| /* |
| * 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<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.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<SkPoint, true>* 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]); |
| } |
