/*
* 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]);
}