/*
* 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 "SkMathPriv.h"
#include "SkPointPriv.h"
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 = 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 = SkTMax(pathBounds.width(), pathBounds.height());
} else {
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 = SkPointPriv::DistanceToLineSegmentBetween(points[1], points[0], points[2]);
if (!SkScalarIsFinite(d)) {
return kMaxPointsPerCurve;
} 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 kMaxPointsPerCurve;
} 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, kMaxPointsPerCurve);
}
}
}
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) {
// You should have called scaleToleranceToSrc, which guarantees this
SkASSERT(tol >= gMinCurveTol);
SkScalar d = SkTMax(
SkPointPriv::DistanceToLineSegmentBetweenSqd(points[1], points[0], points[3]),
SkPointPriv::DistanceToLineSegmentBetweenSqd(points[2], points[0], points[3]));
d = SkScalarSqrt(d);
if (!SkScalarIsFinite(d)) {
return kMaxPointsPerCurve;
} else if (d <= tol) {
return 1;
} else {
SkScalar divSqrt = SkScalarSqrt(d / tol);
if (((SkScalar)SK_MaxS32) <= divSqrt) {
return kMaxPointsPerCurve;
} 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, kMaxPointsPerCurve);
}
}
}
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;
}
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, false)) != 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 = 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 = 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,
SkTArray<SkPoint, true>* 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
// GrAAHairlinePathRenderer.
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,
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 (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);
}
}
void GrPathUtils::convertCubicToQuads(const SkPoint p[4],
SkScalar tolScale,
SkTArray<SkPoint, true>* 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,
SkPathPriv::FirstDirection dir,
SkTArray<SkPoint, true>* 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);
}
}
////////////////////////////////////////////////////////////////////////////////
using ExcludedTerm = GrPathUtils::ExcludedTerm;
ExcludedTerm GrPathUtils::calcCubicInverseTransposePowerBasisMatrix(const SkPoint p[4],
SkMatrix* out) {
GR_STATIC_ASSERT(SK_SCALAR_IS_FLOAT);
// First convert the bezier coordinates p[0..3] 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 Sk4f M3[3] = {Sk4f(-1, 3, -3, 1),
Sk4f(3, -6, 3, 0),
Sk4f(-3, 3, 0, 0)};
// 4th col of M3 = Sk4f(1, 0, 0, 0)};
Sk4f X(p[3].x(), 0, 0, 0);
Sk4f Y(p[3].y(), 0, 0, 0);
for (int i = 2; i >= 0; --i) {
X += M3[i] * p[i].x();
Y += M3[i] * p[i].y();
}
// The matrix is 3x4. In order to invert it, we first need to make it square by throwing out one
// of the middle two rows. We toss the row that leaves us with the largest absolute determinant.
// Since the right column will be [0 0 1], the respective determinants reduce to x0*y2 - y0*x2
// and x0*y1 - y0*x1.
SkScalar dets[4];
Sk4f D = SkNx_shuffle<0,0,2,1>(X) * SkNx_shuffle<2,1,0,0>(Y);
D -= SkNx_shuffle<2,3,0,1>(D);
D.store(dets);
ExcludedTerm skipTerm = SkScalarAbs(dets[0]) > SkScalarAbs(dets[1]) ?
ExcludedTerm::kQuadraticTerm : ExcludedTerm::kLinearTerm;
SkScalar det = dets[ExcludedTerm::kQuadraticTerm == skipTerm ? 0 : 1];
if (0 == det) {
return ExcludedTerm::kNonInvertible;
}
SkScalar rdet = 1 / det;
// 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 |
//
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);
int middleRow = ExcludedTerm::kQuadraticTerm == skipTerm ? 2 : 1;
out->setAll( y[middleRow] * rdet, -x[middleRow] * rdet, z[middleRow] * rdet,
-y[0] * rdet, x[0] * rdet, -z[0] * rdet,
0, 0, 1);
return skipTerm;
}
inline static void calc_serp_kcoeffs(SkScalar tl, SkScalar sl, SkScalar tm, SkScalar sm,
ExcludedTerm skipTerm, SkScalar outCoeffs[3]) {
SkASSERT(ExcludedTerm::kQuadraticTerm == skipTerm || ExcludedTerm::kLinearTerm == skipTerm);
outCoeffs[0] = 0;
outCoeffs[1] = (ExcludedTerm::kLinearTerm == skipTerm) ? sl*sm : -tl*sm - tm*sl;
outCoeffs[2] = tl*tm;
}
inline static void calc_serp_lmcoeffs(SkScalar t, SkScalar s, ExcludedTerm skipTerm,
SkScalar outCoeffs[3]) {
SkASSERT(ExcludedTerm::kQuadraticTerm == skipTerm || ExcludedTerm::kLinearTerm == skipTerm);
outCoeffs[0] = -s*s*s;
outCoeffs[1] = (ExcludedTerm::kLinearTerm == skipTerm) ? 3*s*s*t : -3*s*t*t;
outCoeffs[2] = t*t*t;
}
inline static void calc_loop_kcoeffs(SkScalar td, SkScalar sd, SkScalar te, SkScalar se,
SkScalar tdse, SkScalar tesd, ExcludedTerm skipTerm,
SkScalar outCoeffs[3]) {
SkASSERT(ExcludedTerm::kQuadraticTerm == skipTerm || ExcludedTerm::kLinearTerm == skipTerm);
outCoeffs[0] = 0;
outCoeffs[1] = (ExcludedTerm::kLinearTerm == skipTerm) ? sd*se : -tdse - tesd;
outCoeffs[2] = td*te;
}
inline static void calc_loop_lmcoeffs(SkScalar t2, SkScalar s2, SkScalar t1, SkScalar s1,
SkScalar t2s1, SkScalar t1s2, ExcludedTerm skipTerm,
SkScalar outCoeffs[3]) {
SkASSERT(ExcludedTerm::kQuadraticTerm == skipTerm || ExcludedTerm::kLinearTerm == skipTerm);
outCoeffs[0] = -s2*s2*s1;
outCoeffs[1] = (ExcludedTerm::kLinearTerm == skipTerm) ? s2 * (2*t2s1 + t1s2)
: -t2 * (t2s1 + 2*t1s2);
outCoeffs[2] = t2*t2*t1;
}
// 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], double 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) {
klm->postScale(-1, -1);
}
}
// 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);
}
SkCubicType GrPathUtils::getCubicKLM(const SkPoint src[4], SkMatrix* klm, double tt[2],
double ss[2]) {
double d[4];
SkCubicType type = SkClassifyCubic(src, tt, ss, d);
if (SkCubicType::kLineOrPoint == type) {
calc_line_klm(src, klm);
return SkCubicType::kLineOrPoint;
}
if (SkCubicType::kQuadratic == type) {
calc_quadratic_klm(src, d[3], klm);
return SkCubicType::kQuadratic;
}
SkMatrix CIT;
ExcludedTerm skipTerm = calcCubicInverseTransposePowerBasisMatrix(src, &CIT);
if (ExcludedTerm::kNonInvertible == skipTerm) {
// This could technically also happen if the curve were quadratic, but SkClassifyCubic
// should have detected that case already with tolerance.
calc_line_klm(src, klm);
return SkCubicType::kLineOrPoint;
}
const SkScalar t0 = static_cast<SkScalar>(tt[0]), t1 = static_cast<SkScalar>(tt[1]),
s0 = static_cast<SkScalar>(ss[0]), s1 = static_cast<SkScalar>(ss[1]);
SkMatrix klmCoeffs;
switch (type) {
case SkCubicType::kCuspAtInfinity:
SkASSERT(1 == t1 && 0 == s1); // Infinity.
// fallthru.
case SkCubicType::kLocalCusp:
case SkCubicType::kSerpentine:
calc_serp_kcoeffs(t0, s0, t1, s1, skipTerm, &klmCoeffs[0]);
calc_serp_lmcoeffs(t0, s0, skipTerm, &klmCoeffs[3]);
calc_serp_lmcoeffs(t1, s1, skipTerm, &klmCoeffs[6]);
break;
case SkCubicType::kLoop: {
const SkScalar tdse = t0 * s1;
const SkScalar tesd = t1 * s0;
calc_loop_kcoeffs(t0, s0, t1, s1, tdse, tesd, skipTerm, &klmCoeffs[0]);
calc_loop_lmcoeffs(t0, s0, t1, s1, tdse, tesd, skipTerm, &klmCoeffs[3]);
calc_loop_lmcoeffs(t1, s1, t0, s0, tesd, tdse, skipTerm, &klmCoeffs[6]);
break;
}
default:
SK_ABORT("Unexpected cubic type.");
break;
}
klm->setConcat(klmCoeffs, CIT);
return type;
}
int GrPathUtils::chopCubicAtLoopIntersection(const SkPoint src[4], SkPoint dst[10], SkMatrix* klm,
int* loopIndex) {
SkSTArray<2, SkScalar> chops;
*loopIndex = -1;
double t[2], s[2];
if (SkCubicType::kLoop == GrPathUtils::getCubicKLM(src, klm, t, s)) {
SkScalar t0 = static_cast<SkScalar>(t[0] / s[0]);
SkScalar t1 = static_cast<SkScalar>(t[1] / s[1]);
SkASSERT(t0 <= t1); // Technically t0 != t1 in a loop, but there may be FP error.
if (t0 < 1 && t1 > 0) {
*loopIndex = 0;
if (t0 > 0) {
chops.push_back(t0);
*loopIndex = 1;
}
if (t1 < 1) {
chops.push_back(t1);
*loopIndex = chops.count() - 1;
}
}
}
SkChopCubicAt(src, dst, chops.begin(), chops.count());
return chops.count() + 1;
}