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/*
* Copyright (C) 2008 Apple Inc. All Rights Reserved.
*
* Redistribution and use in source and binary forms, with or without
* modification, are permitted provided that the following conditions
* are met:
* 1. Redistributions of source code must retain the above copyright
* notice, this list of conditions and the following disclaimer.
* 2. Redistributions in binary form must reproduce the above copyright
* notice, this list of conditions and the following disclaimer in the
* documentation and/or other materials provided with the distribution.
*
* THIS SOFTWARE IS PROVIDED BY APPLE INC. ``AS IS'' AND ANY
* EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
* IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR
* PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL APPLE INC. OR
* CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL,
* EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO,
* PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR
* PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY
* OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT
* (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE
* OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
*/
#ifndef UnitBezier_h
#define UnitBezier_h
#include <array>
#include <math.h>
namespace WebCore {
struct UnitBezier {
#define CUBIC_BEZIER_SPLINE_SAMPLES 11
static constexpr double kBezierEpsilon = 1e-7;
static constexpr int kMaxNewtonIterations = 4;
UnitBezier(double p1x, double p1y, double p2x, double p2y)
{
// Calculate the polynomial coefficients, implicit first and last control points are (0,0) and (1,1).
cx = 3.0 * p1x;
bx = 3.0 * (p2x - p1x) - cx;
ax = 1.0 - cx -bx;
cy = 3.0 * p1y;
by = 3.0 * (p2y - p1y) - cy;
ay = 1.0 - cy - by;
// End-point gradients are used to calculate timing function results
// outside the range [0, 1].
//
// There are four possibilities for the gradient at each end:
// (1) the closest control point is not horizontally coincident with regard to
// (0, 0) or (1, 1). In this case the line between the end point and
// the control point is tangent to the bezier at the end point.
// (2) the closest control point is coincident with the end point. In
// this case the line between the end point and the far control
// point is tangent to the bezier at the end point.
// (3) both internal control points are coincident with an endpoint. There
// are two special case that fall into this category:
// CubicBezier(0, 0, 0, 0) and CubicBezier(1, 1, 1, 1). Both are
// equivalent to linear.
// (4) the closest control point is horizontally coincident with the end
// point, but vertically distinct. In this case the gradient at the
// end point is Infinite. However, this causes issues when
// interpolating. As a result, we break down to a simple case of
// 0 gradient under these conditions.
if (p1x > 0)
startGradient = p1y / p1x;
else if (!p1y && p2x > 0)
startGradient = p2y / p2x;
else if (!p1y && !p2y)
startGradient = 1;
else
startGradient = 0;
if (p2x < 1)
endGradient = (p2y - 1) / (p2x - 1);
else if (p2y == 1 && p1x < 1)
endGradient = (p1y - 1) / (p1x - 1);
else if (p2y == 1 && p1y == 1)
endGradient = 1;
else
endGradient = 0;
double deltaT = 1.0 / (CUBIC_BEZIER_SPLINE_SAMPLES - 1);
for (int i = 0; i < CUBIC_BEZIER_SPLINE_SAMPLES; i++)
splineSamples[i] = sampleCurveX(i * deltaT);
}
double sampleCurveX(double t)
{
// `ax t^3 + bx t^2 + cx t' expanded using Horner's rule.
return ((ax * t + bx) * t + cx) * t;
}
double sampleCurveY(double t)
{
return ((ay * t + by) * t + cy) * t;
}
double sampleCurveDerivativeX(double t)
{
return (3.0 * ax * t + 2.0 * bx) * t + cx;
}
// Given an x value, find a parametric value it came from.
double solveCurveX(double x, double epsilon)
{
double t0 = 0.0;
double t1 = 0.0;
double t2 = x;
double x2 = 0.0;
double d2 = 0.0;
int i = 0;
// Linear interpolation of spline curve for initial guess.
double deltaT = 1.0 / (CUBIC_BEZIER_SPLINE_SAMPLES - 1);
for (i = 1; i < CUBIC_BEZIER_SPLINE_SAMPLES; i++) {
if (x <= splineSamples[i]) {
t1 = deltaT * i;
t0 = t1 - deltaT;
t2 = t0 + (t1 - t0) * (x - splineSamples[i - 1]) / (splineSamples[i] - splineSamples[i - 1]);
break;
}
}
// Perform a few iterations of Newton's method -- normally very fast.
// See https://en.wikipedia.org/wiki/Newton%27s_method.
double newtonEpsilon = std::min(kBezierEpsilon, epsilon);
for (i = 0; i < kMaxNewtonIterations; i++) {
x2 = sampleCurveX(t2) - x;
if (std::abs(x2) < newtonEpsilon)
return t2;
d2 = sampleCurveDerivativeX(t2);
if (std::abs(d2) < kBezierEpsilon)
break;
t2 = t2 - x2 / d2;
}
if (std::abs(x2) < epsilon)
return t2;
// Fall back to the bisection method for reliability.
while (t0 < t1) {
x2 = sampleCurveX(t2);
if (std::abs(x2 - x) < epsilon)
return t2;
if (x > x2)
t0 = t2;
else
t1 = t2;
t2 = (t1 + t0) * .5;
}
// Failure.
return t2;
}
double solve(double x, double epsilon)
{
if (x < 0.0)
return 0.0 + startGradient * x;
if (x > 1.0)
return 1.0 + endGradient * (x - 1.0);
return sampleCurveY(solveCurveX(x, epsilon));
}
private:
double ax;
double bx;
double cx;
double ay;
double by;
double cy;
double startGradient;
double endGradient;
std::array<double, CUBIC_BEZIER_SPLINE_SAMPLES> splineSamples;
};
}
#endif
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