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 <math.h> 30 31 namespace WebCore { 32 33 struct UnitBezier { UnitBezierUnitBezier34 UnitBezier(double p1x, double p1y, double p2x, double p2y) 35 { 36 // Calculate the polynomial coefficients, implicit first and last control points are (0,0) and (1,1). 37 cx = 3.0 * p1x; 38 bx = 3.0 * (p2x - p1x) - cx; 39 ax = 1.0 - cx -bx; 40 41 cy = 3.0 * p1y; 42 by = 3.0 * (p2y - p1y) - cy; 43 ay = 1.0 - cy - by; 44 } 45 sampleCurveXUnitBezier46 double sampleCurveX(double t) 47 { 48 // `ax t^3 + bx t^2 + cx t' expanded using Horner's rule. 49 return ((ax * t + bx) * t + cx) * t; 50 } 51 sampleCurveYUnitBezier52 double sampleCurveY(double t) 53 { 54 return ((ay * t + by) * t + cy) * t; 55 } 56 sampleCurveDerivativeXUnitBezier57 double sampleCurveDerivativeX(double t) 58 { 59 return (3.0 * ax * t + 2.0 * bx) * t + cx; 60 } 61 62 // Given an x value, find a parametric value it came from. solveCurveXUnitBezier63 double solveCurveX(double x, double epsilon) 64 { 65 double t0; 66 double t1; 67 double t2; 68 double x2; 69 double d2; 70 int i; 71 72 // First try a few iterations of Newton's method -- normally very fast. 73 for (t2 = x, i = 0; i < 8; i++) { 74 x2 = sampleCurveX(t2) - x; 75 if (fabs (x2) < epsilon) 76 return t2; 77 d2 = sampleCurveDerivativeX(t2); 78 if (fabs(d2) < 1e-6) 79 break; 80 t2 = t2 - x2 / d2; 81 } 82 83 // Fall back to the bisection method for reliability. 84 t0 = 0.0; 85 t1 = 1.0; 86 t2 = x; 87 88 if (t2 < t0) 89 return t0; 90 if (t2 > t1) 91 return t1; 92 93 while (t0 < t1) { 94 x2 = sampleCurveX(t2); 95 if (fabs(x2 - x) < epsilon) 96 return t2; 97 if (x > x2) 98 t0 = t2; 99 else 100 t1 = t2; 101 t2 = (t1 - t0) * .5 + t0; 102 } 103 104 // Failure. 105 return t2; 106 } 107 solveUnitBezier108 double solve(double x, double epsilon) 109 { 110 return sampleCurveY(solveCurveX(x, epsilon)); 111 } 112 113 private: 114 double ax; 115 double bx; 116 double cx; 117 118 double ay; 119 double by; 120 double cy; 121 }; 122 } 123 #endif 124