The Tor Browser: comparison gfx/2d/Path.cpp

--1:000000000000
+:ded075acf0b6
+/* -*- Mode: C++; tab-width: 20; indent-tabs-mode: nil; c-basic-offset: 2 -*-
+* 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 float CubicRoot(float aValue) {
+if (aValue < 0.0) {
+return -CubicRoot(-aValue);
+}
+else {
+return powf(aValue, 1.0f / 3.0f);
+}
+}
+struct BezierControlPoints
+{
+BezierControlPoints() {}
+BezierControlPoints(const Point &aCP1, const Point &aCP2,
+const Point &aCP3, const Point &aCP4)
+: mCP1(aCP1), mCP2(aCP2), mCP3(aCP3), mCP4(aCP4)
+{
+}
+Point mCP1, mCP2, mCP3, mCP4;
+};
+void
+FlattenBezier(const BezierControlPoints &aPoints,
+PathSink *aSink, Float aTolerance);
+Path::Path()
+{
+}
+Path::~Path()
+{
+}
+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);
+mLastMove = 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(mLastMove);
+}
+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,
+Float t)
+{
+MOZ_ASSERT(aSecondSegmentControlPoints);
+*aSecondSegmentControlPoints = aControlPoints;
+Point cp1a = aControlPoints.mCP1 + (aControlPoints.mCP2 - aControlPoints.mCP1) * t;
+Point cp2a = aControlPoints.mCP2 + (aControlPoints.mCP3 - aControlPoints.mCP2) * t;
+Point cp1aa = cp1a + (cp2a - cp1a) * t;
+Point cp3a = aControlPoints.mCP3 + (aControlPoints.mCP4 - aControlPoints.mCP3) * t;
+Point cp2aa = cp2a + (cp3a - cp2a) * t;
+Point 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,
+Float aTolerance)
+{
+/* The algorithm implemented here is based on:
+* http://cis.usouthal.edu/~hain/general/Publications/Bezier/Bezier%20Offset%20Curves.pdf
+*
+* 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;
+Float t = 0;
+while (t < 1.0f) {
+Point cp21 = currentCP.mCP2 - currentCP.mCP3;
+Point cp31 = currentCP.mCP3 - currentCP.mCP1;
+Float s3 = (cp31.x * cp21.y - cp31.y * cp21.x) / hypotf(cp21.x, cp21.y);
+t = 2 * Float(sqrt(aTolerance / (3. * abs(s3))));
+if (t >= 1.0f) {
+aSink->LineTo(aControlPoints.mCP4);
+break;
+}
+Point prevCP2, prevCP3, nextCP1, nextCP2, nextCP3;
+SplitBezier(currentCP, nullptr, &currentCP, t);
+aSink->LineTo(currentCP.mCP1);
+}
+}
+static inline void
+FindInflectionApproximationRange(BezierControlPoints aControlPoints,
+Float *aMin, Float *aMax, Float aT,
+Float aTolerance)
+{
+SplitBezier(aControlPoints, nullptr, &aControlPoints, aT);
+Point cp21 = aControlPoints.mCP2 - aControlPoints.mCP1;
+Point cp41 = aControlPoints.mCP4 - 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.
+// Use the absolute value so that Min and Max will correspond with the
+// minimum and maximum of the range.
+*aMin = aT - CubicRoot(abs(aTolerance / (cp41.x - cp41.y)));
+*aMax = aT + CubicRoot(abs(aTolerance / (cp41.x - cp41.y)));
+return;
+}
+Float s3 = (cp41.x * cp21.y - cp41.y * cp21.x) / hypotf(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 = -1.0f;
+*aMax = 2.0f;
+return;
+}
+Float tf = CubicRoot(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,
+Float *aT1, Float *aT2, uint32_t *aCount)
+{
+// Find inflection points.
+// See www.faculty.idc.ac.il/arik/quality/appendixa.html for an explanation
+// of this approach.
+Point A = aControlPoints.mCP2 - aControlPoints.mCP1;
+Point B = aControlPoints.mCP3 - (aControlPoints.mCP2 * 2) + aControlPoints.mCP1;
+Point C = aControlPoints.mCP4 - (aControlPoints.mCP3 * 3) + (aControlPoints.mCP2 * 3) - aControlPoints.mCP1;
+Float a = Float(B.x) * C.y - Float(B.y) * C.x;
+Float b = Float(A.x) * C.y - Float(A.y) * C.x;
+Float c = Float(A.x) * B.y - Float(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, but we'll let
+// FlattenBezierCurveSegment deal with this.
+*aCount = 0;
+return;
+}
+*aT1 = -c / b;
+*aCount = 1;
+return;
+} else {
+Float 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
+*/
+Float q = sqrtf(discriminant);
+if (b < 0) {
+q = b - q;
+} else {
+q = b + q;
+}
+q *= Float(-1./2);
+*aT1 = q / a;
+*aT2 = c / q;
+if (*aT1 > *aT2) {
+std::swap(*aT1, *aT2);
+}
+*aCount = 2;
+}
+}
+return;
+}
+void
+FlattenBezier(const BezierControlPoints &aControlPoints,
+PathSink *aSink, Float aTolerance)
+{
+Float t1;
+Float 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 || t1 > 1.0) && ((t2 < 0 || t2 > 1.0) || count == 1)) ) {
+FlattenBezierCurveSegment(aControlPoints, aSink, aTolerance);
+return;
+}
+Float 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 (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);
+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);
+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);
+} 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.
+Float 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);
+FlattenBezierCurveSegment(nextCPs, aSink, aTolerance);
+} else {
+// Our approximation range extends beyond the end of the curve.
+aSink->LineTo(aControlPoints.mCP4);
+return;
+}
+}
+}
+}
+}

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comparison: gfx/2d/Path.cpp

gfx/2d/Path.cpp