/* * 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 "src/gpu/geometry/GrPathUtils.h" #include "include/gpu/GrTypes.h" #include "src/core/SkMathPriv.h" #include "src/core/SkPointPriv.h" #include "src/core/SkUtils.h" #include "src/gpu/tessellate/WangsFormula.h" static const SkScalar kMinCurveTol = 0.0001f; static float tolerance_to_wangs_precision(float srcTol) { // You should have called scaleToleranceToSrc, which guarantees this SkASSERT(srcTol >= kMinCurveTol); // The GrPathUtil API defines tolerance as the max distance the linear segment can be from // the real curve. Wang's formula guarantees the linear segments will be within 1/precision // of the true curve, so precision = 1/srcTol return 1.f / srcTol; } uint32_t max_bezier_vertices(uint32_t chopCount) { static constexpr uint32_t kMaxChopsPerCurve = 10; static_assert((1 << kMaxChopsPerCurve) == GrPathUtils::kMaxPointsPerCurve); return 1 << std::min(chopCount, kMaxChopsPerCurve); } 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 = std::max(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 = std::max(pathBounds.width(), pathBounds.height()); } else { srcTol = devTol / stretch; } if (srcTol < kMinCurveTol) { srcTol = kMinCurveTol; } return srcTol; } uint32_t GrPathUtils::quadraticPointCount(const SkPoint points[], SkScalar tol) { return max_bezier_vertices(skgpu::wangs_formula::quadratic_log2( tolerance_to_wangs_precision(tol), points)); } 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) { return max_bezier_vertices(skgpu::wangs_formula::cubic_log2( tolerance_to_wangs_precision(tol), points)); } 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; } 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. lineVec = SkPointPriv::MakeOrthog(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 = std::max(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, SkPathFirstDirection dir, const SkPoint p) { SkVector ap = p - a; SkScalar apXab = ap.cross(ab); if (SkPathFirstDirection::kCW == dir) { if (apXab > 0) { return false; } } else { SkASSERT(SkPathFirstDirection::kCCW == dir); if (apXab < 0) { return false; } } SkVector dp = p - d; SkScalar dpXdc = dp.cross(dc); if (SkPathFirstDirection::kCW == dir) { if (dpXdc < 0) { return false; } } else { SkASSERT(SkPathFirstDirection::kCCW == dir); if (dpXdc > 0) { return false; } } return true; } void convert_noninflect_cubic_to_quads(const SkPoint p[4], SkScalar toleranceSqd, SkTArray* quads, int sublevel = 0, bool preserveFirstTangent = true, bool preserveLastTangent = true) { // 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]; } static const SkScalar kLengthScale = 3 * SK_Scalar1 / 2; static const int kMaxSubdivs = 10; ab.scale(kLengthScale); dc.scale(kLengthScale); // c0 and c1 are extrapolations along vectors ab and dc. SkPoint c0 = p[0] + ab; SkPoint c1 = p[3] + dc; SkScalar dSqd = sublevel > kMaxSubdivs ? 0 : SkPointPriv::DistanceToSqd(c0, c1); if (dSqd < toleranceSqd) { SkPoint newC; if (preserveFirstTangent == preserveLastTangent) { // We used to force a split when both tangents need to be preserved and c0 != c1. // This introduced a large performance regression for tiny paths for no noticeable // quality improvement. However, we aren't quite fulfilling our contract of guaranteeing // the two tangent vectors and this could introduce a missed pixel in // AAHairlinePathRenderer. newC = (c0 + c1) * 0.5f; } else if (preserveFirstTangent) { newC = c0; } else { newC = c1; } SkPoint* pts = quads->push_back_n(3); pts[0] = p[0]; pts[1] = newC; pts[2] = p[3]; return; } SkPoint choppedPts[7]; SkChopCubicAtHalf(p, choppedPts); convert_noninflect_cubic_to_quads( choppedPts + 0, toleranceSqd, quads, sublevel + 1, preserveFirstTangent, false); convert_noninflect_cubic_to_quads( choppedPts + 3, toleranceSqd, quads, sublevel + 1, false, preserveLastTangent); } void convert_noninflect_cubic_to_quads_with_constraint(const SkPoint p[4], SkScalar toleranceSqd, SkPathFirstDirection 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. 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); // c0 and c1 are extrapolations along vectors ab and dc. SkVector c0 = p[0] + ab; SkVector c1 = p[3] + dc; SkScalar dSqd = sublevel > kMaxSubdivs ? 0 : SkPointPriv::DistanceToSqd(c0, c1); if (dSqd < toleranceSqd) { SkPoint cAvg = (c0 + c1) * 0.5f; bool subdivide = false; if (!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 = SkPointPriv::MakeOrthog(ab); SkScalar z0 = -ab.dot(p[0]); dc = SkPointPriv::MakeOrthog(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_with_constraint( choppedPts + 0, toleranceSqd, dir, quads, sublevel + 1); convert_noninflect_cubic_to_quads_with_constraint( choppedPts + 3, toleranceSqd, dir, quads, sublevel + 1); } } // namespace 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; convert_noninflect_cubic_to_quads(cubic, tolSqd, quads); } } void GrPathUtils::convertCubicToQuadsConstrainToTangents(const SkPoint p[4], SkScalar tolScale, SkPathFirstDirection 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_with_constraint(cubic, tolSqd, dir, quads); } } int GrPathUtils::findCubicConvex180Chops(const SkPoint pts[], float T[2], bool* areCusps) { using grvx::float2; SkASSERT(pts); SkASSERT(T); SkASSERT(areCusps); // If a chop falls within a distance of "kEpsilon" from 0 or 1, throw it out. Tangents become // unstable when we chop too close to the boundary. This works out because the tessellation // shaders don't allow more than 2^10 parametric segments, and they snap the beginning and // ending edges at 0 and 1. So if we overstep an inflection or point of 180-degree rotation by a // fraction of a tessellation segment, it just gets snapped. constexpr static float kEpsilon = 1.f / (1 << 11); // Floating-point representation of "1 - 2*kEpsilon". constexpr static uint32_t kIEEE_one_minus_2_epsilon = (127 << 23) - 2 * (1 << (24 - 11)); // Unfortunately we don't have a way to static_assert this, but we can runtime assert that the // kIEEE_one_minus_2_epsilon bits are correct. SkASSERT(sk_bit_cast(kIEEE_one_minus_2_epsilon) == 1 - 2*kEpsilon); float2 p0 = skvx::bit_pun(pts[0]); float2 p1 = skvx::bit_pun(pts[1]); float2 p2 = skvx::bit_pun(pts[2]); float2 p3 = skvx::bit_pun(pts[3]); // Find the cubic's power basis coefficients. These define the bezier curve as: // // |T^3| // Cubic(T) = x,y = |A 3B 3C| * |T^2| + P0 // |. . .| |T | // // And the tangent direction (scaled by a uniform 1/3) will be: // // |T^2| // Tangent_Direction(T) = dx,dy = |A 2B C| * |T | // |. . .| |1 | // float2 C = p1 - p0; float2 D = p2 - p1; float2 E = p3 - p0; float2 B = D - C; float2 A = -3*D + E; // Now find the cubic's inflection function. There are inflections where F' x F'' == 0. // We formulate this as a quadratic equation: F' x F'' == aT^2 + bT + c == 0. // See: https://www.microsoft.com/en-us/research/wp-content/uploads/2005/01/p1000-loop.pdf // NOTE: We only need the roots, so a uniform scale factor does not affect the solution. float a = grvx::cross(A,B); float b = grvx::cross(A,C); float c = grvx::cross(B,C); float b_over_minus_2 = -.5f * b; float discr_over_4 = b_over_minus_2*b_over_minus_2 - a*c; // If -cuspThreshold <= discr_over_4 <= cuspThreshold, it means the two roots are within // kEpsilon of one another (in parametric space). This is close enough for our purposes to // consider them a single cusp. float cuspThreshold = a * (kEpsilon/2); cuspThreshold *= cuspThreshold; if (discr_over_4 < -cuspThreshold) { // The curve does not inflect or cusp. This means it might rotate more than 180 degrees // instead. Chop were rotation == 180 deg. (This is the 2nd root where the tangent is // parallel to tan0.) // // Tangent_Direction(T) x tan0 == 0 // (AT^2 x tan0) + (2BT x tan0) + (C x tan0) == 0 // (A x C)T^2 + (2B x C)T + (C x C) == 0 [[because tan0 == P1 - P0 == C]] // bT^2 + 2cT + 0 == 0 [[because A x C == b, B x C == c]] // T = [0, -2c/b] // // NOTE: if C == 0, then C != tan0. But this is fine because the curve is definitely // convex-180 if any points are colocated, and T[0] will equal NaN which returns 0 chops. *areCusps = false; float root = sk_ieee_float_divide(c, b_over_minus_2); // Is "root" inside the range [kEpsilon, 1 - kEpsilon)? if (sk_bit_cast(root - kEpsilon) < kIEEE_one_minus_2_epsilon) { T[0] = root; return 1; } return 0; } *areCusps = (discr_over_4 <= cuspThreshold); if (*areCusps) { // The two roots are close enough that we can consider them a single cusp. if (a != 0 || b_over_minus_2 != 0 || c != 0) { // Pick the average of both roots. float root = sk_ieee_float_divide(b_over_minus_2, a); // Is "root" inside the range [kEpsilon, 1 - kEpsilon)? if (sk_bit_cast(root - kEpsilon) < kIEEE_one_minus_2_epsilon) { T[0] = root; return 1; } return 0; } // The curve is a flat line. The standard inflection function doesn't detect cusps from flat // lines. Find cusps by searching instead for points where the tangent is perpendicular to // tan0. This will find any cusp point. // // dot(tan0, Tangent_Direction(T)) == 0 // // |T^2| // tan0 * |A 2B C| * |T | == 0 // |. . .| |1 | // float2 tan0 = skvx::if_then_else(C != 0, C, p2 - p0); a = grvx::dot(tan0, A); b_over_minus_2 = -grvx::dot(tan0, B); c = grvx::dot(tan0, C); discr_over_4 = std::max(b_over_minus_2*b_over_minus_2 - a*c, 0.f); } // Solve our quadratic equation to find where to chop. See the quadratic formula from // Numerical Recipes in C. float q = sqrtf(discr_over_4); q = copysignf(q, b_over_minus_2); q = q + b_over_minus_2; float2 roots = float2{q,c} / float2{a,q}; auto inside = (roots > kEpsilon) & (roots < (1 - kEpsilon)); if (inside[0]) { if (inside[1] && roots[0] != roots[1]) { if (roots[0] > roots[1]) { roots = skvx::shuffle<1,0>(roots); // Sort. } roots.store(T); return 2; } T[0] = roots[0]; return 1; } if (inside[1]) { T[0] = roots[1]; return 1; } return 0; }