WebCore/UnitBezier.h

   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 {
  34         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
  46         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
  52         double sampleCurveY(double t)
  53         {
  54             return ((ay * t + by) * t + cy) * t;
  55         }
  56
  57         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.
  63         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
 108         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