/* * 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 "SkMathPriv.h" #include "SkPointPriv.h" static const SkScalar gMinCurveTol = 0.0001f; 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(); 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)); } } SkScalar srcTol = 0; if (stretch <= 0) { // We have degenerate bounds or some degenerate matrix. Thus we set the tolerance to be the // max of the path pathBounds width and height. srcTol = SkTMax(pathBounds.width(), pathBounds.height()); } else { srcTol = devTol / stretch; } if (srcTol < gMinCurveTol) { srcTol = gMinCurveTol; } return srcTol; } uint32_t GrPathUtils::quadraticPointCount(const SkPoint points[], SkScalar tol) { // You should have called scaleToleranceToSrc, which guarantees this SkASSERT(tol >= gMinCurveTol); SkScalar d = SkPointPriv::DistanceToLineSegmentBetween(points[1], points[0], points[2]); if (!SkScalarIsFinite(d)) { return kMaxPointsPerCurve; } else 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); SkScalar divSqrt = SkScalarSqrt(d / tol); if (((SkScalar)SK_MaxS32) <= divSqrt) { return kMaxPointsPerCurve; } else { int temp = SkScalarCeilToInt(divSqrt); 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, kMaxPointsPerCurve); } } } uint32_t GrPathUtils::generateQuadraticPoints(const SkPoint& p0, const SkPoint& p1, const SkPoint& p2, SkScalar tolSqd, SkPoint** points, uint32_t pointsLeft) { if (pointsLeft < 2 || (SkPointPriv::DistanceToLineSegmentBetweenSqd(p1, 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) { // You should have called scaleToleranceToSrc, which guarantees this SkASSERT(tol >= gMinCurveTol); SkScalar d = SkTMax( SkPointPriv::DistanceToLineSegmentBetweenSqd(points[1], points[0], points[3]), SkPointPriv::DistanceToLineSegmentBetweenSqd(points[2], points[0], points[3])); d = SkScalarSqrt(d); if (!SkScalarIsFinite(d)) { return kMaxPointsPerCurve; } else if (d <= tol) { return 1; } else { SkScalar divSqrt = SkScalarSqrt(d / tol); if (((SkScalar)SK_MaxS32) <= divSqrt) { return kMaxPointsPerCurve; } else { int temp = SkScalarCeilToInt(SkScalarSqrt(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, kMaxPointsPerCurve); } } } 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 || (SkPointPriv::DistanceToLineSegmentBetweenSqd(p1, p0, p3) < tolSqd && SkPointPriv::DistanceToLineSegmentBetweenSqd(p2, 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) { // You should have called scaleToleranceToSrc, which guarantees this SkASSERT(tol >= gMinCurveTol); int pointCount = 0; *subpaths = 1; bool first = true; SkPath::Iter iter(path, false); SkPath::Verb verb; SkPoint pts[4]; while ((verb = iter.next(pts, false)) != SkPath::kDone_Verb) { switch (verb) { case SkPath::kLine_Verb: pointCount += 1; break; case SkPath::kConic_Verb: { SkScalar weight = iter.conicWeight(); SkAutoConicToQuads converter; const SkPoint* quadPts = converter.computeQuads(pts, weight, tol); for (int i = 0; i < converter.countQuads(); ++i) { pointCount += quadraticPointCount(quadPts + 2*i, tol); } } 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 = SkPointPriv::DistanceToSqd(qPts[0], qPts[1]); int maxEdge = 0; SkScalar d = SkPointPriv::DistanceToSqd(qPts[1], qPts[2]); if (d > maxD) { maxD = d; maxEdge = 1; } d = SkPointPriv::DistanceToSqd(qPts[2], 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. SkPointPriv::SetOrthog(&lineVec, lineVec, SkPointPriv::kLeft_Side); // 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 a2, a3, a4, a5, a6, a7, a8; 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); // kMPersp0 & kMPersp1 should algebraically be zero m[SkMatrix::kMPersp0] = 0.0f; m[SkMatrix::kMPersp1] = 0.0f; m[SkMatrix::kMPersp2] = (float)((a2 + a5 + a8)*scale); // 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*y0 - x0*y2) // l = (y1 - y0, x0 - x1, x1*y0 - x0*y1) * 2*w // m = (y2 - y1, x1 - x2, x2*y1 - x1*y2) * 2*w void GrPathUtils::getConicKLM(const SkPoint p[3], const SkScalar weight, SkMatrix* out) { SkMatrix& klm = *out; 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].fY - p[0].fX * p[2].fY; 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, SkPathPriv::FirstDirection dir, const SkPoint p) { SkVector ap = p - a; SkScalar apXab = ap.cross(ab); if (SkPathPriv::kCW_FirstDirection == dir) { if (apXab > 0) { return false; } } else { SkASSERT(SkPathPriv::kCCW_FirstDirection == dir); if (apXab < 0) { return false; } } SkVector dp = p - d; SkScalar dpXdc = dp.cross(dc); if (SkPathPriv::kCW_FirstDirection == dir) { if (dpXdc < 0) { return false; } } else { SkASSERT(SkPathPriv::kCCW_FirstDirection == dir); if (dpXdc > 0) { return false; } } return true; } void convert_noninflect_cubic_to_quads(const SkPoint p[4], SkScalar toleranceSqd, bool constrainWithinTangents, SkPathPriv::FirstDirection 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 (SkPointPriv::LengthSqd(ab) < SK_ScalarNearlyZero) { if (SkPointPriv::LengthSqd(dc) < SK_ScalarNearlyZero) { 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 (SkPointPriv::LengthSqd(dc) < SK_ScalarNearlyZero) { 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 = SkPointPriv::LengthSqd(dc) < SK_ScalarNearlyZero || SkPointPriv::LengthSqd(ab) < SK_ScalarNearlyZero; if (!doQuads) { SkScalar invDALengthSqd = SkPointPriv::LengthSqd(da); 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 (detABSqd * invDALengthSqd < toleranceSqd && 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 : SkPointPriv::DistanceToSqd(c0, 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. SkPointPriv::SetOrthog(&ab, ab); SkScalar z0 = -ab.dot(p[0]); SkPointPriv::SetOrthog(&dc, dc); SkScalar z1 = -dc.dot(p[3]); cAvg.fX = ab.fY * z1 - z0 * dc.fY; cAvg.fY = z0 * dc.fX - ab.fX * z1; SkScalar z = ab.fX * dc.fY - ab.fY * dc.fX; z = SkScalarInvert(z); cAvg.fX *= z; cAvg.fY *= z; if (sublevel <= kMaxSubdivs) { SkScalar d0Sqd = SkPointPriv::DistanceToSqd(c0, cAvg); SkScalar d1Sqd = SkPointPriv::DistanceToSqd(c1, 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(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, SkTArray* quads) { if (!p[0].isFinite() || !p[1].isFinite() || !p[2].isFinite() || !p[3].isFinite()) { return; } if (!SkScalarIsFinite(tolScale)) { return; } SkPoint chopped[10]; int count = SkChopCubicAtInflections(p, chopped); const SkScalar tolSqd = SkScalarSquare(tolScale); for (int i = 0; i < count; ++i) { SkPoint* cubic = chopped + 3*i; // The direction param is ignored if the third param is false. convert_noninflect_cubic_to_quads(cubic, tolSqd, false, SkPathPriv::kCCW_FirstDirection, quads); } } void GrPathUtils::convertCubicToQuadsConstrainToTangents(const SkPoint p[4], SkScalar tolScale, SkPathPriv::FirstDirection dir, SkTArray* quads) { if (!p[0].isFinite() || !p[1].isFinite() || !p[2].isFinite() || !p[3].isFinite()) { return; } if (!SkScalarIsFinite(tolScale)) { return; } SkPoint chopped[10]; int count = SkChopCubicAtInflections(p, chopped); const SkScalar tolSqd = SkScalarSquare(tolScale); for (int i = 0; i < count; ++i) { SkPoint* cubic = chopped + 3*i; convert_noninflect_cubic_to_quads(cubic, tolSqd, true, dir, quads); } } //////////////////////////////////////////////////////////////////////////////// using ExcludedTerm = GrPathUtils::ExcludedTerm; ExcludedTerm GrPathUtils::calcCubicInverseTransposePowerBasisMatrix(const SkPoint p[4], SkMatrix* out) { GR_STATIC_ASSERT(SK_SCALAR_IS_FLOAT); // First convert the bezier coordinates p[0..3] to power basis coefficients X,Y(,W=[0 0 0 1]). // M3 is the matrix that does this conversion. The homogeneous equation for the cubic becomes: // // | X Y 0 | // C(t,s) = [t^3 t^2*s t*s^2 s^3] * | . . 0 | // | . . 0 | // | . . 1 | // const Sk4f M3[3] = {Sk4f(-1, 3, -3, 1), Sk4f(3, -6, 3, 0), Sk4f(-3, 3, 0, 0)}; // 4th col of M3 = Sk4f(1, 0, 0, 0)}; Sk4f X(p[3].x(), 0, 0, 0); Sk4f Y(p[3].y(), 0, 0, 0); for (int i = 2; i >= 0; --i) { X += M3[i] * p[i].x(); Y += M3[i] * p[i].y(); } // The matrix is 3x4. In order to invert it, we first need to make it square by throwing out one // of the middle two rows. We toss the row that leaves us with the largest absolute determinant. // Since the right column will be [0 0 1], the respective determinants reduce to x0*y2 - y0*x2 // and x0*y1 - y0*x1. SkScalar dets[4]; Sk4f D = SkNx_shuffle<0,0,2,1>(X) * SkNx_shuffle<2,1,0,0>(Y); D -= SkNx_shuffle<2,3,0,1>(D); D.store(dets); ExcludedTerm skipTerm = SkScalarAbs(dets[0]) > SkScalarAbs(dets[1]) ? ExcludedTerm::kQuadraticTerm : ExcludedTerm::kLinearTerm; SkScalar det = dets[ExcludedTerm::kQuadraticTerm == skipTerm ? 0 : 1]; if (0 == det) { return ExcludedTerm::kNonInvertible; } SkScalar rdet = 1 / det; // Compute the inverse-transpose of the power basis matrix with the 'skipRow'th row removed. // Since W=[0 0 0 1], it follows that our corresponding solution will be equal to: // // | y1 -x1 x1*y2 - y1*x2 | // 1/det * | -y0 x0 -x0*y2 + y0*x2 | // | 0 0 det | // SkScalar x[4], y[4], z[4]; X.store(x); Y.store(y); (X * SkNx_shuffle<3,3,3,3>(Y) - Y * SkNx_shuffle<3,3,3,3>(X)).store(z); int middleRow = ExcludedTerm::kQuadraticTerm == skipTerm ? 2 : 1; out->setAll( y[middleRow] * rdet, -x[middleRow] * rdet, z[middleRow] * rdet, -y[0] * rdet, x[0] * rdet, -z[0] * rdet, 0, 0, 1); return skipTerm; } inline static void calc_serp_kcoeffs(SkScalar tl, SkScalar sl, SkScalar tm, SkScalar sm, ExcludedTerm skipTerm, SkScalar outCoeffs[3]) { SkASSERT(ExcludedTerm::kQuadraticTerm == skipTerm || ExcludedTerm::kLinearTerm == skipTerm); outCoeffs[0] = 0; outCoeffs[1] = (ExcludedTerm::kLinearTerm == skipTerm) ? sl*sm : -tl*sm - tm*sl; outCoeffs[2] = tl*tm; } inline static void calc_serp_lmcoeffs(SkScalar t, SkScalar s, ExcludedTerm skipTerm, SkScalar outCoeffs[3]) { SkASSERT(ExcludedTerm::kQuadraticTerm == skipTerm || ExcludedTerm::kLinearTerm == skipTerm); outCoeffs[0] = -s*s*s; outCoeffs[1] = (ExcludedTerm::kLinearTerm == skipTerm) ? 3*s*s*t : -3*s*t*t; outCoeffs[2] = t*t*t; } inline static void calc_loop_kcoeffs(SkScalar td, SkScalar sd, SkScalar te, SkScalar se, SkScalar tdse, SkScalar tesd, ExcludedTerm skipTerm, SkScalar outCoeffs[3]) { SkASSERT(ExcludedTerm::kQuadraticTerm == skipTerm || ExcludedTerm::kLinearTerm == skipTerm); outCoeffs[0] = 0; outCoeffs[1] = (ExcludedTerm::kLinearTerm == skipTerm) ? sd*se : -tdse - tesd; outCoeffs[2] = td*te; } inline static void calc_loop_lmcoeffs(SkScalar t2, SkScalar s2, SkScalar t1, SkScalar s1, SkScalar t2s1, SkScalar t1s2, ExcludedTerm skipTerm, SkScalar outCoeffs[3]) { SkASSERT(ExcludedTerm::kQuadraticTerm == skipTerm || ExcludedTerm::kLinearTerm == skipTerm); outCoeffs[0] = -s2*s2*s1; outCoeffs[1] = (ExcludedTerm::kLinearTerm == skipTerm) ? s2 * (2*t2s1 + t1s2) : -t2 * (t2s1 + 2*t1s2); outCoeffs[2] = t2*t2*t1; } // For the case when a cubic bezier is actually a quadratic. We duplicate k in l so that the // implicit becomes: // // k^3 - l*m == k^3 - l*k == k * (k^2 - l) // // In the quadratic case we can simply assign fixed values at each control point: // // | ..K.. | | pts[0] pts[1] pts[2] pts[3] | | 0 1/3 2/3 1 | // | ..L.. | * | . . . . | == | 0 0 1/3 1 | // | ..K.. | | 1 1 1 1 | | 0 1/3 2/3 1 | // static void calc_quadratic_klm(const SkPoint pts[4], double d3, SkMatrix* klm) { SkMatrix klmAtPts; klmAtPts.setAll(0, 1.f/3, 1, 0, 0, 1, 0, 1.f/3, 1); SkMatrix inversePts; inversePts.setAll(pts[0].x(), pts[1].x(), pts[3].x(), pts[0].y(), pts[1].y(), pts[3].y(), 1, 1, 1); SkAssertResult(inversePts.invert(&inversePts)); klm->setConcat(klmAtPts, inversePts); // If d3 > 0 we need to flip the orientation of our curve // This is done by negating the k and l values if (d3 > 0) { klm->postScale(-1, -1); } } // For the case when a cubic bezier is actually a line. We set K=0, L=1, M=-line, which results in // the following implicit: // // k^3 - l*m == 0^3 - 1*(-line) == -(-line) == line // static void calc_line_klm(const SkPoint pts[4], SkMatrix* klm) { SkScalar ny = pts[0].x() - pts[3].x(); SkScalar nx = pts[3].y() - pts[0].y(); SkScalar k = nx * pts[0].x() + ny * pts[0].y(); klm->setAll( 0, 0, 0, 0, 0, 1, -nx, -ny, k); } SkCubicType GrPathUtils::getCubicKLM(const SkPoint src[4], SkMatrix* klm, double tt[2], double ss[2]) { double d[4]; SkCubicType type = SkClassifyCubic(src, tt, ss, d); if (SkCubicType::kLineOrPoint == type) { calc_line_klm(src, klm); return SkCubicType::kLineOrPoint; } if (SkCubicType::kQuadratic == type) { calc_quadratic_klm(src, d[3], klm); return SkCubicType::kQuadratic; } SkMatrix CIT; ExcludedTerm skipTerm = calcCubicInverseTransposePowerBasisMatrix(src, &CIT); if (ExcludedTerm::kNonInvertible == skipTerm) { // This could technically also happen if the curve were quadratic, but SkClassifyCubic // should have detected that case already with tolerance. calc_line_klm(src, klm); return SkCubicType::kLineOrPoint; } const SkScalar t0 = static_cast(tt[0]), t1 = static_cast(tt[1]), s0 = static_cast(ss[0]), s1 = static_cast(ss[1]); SkMatrix klmCoeffs; switch (type) { case SkCubicType::kCuspAtInfinity: SkASSERT(1 == t1 && 0 == s1); // Infinity. // fallthru. case SkCubicType::kLocalCusp: case SkCubicType::kSerpentine: calc_serp_kcoeffs(t0, s0, t1, s1, skipTerm, &klmCoeffs[0]); calc_serp_lmcoeffs(t0, s0, skipTerm, &klmCoeffs[3]); calc_serp_lmcoeffs(t1, s1, skipTerm, &klmCoeffs[6]); break; case SkCubicType::kLoop: { const SkScalar tdse = t0 * s1; const SkScalar tesd = t1 * s0; calc_loop_kcoeffs(t0, s0, t1, s1, tdse, tesd, skipTerm, &klmCoeffs[0]); calc_loop_lmcoeffs(t0, s0, t1, s1, tdse, tesd, skipTerm, &klmCoeffs[3]); calc_loop_lmcoeffs(t1, s1, t0, s0, tesd, tdse, skipTerm, &klmCoeffs[6]); break; } default: SK_ABORT("Unexpected cubic type."); break; } klm->setConcat(klmCoeffs, CIT); return type; } int GrPathUtils::chopCubicAtLoopIntersection(const SkPoint src[4], SkPoint dst[10], SkMatrix* klm, int* loopIndex) { SkSTArray<2, SkScalar> chops; *loopIndex = -1; double t[2], s[2]; if (SkCubicType::kLoop == GrPathUtils::getCubicKLM(src, klm, t, s)) { SkScalar t0 = static_cast(t[0] / s[0]); SkScalar t1 = static_cast(t[1] / s[1]); SkASSERT(t0 <= t1); // Technically t0 != t1 in a loop, but there may be FP error. if (t0 < 1 && t1 > 0) { *loopIndex = 0; if (t0 > 0) { chops.push_back(t0); *loopIndex = 1; } if (t1 < 1) { chops.push_back(t1); *loopIndex = chops.count() - 1; } } } SkChopCubicAt(src, dst, chops.begin(), chops.count()); return chops.count() + 1; }