| /* |
| * 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 "src/pathops/SkIntersections.h" |
| #include "src/pathops/SkLineParameters.h" |
| #include "src/pathops/SkPathOpsCubic.h" |
| #include "src/pathops/SkPathOpsCurve.h" |
| #include "src/pathops/SkPathOpsQuad.h" |
| #include "src/pathops/SkPathOpsRect.h" |
| |
| // from blackpawn.com/texts/pointinpoly |
| static bool pointInTriangle(const SkDPoint fPts[3], const SkDPoint& test) { |
| SkDVector v0 = fPts[2] - fPts[0]; |
| SkDVector v1 = fPts[1] - fPts[0]; |
| SkDVector v2 = test - fPts[0]; |
| double dot00 = v0.dot(v0); |
| double dot01 = v0.dot(v1); |
| double dot02 = v0.dot(v2); |
| double dot11 = v1.dot(v1); |
| double dot12 = v1.dot(v2); |
| // Compute barycentric coordinates |
| double denom = dot00 * dot11 - dot01 * dot01; |
| double u = dot11 * dot02 - dot01 * dot12; |
| double v = dot00 * dot12 - dot01 * dot02; |
| // Check if point is in triangle |
| if (denom >= 0) { |
| return u >= 0 && v >= 0 && u + v < denom; |
| } |
| return u <= 0 && v <= 0 && u + v > denom; |
| } |
| |
| static bool matchesEnd(const SkDPoint fPts[3], const SkDPoint& test) { |
| return fPts[0] == test || fPts[2] == test; |
| } |
| |
| /* started with at_most_end_pts_in_common from SkDQuadIntersection.cpp */ |
| // Do a quick reject by rotating all points relative to a line formed by |
| // a pair of one quad's points. If the 2nd quad's points |
| // are on the line or on the opposite side from the 1st quad's 'odd man', the |
| // curves at most intersect at the endpoints. |
| /* if returning true, check contains true if quad's hull collapsed, making the cubic linear |
| if returning false, check contains true if the the quad pair have only the end point in common |
| */ |
| bool SkDQuad::hullIntersects(const SkDQuad& q2, bool* isLinear) const { |
| bool linear = true; |
| for (int oddMan = 0; oddMan < kPointCount; ++oddMan) { |
| const SkDPoint* endPt[2]; |
| this->otherPts(oddMan, endPt); |
| double origX = endPt[0]->fX; |
| double origY = endPt[0]->fY; |
| double adj = endPt[1]->fX - origX; |
| double opp = endPt[1]->fY - origY; |
| double sign = (fPts[oddMan].fY - origY) * adj - (fPts[oddMan].fX - origX) * opp; |
| if (approximately_zero(sign)) { |
| continue; |
| } |
| linear = false; |
| bool foundOutlier = false; |
| for (int n = 0; n < kPointCount; ++n) { |
| double test = (q2[n].fY - origY) * adj - (q2[n].fX - origX) * opp; |
| if (test * sign > 0 && !precisely_zero(test)) { |
| foundOutlier = true; |
| break; |
| } |
| } |
| if (!foundOutlier) { |
| return false; |
| } |
| } |
| if (linear && !matchesEnd(fPts, q2.fPts[0]) && !matchesEnd(fPts, q2.fPts[2])) { |
| // if the end point of the opposite quad is inside the hull that is nearly a line, |
| // then representing the quad as a line may cause the intersection to be missed. |
| // Check to see if the endpoint is in the triangle. |
| if (pointInTriangle(fPts, q2.fPts[0]) || pointInTriangle(fPts, q2.fPts[2])) { |
| linear = false; |
| } |
| } |
| *isLinear = linear; |
| return true; |
| } |
| |
| bool SkDQuad::hullIntersects(const SkDConic& conic, bool* isLinear) const { |
| return conic.hullIntersects(*this, isLinear); |
| } |
| |
| bool SkDQuad::hullIntersects(const SkDCubic& cubic, bool* isLinear) const { |
| return cubic.hullIntersects(*this, isLinear); |
| } |
| |
| /* bit twiddling for finding the off curve index (x&~m is the pair in [0,1,2] excluding oddMan) |
| oddMan opp x=oddMan^opp x=x-oddMan m=x>>2 x&~m |
| 0 1 1 1 0 1 |
| 2 2 2 0 2 |
| 1 1 0 -1 -1 0 |
| 2 3 2 0 2 |
| 2 1 3 1 0 1 |
| 2 0 -2 -1 0 |
| */ |
| void SkDQuad::otherPts(int oddMan, const SkDPoint* endPt[2]) const { |
| for (int opp = 1; opp < kPointCount; ++opp) { |
| int end = (oddMan ^ opp) - oddMan; // choose a value not equal to oddMan |
| end &= ~(end >> 2); // if the value went negative, set it to zero |
| endPt[opp - 1] = &fPts[end]; |
| } |
| } |
| |
| int SkDQuad::AddValidTs(double s[], int realRoots, double* t) { |
| int foundRoots = 0; |
| for (int index = 0; index < realRoots; ++index) { |
| double tValue = s[index]; |
| if (approximately_zero_or_more(tValue) && approximately_one_or_less(tValue)) { |
| if (approximately_less_than_zero(tValue)) { |
| tValue = 0; |
| } else if (approximately_greater_than_one(tValue)) { |
| tValue = 1; |
| } |
| for (int idx2 = 0; idx2 < foundRoots; ++idx2) { |
| if (approximately_equal(t[idx2], tValue)) { |
| goto nextRoot; |
| } |
| } |
| t[foundRoots++] = tValue; |
| } |
| nextRoot: |
| {} |
| } |
| return foundRoots; |
| } |
| |
| // note: caller expects multiple results to be sorted smaller first |
| // note: http://en.wikipedia.org/wiki/Loss_of_significance has an interesting |
| // analysis of the quadratic equation, suggesting why the following looks at |
| // the sign of B -- and further suggesting that the greatest loss of precision |
| // is in b squared less two a c |
| int SkDQuad::RootsValidT(double A, double B, double C, double t[2]) { |
| double s[2]; |
| int realRoots = RootsReal(A, B, C, s); |
| int foundRoots = AddValidTs(s, realRoots, t); |
| return foundRoots; |
| } |
| |
| static int handle_zero(const double B, const double C, double s[2]) { |
| if (approximately_zero(B)) { |
| s[0] = 0; |
| return C == 0; |
| } |
| s[0] = -C / B; |
| return 1; |
| } |
| |
| /* |
| Numeric Solutions (5.6) suggests to solve the quadratic by computing |
| Q = -1/2(B + sgn(B)Sqrt(B^2 - 4 A C)) |
| and using the roots |
| t1 = Q / A |
| t2 = C / Q |
| */ |
| // this does not discard real roots <= 0 or >= 1 |
| int SkDQuad::RootsReal(const double A, const double B, const double C, double s[2]) { |
| if (!A) { |
| return handle_zero(B, C, s); |
| } |
| const double p = B / (2 * A); |
| const double q = C / A; |
| if (approximately_zero(A) && (approximately_zero_inverse(p) || approximately_zero_inverse(q))) { |
| return handle_zero(B, C, s); |
| } |
| /* normal form: x^2 + px + q = 0 */ |
| const double p2 = p * p; |
| if (!AlmostDequalUlps(p2, q) && p2 < q) { |
| return 0; |
| } |
| double sqrt_D = 0; |
| if (p2 > q) { |
| sqrt_D = sqrt(p2 - q); |
| } |
| s[0] = sqrt_D - p; |
| s[1] = -sqrt_D - p; |
| return 1 + !AlmostDequalUlps(s[0], s[1]); |
| } |
| |
| bool SkDQuad::isLinear(int startIndex, int endIndex) const { |
| SkLineParameters lineParameters; |
| lineParameters.quadEndPoints(*this, startIndex, endIndex); |
| // FIXME: maybe it's possible to avoid this and compare non-normalized |
| lineParameters.normalize(); |
| double distance = lineParameters.controlPtDistance(*this); |
| double tiniest = std::min(std::min(std::min(std::min(std::min(fPts[0].fX, fPts[0].fY), |
| fPts[1].fX), fPts[1].fY), fPts[2].fX), fPts[2].fY); |
| double largest = std::max(std::max(std::max(std::max(std::max(fPts[0].fX, fPts[0].fY), |
| fPts[1].fX), fPts[1].fY), fPts[2].fX), fPts[2].fY); |
| largest = std::max(largest, -tiniest); |
| return approximately_zero_when_compared_to(distance, largest); |
| } |
| |
| SkDVector SkDQuad::dxdyAtT(double t) const { |
| double a = t - 1; |
| double b = 1 - 2 * t; |
| double c = t; |
| SkDVector result = { a * fPts[0].fX + b * fPts[1].fX + c * fPts[2].fX, |
| a * fPts[0].fY + b * fPts[1].fY + c * fPts[2].fY }; |
| if (result.fX == 0 && result.fY == 0) { |
| if (zero_or_one(t)) { |
| result = fPts[2] - fPts[0]; |
| } else { |
| // incomplete |
| SkDebugf("!q"); |
| } |
| } |
| return result; |
| } |
| |
| // OPTIMIZE: assert if caller passes in t == 0 / t == 1 ? |
| SkDPoint SkDQuad::ptAtT(double t) const { |
| if (0 == t) { |
| return fPts[0]; |
| } |
| if (1 == t) { |
| return fPts[2]; |
| } |
| double one_t = 1 - t; |
| double a = one_t * one_t; |
| double b = 2 * one_t * t; |
| double c = t * t; |
| SkDPoint result = { a * fPts[0].fX + b * fPts[1].fX + c * fPts[2].fX, |
| a * fPts[0].fY + b * fPts[1].fY + c * fPts[2].fY }; |
| return result; |
| } |
| |
| static double interp_quad_coords(const double* src, double t) { |
| if (0 == t) { |
| return src[0]; |
| } |
| if (1 == t) { |
| return src[4]; |
| } |
| double ab = SkDInterp(src[0], src[2], t); |
| double bc = SkDInterp(src[2], src[4], t); |
| double abc = SkDInterp(ab, bc, t); |
| return abc; |
| } |
| |
| bool SkDQuad::monotonicInX() const { |
| return between(fPts[0].fX, fPts[1].fX, fPts[2].fX); |
| } |
| |
| bool SkDQuad::monotonicInY() const { |
| return between(fPts[0].fY, fPts[1].fY, fPts[2].fY); |
| } |
| |
| /* |
| Given a quadratic q, t1, and t2, find a small quadratic segment. |
| |
| The new quadratic is defined by A, B, and C, where |
| A = c[0]*(1 - t1)*(1 - t1) + 2*c[1]*t1*(1 - t1) + c[2]*t1*t1 |
| C = c[3]*(1 - t1)*(1 - t1) + 2*c[2]*t1*(1 - t1) + c[1]*t1*t1 |
| |
| To find B, compute the point halfway between t1 and t2: |
| |
| q(at (t1 + t2)/2) == D |
| |
| Next, compute where D must be if we know the value of B: |
| |
| _12 = A/2 + B/2 |
| 12_ = B/2 + C/2 |
| 123 = A/4 + B/2 + C/4 |
| = D |
| |
| Group the known values on one side: |
| |
| B = D*2 - A/2 - C/2 |
| */ |
| |
| // OPTIMIZE? : special case t1 = 1 && t2 = 0 |
| SkDQuad SkDQuad::subDivide(double t1, double t2) const { |
| if (0 == t1 && 1 == t2) { |
| return *this; |
| } |
| SkDQuad dst; |
| double ax = dst[0].fX = interp_quad_coords(&fPts[0].fX, t1); |
| double ay = dst[0].fY = interp_quad_coords(&fPts[0].fY, t1); |
| double dx = interp_quad_coords(&fPts[0].fX, (t1 + t2) / 2); |
| double dy = interp_quad_coords(&fPts[0].fY, (t1 + t2) / 2); |
| double cx = dst[2].fX = interp_quad_coords(&fPts[0].fX, t2); |
| double cy = dst[2].fY = interp_quad_coords(&fPts[0].fY, t2); |
| /* bx = */ dst[1].fX = 2 * dx - (ax + cx) / 2; |
| /* by = */ dst[1].fY = 2 * dy - (ay + cy) / 2; |
| return dst; |
| } |
| |
| void SkDQuad::align(int endIndex, SkDPoint* dstPt) const { |
| if (fPts[endIndex].fX == fPts[1].fX) { |
| dstPt->fX = fPts[endIndex].fX; |
| } |
| if (fPts[endIndex].fY == fPts[1].fY) { |
| dstPt->fY = fPts[endIndex].fY; |
| } |
| } |
| |
| SkDPoint SkDQuad::subDivide(const SkDPoint& a, const SkDPoint& c, double t1, double t2) const { |
| SkASSERT(t1 != t2); |
| SkDPoint b; |
| SkDQuad sub = subDivide(t1, t2); |
| SkDLine b0 = {{a, sub[1] + (a - sub[0])}}; |
| SkDLine b1 = {{c, sub[1] + (c - sub[2])}}; |
| SkIntersections i; |
| i.intersectRay(b0, b1); |
| if (i.used() == 1 && i[0][0] >= 0 && i[1][0] >= 0) { |
| b = i.pt(0); |
| } else { |
| SkASSERT(i.used() <= 2); |
| return SkDPoint::Mid(b0[1], b1[1]); |
| } |
| if (t1 == 0 || t2 == 0) { |
| align(0, &b); |
| } |
| if (t1 == 1 || t2 == 1) { |
| align(2, &b); |
| } |
| if (AlmostBequalUlps(b.fX, a.fX)) { |
| b.fX = a.fX; |
| } else if (AlmostBequalUlps(b.fX, c.fX)) { |
| b.fX = c.fX; |
| } |
| if (AlmostBequalUlps(b.fY, a.fY)) { |
| b.fY = a.fY; |
| } else if (AlmostBequalUlps(b.fY, c.fY)) { |
| b.fY = c.fY; |
| } |
| return b; |
| } |
| |
| /* classic one t subdivision */ |
| static void interp_quad_coords(const double* src, double* dst, double t) { |
| double ab = SkDInterp(src[0], src[2], t); |
| double bc = SkDInterp(src[2], src[4], t); |
| dst[0] = src[0]; |
| dst[2] = ab; |
| dst[4] = SkDInterp(ab, bc, t); |
| dst[6] = bc; |
| dst[8] = src[4]; |
| } |
| |
| SkDQuadPair SkDQuad::chopAt(double t) const |
| { |
| SkDQuadPair dst; |
| interp_quad_coords(&fPts[0].fX, &dst.pts[0].fX, t); |
| interp_quad_coords(&fPts[0].fY, &dst.pts[0].fY, t); |
| return dst; |
| } |
| |
| static int valid_unit_divide(double numer, double denom, double* ratio) |
| { |
| if (numer < 0) { |
| numer = -numer; |
| denom = -denom; |
| } |
| if (denom == 0 || numer == 0 || numer >= denom) { |
| return 0; |
| } |
| double r = numer / denom; |
| if (r == 0) { // catch underflow if numer <<<< denom |
| return 0; |
| } |
| *ratio = r; |
| return 1; |
| } |
| |
| /** Quad'(t) = At + B, where |
| A = 2(a - 2b + c) |
| B = 2(b - a) |
| Solve for t, only if it fits between 0 < t < 1 |
| */ |
| int SkDQuad::FindExtrema(const double src[], double tValue[1]) { |
| /* At + B == 0 |
| t = -B / A |
| */ |
| double a = src[0]; |
| double b = src[2]; |
| double c = src[4]; |
| return valid_unit_divide(a - b, a - b - b + c, tValue); |
| } |
| |
| /* Parameterization form, given A*t*t + 2*B*t*(1-t) + C*(1-t)*(1-t) |
| * |
| * a = A - 2*B + C |
| * b = 2*B - 2*C |
| * c = C |
| */ |
| void SkDQuad::SetABC(const double* quad, double* a, double* b, double* c) { |
| *a = quad[0]; // a = A |
| *b = 2 * quad[2]; // b = 2*B |
| *c = quad[4]; // c = C |
| *b -= *c; // b = 2*B - C |
| *a -= *b; // a = A - 2*B + C |
| *b -= *c; // b = 2*B - 2*C |
| } |
| |
| int SkTQuad::intersectRay(SkIntersections* i, const SkDLine& line) const { |
| return i->intersectRay(fQuad, line); |
| } |
| |
| bool SkTQuad::hullIntersects(const SkDConic& conic, bool* isLinear) const { |
| return conic.hullIntersects(fQuad, isLinear); |
| } |
| |
| bool SkTQuad::hullIntersects(const SkDCubic& cubic, bool* isLinear) const { |
| return cubic.hullIntersects(fQuad, isLinear); |
| } |
| |
| void SkTQuad::setBounds(SkDRect* rect) const { |
| rect->setBounds(fQuad); |
| } |
