| #include "CurveIntersection.h" |
| #include "Intersections.h" |
| #include "IntersectionUtilities.h" |
| #include "LineIntersection.h" |
| |
| 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; |
| } |
| 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) { |
| 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); |
| Quadratic smallResult; |
| if (reduceOrder(smaller, smallResult) <= 2) { |
| Quadratic largeResult; |
| if (reduceOrder(larger, largeResult) <= 2) { |
| double smallT[2], largeT[2]; |
| const _Line& smallLine = (const _Line&) smallResult; |
| const _Line& largeLine = (const _Line&) largeResult; |
| // 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: |
| |
| 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; |
| }; |
| |
| 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.fT[0][0] = t; |
| i.fT[1][0] = 0; |
| i.fUsed++; |
| } |
| if ((t = axialIntersect(q1, q2[2], useVertical)) >= 0) { |
| i.fT[0][i.fUsed] = t; |
| i.fT[1][i.fUsed] = 1; |
| i.fUsed++; |
| } |
| useVertical = fabs(q2[0].x - q2[2].x) < fabs(q2[0].y - q2[2].y); |
| if ((t = axialIntersect(q2, q1[0], useVertical)) >= 0) { |
| i.fT[0][i.fUsed] = 0; |
| i.fT[1][i.fUsed] = t; |
| i.fUsed++; |
| } |
| if ((t = axialIntersect(q2, q1[2], useVertical)) >= 0) { |
| i.fT[0][i.fUsed] = 1; |
| i.fT[1][i.fUsed] = t; |
| i.fUsed++; |
| } |
| assert(i.fUsed <= 2); |
| return i.fUsed > 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. |
| */ |