1/* 2 * Copyright (C) 2008 Apple Inc. All Rights Reserved. 3 * 4 * Redistribution and use in source and binary forms, with or without 5 * modification, are permitted provided that the following conditions 6 * are met: 7 * 1. Redistributions of source code must retain the above copyright 8 * notice, this list of conditions and the following disclaimer. 9 * 2. Redistributions in binary form must reproduce the above copyright 10 * notice, this list of conditions and the following disclaimer in the 11 * documentation and/or other materials provided with the distribution. 12 * 13 * THIS SOFTWARE IS PROVIDED BY APPLE INC. ``AS IS'' AND ANY 14 * EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE 15 * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR 16 * PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL APPLE INC. OR 17 * CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, 18 * EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, 19 * PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR 20 * PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY 21 * OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT 22 * (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE 23 * OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. 24 */ 25 26#ifndef UnitBezier_h 27#define UnitBezier_h 28 29#include "platform/PlatformExport.h" 30#include "wtf/Assertions.h" 31#include <math.h> 32 33namespace blink { 34 35struct UnitBezier { 36 UnitBezier(double p1x, double p1y, double p2x, double p2y) 37 { 38 // Calculate the polynomial coefficients, implicit first and last control points are (0,0) and (1,1). 39 cx = 3.0 * p1x; 40 bx = 3.0 * (p2x - p1x) - cx; 41 ax = 1.0 - cx -bx; 42 43 cy = 3.0 * p1y; 44 by = 3.0 * (p2y - p1y) - cy; 45 ay = 1.0 - cy - by; 46 47 // End-point gradients are used to calculate timing function results 48 // outside the range [0, 1]. 49 // 50 // There are three possibilities for the gradient at each end: 51 // (1) the closest control point is not horizontally coincident with regard to 52 // (0, 0) or (1, 1). In this case the line between the end point and 53 // the control point is tangent to the bezier at the end point. 54 // (2) the closest control point is coincident with the end point. In 55 // this case the line between the end point and the far control 56 // point is tangent to the bezier at the end point. 57 // (3) the closest control point is horizontally coincident with the end 58 // point, but vertically distinct. In this case the gradient at the 59 // end point is Infinite. However, this causes issues when 60 // interpolating. As a result, we break down to a simple case of 61 // 0 gradient under these conditions. 62 63 if (p1x > 0) 64 m_startGradient = p1y / p1x; 65 else if (!p1y && p2x > 0) 66 m_startGradient = p2y / p2x; 67 else 68 m_startGradient = 0; 69 70 if (p2x < 1) 71 m_endGradient = (p2y - 1) / (p2x - 1); 72 else if (p2x == 1 && p1x < 1) 73 m_endGradient = (p1y - 1) / (p1x - 1); 74 else 75 m_endGradient = 0; 76 } 77 78 double sampleCurveX(double t) 79 { 80 // `ax t^3 + bx t^2 + cx t' expanded using Horner's rule. 81 return ((ax * t + bx) * t + cx) * t; 82 } 83 84 double sampleCurveY(double t) 85 { 86 return ((ay * t + by) * t + cy) * t; 87 } 88 89 double sampleCurveDerivativeX(double t) 90 { 91 return (3.0 * ax * t + 2.0 * bx) * t + cx; 92 } 93 94 // Given an x value, find a parametric value it came from. 95 double solveCurveX(double x, double epsilon) 96 { 97 ASSERT(x >= 0.0); 98 ASSERT(x <= 1.0); 99 100 double t0; 101 double t1; 102 double t2; 103 double x2; 104 double d2; 105 int i; 106 107 // First try a few iterations of Newton's method -- normally very fast. 108 for (t2 = x, i = 0; i < 8; i++) { 109 x2 = sampleCurveX(t2) - x; 110 if (fabs (x2) < epsilon) 111 return t2; 112 d2 = sampleCurveDerivativeX(t2); 113 if (fabs(d2) < 1e-6) 114 break; 115 t2 = t2 - x2 / d2; 116 } 117 118 // Fall back to the bisection method for reliability. 119 t0 = 0.0; 120 t1 = 1.0; 121 t2 = x; 122 123 while (t0 < t1) { 124 x2 = sampleCurveX(t2); 125 if (fabs(x2 - x) < epsilon) 126 return t2; 127 if (x > x2) 128 t0 = t2; 129 else 130 t1 = t2; 131 t2 = (t1 - t0) * .5 + t0; 132 } 133 134 // Failure. 135 return t2; 136 } 137 138 // Evaluates y at the given x. The epsilon parameter provides a hint as to the required 139 // accuracy and is not guaranteed. 140 double solve(double x, double epsilon) 141 { 142 if (x < 0.0) 143 return 0.0 + m_startGradient * x; 144 if (x > 1.0) 145 return 1.0 + m_endGradient * (x - 1.0); 146 return sampleCurveY(solveCurveX(x, epsilon)); 147 } 148 149private: 150 double ax; 151 double bx; 152 double cx; 153 154 double ay; 155 double by; 156 double cy; 157 158 double m_startGradient; 159 double m_endGradient; 160}; 161 162} // namespace blink 163 164#endif // UnitBezier_h 165