } |
| assert(i.fCoincidentUsed <= 2); |
| return i.fCoincidentUsed > 0; |
| } |
| QuadraticIntersections q(q1, q2, i); |
| return q.intersect(); |
| } |
| |
| |
| // Another approach is to start with the implicit form of one curve and solve |
| // 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 |
| /* |
| given x^4 + ax^3 + bx^2 + cx + d |
| the resolvent cubic is x^3 - 2bx^2 + (b^2 + ac - 4d)x + (c^2 + a^2d - abc) |
| use the cubic formula (CubicRoots.cpp) to find the radical expressions t1, t2, and t3. |
| |
| (x - r1 r2) (x - r3 r4) = x^2 - (t2 + t3 - t1) / 2 x + d |
| s = r1*r2 = ((t2 + t3 - t1) + sqrt((t2 + t3 - t1)^2 - 16*d)) / 4 |
| t = r3*r4 = ((t2 + t3 - t1) - sqrt((t2 + t3 - t1)^2 - 16*d)) / 4 |
| |
| u = r1+r2 = (-a + sqrt(a^2 - 4*t1)) / 2 |
| v = r3+r4 = (-a - sqrt(a^2 - 4*t1)) / 2 |
| |
| r1 = (u + sqrt(u^2 - 4*s)) / 2 |
| r2 = (u - sqrt(u^2 - 4*s)) / 2 |
| r3 = (v + sqrt(v^2 - 4*t)) / 2 |
| r4 = (v - sqrt(v^2 - 4*t)) / 2 |
| */ |
| |
| |
| /* square root of complex number |
| http://en.wikipedia.org/wiki/Square_root#Square_roots_of_negative_and_complex_numbers |
| Algebraic formula |
| When the number is expressed using Cartesian coordinates the following formula |
| can be used for the principal square root:[5][6] |
| |
| sqrt(x + iy) = sqrt((r + x) / 2) +/- i*sqrt((r - x) / 2) |
| |
| where the sign of the imaginary part of the root is taken to be same as the sign |
| of the imaginary part of the original number, and |
| |
| r = abs(x + iy) = sqrt(x^2 + y^2) |
| |
| is the absolute value or modulus of the original number. The real part of the |
| principal value is always non-negative. |
| The other square root is simply –1 times the principal square root; in other |
| words, the two square roots of a number sum to 0. |
| */ |