static inline bool is_cubic_nearly_quadratic(const Sk2f& p0, const Sk2f& p1, const Sk2f& p2, const Sk2f& p3, const Sk2f& tan0, const Sk2f& tan1, Sk2f* c) { Sk2f c1 = SkNx_fma(Sk2f(1.5f), tan0, p0); Sk2f c2 = SkNx_fma(Sk2f(-1.5f), tan1, p3); *c = (c1 + c2) * .5f; // Hopefully optimized out if not used? return ((c1 - c2).abs() <= 1).allTrue(); }
void GrCCPathParser::parsePath(const SkMatrix& m, const SkPath& path, SkRect* devBounds, SkRect* devBounds45) { const SkPoint* pts = SkPathPriv::PointData(path); int numPts = path.countPoints(); SkASSERT(numPts + 1 <= fLocalDevPtsBuffer.count()); if (!numPts) { devBounds->setEmpty(); devBounds45->setEmpty(); this->parsePath(path, nullptr); return; } // m45 transforms path points into "45 degree" device space. A bounding box in this space gives // the circumscribing octagon's diagonals. We could use SK_ScalarRoot2Over2, but an orthonormal // transform is not necessary as long as the shader uses the correct inverse. SkMatrix m45; m45.setSinCos(1, 1); m45.preConcat(m); // X,Y,T are two parallel view matrices that accumulate two bounding boxes as they map points: // device-space bounds and "45 degree" device-space bounds (| 1 -1 | * devCoords). // | 1 1 | Sk4f X = Sk4f(m.getScaleX(), m.getSkewY(), m45.getScaleX(), m45.getSkewY()); Sk4f Y = Sk4f(m.getSkewX(), m.getScaleY(), m45.getSkewX(), m45.getScaleY()); Sk4f T = Sk4f(m.getTranslateX(), m.getTranslateY(), m45.getTranslateX(), m45.getTranslateY()); // Map the path's points to device space and accumulate bounding boxes. Sk4f devPt = SkNx_fma(Y, Sk4f(pts[0].y()), T); devPt = SkNx_fma(X, Sk4f(pts[0].x()), devPt); Sk4f topLeft = devPt; Sk4f bottomRight = devPt; // Store all 4 values [dev.x, dev.y, dev45.x, dev45.y]. We are only interested in the first two, // and will overwrite [dev45.x, dev45.y] with the next point. This is why the dst buffer must // be at least one larger than the number of points. devPt.store(&fLocalDevPtsBuffer[0]); for (int i = 1; i < numPts; ++i) { devPt = SkNx_fma(Y, Sk4f(pts[i].y()), T); devPt = SkNx_fma(X, Sk4f(pts[i].x()), devPt); topLeft = Sk4f::Min(topLeft, devPt); bottomRight = Sk4f::Max(bottomRight, devPt); devPt.store(&fLocalDevPtsBuffer[i]); } SkPoint topLeftPts[2], bottomRightPts[2]; topLeft.store(topLeftPts); bottomRight.store(bottomRightPts); devBounds->setLTRB(topLeftPts[0].x(), topLeftPts[0].y(), bottomRightPts[0].x(), bottomRightPts[0].y()); devBounds45->setLTRB(topLeftPts[1].x(), topLeftPts[1].y(), bottomRightPts[1].x(), bottomRightPts[1].y()); this->parsePath(path, fLocalDevPtsBuffer.get()); }
inline void GrCCFillGeometry::appendQuadratics(const Sk2f& p0, const Sk2f& p1, const Sk2f& p2) { Sk2f tan0 = p1 - p0; Sk2f tan1 = p2 - p1; // This should almost always be this case for well-behaved curves in the real world. if (is_convex_curve_monotonic(p0, tan0, p2, tan1)) { this->appendMonotonicQuadratic(p0, p1, p2); return; } // Chop the curve into two segments with equal curvature. To do this we find the T value whose // tangent angle is halfway between tan0 and tan1. Sk2f n = normalize(tan0) - normalize(tan1); // The midtangent can be found where (dQ(t) dot n) = 0: // // 0 = (dQ(t) dot n) = | 2*t 1 | * | p0 - 2*p1 + p2 | * | n | // | -2*p0 + 2*p1 | | . | // // = | 2*t 1 | * | tan1 - tan0 | * | n | // | 2*tan0 | | . | // // = 2*t * ((tan1 - tan0) dot n) + (2*tan0 dot n) // // t = (tan0 dot n) / ((tan0 - tan1) dot n) Sk2f dQ1n = (tan0 - tan1) * n; Sk2f dQ0n = tan0 * n; Sk2f t = (dQ0n + SkNx_shuffle<1,0>(dQ0n)) / (dQ1n + SkNx_shuffle<1,0>(dQ1n)); t = Sk2f::Min(Sk2f::Max(t, 0), 1); // Clamp for FP error. Sk2f p01 = SkNx_fma(t, tan0, p0); Sk2f p12 = SkNx_fma(t, tan1, p1); Sk2f p012 = lerp(p01, p12, t); this->appendMonotonicQuadratic(p0, p01, p012); this->appendMonotonicQuadratic(p012, p12, p2); }
void GrCCFillGeometry::cubicTo(const SkPoint P[4], float inflectPad, float loopIntersectPad) { SkASSERT(fBuildingContour); SkASSERT(P[0] == fPoints.back()); // Don't crunch on the curve or inflate geometry if it is nearly flat (or just very small). // Flat curves can break the math below. if (are_collinear(P)) { Sk2f p0 = Sk2f::Load(P); Sk2f p3 = Sk2f::Load(P+3); this->appendLine(p0, p3); return; } Sk2f p0 = Sk2f::Load(P); Sk2f p1 = Sk2f::Load(P+1); Sk2f p2 = Sk2f::Load(P+2); Sk2f p3 = Sk2f::Load(P+3); // Also detect near-quadratics ahead of time. Sk2f tan0, tan1, c; get_cubic_tangents(p0, p1, p2, p3, &tan0, &tan1); if (is_cubic_nearly_quadratic(p0, p1, p2, p3, tan0, tan1, &c)) { this->appendQuadratics(p0, c, p3); return; } double tt[2], ss[2], D[4]; fCurrCubicType = SkClassifyCubic(P, tt, ss, D); SkASSERT(!SkCubicIsDegenerate(fCurrCubicType)); Sk2f t = Sk2f(static_cast<float>(tt[0]), static_cast<float>(tt[1])); Sk2f s = Sk2f(static_cast<float>(ss[0]), static_cast<float>(ss[1])); ExcludedTerm skipTerm = (std::abs(D[2]) > std::abs(D[1])) ? ExcludedTerm::kQuadraticTerm : ExcludedTerm::kLinearTerm; Sk2f C0 = SkNx_fma(Sk2f(3), p1 - p2, p3 - p0); Sk2f C1 = (ExcludedTerm::kLinearTerm == skipTerm ? SkNx_fma(Sk2f(-2), p1, p0 + p2) : p1 - p0) * 3; Sk2f C0x1 = C0 * SkNx_shuffle<1,0>(C1); float Cdet = C0x1[0] - C0x1[1]; SkSTArray<4, float> chops; if (SkCubicType::kLoop != fCurrCubicType) { find_chops_around_inflection_points(inflectPad, t, s, C0, C1, skipTerm, Cdet, &chops); } else { find_chops_around_loop_intersection(loopIntersectPad, t, s, C0, C1, skipTerm, Cdet, &chops); } if (4 == chops.count() && chops[1] >= chops[2]) { // This just the means the KLM roots are so close that their paddings overlap. We will // approximate the entire middle section, but still have it chopped midway. For loops this // chop guarantees the append code only sees convex segments. Otherwise, it means we are (at // least almost) a cusp and the chop makes sure we get a sharp point. Sk2f ts = t * SkNx_shuffle<1,0>(s); chops[1] = chops[2] = (ts[0] + ts[1]) / (2*s[0]*s[1]); } #ifdef SK_DEBUG for (int i = 1; i < chops.count(); ++i) { SkASSERT(chops[i] >= chops[i - 1]); } #endif this->appendCubics(AppendCubicMode::kLiteral, p0, p1, p2, p3, chops.begin(), chops.count()); }
// Finds where to chop a non-loop around its intersection point. The resulting cubic segments will // be chopped such that a box of radius 'padRadius', centered at any point along the curve segment, // is guaranteed to not cross the tangent lines at the intersection point (a.k.a lines L & M). // // 'chops' will be filled with 0, 2, or 4 T values. The segments between T0..T1 and T2..T3 must be // drawn with quadratic splines instead of cubics. // // A loop intersection falls at two different T values, so this method takes Sk2f and computes the // padding for both in SIMD. static inline void find_chops_around_loop_intersection(float padRadius, Sk2f t2, Sk2f s2, const Sk2f& C0, const Sk2f& C1, ExcludedTerm skipTerm, float Cdet, SkSTArray<4, float>* chops) { SkASSERT(chops->empty()); SkASSERT(padRadius >= 0); padRadius /= std::abs(Cdet); // Scale this single value rather than all of C^-1 later on. // The parametric functions for distance from lines L & M are: // // l(T) = (T - Td)^2 * (T - Te) // m(T) = (T - Td) * (T - Te)^2 // // See "Resolution Independent Curve Rendering using Programmable Graphics Hardware", // 4.3 Finding klmn: // // https://www.microsoft.com/en-us/research/wp-content/uploads/2005/01/p1000-loop.pdf Sk2f T2 = t2/s2; // T2 is the double root of l(T). Sk2f T1 = SkNx_shuffle<1,0>(T2); // T1 is the other root of l(T). // Convert l(T), m(T) to power-basis form: // // | 1 1 | // |l(T) m(T)| = |T^3 T^2 T 1| * | l2 m2 | // | l1 m1 | // | l0 m0 | // // From here on we use Sk2f with "L" names, but the second lane will be for line M. Sk2f l2 = SkNx_fma(Sk2f(-2), T2, -T1); Sk2f l1 = T2 * SkNx_fma(Sk2f(2), T1, T2); Sk2f l0 = -T2*T2*T1; // The equation for line L can be found as follows: // // L = C^-1 * (l excluding skipTerm) // // (See comments for GrPathUtils::calcCubicInverseTransposePowerBasisMatrix.) // We are only interested in the normal to L, so only need the upper 2x2 of C^-1. And rather // than divide by determinant(C) here, we have already performed this divide on padRadius. Sk2f l2or1 = (ExcludedTerm::kLinearTerm == skipTerm) ? l2 : l1; Sk2f Lx = -C0[1]*l2or1 + C1[1]; // l3 is always 1. Sk2f Ly = C0[0]*l2or1 - C1[0]; // A box of radius "padRadius" is touching line L if "center dot L" is less than the Manhattan // with of L. (See rationale in are_collinear.) Sk2f Lwidth = Lx.abs() + Ly.abs(); Sk2f pad = Lwidth * padRadius; // Is l(T=0) outside the padding around line L? Sk2f lT0 = l0; // l(T=0) = |0 0 0 1| dot |1 l2 l1 l0| = l0 Sk2f outsideT0 = lT0.abs() - pad; // Is l(T=1) outside the padding around line L? Sk2f lT1 = (Sk2f(1) + l2 + l1 + l0).abs(); // l(T=1) = |1 1 1 1| dot |1 l2 l1 l0| Sk2f outsideT1 = lT1.abs() - pad; // Values for solving the cubic. Sk2f p, q, qqq, discr, numRoots, D; bool hasDiscr = false; // Values for calculating one root (rarely needed). Sk2f R, QQ; bool hasOneRootVals = false; // Values for calculating three roots. Sk2f P, cosTheta3; bool hasThreeRootVals = false; // Solve for the T values where l(T) = +pad and m(T) = -pad. for (int i = 0; i < 2; ++i) { float T = T2[i]; // T is the point we are chopping around. if ((T < 0 && outsideT0[i] >= 0) || (T > 1 && outsideT1[i] >= 0)) { // The padding around T is completely out of range. No point solving for it. continue; } if (!hasDiscr) { p = Sk2f(+.5f, -.5f) * pad; q = (1.f/3) * (T2 - T1); qqq = q*q*q; discr = qqq*p*2 + p*p; numRoots = (discr < 0).thenElse(3, 1); D = T2 - q; hasDiscr = true; } if (1 == numRoots[i]) { if (!hasOneRootVals) { Sk2f r = qqq + p; Sk2f s = r.abs() + discr.sqrt(); R = (r > 0).thenElse(-s, s); QQ = q*q; hasOneRootVals = true; } float A = cbrtf(R[i]); float B = A != 0 ? QQ[i]/A : 0; // When there is only one root, ine L chops from root..1, line M chops from 0..root. if (1 == i) { chops->push_back(0); } chops->push_back(A + B + D[i]); if (0 == i) { chops->push_back(1); } continue; } if (!hasThreeRootVals) { P = q.abs() * -2; cosTheta3 = (q >= 0).thenElse(1, -1) + p / qqq.abs(); hasThreeRootVals = true; } static constexpr float k2PiOver3 = 2 * SK_ScalarPI / 3; float theta = std::acos(cosTheta3[i]) * (1.f/3); float roots[3] = {P[i] * std::cos(theta) + D[i], P[i] * std::cos(theta + k2PiOver3) + D[i], P[i] * std::cos(theta - k2PiOver3) + D[i]}; // Sort the three roots. swap_if_greater(roots[0], roots[1]); swap_if_greater(roots[1], roots[2]); swap_if_greater(roots[0], roots[1]); // Line L chops around the first 2 roots, line M chops around the second 2. chops->push_back_n(2, &roots[i]); } }
template<int N> static inline SkNx<N,float> lerp(const SkNx<N,float>& a, const SkNx<N,float>& b, const SkNx<N,float>& t) { return SkNx_fma(t, b - a, a); }