/* * Copyright 2011 Google Inc. * * Use of this source code is governed by a BSD-style license that can be * found in the LICENSE file. */ #include "GrPathUtils.h" #include "GrTypes.h" #include "SkGeometry.h" SkScalar GrPathUtils::scaleToleranceToSrc(SkScalar devTol, const SkMatrix& viewM, const SkRect& pathBounds) { // In order to tesselate the path we get a bound on how much the matrix can // scale when mapping to screen coordinates. SkScalar stretch = viewM.getMaxScale(); SkScalar srcTol = devTol; if (stretch < 0) { // take worst case mapRadius amoung four corners. // (less than perfect) for (int i = 0; i < 4; ++i) { SkMatrix mat; mat.setTranslate((i % 2) ? pathBounds.fLeft : pathBounds.fRight, (i < 2) ? pathBounds.fTop : pathBounds.fBottom); mat.postConcat(viewM); stretch = SkMaxScalar(stretch, mat.mapRadius(SK_Scalar1)); } } srcTol = SkScalarDiv(srcTol, stretch); return srcTol; } static const int MAX_POINTS_PER_CURVE = 1 << 10; static const SkScalar gMinCurveTol = 0.0001f; uint32_t GrPathUtils::quadraticPointCount(const SkPoint points[], SkScalar tol) { if (tol < gMinCurveTol) { tol = gMinCurveTol; } SkASSERT(tol > 0); SkScalar d = points[1].distanceToLineSegmentBetween(points[0], points[2]); if (d <= tol) { return 1; } else { // Each time we subdivide, d should be cut in 4. So we need to // subdivide x = log4(d/tol) times. x subdivisions creates 2^(x) // points. // 2^(log4(x)) = sqrt(x); int temp = SkScalarCeilToInt(SkScalarSqrt(SkScalarDiv(d, tol))); int pow2 = GrNextPow2(temp); // Because of NaNs & INFs we can wind up with a degenerate temp // such that pow2 comes out negative. Also, our point generator // will always output at least one pt. if (pow2 < 1) { pow2 = 1; } return SkTMin(pow2, MAX_POINTS_PER_CURVE); } } uint32_t GrPathUtils::generateQuadraticPoints(const SkPoint& p0, const SkPoint& p1, const SkPoint& p2, SkScalar tolSqd, SkPoint** points, uint32_t pointsLeft) { if (pointsLeft < 2 || (p1.distanceToLineSegmentBetweenSqd(p0, p2)) < tolSqd) { (*points)[0] = p2; *points += 1; return 1; } SkPoint q[] = { { SkScalarAve(p0.fX, p1.fX), SkScalarAve(p0.fY, p1.fY) }, { SkScalarAve(p1.fX, p2.fX), SkScalarAve(p1.fY, p2.fY) }, }; SkPoint r = { SkScalarAve(q[0].fX, q[1].fX), SkScalarAve(q[0].fY, q[1].fY) }; pointsLeft >>= 1; uint32_t a = generateQuadraticPoints(p0, q[0], r, tolSqd, points, pointsLeft); uint32_t b = generateQuadraticPoints(r, q[1], p2, tolSqd, points, pointsLeft); return a + b; } uint32_t GrPathUtils::cubicPointCount(const SkPoint points[], SkScalar tol) { if (tol < gMinCurveTol) { tol = gMinCurveTol; } SkASSERT(tol > 0); SkScalar d = SkTMax( points[1].distanceToLineSegmentBetweenSqd(points[0], points[3]), points[2].distanceToLineSegmentBetweenSqd(points[0], points[3])); d = SkScalarSqrt(d); if (d <= tol) { return 1; } else { int temp = SkScalarCeilToInt(SkScalarSqrt(SkScalarDiv(d, tol))); int pow2 = GrNextPow2(temp); // Because of NaNs & INFs we can wind up with a degenerate temp // such that pow2 comes out negative. Also, our point generator // will always output at least one pt. if (pow2 < 1) { pow2 = 1; } return SkTMin(pow2, MAX_POINTS_PER_CURVE); } } uint32_t GrPathUtils::generateCubicPoints(const SkPoint& p0, const SkPoint& p1, const SkPoint& p2, const SkPoint& p3, SkScalar tolSqd, SkPoint** points, uint32_t pointsLeft) { if (pointsLeft < 2 || (p1.distanceToLineSegmentBetweenSqd(p0, p3) < tolSqd && p2.distanceToLineSegmentBetweenSqd(p0, p3) < tolSqd)) { (*points)[0] = p3; *points += 1; return 1; } SkPoint q[] = { { SkScalarAve(p0.fX, p1.fX), SkScalarAve(p0.fY, p1.fY) }, { SkScalarAve(p1.fX, p2.fX), SkScalarAve(p1.fY, p2.fY) }, { SkScalarAve(p2.fX, p3.fX), SkScalarAve(p2.fY, p3.fY) } }; SkPoint r[] = { { SkScalarAve(q[0].fX, q[1].fX), SkScalarAve(q[0].fY, q[1].fY) }, { SkScalarAve(q[1].fX, q[2].fX), SkScalarAve(q[1].fY, q[2].fY) } }; SkPoint s = { SkScalarAve(r[0].fX, r[1].fX), SkScalarAve(r[0].fY, r[1].fY) }; pointsLeft >>= 1; uint32_t a = generateCubicPoints(p0, q[0], r[0], s, tolSqd, points, pointsLeft); uint32_t b = generateCubicPoints(s, r[1], q[2], p3, tolSqd, points, pointsLeft); return a + b; } int GrPathUtils::worstCasePointCount(const SkPath& path, int* subpaths, SkScalar tol) { if (tol < gMinCurveTol) { tol = gMinCurveTol; } SkASSERT(tol > 0); int pointCount = 0; *subpaths = 1; bool first = true; SkPath::Iter iter(path, false); SkPath::Verb verb; SkPoint pts[4]; while ((verb = iter.next(pts)) != SkPath::kDone_Verb) { switch (verb) { case SkPath::kLine_Verb: pointCount += 1; break; case SkPath::kQuad_Verb: pointCount += quadraticPointCount(pts, tol); break; case SkPath::kCubic_Verb: pointCount += cubicPointCount(pts, tol); break; case SkPath::kMove_Verb: pointCount += 1; if (!first) { ++(*subpaths); } break; default: break; } first = false; } return pointCount; } void GrPathUtils::QuadUVMatrix::set(const SkPoint qPts[3]) { SkMatrix m; // We want M such that M * xy_pt = uv_pt // We know M * control_pts = [0 1/2 1] // [0 0 1] // [1 1 1] // And control_pts = [x0 x1 x2] // [y0 y1 y2] // [1 1 1 ] // We invert the control pt matrix and post concat to both sides to get M. // Using the known form of the control point matrix and the result, we can // optimize and improve precision. double x0 = qPts[0].fX; double y0 = qPts[0].fY; double x1 = qPts[1].fX; double y1 = qPts[1].fY; double x2 = qPts[2].fX; double y2 = qPts[2].fY; double det = x0*y1 - y0*x1 + x2*y0 - y2*x0 + x1*y2 - y1*x2; if (!sk_float_isfinite(det) || SkScalarNearlyZero((float)det, SK_ScalarNearlyZero * SK_ScalarNearlyZero)) { // The quad is degenerate. Hopefully this is rare. Find the pts that are // farthest apart to compute a line (unless it is really a pt). SkScalar maxD = qPts[0].distanceToSqd(qPts[1]); int maxEdge = 0; SkScalar d = qPts[1].distanceToSqd(qPts[2]); if (d > maxD) { maxD = d; maxEdge = 1; } d = qPts[2].distanceToSqd(qPts[0]); if (d > maxD) { maxD = d; maxEdge = 2; } // We could have a tolerance here, not sure if it would improve anything if (maxD > 0) { // Set the matrix to give (u = 0, v = distance_to_line) SkVector lineVec = qPts[(maxEdge + 1)%3] - qPts[maxEdge]; // when looking from the point 0 down the line we want positive // distances to be to the left. This matches the non-degenerate // case. lineVec.setOrthog(lineVec, SkPoint::kLeft_Side); lineVec.dot(qPts[0]); // first row fM[0] = 0; fM[1] = 0; fM[2] = 0; // second row fM[3] = lineVec.fX; fM[4] = lineVec.fY; fM[5] = -lineVec.dot(qPts[maxEdge]); } else { // It's a point. It should cover zero area. Just set the matrix such // that (u, v) will always be far away from the quad. fM[0] = 0; fM[1] = 0; fM[2] = 100.f; fM[3] = 0; fM[4] = 0; fM[5] = 100.f; } } else { double scale = 1.0/det; // compute adjugate matrix double a0, a1, a2, a3, a4, a5, a6, a7, a8; a0 = y1-y2; a1 = x2-x1; a2 = x1*y2-x2*y1; a3 = y2-y0; a4 = x0-x2; a5 = x2*y0-x0*y2; a6 = y0-y1; a7 = x1-x0; a8 = x0*y1-x1*y0; // this performs the uv_pts*adjugate(control_pts) multiply, // then does the scale by 1/det afterwards to improve precision m[SkMatrix::kMScaleX] = (float)((0.5*a3 + a6)*scale); m[SkMatrix::kMSkewX] = (float)((0.5*a4 + a7)*scale); m[SkMatrix::kMTransX] = (float)((0.5*a5 + a8)*scale); m[SkMatrix::kMSkewY] = (float)(a6*scale); m[SkMatrix::kMScaleY] = (float)(a7*scale); m[SkMatrix::kMTransY] = (float)(a8*scale); m[SkMatrix::kMPersp0] = (float)((a0 + a3 + a6)*scale); m[SkMatrix::kMPersp1] = (float)((a1 + a4 + a7)*scale); m[SkMatrix::kMPersp2] = (float)((a2 + a5 + a8)*scale); // The matrix should not have perspective. SkDEBUGCODE(static const SkScalar gTOL = 1.f / 100.f); SkASSERT(SkScalarAbs(m.get(SkMatrix::kMPersp0)) < gTOL); SkASSERT(SkScalarAbs(m.get(SkMatrix::kMPersp1)) < gTOL); // It may not be normalized to have 1.0 in the bottom right float m33 = m.get(SkMatrix::kMPersp2); if (1.f != m33) { m33 = 1.f / m33; fM[0] = m33 * m.get(SkMatrix::kMScaleX); fM[1] = m33 * m.get(SkMatrix::kMSkewX); fM[2] = m33 * m.get(SkMatrix::kMTransX); fM[3] = m33 * m.get(SkMatrix::kMSkewY); fM[4] = m33 * m.get(SkMatrix::kMScaleY); fM[5] = m33 * m.get(SkMatrix::kMTransY); } else { fM[0] = m.get(SkMatrix::kMScaleX); fM[1] = m.get(SkMatrix::kMSkewX); fM[2] = m.get(SkMatrix::kMTransX); fM[3] = m.get(SkMatrix::kMSkewY); fM[4] = m.get(SkMatrix::kMScaleY); fM[5] = m.get(SkMatrix::kMTransY); } } } //////////////////////////////////////////////////////////////////////////////// // k = (y2 - y0, x0 - x2, (x2 - x0)*y0 - (y2 - y0)*x0 ) // l = (2*w * (y1 - y0), 2*w * (x0 - x1), 2*w * (x1*y0 - x0*y1)) // m = (2*w * (y2 - y1), 2*w * (x1 - x2), 2*w * (x2*y1 - x1*y2)) void GrPathUtils::getConicKLM(const SkPoint p[3], const SkScalar weight, SkScalar klm[9]) { const SkScalar w2 = 2.f * weight; klm[0] = p[2].fY - p[0].fY; klm[1] = p[0].fX - p[2].fX; klm[2] = (p[2].fX - p[0].fX) * p[0].fY - (p[2].fY - p[0].fY) * p[0].fX; klm[3] = w2 * (p[1].fY - p[0].fY); klm[4] = w2 * (p[0].fX - p[1].fX); klm[5] = w2 * (p[1].fX * p[0].fY - p[0].fX * p[1].fY); klm[6] = w2 * (p[2].fY - p[1].fY); klm[7] = w2 * (p[1].fX - p[2].fX); klm[8] = w2 * (p[2].fX * p[1].fY - p[1].fX * p[2].fY); // scale the max absolute value of coeffs to 10 SkScalar scale = 0.f; for (int i = 0; i < 9; ++i) { scale = SkMaxScalar(scale, SkScalarAbs(klm[i])); } SkASSERT(scale > 0.f); scale = 10.f / scale; for (int i = 0; i < 9; ++i) { klm[i] *= scale; } } //////////////////////////////////////////////////////////////////////////////// namespace { // a is the first control point of the cubic. // ab is the vector from a to the second control point. // dc is the vector from the fourth to the third control point. // d is the fourth control point. // p is the candidate quadratic control point. // this assumes that the cubic doesn't inflect and is simple bool is_point_within_cubic_tangents(const SkPoint& a, const SkVector& ab, const SkVector& dc, const SkPoint& d, SkPath::Direction dir, const SkPoint p) { SkVector ap = p - a; SkScalar apXab = ap.cross(ab); if (SkPath::kCW_Direction == dir) { if (apXab > 0) { return false; } } else { SkASSERT(SkPath::kCCW_Direction == dir); if (apXab < 0) { return false; } } SkVector dp = p - d; SkScalar dpXdc = dp.cross(dc); if (SkPath::kCW_Direction == dir) { if (dpXdc < 0) { return false; } } else { SkASSERT(SkPath::kCCW_Direction == dir); if (dpXdc > 0) { return false; } } return true; } void convert_noninflect_cubic_to_quads(const SkPoint p[4], SkScalar toleranceSqd, bool constrainWithinTangents, SkPath::Direction dir, SkTArray* quads, int sublevel = 0) { // Notation: Point a is always p[0]. Point b is p[1] unless p[1] == p[0], in which case it is // p[2]. Point d is always p[3]. Point c is p[2] unless p[2] == p[3], in which case it is p[1]. SkVector ab = p[1] - p[0]; SkVector dc = p[2] - p[3]; if (ab.isZero()) { if (dc.isZero()) { SkPoint* degQuad = quads->push_back_n(3); degQuad[0] = p[0]; degQuad[1] = p[0]; degQuad[2] = p[3]; return; } ab = p[2] - p[0]; } if (dc.isZero()) { dc = p[1] - p[3]; } // When the ab and cd tangents are degenerate or nearly parallel with vector from d to a the // constraint that the quad point falls between the tangents becomes hard to enforce and we are // likely to hit the max subdivision count. However, in this case the cubic is approaching a // line and the accuracy of the quad point isn't so important. We check if the two middle cubic // control points are very close to the baseline vector. If so then we just pick quadratic // points on the control polygon. if (constrainWithinTangents) { SkVector da = p[0] - p[3]; bool doQuads = dc.lengthSqd() < SK_ScalarNearlyZero || ab.lengthSqd() < SK_ScalarNearlyZero; if (!doQuads) { SkScalar invDALengthSqd = da.lengthSqd(); if (invDALengthSqd > SK_ScalarNearlyZero) { invDALengthSqd = SkScalarInvert(invDALengthSqd); // cross(ab, da)^2/length(da)^2 == sqd distance from b to line from d to a. // same goes for point c using vector cd. SkScalar detABSqd = ab.cross(da); detABSqd = SkScalarSquare(detABSqd); SkScalar detDCSqd = dc.cross(da); detDCSqd = SkScalarSquare(detDCSqd); if (SkScalarMul(detABSqd, invDALengthSqd) < toleranceSqd && SkScalarMul(detDCSqd, invDALengthSqd) < toleranceSqd) { doQuads = true; } } } if (doQuads) { SkPoint b = p[0] + ab; SkPoint c = p[3] + dc; SkPoint mid = b + c; mid.scale(SK_ScalarHalf); // Insert two quadratics to cover the case when ab points away from d and/or dc // points away from a. if (SkVector::DotProduct(da, dc) < 0 || SkVector::DotProduct(ab,da) > 0) { SkPoint* qpts = quads->push_back_n(6); qpts[0] = p[0]; qpts[1] = b; qpts[2] = mid; qpts[3] = mid; qpts[4] = c; qpts[5] = p[3]; } else { SkPoint* qpts = quads->push_back_n(3); qpts[0] = p[0]; qpts[1] = mid; qpts[2] = p[3]; } return; } } static const SkScalar kLengthScale = 3 * SK_Scalar1 / 2; static const int kMaxSubdivs = 10; ab.scale(kLengthScale); dc.scale(kLengthScale); // e0 and e1 are extrapolations along vectors ab and dc. SkVector c0 = p[0]; c0 += ab; SkVector c1 = p[3]; c1 += dc; SkScalar dSqd = sublevel > kMaxSubdivs ? 0 : c0.distanceToSqd(c1); if (dSqd < toleranceSqd) { SkPoint cAvg = c0; cAvg += c1; cAvg.scale(SK_ScalarHalf); bool subdivide = false; if (constrainWithinTangents && !is_point_within_cubic_tangents(p[0], ab, dc, p[3], dir, cAvg)) { // choose a new cAvg that is the intersection of the two tangent lines. ab.setOrthog(ab); SkScalar z0 = -ab.dot(p[0]); dc.setOrthog(dc); SkScalar z1 = -dc.dot(p[3]); cAvg.fX = SkScalarMul(ab.fY, z1) - SkScalarMul(z0, dc.fY); cAvg.fY = SkScalarMul(z0, dc.fX) - SkScalarMul(ab.fX, z1); SkScalar z = SkScalarMul(ab.fX, dc.fY) - SkScalarMul(ab.fY, dc.fX); z = SkScalarInvert(z); cAvg.fX *= z; cAvg.fY *= z; if (sublevel <= kMaxSubdivs) { SkScalar d0Sqd = c0.distanceToSqd(cAvg); SkScalar d1Sqd = c1.distanceToSqd(cAvg); // We need to subdivide if d0 + d1 > tolerance but we have the sqd values. We know // the distances and tolerance can't be negative. // (d0 + d1)^2 > toleranceSqd // d0Sqd + 2*d0*d1 + d1Sqd > toleranceSqd SkScalar d0d1 = SkScalarSqrt(SkScalarMul(d0Sqd, d1Sqd)); subdivide = 2 * d0d1 + d0Sqd + d1Sqd > toleranceSqd; } } if (!subdivide) { SkPoint* pts = quads->push_back_n(3); pts[0] = p[0]; pts[1] = cAvg; pts[2] = p[3]; return; } } SkPoint choppedPts[7]; SkChopCubicAtHalf(p, choppedPts); convert_noninflect_cubic_to_quads(choppedPts + 0, toleranceSqd, constrainWithinTangents, dir, quads, sublevel + 1); convert_noninflect_cubic_to_quads(choppedPts + 3, toleranceSqd, constrainWithinTangents, dir, quads, sublevel + 1); } } void GrPathUtils::convertCubicToQuads(const SkPoint p[4], SkScalar tolScale, bool constrainWithinTangents, SkPath::Direction dir, SkTArray* quads) { SkPoint chopped[10]; int count = SkChopCubicAtInflections(p, chopped); // base tolerance is 1 pixel. static const SkScalar kTolerance = SK_Scalar1; const SkScalar tolSqd = SkScalarSquare(SkScalarMul(tolScale, kTolerance)); for (int i = 0; i < count; ++i) { SkPoint* cubic = chopped + 3*i; convert_noninflect_cubic_to_quads(cubic, tolSqd, constrainWithinTangents, dir, quads); } } //////////////////////////////////////////////////////////////////////////////// enum CubicType { kSerpentine_CubicType, kCusp_CubicType, kLoop_CubicType, kQuadratic_CubicType, kLine_CubicType, kPoint_CubicType }; // discr(I) = d0^2 * (3*d1^2 - 4*d0*d2) // Classification: // discr(I) > 0 Serpentine // discr(I) = 0 Cusp // discr(I) < 0 Loop // d0 = d1 = 0 Quadratic // d0 = d1 = d2 = 0 Line // p0 = p1 = p2 = p3 Point static CubicType classify_cubic(const SkPoint p[4], const SkScalar d[3]) { if (p[0] == p[1] && p[0] == p[2] && p[0] == p[3]) { return kPoint_CubicType; } const SkScalar discr = d[0] * d[0] * (3.f * d[1] * d[1] - 4.f * d[0] * d[2]); if (discr > SK_ScalarNearlyZero) { return kSerpentine_CubicType; } else if (discr < -SK_ScalarNearlyZero) { return kLoop_CubicType; } else { if (0.f == d[0] && 0.f == d[1]) { return (0.f == d[2] ? kLine_CubicType : kQuadratic_CubicType); } else { return kCusp_CubicType; } } } // Assumes the third component of points is 1. // Calcs p0 . (p1 x p2) static SkScalar calc_dot_cross_cubic(const SkPoint& p0, const SkPoint& p1, const SkPoint& p2) { const SkScalar xComp = p0.fX * (p1.fY - p2.fY); const SkScalar yComp = p0.fY * (p2.fX - p1.fX); const SkScalar wComp = p1.fX * p2.fY - p1.fY * p2.fX; return (xComp + yComp + wComp); } // Solves linear system to extract klm // P.K = k (similarly for l, m) // Where P is matrix of control points // K is coefficients for the line K // k is vector of values of K evaluated at the control points // Solving for K, thus K = P^(-1) . k static void calc_cubic_klm(const SkPoint p[4], const SkScalar controlK[4], const SkScalar controlL[4], const SkScalar controlM[4], SkScalar k[3], SkScalar l[3], SkScalar m[3]) { SkMatrix matrix; matrix.setAll(p[0].fX, p[0].fY, 1.f, p[1].fX, p[1].fY, 1.f, p[2].fX, p[2].fY, 1.f); SkMatrix inverse; if (matrix.invert(&inverse)) { inverse.mapHomogeneousPoints(k, controlK, 1); inverse.mapHomogeneousPoints(l, controlL, 1); inverse.mapHomogeneousPoints(m, controlM, 1); } } static void set_serp_klm(const SkScalar d[3], SkScalar k[4], SkScalar l[4], SkScalar m[4]) { SkScalar tempSqrt = SkScalarSqrt(9.f * d[1] * d[1] - 12.f * d[0] * d[2]); SkScalar ls = 3.f * d[1] - tempSqrt; SkScalar lt = 6.f * d[0]; SkScalar ms = 3.f * d[1] + tempSqrt; SkScalar mt = 6.f * d[0]; k[0] = ls * ms; k[1] = (3.f * ls * ms - ls * mt - lt * ms) / 3.f; k[2] = (lt * (mt - 2.f * ms) + ls * (3.f * ms - 2.f * mt)) / 3.f; k[3] = (lt - ls) * (mt - ms); l[0] = ls * ls * ls; const SkScalar lt_ls = lt - ls; l[1] = ls * ls * lt_ls * -1.f; l[2] = lt_ls * lt_ls * ls; l[3] = -1.f * lt_ls * lt_ls * lt_ls; m[0] = ms * ms * ms; const SkScalar mt_ms = mt - ms; m[1] = ms * ms * mt_ms * -1.f; m[2] = mt_ms * mt_ms * ms; m[3] = -1.f * mt_ms * mt_ms * mt_ms; // If d0 < 0 we need to flip the orientation of our curve // This is done by negating the k and l values // We want negative distance values to be on the inside if ( d[0] > 0) { for (int i = 0; i < 4; ++i) { k[i] = -k[i]; l[i] = -l[i]; } } } static void set_loop_klm(const SkScalar d[3], SkScalar k[4], SkScalar l[4], SkScalar m[4]) { SkScalar tempSqrt = SkScalarSqrt(4.f * d[0] * d[2] - 3.f * d[1] * d[1]); SkScalar ls = d[1] - tempSqrt; SkScalar lt = 2.f * d[0]; SkScalar ms = d[1] + tempSqrt; SkScalar mt = 2.f * d[0]; k[0] = ls * ms; k[1] = (3.f * ls*ms - ls * mt - lt * ms) / 3.f; k[2] = (lt * (mt - 2.f * ms) + ls * (3.f * ms - 2.f * mt)) / 3.f; k[3] = (lt - ls) * (mt - ms); l[0] = ls * ls * ms; l[1] = (ls * (ls * (mt - 3.f * ms) + 2.f * lt * ms))/-3.f; l[2] = ((lt - ls) * (ls * (2.f * mt - 3.f * ms) + lt * ms))/3.f; l[3] = -1.f * (lt - ls) * (lt - ls) * (mt - ms); m[0] = ls * ms * ms; m[1] = (ms * (ls * (2.f * mt - 3.f * ms) + lt * ms))/-3.f; m[2] = ((mt - ms) * (ls * (mt - 3.f * ms) + 2.f * lt * ms))/3.f; m[3] = -1.f * (lt - ls) * (mt - ms) * (mt - ms); // If (d0 < 0 && sign(k1) > 0) || (d0 > 0 && sign(k1) < 0), // we need to flip the orientation of our curve. // This is done by negating the k and l values if ( (d[0] < 0 && k[1] > 0) || (d[0] > 0 && k[1] < 0)) { for (int i = 0; i < 4; ++i) { k[i] = -k[i]; l[i] = -l[i]; } } } static void set_cusp_klm(const SkScalar d[3], SkScalar k[4], SkScalar l[4], SkScalar m[4]) { const SkScalar ls = d[2]; const SkScalar lt = 3.f * d[1]; k[0] = ls; k[1] = ls - lt / 3.f; k[2] = ls - 2.f * lt / 3.f; k[3] = ls - lt; l[0] = ls * ls * ls; const SkScalar ls_lt = ls - lt; l[1] = ls * ls * ls_lt; l[2] = ls_lt * ls_lt * ls; l[3] = ls_lt * ls_lt * ls_lt; m[0] = 1.f; m[1] = 1.f; m[2] = 1.f; m[3] = 1.f; } // For the case when a cubic is actually a quadratic // M = // 0 0 0 // 1/3 0 1/3 // 2/3 1/3 2/3 // 1 1 1 static void set_quadratic_klm(const SkScalar d[3], SkScalar k[4], SkScalar l[4], SkScalar m[4]) { k[0] = 0.f; k[1] = 1.f/3.f; k[2] = 2.f/3.f; k[3] = 1.f; l[0] = 0.f; l[1] = 0.f; l[2] = 1.f/3.f; l[3] = 1.f; m[0] = 0.f; m[1] = 1.f/3.f; m[2] = 2.f/3.f; m[3] = 1.f; // If d2 < 0 we need to flip the orientation of our curve // This is done by negating the k and l values if ( d[2] > 0) { for (int i = 0; i < 4; ++i) { k[i] = -k[i]; l[i] = -l[i]; } } } // Calc coefficients of I(s,t) where roots of I are inflection points of curve // I(s,t) = t*(3*d0*s^2 - 3*d1*s*t + d2*t^2) // d0 = a1 - 2*a2+3*a3 // d1 = -a2 + 3*a3 // d2 = 3*a3 // a1 = p0 . (p3 x p2) // a2 = p1 . (p0 x p3) // a3 = p2 . (p1 x p0) // Places the values of d1, d2, d3 in array d passed in static void calc_cubic_inflection_func(const SkPoint p[4], SkScalar d[3]) { SkScalar a1 = calc_dot_cross_cubic(p[0], p[3], p[2]); SkScalar a2 = calc_dot_cross_cubic(p[1], p[0], p[3]); SkScalar a3 = calc_dot_cross_cubic(p[2], p[1], p[0]); // need to scale a's or values in later calculations will grow to high SkScalar max = SkScalarAbs(a1); max = SkMaxScalar(max, SkScalarAbs(a2)); max = SkMaxScalar(max, SkScalarAbs(a3)); max = 1.f/max; a1 = a1 * max; a2 = a2 * max; a3 = a3 * max; d[2] = 3.f * a3; d[1] = d[2] - a2; d[0] = d[1] - a2 + a1; } int GrPathUtils::chopCubicAtLoopIntersection(const SkPoint src[4], SkPoint dst[10], SkScalar klm[9], SkScalar klm_rev[3]) { // Variable to store the two parametric values at the loop double point SkScalar smallS = 0.f; SkScalar largeS = 0.f; SkScalar d[3]; calc_cubic_inflection_func(src, d); CubicType cType = classify_cubic(src, d); int chop_count = 0; if (kLoop_CubicType == cType) { SkScalar tempSqrt = SkScalarSqrt(4.f * d[0] * d[2] - 3.f * d[1] * d[1]); SkScalar ls = d[1] - tempSqrt; SkScalar lt = 2.f * d[0]; SkScalar ms = d[1] + tempSqrt; SkScalar mt = 2.f * d[0]; ls = ls / lt; ms = ms / mt; // need to have t values sorted since this is what is expected by SkChopCubicAt if (ls <= ms) { smallS = ls; largeS = ms; } else { smallS = ms; largeS = ls; } SkScalar chop_ts[2]; if (smallS > 0.f && smallS < 1.f) { chop_ts[chop_count++] = smallS; } if (largeS > 0.f && largeS < 1.f) { chop_ts[chop_count++] = largeS; } if(dst) { SkChopCubicAt(src, dst, chop_ts, chop_count); } } else { if (dst) { memcpy(dst, src, sizeof(SkPoint) * 4); } } if (klm && klm_rev) { // Set klm_rev to to match the sub_section of cubic that needs to have its orientation // flipped. This will always be the section that is the "loop" if (2 == chop_count) { klm_rev[0] = 1.f; klm_rev[1] = -1.f; klm_rev[2] = 1.f; } else if (1 == chop_count) { if (smallS < 0.f) { klm_rev[0] = -1.f; klm_rev[1] = 1.f; } else { klm_rev[0] = 1.f; klm_rev[1] = -1.f; } } else { if (smallS < 0.f && largeS > 1.f) { klm_rev[0] = -1.f; } else { klm_rev[0] = 1.f; } } SkScalar controlK[4]; SkScalar controlL[4]; SkScalar controlM[4]; if (kSerpentine_CubicType == cType || (kCusp_CubicType == cType && 0.f != d[0])) { set_serp_klm(d, controlK, controlL, controlM); } else if (kLoop_CubicType == cType) { set_loop_klm(d, controlK, controlL, controlM); } else if (kCusp_CubicType == cType) { SkASSERT(0.f == d[0]); set_cusp_klm(d, controlK, controlL, controlM); } else if (kQuadratic_CubicType == cType) { set_quadratic_klm(d, controlK, controlL, controlM); } calc_cubic_klm(src, controlK, controlL, controlM, klm, &klm[3], &klm[6]); } return chop_count + 1; } void GrPathUtils::getCubicKLM(const SkPoint p[4], SkScalar klm[9]) { SkScalar d[3]; calc_cubic_inflection_func(p, d); CubicType cType = classify_cubic(p, d); SkScalar controlK[4]; SkScalar controlL[4]; SkScalar controlM[4]; if (kSerpentine_CubicType == cType || (kCusp_CubicType == cType && 0.f != d[0])) { set_serp_klm(d, controlK, controlL, controlM); } else if (kLoop_CubicType == cType) { set_loop_klm(d, controlK, controlL, controlM); } else if (kCusp_CubicType == cType) { SkASSERT(0.f == d[0]); set_cusp_klm(d, controlK, controlL, controlM); } else if (kQuadratic_CubicType == cType) { set_quadratic_klm(d, controlK, controlL, controlM); } calc_cubic_klm(p, controlK, controlL, controlM, klm, &klm[3], &klm[6]); }