| /* |
| * 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" |
| #include "SkMathPriv.h" |
| |
| static const int MAX_POINTS_PER_CURVE = 1 << 10; |
| 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 = 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 = points[1].distanceToLineSegmentBetween(points[0], points[2]); |
| if (!SkScalarIsFinite(d)) { |
| return MAX_POINTS_PER_CURVE; |
| } 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 MAX_POINTS_PER_CURVE; |
| } 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, 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) { |
| // You should have called scaleToleranceToSrc, which guarantees this |
| SkASSERT(tol >= gMinCurveTol); |
| |
| SkScalar d = SkTMax( |
| points[1].distanceToLineSegmentBetweenSqd(points[0], points[3]), |
| points[2].distanceToLineSegmentBetweenSqd(points[0], points[3])); |
| d = SkScalarSqrt(d); |
| if (!SkScalarIsFinite(d)) { |
| return MAX_POINTS_PER_CURVE; |
| } else if (d <= tol) { |
| return 1; |
| } else { |
| SkScalar divSqrt = SkScalarSqrt(d / tol); |
| if (((SkScalar)SK_MaxS32) <= divSqrt) { |
| return MAX_POINTS_PER_CURVE; |
| } 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, 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) { |
| // 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)) != 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 = 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); |
| // 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<SkPoint, true>* 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.lengthSqd() < SK_ScalarNearlyZero) { |
| if (dc.lengthSqd() < 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 (dc.lengthSqd() < 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 = 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 (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 : 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 = 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 = 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(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<SkPoint, true>* quads) { |
| 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<SkPoint, true>* quads) { |
| 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); |
| } |
| } |
| |
| //////////////////////////////////////////////////////////////////////////////// |
| |
| /** |
| * Computes an SkMatrix that can find the cubic KLM functionals as follows: |
| * |
| * | ..K.. | | ..kcoeffs.. | |
| * | ..L.. | = | ..lcoeffs.. | * inverse_transpose_power_basis_matrix |
| * | ..M.. | | ..mcoeffs.. | |
| * |
| * 'kcoeffs' are the power basis coefficients to a scalar valued cubic function that returns the |
| * signed distance to line K from a given point on the curve: |
| * |
| * k(t,s) = C(t,s) * K [C(t,s) is defined in the following comment] |
| * |
| * The same applies for lcoeffs and mcoeffs. These are found separately, depending on the type of |
| * curve. There are 4 coefficients but 3 rows in the matrix, so in order to do this calculation the |
| * caller must first remove a specific column of coefficients. |
| * |
| * @return which column of klm coefficients to exclude from the calculation. |
| */ |
| static int calc_inverse_transpose_power_basis_matrix(const SkPoint pts[4], SkMatrix* out) { |
| using SkScalar4 = SkNx<4, SkScalar>; |
| |
| // First we convert the bezier coordinates 'pts' 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 SkScalar4 M3[3] = {SkScalar4(-1, 3, -3, 1), |
| SkScalar4(3, -6, 3, 0), |
| SkScalar4(-3, 3, 0, 0)}; |
| // 4th column of M3 = SkScalar4(1, 0, 0, 0)}; |
| SkScalar4 X(pts[3].x(), 0, 0, 0); |
| SkScalar4 Y(pts[3].y(), 0, 0, 0); |
| for (int i = 2; i >= 0; --i) { |
| X += M3[i] * pts[i].x(); |
| Y += M3[i] * pts[i].y(); |
| } |
| |
| // The matrix is 3x4. In order to invert it, we first need to make it square by throwing out one |
| // of the top three rows. We toss the row that leaves us with the largest absolute determinant. |
| // Since the right column will be [0 0 1], the determinant reduces to x0*y1 - y0*x1. |
| SkScalar absDet[4]; |
| const SkScalar4 DETX1 = SkNx_shuffle<1,0,0,3>(X), DETY1 = SkNx_shuffle<1,0,0,3>(Y); |
| const SkScalar4 DETX2 = SkNx_shuffle<2,2,1,3>(X), DETY2 = SkNx_shuffle<2,2,1,3>(Y); |
| const SkScalar4 DET = DETX1 * DETY2 - DETY1 * DETX2; |
| DET.abs().store(absDet); |
| const int skipRow = absDet[0] > absDet[2] ? (absDet[0] > absDet[1] ? 0 : 1) |
| : (absDet[1] > absDet[2] ? 1 : 2); |
| const SkScalar rdet = 1 / DET[skipRow]; |
| const int row0 = (0 != skipRow) ? 0 : 1; |
| const int row1 = (2 == skipRow) ? 1 : 2; |
| |
| // 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 | |
| // |
| const SkScalar4 R(rdet, rdet, rdet, 1); |
| X *= R; |
| Y *= R; |
| |
| 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); |
| |
| out->setAll( y[row1], -x[row1], z[row1], |
| -y[row0], x[row0], -z[row0], |
| 0, 0, 1); |
| |
| return skipRow; |
| } |
| |
| static void negate_kl(SkMatrix* klm) { |
| // We could use klm->postScale(-1, -1), but it ends up doing a full matrix multiply. |
| for (int i = 0; i < 6; ++i) { |
| (*klm)[i] = -(*klm)[i]; |
| } |
| } |
| |
| static void calc_serp_klm(const SkPoint pts[4], const SkScalar d[4], SkMatrix* klm) { |
| SkMatrix CIT; |
| int skipCol = calc_inverse_transpose_power_basis_matrix(pts, &CIT); |
| |
| SkASSERT(d[0] >= 0); |
| const SkScalar root = SkScalarSqrt(3 * d[0]); |
| |
| const SkScalar tl = 3 * d[2] + root; |
| const SkScalar sl = 6 * d[1]; |
| const SkScalar tm = 3 * d[2] - root; |
| const SkScalar sm = 6 * d[1]; |
| |
| SkMatrix klmCoeffs; |
| int col = 0; |
| if (0 != skipCol) { |
| klmCoeffs[0] = 0; |
| klmCoeffs[3] = -sl * sl * sl; |
| klmCoeffs[6] = -sm * sm * sm; |
| ++col; |
| } |
| if (1 != skipCol) { |
| klmCoeffs[col + 0] = sl * sm; |
| klmCoeffs[col + 3] = 3 * sl * sl * tl; |
| klmCoeffs[col + 6] = 3 * sm * sm * tm; |
| ++col; |
| } |
| if (2 != skipCol) { |
| klmCoeffs[col + 0] = -tl * sm - tm * sl; |
| klmCoeffs[col + 3] = -3 * sl * tl * tl; |
| klmCoeffs[col + 6] = -3 * sm * tm * tm; |
| ++col; |
| } |
| |
| SkASSERT(2 == col); |
| klmCoeffs[2] = tl * tm; |
| klmCoeffs[5] = tl * tl * tl; |
| klmCoeffs[8] = tm * tm * tm; |
| |
| klm->setConcat(klmCoeffs, CIT); |
| |
| // If d1 > 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[1] > 0) { |
| negate_kl(klm); |
| } |
| } |
| |
| static void calc_loop_klm(const SkPoint pts[4], SkScalar d1, SkScalar td, SkScalar sd, |
| SkScalar te, SkScalar se, SkMatrix* klm) { |
| SkMatrix CIT; |
| int skipCol = calc_inverse_transpose_power_basis_matrix(pts, &CIT); |
| |
| const SkScalar tesd = te * sd; |
| const SkScalar tdse = td * se; |
| |
| SkMatrix klmCoeffs; |
| int col = 0; |
| if (0 != skipCol) { |
| klmCoeffs[0] = 0; |
| klmCoeffs[3] = -sd * sd * se; |
| klmCoeffs[6] = -se * se * sd; |
| ++col; |
| } |
| if (1 != skipCol) { |
| klmCoeffs[col + 0] = sd * se; |
| klmCoeffs[col + 3] = sd * (2 * tdse + tesd); |
| klmCoeffs[col + 6] = se * (2 * tesd + tdse); |
| ++col; |
| } |
| if (2 != skipCol) { |
| klmCoeffs[col + 0] = -tdse - tesd; |
| klmCoeffs[col + 3] = -td * (tdse + 2 * tesd); |
| klmCoeffs[col + 6] = -te * (tesd + 2 * tdse); |
| ++col; |
| } |
| |
| SkASSERT(2 == col); |
| klmCoeffs[2] = td * te; |
| klmCoeffs[5] = td * td * te; |
| klmCoeffs[8] = te * te * td; |
| |
| klm->setConcat(klmCoeffs, CIT); |
| |
| // For the general loop curve, we flip the orientation in the same pattern as the serp case |
| // above. Thus we only check d1. Technically we should check the value of the hessian as well |
| // cause we care about the sign of d1*Hessian. However, the Hessian is always negative outside |
| // the loop section and positive inside. We take care of the flipping for the loop sections |
| // later on. |
| if (d1 > 0) { |
| negate_kl(klm); |
| } |
| } |
| |
| // For the case when we have a cusp at a parameter value of infinity (discr == 0, d1 == 0). |
| static void calc_inf_cusp_klm(const SkPoint pts[4], SkScalar d2, SkScalar d3, SkMatrix* klm) { |
| SkMatrix CIT; |
| int skipCol = calc_inverse_transpose_power_basis_matrix(pts, &CIT); |
| |
| const SkScalar tn = d3; |
| const SkScalar sn = 3 * d2; |
| |
| SkMatrix klmCoeffs; |
| int col = 0; |
| if (0 != skipCol) { |
| klmCoeffs[0] = 0; |
| klmCoeffs[3] = -sn * sn * sn; |
| ++col; |
| } |
| if (1 != skipCol) { |
| klmCoeffs[col + 0] = 0; |
| klmCoeffs[col + 3] = 3 * sn * sn * tn; |
| ++col; |
| } |
| if (2 != skipCol) { |
| klmCoeffs[col + 0] = -sn; |
| klmCoeffs[col + 3] = -3 * sn * tn * tn; |
| ++col; |
| } |
| |
| SkASSERT(2 == col); |
| klmCoeffs[2] = tn; |
| klmCoeffs[5] = tn * tn * tn; |
| |
| klmCoeffs[6] = 0; |
| klmCoeffs[7] = 0; |
| klmCoeffs[8] = 1; |
| |
| klm->setConcat(klmCoeffs, CIT); |
| } |
| |
| // 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], SkScalar 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) { |
| negate_kl(klm); |
| } |
| } |
| |
| // 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); |
| } |
| |
| int GrPathUtils::chopCubicAtLoopIntersection(const SkPoint src[4], SkPoint dst[10], SkMatrix* klm, |
| int* loopIndex) { |
| // Variables to store the two parametric values at the loop double point. |
| SkScalar t1 = 0, t2 = 0; |
| |
| // Homogeneous parametric values at the loop double point. |
| SkScalar td, sd, te, se; |
| |
| SkScalar d[4]; |
| SkCubicType cType = SkClassifyCubic(src, d); |
| |
| int chop_count = 0; |
| if (SkCubicType::kLoop == cType) { |
| SkASSERT(d[0] < 0); |
| const SkScalar tempSqrt = SkScalarSqrt(-d[0]); |
| td = d[2] + tempSqrt; |
| sd = 2.f * d[1]; |
| te = d[2] - tempSqrt; |
| se = 2.f * d[1]; |
| |
| t1 = td / sd; |
| t2 = te / se; |
| // need to have t values sorted since this is what is expected by SkChopCubicAt |
| if (t1 > t2) { |
| SkTSwap(t1, t2); |
| } |
| |
| SkScalar chop_ts[2]; |
| if (t1 > 0.f && t1 < 1.f) { |
| chop_ts[chop_count++] = t1; |
| } |
| if (t2 > 0.f && t2 < 1.f) { |
| chop_ts[chop_count++] = t2; |
| } |
| if(dst) { |
| SkChopCubicAt(src, dst, chop_ts, chop_count); |
| } |
| } else { |
| if (dst) { |
| memcpy(dst, src, sizeof(SkPoint) * 4); |
| } |
| } |
| |
| if (loopIndex) { |
| if (2 == chop_count) { |
| *loopIndex = 1; |
| } else if (1 == chop_count) { |
| if (t1 < 0.f) { |
| *loopIndex = 0; |
| } else { |
| *loopIndex = 1; |
| } |
| } else { |
| if (t1 < 0.f && t2 > 1.f) { |
| *loopIndex = 0; |
| } else { |
| *loopIndex = -1; |
| } |
| } |
| } |
| |
| if (klm) { |
| switch (cType) { |
| case SkCubicType::kSerpentine: |
| case SkCubicType::kLocalCusp: |
| calc_serp_klm(src, d, klm); |
| break; |
| case SkCubicType::kLoop: |
| calc_loop_klm(src, d[1], td, sd, te, se, klm); |
| break; |
| case SkCubicType::kInfiniteCusp: |
| calc_inf_cusp_klm(src, d[2], d[3], klm); |
| break; |
| case SkCubicType::kQuadratic: |
| calc_quadratic_klm(src, d[3], klm); |
| break; |
| case SkCubicType::kLineOrPoint: |
| calc_line_klm(src, klm); |
| break; |
| }; |
| } |
| return chop_count + 1; |
| } |
