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      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
    124