michael@0: /* -*- Mode: C++; tab-width: 20; indent-tabs-mode: nil; c-basic-offset: 2 -*-
michael@0:  * This Source Code Form is subject to the terms of the Mozilla Public
michael@0:  * License, v. 2.0. If a copy of the MPL was not distributed with this
michael@0:  * file, You can obtain one at http://mozilla.org/MPL/2.0/. */
michael@0: 
michael@0: #include "2D.h"
michael@0: #include "PathAnalysis.h"
michael@0: #include "PathHelpers.h"
michael@0: 
michael@0: namespace mozilla {
michael@0: namespace gfx {
michael@0: 
michael@0: static float CubicRoot(float aValue) {
michael@0:   if (aValue < 0.0) {
michael@0:     return -CubicRoot(-aValue);
michael@0:   }
michael@0:   else {
michael@0:     return powf(aValue, 1.0f / 3.0f);
michael@0:   }
michael@0: }
michael@0: 
michael@0: struct BezierControlPoints
michael@0: {
michael@0:   BezierControlPoints() {}
michael@0:   BezierControlPoints(const Point &aCP1, const Point &aCP2,
michael@0:                       const Point &aCP3, const Point &aCP4)
michael@0:     : mCP1(aCP1), mCP2(aCP2), mCP3(aCP3), mCP4(aCP4)
michael@0:   {
michael@0:   }
michael@0: 
michael@0:   Point mCP1, mCP2, mCP3, mCP4;
michael@0: };
michael@0: 
michael@0: void
michael@0: FlattenBezier(const BezierControlPoints &aPoints,
michael@0:               PathSink *aSink, Float aTolerance);
michael@0: 
michael@0: 
michael@0: Path::Path()
michael@0: {
michael@0: }
michael@0: 
michael@0: Path::~Path()
michael@0: {
michael@0: }
michael@0: 
michael@0: Float
michael@0: Path::ComputeLength()
michael@0: {
michael@0:   EnsureFlattenedPath();
michael@0:   return mFlattenedPath->ComputeLength();
michael@0: }
michael@0: 
michael@0: Point
michael@0: Path::ComputePointAtLength(Float aLength, Point* aTangent)
michael@0: {
michael@0:   EnsureFlattenedPath();
michael@0:   return mFlattenedPath->ComputePointAtLength(aLength, aTangent);
michael@0: }
michael@0: 
michael@0: void
michael@0: Path::EnsureFlattenedPath()
michael@0: {
michael@0:   if (!mFlattenedPath) {
michael@0:     mFlattenedPath = new FlattenedPath();
michael@0:     StreamToSink(mFlattenedPath);
michael@0:   }
michael@0: }
michael@0: 
michael@0: // This is the maximum deviation we allow (with an additional ~20% margin of
michael@0: // error) of the approximation from the actual Bezier curve.
michael@0: const Float kFlatteningTolerance = 0.0001f;
michael@0: 
michael@0: void
michael@0: FlattenedPath::MoveTo(const Point &aPoint)
michael@0: {
michael@0:   MOZ_ASSERT(!mCalculatedLength);
michael@0:   FlatPathOp op;
michael@0:   op.mType = FlatPathOp::OP_MOVETO;
michael@0:   op.mPoint = aPoint;
michael@0:   mPathOps.push_back(op);
michael@0: 
michael@0:   mLastMove = aPoint;
michael@0: }
michael@0: 
michael@0: void
michael@0: FlattenedPath::LineTo(const Point &aPoint)
michael@0: {
michael@0:   MOZ_ASSERT(!mCalculatedLength);
michael@0:   FlatPathOp op;
michael@0:   op.mType = FlatPathOp::OP_LINETO;
michael@0:   op.mPoint = aPoint;
michael@0:   mPathOps.push_back(op);
michael@0: }
michael@0: 
michael@0: void
michael@0: FlattenedPath::BezierTo(const Point &aCP1,
michael@0:                         const Point &aCP2,
michael@0:                         const Point &aCP3)
michael@0: {
michael@0:   MOZ_ASSERT(!mCalculatedLength);
michael@0:   FlattenBezier(BezierControlPoints(CurrentPoint(), aCP1, aCP2, aCP3), this, kFlatteningTolerance);
michael@0: }
michael@0: 
michael@0: void
michael@0: FlattenedPath::QuadraticBezierTo(const Point &aCP1,
michael@0:                                  const Point &aCP2)
michael@0: {
michael@0:   MOZ_ASSERT(!mCalculatedLength);
michael@0:   // We need to elevate the degree of this quadratic Bézier to cubic, so we're
michael@0:   // going to add an intermediate control point, and recompute control point 1.
michael@0:   // The first and last control points remain the same.
michael@0:   // This formula can be found on http://fontforge.sourceforge.net/bezier.html
michael@0:   Point CP0 = CurrentPoint();
michael@0:   Point CP1 = (CP0 + aCP1 * 2.0) / 3.0;
michael@0:   Point CP2 = (aCP2 + aCP1 * 2.0) / 3.0;
michael@0:   Point CP3 = aCP2;
michael@0: 
michael@0:   BezierTo(CP1, CP2, CP3);
michael@0: }
michael@0: 
michael@0: void
michael@0: FlattenedPath::Close()
michael@0: {
michael@0:   MOZ_ASSERT(!mCalculatedLength);
michael@0:   LineTo(mLastMove);
michael@0: }
michael@0: 
michael@0: void
michael@0: FlattenedPath::Arc(const Point &aOrigin, float aRadius, float aStartAngle,
michael@0:                    float aEndAngle, bool aAntiClockwise)
michael@0: {
michael@0:   ArcToBezier(this, aOrigin, Size(aRadius, aRadius), aStartAngle, aEndAngle, aAntiClockwise);
michael@0: }
michael@0: 
michael@0: Float
michael@0: FlattenedPath::ComputeLength()
michael@0: {
michael@0:   if (!mCalculatedLength) {
michael@0:     Point currentPoint;
michael@0: 
michael@0:     for (uint32_t i = 0; i < mPathOps.size(); i++) {
michael@0:       if (mPathOps[i].mType == FlatPathOp::OP_MOVETO) {
michael@0:         currentPoint = mPathOps[i].mPoint;
michael@0:       } else {
michael@0:         mCachedLength += Distance(currentPoint, mPathOps[i].mPoint);
michael@0:         currentPoint = mPathOps[i].mPoint;
michael@0:       }
michael@0:     }
michael@0: 
michael@0:     mCalculatedLength =  true;
michael@0:   }
michael@0: 
michael@0:   return mCachedLength;
michael@0: }
michael@0: 
michael@0: Point
michael@0: FlattenedPath::ComputePointAtLength(Float aLength, Point *aTangent)
michael@0: {
michael@0:   // We track the last point that -wasn't- in the same place as the current
michael@0:   // point so if we pass the edge of the path with a bunch of zero length
michael@0:   // paths we still get the correct tangent vector.
michael@0:   Point lastPointSinceMove;
michael@0:   Point currentPoint;
michael@0:   for (uint32_t i = 0; i < mPathOps.size(); i++) {
michael@0:     if (mPathOps[i].mType == FlatPathOp::OP_MOVETO) {
michael@0:       if (Distance(currentPoint, mPathOps[i].mPoint)) {
michael@0:         lastPointSinceMove = currentPoint;
michael@0:       }
michael@0:       currentPoint = mPathOps[i].mPoint;
michael@0:     } else {
michael@0:       Float segmentLength = Distance(currentPoint, mPathOps[i].mPoint);
michael@0: 
michael@0:       if (segmentLength) {
michael@0:         lastPointSinceMove = currentPoint;
michael@0:         if (segmentLength > aLength) {
michael@0:           Point currentVector = mPathOps[i].mPoint - currentPoint;
michael@0:           Point tangent = currentVector / segmentLength;
michael@0:           if (aTangent) {
michael@0:             *aTangent = tangent;
michael@0:           }
michael@0:           return currentPoint + tangent * aLength;
michael@0:         }
michael@0:       }
michael@0: 
michael@0:       aLength -= segmentLength;
michael@0:       currentPoint = mPathOps[i].mPoint;
michael@0:     }
michael@0:   }
michael@0: 
michael@0:   Point currentVector = currentPoint - lastPointSinceMove;
michael@0:   if (aTangent) {
michael@0:     if (hypotf(currentVector.x, currentVector.y)) {
michael@0:       *aTangent = currentVector / hypotf(currentVector.x, currentVector.y);
michael@0:     } else {
michael@0:       *aTangent = Point();
michael@0:     }
michael@0:   }
michael@0:   return currentPoint;
michael@0: }
michael@0: 
michael@0: // This function explicitly permits aControlPoints to refer to the same object
michael@0: // as either of the other arguments.
michael@0: static void 
michael@0: SplitBezier(const BezierControlPoints &aControlPoints,
michael@0:             BezierControlPoints *aFirstSegmentControlPoints,
michael@0:             BezierControlPoints *aSecondSegmentControlPoints,
michael@0:             Float t)
michael@0: {
michael@0:   MOZ_ASSERT(aSecondSegmentControlPoints);
michael@0:   
michael@0:   *aSecondSegmentControlPoints = aControlPoints;
michael@0: 
michael@0:   Point cp1a = aControlPoints.mCP1 + (aControlPoints.mCP2 - aControlPoints.mCP1) * t;
michael@0:   Point cp2a = aControlPoints.mCP2 + (aControlPoints.mCP3 - aControlPoints.mCP2) * t;
michael@0:   Point cp1aa = cp1a + (cp2a - cp1a) * t;
michael@0:   Point cp3a = aControlPoints.mCP3 + (aControlPoints.mCP4 - aControlPoints.mCP3) * t;
michael@0:   Point cp2aa = cp2a + (cp3a - cp2a) * t;
michael@0:   Point cp1aaa = cp1aa + (cp2aa - cp1aa) * t;
michael@0:   aSecondSegmentControlPoints->mCP4 = aControlPoints.mCP4;
michael@0: 
michael@0:   if(aFirstSegmentControlPoints) {
michael@0:     aFirstSegmentControlPoints->mCP1 = aControlPoints.mCP1;
michael@0:     aFirstSegmentControlPoints->mCP2 = cp1a;
michael@0:     aFirstSegmentControlPoints->mCP3 = cp1aa;
michael@0:     aFirstSegmentControlPoints->mCP4 = cp1aaa;
michael@0:   }
michael@0:   aSecondSegmentControlPoints->mCP1 = cp1aaa;
michael@0:   aSecondSegmentControlPoints->mCP2 = cp2aa;
michael@0:   aSecondSegmentControlPoints->mCP3 = cp3a;
michael@0: }
michael@0: 
michael@0: static void
michael@0: FlattenBezierCurveSegment(const BezierControlPoints &aControlPoints,
michael@0:                           PathSink *aSink,
michael@0:                           Float aTolerance)
michael@0: {
michael@0:   /* The algorithm implemented here is based on:
michael@0:    * http://cis.usouthal.edu/~hain/general/Publications/Bezier/Bezier%20Offset%20Curves.pdf
michael@0:    *
michael@0:    * The basic premise is that for a small t the third order term in the
michael@0:    * equation of a cubic bezier curve is insignificantly small. This can
michael@0:    * then be approximated by a quadratic equation for which the maximum
michael@0:    * difference from a linear approximation can be much more easily determined.
michael@0:    */
michael@0:   BezierControlPoints currentCP = aControlPoints;
michael@0: 
michael@0:   Float t = 0;
michael@0:   while (t < 1.0f) {
michael@0:     Point cp21 = currentCP.mCP2 - currentCP.mCP3;
michael@0:     Point cp31 = currentCP.mCP3 - currentCP.mCP1;
michael@0: 
michael@0:     Float s3 = (cp31.x * cp21.y - cp31.y * cp21.x) / hypotf(cp21.x, cp21.y);
michael@0: 
michael@0:     t = 2 * Float(sqrt(aTolerance / (3. * abs(s3))));
michael@0: 
michael@0:     if (t >= 1.0f) {
michael@0:       aSink->LineTo(aControlPoints.mCP4);
michael@0:       break;
michael@0:     }
michael@0: 
michael@0:     Point prevCP2, prevCP3, nextCP1, nextCP2, nextCP3;
michael@0:     SplitBezier(currentCP, nullptr, &currentCP, t);
michael@0: 
michael@0:     aSink->LineTo(currentCP.mCP1);
michael@0:   }
michael@0: }
michael@0: 
michael@0: static inline void
michael@0: FindInflectionApproximationRange(BezierControlPoints aControlPoints,
michael@0:                                  Float *aMin, Float *aMax, Float aT,
michael@0:                                  Float aTolerance)
michael@0: {
michael@0:     SplitBezier(aControlPoints, nullptr, &aControlPoints, aT);
michael@0: 
michael@0:     Point cp21 = aControlPoints.mCP2 - aControlPoints.mCP1;
michael@0:     Point cp41 = aControlPoints.mCP4 - aControlPoints.mCP1;
michael@0: 
michael@0:     if (cp21.x == 0 && cp21.y == 0) {
michael@0:       // In this case s3 becomes lim[n->0] (cp41.x * n) / n - (cp41.y * n) / n = cp41.x - cp41.y.
michael@0: 
michael@0:       // Use the absolute value so that Min and Max will correspond with the
michael@0:       // minimum and maximum of the range.
michael@0:       *aMin = aT - CubicRoot(abs(aTolerance / (cp41.x - cp41.y)));
michael@0:       *aMax = aT + CubicRoot(abs(aTolerance / (cp41.x - cp41.y)));
michael@0:       return;
michael@0:     }
michael@0: 
michael@0:     Float s3 = (cp41.x * cp21.y - cp41.y * cp21.x) / hypotf(cp21.x, cp21.y);
michael@0: 
michael@0:     if (s3 == 0) {
michael@0:       // This means within the precision we have it can be approximated
michael@0:       // infinitely by a linear segment. Deal with this by specifying the
michael@0:       // approximation range as extending beyond the entire curve.
michael@0:       *aMin = -1.0f;
michael@0:       *aMax = 2.0f;
michael@0:       return;
michael@0:     }
michael@0: 
michael@0:     Float tf = CubicRoot(abs(aTolerance / s3));
michael@0: 
michael@0:     *aMin = aT - tf * (1 - aT);
michael@0:     *aMax = aT + tf * (1 - aT);
michael@0: }
michael@0: 
michael@0: /* Find the inflection points of a bezier curve. Will return false if the
michael@0:  * curve is degenerate in such a way that it is best approximated by a straight
michael@0:  * line.
michael@0:  *
michael@0:  * The below algorithm was written by Jeff Muizelaar <jmuizelaar@mozilla.com>, explanation follows:
michael@0:  *
michael@0:  * The lower inflection point is returned in aT1, the higher one in aT2. In the
michael@0:  * case of a single inflection point this will be in aT1.
michael@0:  *
michael@0:  * The method is inspired by the algorithm in "analysis of in?ection points for planar cubic bezier curve"
michael@0:  *
michael@0:  * Here are some differences between this algorithm and versions discussed elsewhere in the literature:
michael@0:  *
michael@0:  * zhang et. al compute a0, d0 and e0 incrementally using the follow formula:
michael@0:  *
michael@0:  * Point a0 = CP2 - CP1
michael@0:  * Point a1 = CP3 - CP2
michael@0:  * Point a2 = CP4 - CP1
michael@0:  *
michael@0:  * Point d0 = a1 - a0
michael@0:  * Point d1 = a2 - a1
michael@0:  
michael@0:  * Point e0 = d1 - d0
michael@0:  *
michael@0:  * this avoids any multiplications and may or may not be faster than the approach take below.
michael@0:  *
michael@0:  * "fast, precise flattening of cubic bezier path and ofset curves" by hain et. al
michael@0:  * Point a = CP1 + 3 * CP2 - 3 * CP3 + CP4
michael@0:  * Point b = 3 * CP1 - 6 * CP2 + 3 * CP3
michael@0:  * Point c = -3 * CP1 + 3 * CP2
michael@0:  * Point d = CP1
michael@0:  * the a, b, c, d can be expressed in terms of a0, d0 and e0 defined above as:
michael@0:  * c = 3 * a0
michael@0:  * b = 3 * d0
michael@0:  * a = e0
michael@0:  *
michael@0:  *
michael@0:  * a = 3a = a.y * b.x - a.x * b.y
michael@0:  * b = 3b = a.y * c.x - a.x * c.y
michael@0:  * c = 9c = b.y * c.x - b.x * c.y
michael@0:  *
michael@0:  * The additional multiples of 3 cancel each other out as show below:
michael@0:  *
michael@0:  * x = (-b + sqrt(b * b - 4 * a * c)) / (2 * a)
michael@0:  * x = (-3 * b + sqrt(3 * b * 3 * b - 4 * a * 3 * 9 * c / 3)) / (2 * 3 * a)
michael@0:  * x = 3 * (-b + sqrt(b * b - 4 * a * c)) / (2 * 3 * a)
michael@0:  * x = (-b + sqrt(b * b - 4 * a * c)) / (2 * a)
michael@0:  *
michael@0:  * I haven't looked into whether the formulation of the quadratic formula in
michael@0:  * hain has any numerical advantages over the one used below.
michael@0:  */
michael@0: static inline void
michael@0: FindInflectionPoints(const BezierControlPoints &aControlPoints,
michael@0:                      Float *aT1, Float *aT2, uint32_t *aCount)
michael@0: {
michael@0:   // Find inflection points.
michael@0:   // See www.faculty.idc.ac.il/arik/quality/appendixa.html for an explanation
michael@0:   // of this approach.
michael@0:   Point A = aControlPoints.mCP2 - aControlPoints.mCP1;
michael@0:   Point B = aControlPoints.mCP3 - (aControlPoints.mCP2 * 2) + aControlPoints.mCP1;
michael@0:   Point C = aControlPoints.mCP4 - (aControlPoints.mCP3 * 3) + (aControlPoints.mCP2 * 3) - aControlPoints.mCP1;
michael@0: 
michael@0:   Float a = Float(B.x) * C.y - Float(B.y) * C.x;
michael@0:   Float b = Float(A.x) * C.y - Float(A.y) * C.x;
michael@0:   Float c = Float(A.x) * B.y - Float(A.y) * B.x;
michael@0: 
michael@0:   if (a == 0) {
michael@0:     // Not a quadratic equation.
michael@0:     if (b == 0) {
michael@0:       // Instead of a linear acceleration change we have a constant
michael@0:       // acceleration change. This means the equation has no solution
michael@0:       // and there are no inflection points, unless the constant is 0.
michael@0:       // In that case the curve is a straight line, but we'll let
michael@0:       // FlattenBezierCurveSegment deal with this.
michael@0:       *aCount = 0;
michael@0:       return;
michael@0:     }
michael@0:     *aT1 = -c / b;
michael@0:     *aCount = 1;
michael@0:     return;
michael@0:   } else {
michael@0:     Float discriminant = b * b - 4 * a * c;
michael@0: 
michael@0:     if (discriminant < 0) {
michael@0:       // No inflection points.
michael@0:       *aCount = 0;
michael@0:     } else if (discriminant == 0) {
michael@0:       *aCount = 1;
michael@0:       *aT1 = -b / (2 * a);
michael@0:     } else {
michael@0:       /* Use the following formula for computing the roots:
michael@0:        *
michael@0:        * q = -1/2 * (b + sign(b) * sqrt(b^2 - 4ac))
michael@0:        * t1 = q / a
michael@0:        * t2 = c / q
michael@0:        */
michael@0:       Float q = sqrtf(discriminant);
michael@0:       if (b < 0) {
michael@0:         q = b - q;
michael@0:       } else {
michael@0:         q = b + q;
michael@0:       }
michael@0:       q *= Float(-1./2);
michael@0: 
michael@0:       *aT1 = q / a;
michael@0:       *aT2 = c / q;
michael@0:       if (*aT1 > *aT2) {
michael@0:         std::swap(*aT1, *aT2);
michael@0:       }
michael@0:       *aCount = 2;
michael@0:     }
michael@0:   }
michael@0: 
michael@0:   return;
michael@0: }
michael@0: 
michael@0: void
michael@0: FlattenBezier(const BezierControlPoints &aControlPoints,
michael@0:               PathSink *aSink, Float aTolerance)
michael@0: {
michael@0:   Float t1;
michael@0:   Float t2;
michael@0:   uint32_t count;
michael@0: 
michael@0:   FindInflectionPoints(aControlPoints, &t1, &t2, &count);
michael@0: 
michael@0:   // Check that at least one of the inflection points is inside [0..1]
michael@0:   if (count == 0 || ((t1 < 0 || t1 > 1.0) && ((t2 < 0 || t2 > 1.0) || count == 1)) ) {
michael@0:     FlattenBezierCurveSegment(aControlPoints, aSink, aTolerance);
michael@0:     return;
michael@0:   }
michael@0: 
michael@0:   Float t1min = t1, t1max = t1, t2min = t2, t2max = t2;
michael@0: 
michael@0:   BezierControlPoints remainingCP = aControlPoints;
michael@0: 
michael@0:   // For both inflection points, calulate the range where they can be linearly
michael@0:   // approximated if they are positioned within [0,1]
michael@0:   if (count > 0 && t1 >= 0 && t1 < 1.0) {
michael@0:     FindInflectionApproximationRange(aControlPoints, &t1min, &t1max, t1, aTolerance);
michael@0:   }
michael@0:   if (count > 1 && t2 >= 0 && t2 < 1.0) {
michael@0:     FindInflectionApproximationRange(aControlPoints, &t2min, &t2max, t2, aTolerance);
michael@0:   }
michael@0:   BezierControlPoints nextCPs = aControlPoints;
michael@0:   BezierControlPoints prevCPs;
michael@0: 
michael@0:   // Process ranges. [t1min, t1max] and [t2min, t2max] are approximated by line
michael@0:   // segments.
michael@0:   if (t1min > 0) {
michael@0:     // Flatten the Bezier up until the first inflection point's approximation
michael@0:     // point.
michael@0:     SplitBezier(aControlPoints, &prevCPs,
michael@0:                 &remainingCP, t1min);
michael@0:     FlattenBezierCurveSegment(prevCPs, aSink, aTolerance);
michael@0:   }
michael@0:   if (t1max >= 0 && t1max < 1.0 && (count == 1 || t2min > t1max)) {
michael@0:     // The second inflection point's approximation range begins after the end
michael@0:     // of the first, approximate the first inflection point by a line and
michael@0:     // subsequently flatten up until the end or the next inflection point.
michael@0:     SplitBezier(aControlPoints, nullptr, &nextCPs, t1max);
michael@0: 
michael@0:     aSink->LineTo(nextCPs.mCP1);
michael@0: 
michael@0:     if (count == 1 || (count > 1 && t2min >= 1.0)) {
michael@0:       // No more inflection points to deal with, flatten the rest of the curve.
michael@0:       FlattenBezierCurveSegment(nextCPs, aSink, aTolerance);
michael@0:     }
michael@0:   } else if (count > 1 && t2min > 1.0) {
michael@0:     // We've already concluded t2min <= t1max, so if this is true the
michael@0:     // approximation range for the first inflection point runs past the
michael@0:     // end of the curve, draw a line to the end and we're done.
michael@0:     aSink->LineTo(aControlPoints.mCP4);
michael@0:     return;
michael@0:   }
michael@0: 
michael@0:   if (count > 1 && t2min < 1.0 && t2max > 0) {
michael@0:     if (t2min > 0 && t2min < t1max) {
michael@0:       // In this case the t2 approximation range starts inside the t1
michael@0:       // approximation range.
michael@0:       SplitBezier(aControlPoints, nullptr, &nextCPs, t1max);
michael@0:       aSink->LineTo(nextCPs.mCP1);
michael@0:     } else if (t2min > 0 && t1max > 0) {
michael@0:       SplitBezier(aControlPoints, nullptr, &nextCPs, t1max);
michael@0: 
michael@0:       // Find a control points describing the portion of the curve between t1max and t2min.
michael@0:       Float t2mina = (t2min - t1max) / (1 - t1max);
michael@0:       SplitBezier(nextCPs, &prevCPs, &nextCPs, t2mina);
michael@0:       FlattenBezierCurveSegment(prevCPs, aSink, aTolerance);
michael@0:     } else if (t2min > 0) {
michael@0:       // We have nothing interesting before t2min, find that bit and flatten it.
michael@0:       SplitBezier(aControlPoints, &prevCPs, &nextCPs, t2min);
michael@0:       FlattenBezierCurveSegment(prevCPs, aSink, aTolerance);
michael@0:     }
michael@0:     if (t2max < 1.0) {
michael@0:       // Flatten the portion of the curve after t2max
michael@0:       SplitBezier(aControlPoints, nullptr, &nextCPs, t2max);
michael@0: 
michael@0:       // Draw a line to the start, this is the approximation between t2min and
michael@0:       // t2max.
michael@0:       aSink->LineTo(nextCPs.mCP1);
michael@0:       FlattenBezierCurveSegment(nextCPs, aSink, aTolerance);
michael@0:     } else {
michael@0:       // Our approximation range extends beyond the end of the curve.
michael@0:       aSink->LineTo(aControlPoints.mCP4);
michael@0:       return;
michael@0:     }
michael@0:   }
michael@0: }
michael@0: 
michael@0: }
michael@0: }