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/* -*- Mode: C++; tab-width: 8; indent-tabs-mode: nil; c-basic-offset: 2 -*- */
/* vim: set ts=8 sts=2 et sw=2 tw=80: */
/* This Source Code Form is subject to the terms of the Mozilla Public
* License, v. 2.0. If a copy of the MPL was not distributed with this
* file, You can obtain one at http://mozilla.org/MPL/2.0/. */
#include "2D.h"
#include "PathAnalysis.h"
#include "PathHelpers.h"
namespace mozilla {
namespace gfx {
static double CubicRoot(double aValue) {
if (aValue < 0.0) {
return -CubicRoot(-aValue);
} else {
return pow(aValue, 1.0 / 3.0);
}
}
struct PointD : public BasePoint<double, PointD> {
typedef BasePoint<double, PointD> Super;
PointD() : Super() {}
PointD(double aX, double aY) : Super(aX, aY) {}
MOZ_IMPLICIT PointD(const Point& aPoint) : Super(aPoint.x, aPoint.y) {}
Point ToPoint() const {
return Point(static_cast<Float>(x), static_cast<Float>(y));
}
};
struct BezierControlPoints {
BezierControlPoints() = default;
BezierControlPoints(const PointD& aCP1, const PointD& aCP2,
const PointD& aCP3, const PointD& aCP4)
: mCP1(aCP1), mCP2(aCP2), mCP3(aCP3), mCP4(aCP4) {}
PointD mCP1, mCP2, mCP3, mCP4;
};
void FlattenBezier(const BezierControlPoints& aPoints, PathSink* aSink,
double aTolerance);
Path::Path() = default;
Path::~Path() = default;
Float Path::ComputeLength() {
EnsureFlattenedPath();
return mFlattenedPath->ComputeLength();
}
Point Path::ComputePointAtLength(Float aLength, Point* aTangent) {
EnsureFlattenedPath();
return mFlattenedPath->ComputePointAtLength(aLength, aTangent);
}
void Path::EnsureFlattenedPath() {
if (!mFlattenedPath) {
mFlattenedPath = new FlattenedPath();
StreamToSink(mFlattenedPath);
}
}
// This is the maximum deviation we allow (with an additional ~20% margin of
// error) of the approximation from the actual Bezier curve.
const Float kFlatteningTolerance = 0.0001f;
void FlattenedPath::MoveTo(const Point& aPoint) {
MOZ_ASSERT(!mCalculatedLength);
FlatPathOp op;
op.mType = FlatPathOp::OP_MOVETO;
op.mPoint = aPoint;
mPathOps.push_back(op);
mBeginPoint = aPoint;
}
void FlattenedPath::LineTo(const Point& aPoint) {
MOZ_ASSERT(!mCalculatedLength);
FlatPathOp op;
op.mType = FlatPathOp::OP_LINETO;
op.mPoint = aPoint;
mPathOps.push_back(op);
}
void FlattenedPath::BezierTo(const Point& aCP1, const Point& aCP2,
const Point& aCP3) {
MOZ_ASSERT(!mCalculatedLength);
FlattenBezier(BezierControlPoints(CurrentPoint(), aCP1, aCP2, aCP3), this,
kFlatteningTolerance);
}
void FlattenedPath::QuadraticBezierTo(const Point& aCP1, const Point& aCP2) {
MOZ_ASSERT(!mCalculatedLength);
// We need to elevate the degree of this quadratic B�zier to cubic, so we're
// going to add an intermediate control point, and recompute control point 1.
// The first and last control points remain the same.
// This formula can be found on http://fontforge.sourceforge.net/bezier.html
Point CP0 = CurrentPoint();
Point CP1 = (CP0 + aCP1 * 2.0) / 3.0;
Point CP2 = (aCP2 + aCP1 * 2.0) / 3.0;
Point CP3 = aCP2;
BezierTo(CP1, CP2, CP3);
}
void FlattenedPath::Close() {
MOZ_ASSERT(!mCalculatedLength);
LineTo(mBeginPoint);
}
void FlattenedPath::Arc(const Point& aOrigin, float aRadius, float aStartAngle,
float aEndAngle, bool aAntiClockwise) {
ArcToBezier(this, aOrigin, Size(aRadius, aRadius), aStartAngle, aEndAngle,
aAntiClockwise);
}
Float FlattenedPath::ComputeLength() {
if (!mCalculatedLength) {
Point currentPoint;
for (uint32_t i = 0; i < mPathOps.size(); i++) {
if (mPathOps[i].mType == FlatPathOp::OP_MOVETO) {
currentPoint = mPathOps[i].mPoint;
} else {
mCachedLength += Distance(currentPoint, mPathOps[i].mPoint);
currentPoint = mPathOps[i].mPoint;
}
}
mCalculatedLength = true;
}
return mCachedLength;
}
Point FlattenedPath::ComputePointAtLength(Float aLength, Point* aTangent) {
// We track the last point that -wasn't- in the same place as the current
// point so if we pass the edge of the path with a bunch of zero length
// paths we still get the correct tangent vector.
Point lastPointSinceMove;
Point currentPoint;
for (uint32_t i = 0; i < mPathOps.size(); i++) {
if (mPathOps[i].mType == FlatPathOp::OP_MOVETO) {
if (Distance(currentPoint, mPathOps[i].mPoint)) {
lastPointSinceMove = currentPoint;
}
currentPoint = mPathOps[i].mPoint;
} else {
Float segmentLength = Distance(currentPoint, mPathOps[i].mPoint);
if (segmentLength) {
lastPointSinceMove = currentPoint;
if (segmentLength > aLength) {
Point currentVector = mPathOps[i].mPoint - currentPoint;
Point tangent = currentVector / segmentLength;
if (aTangent) {
*aTangent = tangent;
}
return currentPoint + tangent * aLength;
}
}
aLength -= segmentLength;
currentPoint = mPathOps[i].mPoint;
}
}
Point currentVector = currentPoint - lastPointSinceMove;
if (aTangent) {
if (hypotf(currentVector.x, currentVector.y)) {
*aTangent = currentVector / hypotf(currentVector.x, currentVector.y);
} else {
*aTangent = Point();
}
}
return currentPoint;
}
// This function explicitly permits aControlPoints to refer to the same object
// as either of the other arguments.
static void SplitBezier(const BezierControlPoints& aControlPoints,
BezierControlPoints* aFirstSegmentControlPoints,
BezierControlPoints* aSecondSegmentControlPoints,
double t) {
MOZ_ASSERT(aSecondSegmentControlPoints);
*aSecondSegmentControlPoints = aControlPoints;
PointD cp1a =
aControlPoints.mCP1 + (aControlPoints.mCP2 - aControlPoints.mCP1) * t;
PointD cp2a =
aControlPoints.mCP2 + (aControlPoints.mCP3 - aControlPoints.mCP2) * t;
PointD cp1aa = cp1a + (cp2a - cp1a) * t;
PointD cp3a =
aControlPoints.mCP3 + (aControlPoints.mCP4 - aControlPoints.mCP3) * t;
PointD cp2aa = cp2a + (cp3a - cp2a) * t;
PointD cp1aaa = cp1aa + (cp2aa - cp1aa) * t;
aSecondSegmentControlPoints->mCP4 = aControlPoints.mCP4;
if (aFirstSegmentControlPoints) {
aFirstSegmentControlPoints->mCP1 = aControlPoints.mCP1;
aFirstSegmentControlPoints->mCP2 = cp1a;
aFirstSegmentControlPoints->mCP3 = cp1aa;
aFirstSegmentControlPoints->mCP4 = cp1aaa;
}
aSecondSegmentControlPoints->mCP1 = cp1aaa;
aSecondSegmentControlPoints->mCP2 = cp2aa;
aSecondSegmentControlPoints->mCP3 = cp3a;
}
static void FlattenBezierCurveSegment(const BezierControlPoints& aControlPoints,
PathSink* aSink, double aTolerance) {
/* The algorithm implemented here is based on:
*
* The basic premise is that for a small t the third order term in the
* equation of a cubic bezier curve is insignificantly small. This can
* then be approximated by a quadratic equation for which the maximum
* difference from a linear approximation can be much more easily determined.
*/
BezierControlPoints currentCP = aControlPoints;
double t = 0;
double currentTolerance = aTolerance;
while (t < 1.0) {
PointD cp21 = currentCP.mCP2 - currentCP.mCP1;
PointD cp31 = currentCP.mCP3 - currentCP.mCP1;
/* To remove divisions and check for divide-by-zero, this is optimized from:
* Float s3 = (cp31.x * cp21.y - cp31.y * cp21.x) / hypotf(cp21.x, cp21.y);
* t = 2 * Float(sqrt(aTolerance / (3. * std::abs(s3))));
*/
double cp21x31 = cp31.x * cp21.y - cp31.y * cp21.x;
double h = hypot(cp21.x, cp21.y);
if (cp21x31 * h == 0) {
break;
}
double s3inv = h / cp21x31;
t = 2 * sqrt(currentTolerance * std::abs(s3inv) / 3.);
currentTolerance *= 1 + aTolerance;
// Increase tolerance every iteration to prevent this loop from executing
// too many times. This approximates the length of large curves more
// roughly. In practice, aTolerance is the constant kFlatteningTolerance
// which has value 0.0001. With this value, it takes 6,932 splits to double
// currentTolerance (to 0.0002) and 23,028 splits to increase
// currentTolerance by an order of magnitude (to 0.001).
if (t >= 1.0) {
break;
}
SplitBezier(currentCP, nullptr, &currentCP, t);
aSink->LineTo(currentCP.mCP1.ToPoint());
}
aSink->LineTo(currentCP.mCP4.ToPoint());
}
static inline void FindInflectionApproximationRange(
BezierControlPoints aControlPoints, double* aMin, double* aMax, double aT,
double aTolerance) {
SplitBezier(aControlPoints, nullptr, &aControlPoints, aT);
PointD cp21 = aControlPoints.mCP2 - aControlPoints.mCP1;
PointD cp41 = aControlPoints.mCP4 - aControlPoints.mCP1;
if (cp21.x == 0. && cp21.y == 0.) {
cp21 = aControlPoints.mCP3 - aControlPoints.mCP1;
}
if (cp21.x == 0. && cp21.y == 0.) {
// In this case s3 becomes lim[n->0] (cp41.x * n) / n - (cp41.y * n) / n =
// cp41.x - cp41.y.
double s3 = cp41.x - cp41.y;
// Use the absolute value so that Min and Max will correspond with the
// minimum and maximum of the range.
if (s3 == 0) {
*aMin = -1.0;
*aMax = 2.0;
} else {
double r = CubicRoot(std::abs(aTolerance / s3));
*aMin = aT - r;
*aMax = aT + r;
}
return;
}
double s3 = (cp41.x * cp21.y - cp41.y * cp21.x) / hypot(cp21.x, cp21.y);
if (s3 == 0) {
// This means within the precision we have it can be approximated
// infinitely by a linear segment. Deal with this by specifying the
// approximation range as extending beyond the entire curve.
*aMin = -1.0;
*aMax = 2.0;
return;
}
double tf = CubicRoot(std::abs(aTolerance / s3));
*aMin = aT - tf * (1 - aT);
*aMax = aT + tf * (1 - aT);
}
/* Find the inflection points of a bezier curve. Will return false if the
* curve is degenerate in such a way that it is best approximated by a straight
* line.
*
* The below algorithm was written by Jeff Muizelaar <jmuizelaar@mozilla.com>,
* explanation follows:
*
* The lower inflection point is returned in aT1, the higher one in aT2. In the
* case of a single inflection point this will be in aT1.
*
* The method is inspired by the algorithm in "analysis of in?ection points for
* planar cubic bezier curve"
*
* Here are some differences between this algorithm and versions discussed
* elsewhere in the literature:
*
* zhang et. al compute a0, d0 and e0 incrementally using the follow formula:
*
* Point a0 = CP2 - CP1
* Point a1 = CP3 - CP2
* Point a2 = CP4 - CP1
*
* Point d0 = a1 - a0
* Point d1 = a2 - a1
* Point e0 = d1 - d0
*
* this avoids any multiplications and may or may not be faster than the
* approach take below.
*
* "fast, precise flattening of cubic bezier path and ofset curves" by hain et.
* al
* Point a = CP1 + 3 * CP2 - 3 * CP3 + CP4
* Point b = 3 * CP1 - 6 * CP2 + 3 * CP3
* Point c = -3 * CP1 + 3 * CP2
* Point d = CP1
* the a, b, c, d can be expressed in terms of a0, d0 and e0 defined above as:
* c = 3 * a0
* b = 3 * d0
* a = e0
*
*
* a = 3a = a.y * b.x - a.x * b.y
* b = 3b = a.y * c.x - a.x * c.y
* c = 9c = b.y * c.x - b.x * c.y
*
* The additional multiples of 3 cancel each other out as show below:
*
* x = (-b + sqrt(b * b - 4 * a * c)) / (2 * a)
* x = (-3 * b + sqrt(3 * b * 3 * b - 4 * a * 3 * 9 * c / 3)) / (2 * 3 * a)
* x = 3 * (-b + sqrt(b * b - 4 * a * c)) / (2 * 3 * a)
* x = (-b + sqrt(b * b - 4 * a * c)) / (2 * a)
*
* I haven't looked into whether the formulation of the quadratic formula in
* hain has any numerical advantages over the one used below.
*/
static inline void FindInflectionPoints(
const BezierControlPoints& aControlPoints, double* aT1, double* aT2,
uint32_t* aCount) {
// Find inflection points.
// See www.faculty.idc.ac.il/arik/quality/appendixa.html for an explanation
// of this approach.
PointD A = aControlPoints.mCP2 - aControlPoints.mCP1;
PointD B =
aControlPoints.mCP3 - (aControlPoints.mCP2 * 2) + aControlPoints.mCP1;
PointD C = aControlPoints.mCP4 - (aControlPoints.mCP3 * 3) +
(aControlPoints.mCP2 * 3) - aControlPoints.mCP1;
double a = B.x * C.y - B.y * C.x;
double b = A.x * C.y - A.y * C.x;
double c = A.x * B.y - A.y * B.x;
if (a == 0) {
// Not a quadratic equation.
if (b == 0) {
// Instead of a linear acceleration change we have a constant
// acceleration change. This means the equation has no solution
// and there are no inflection points, unless the constant is 0.
// In that case the curve is a straight line, essentially that means
// the easiest way to deal with is is by saying there's an inflection
// point at t == 0. The inflection point approximation range found will
// automatically extend into infinity.
if (c == 0) {
*aCount = 1;
*aT1 = 0;
return;
}
*aCount = 0;
return;
}
*aT1 = -c / b;
*aCount = 1;
return;
} else {
double discriminant = b * b - 4 * a * c;
if (discriminant < 0) {
// No inflection points.
*aCount = 0;
} else if (discriminant == 0) {
*aCount = 1;
*aT1 = -b / (2 * a);
} else {
/* Use the following formula for computing the roots:
*
* q = -1/2 * (b + sign(b) * sqrt(b^2 - 4ac))
* t1 = q / a
* t2 = c / q
*/
double q = sqrt(discriminant);
if (b < 0) {
q = b - q;
} else {
q = b + q;
}
q *= -1. / 2;
*aT1 = q / a;
*aT2 = c / q;
if (*aT1 > *aT2) {
std::swap(*aT1, *aT2);
}
*aCount = 2;
}
}
}
void FlattenBezier(const BezierControlPoints& aControlPoints, PathSink* aSink,
double aTolerance) {
double t1;
double t2;
uint32_t count;
FindInflectionPoints(aControlPoints, &t1, &t2, &count);
// Check that at least one of the inflection points is inside [0..1]
if (count == 0 ||
((t1 < 0.0 || t1 >= 1.0) && (count == 1 || (t2 < 0.0 || t2 >= 1.0)))) {
FlattenBezierCurveSegment(aControlPoints, aSink, aTolerance);
return;
}
double t1min = t1, t1max = t1, t2min = t2, t2max = t2;
BezierControlPoints remainingCP = aControlPoints;
// For both inflection points, calulate the range where they can be linearly
// approximated if they are positioned within [0,1]
if (count > 0 && t1 >= 0 && t1 < 1.0) {
FindInflectionApproximationRange(aControlPoints, &t1min, &t1max, t1,
aTolerance);
}
if (count > 1 && t2 >= 0 && t2 < 1.0) {
FindInflectionApproximationRange(aControlPoints, &t2min, &t2max, t2,
aTolerance);
}
BezierControlPoints nextCPs = aControlPoints;
BezierControlPoints prevCPs;
// Process ranges. [t1min, t1max] and [t2min, t2max] are approximated by line
// segments.
if (count == 1 && t1min <= 0 && t1max >= 1.0) {
// The whole range can be approximated by a line segment.
aSink->LineTo(aControlPoints.mCP4.ToPoint());
return;
}
if (t1min > 0) {
// Flatten the Bezier up until the first inflection point's approximation
// point.
SplitBezier(aControlPoints, &prevCPs, &remainingCP, t1min);
FlattenBezierCurveSegment(prevCPs, aSink, aTolerance);
}
if (t1max >= 0 && t1max < 1.0 && (count == 1 || t2min > t1max)) {
// The second inflection point's approximation range begins after the end
// of the first, approximate the first inflection point by a line and
// subsequently flatten up until the end or the next inflection point.
SplitBezier(aControlPoints, nullptr, &nextCPs, t1max);
aSink->LineTo(nextCPs.mCP1.ToPoint());
if (count == 1 || (count > 1 && t2min >= 1.0)) {
// No more inflection points to deal with, flatten the rest of the curve.
FlattenBezierCurveSegment(nextCPs, aSink, aTolerance);
}
} else if (count > 1 && t2min > 1.0) {
// We've already concluded t2min <= t1max, so if this is true the
// approximation range for the first inflection point runs past the
// end of the curve, draw a line to the end and we're done.
aSink->LineTo(aControlPoints.mCP4.ToPoint());
return;
}
if (count > 1 && t2min < 1.0 && t2max > 0) {
if (t2min > 0 && t2min < t1max) {
// In this case the t2 approximation range starts inside the t1
// approximation range.
SplitBezier(aControlPoints, nullptr, &nextCPs, t1max);
aSink->LineTo(nextCPs.mCP1.ToPoint());
} else if (t2min > 0 && t1max > 0) {
SplitBezier(aControlPoints, nullptr, &nextCPs, t1max);
// Find a control points describing the portion of the curve between t1max
// and t2min.
double t2mina = (t2min - t1max) / (1 - t1max);
SplitBezier(nextCPs, &prevCPs, &nextCPs, t2mina);
FlattenBezierCurveSegment(prevCPs, aSink, aTolerance);
} else if (t2min > 0) {
// We have nothing interesting before t2min, find that bit and flatten it.
SplitBezier(aControlPoints, &prevCPs, &nextCPs, t2min);
FlattenBezierCurveSegment(prevCPs, aSink, aTolerance);
}
if (t2max < 1.0) {
// Flatten the portion of the curve after t2max
SplitBezier(aControlPoints, nullptr, &nextCPs, t2max);
// Draw a line to the start, this is the approximation between t2min and
// t2max.
aSink->LineTo(nextCPs.mCP1.ToPoint());
FlattenBezierCurveSegment(nextCPs, aSink, aTolerance);
} else {
// Our approximation range extends beyond the end of the curve.
aSink->LineTo(aControlPoints.mCP4.ToPoint());
return;
}
}
}
} // namespace gfx
} // namespace mozilla