Source/WebCore/platform/graphics/UnitBezier.h

/*
 * 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 <math.h>

namespace WebCore {

    struct UnitBezier {
        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;
        }
        
        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;
            double t1;
            double t2;
            double x2;
            double d2;
            int i;

            // First try a few iterations of Newton's method -- normally very fast.
            for (t2 = x, i = 0; i < 8; i++) {
                x2 = sampleCurveX(t2) - x;
                if (fabs (x2) < epsilon)
                    return t2;
                d2 = sampleCurveDerivativeX(t2);
                if (fabs(d2) < 1e-6)
                    break;
                t2 = t2 - x2 / d2;
            }

            // Fall back to the bisection method for reliability.
            t0 = 0.0;
            t1 = 1.0;
            t2 = x;

            if (t2 < t0)
                return t0;
            if (t2 > t1)
                return t1;

            while (t0 < t1) {
                x2 = sampleCurveX(t2);
                if (fabs(x2 - x) < epsilon)
                    return t2;
                if (x > x2)
                    t0 = t2;
                else
                    t1 = t2;
                t2 = (t1 - t0) * .5 + t0;
            }

            // Failure.
            return t2;
        }

        double solve(double x, double epsilon)
        {
            return sampleCurveY(solveCurveX(x, epsilon));
        }
        
    private:
        double ax;
        double bx;
        double cx;
        
        double ay;
        double by;
        double cy;
    };
}
#endif