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34 UnitBezier(double p1x, double p1y, double p2x, double p2y)
36 // Calculate the polynomial coefficients, implicit first and last control points are (0,0) and (1,1).
38 bx = 3.0 * (p2x - p1x) - cx;
42 by = 3.0 * (p2y - p1y) - cy;
46 double sampleCurveX(double t)
48 // `ax t^3 + bx t^2 + cx t' expanded using Horner's rule.
49 return ((ax * t + bx) * t + cx) * t;
52 double sampleCurveY(double t)
54 return ((ay * t + by) * t + cy) * t;
57 double sampleCurveDerivativeX(double t)
59 return (3.0 * ax * t + 2.0 * bx) * t + cx;
62 // Given an x value, find a parametric value it came from.
63 double solveCurveX(double x, double epsilon)
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)
77 d2 = sampleCurveDerivativeX(t2);
83 // Fall back to the bisection method for reliability.
94 x2 = sampleCurveX(t2);
95 if (fabs(x2 - x) < epsilon)
101 t2 = (t1 - t0) * .5 + t0;
108 double solve(double x, double epsilon)
110 return sampleCurveY(solveCurveX(x, epsilon));