Fitting Splines to a Parametric Function

Description

Preface......Page 3 Contents......Page 6 Introduction......Page 8 2.1 Definition of Error Function......Page 10 2.2 Character of Solution......Page 11 2.3 Optimization of F(a, u)......Page 12 References......Page 18 General Properties of Splines......Page 19 References......Page 23 Cubic Bézier......Page 24 4.1 Fitting a Function with a Double Inflection Point......Page 25 4.2 Initializing a Cubic Bézier......Page 26 4.3 Optimizing the Fit......Page 27 4.5 Character of Coalescing Solutions......Page 29 References......Page 31 5.1 Center of Mass Fit......Page 32 5.2 Example of Two Types of Merge/Crossover......Page 35 5.3 Response to Change in g(t)......Page 37 5.4 Distinguishing Between Type 1 and Type 2 Events......Page 39 References......Page 41 5-Point B-Spline......Page 42 6.2 Basis Functions of a 5-Point B-Spline......Page 43 6.3 Decomposition into Two Bézier Segments......Page 44 6.4 ODF Results for a 5-Point B-Spline......Page 46 7.1 Initializing a 6-Point B-Spline......Page 48 7.2 Basis Functions of a 6-Point B-Spline......Page 49 7.3 Decomposition into Three Bézier Segments......Page 50 7.4 ODF Results for a 6-Point B-Spline......Page 51 8.1 Initializing a Quartic Bézier......Page 54 8.2 ODF Results for a Quartic Bézier......Page 55 8.3 Enumeration of Solutions......Page 57 8.4 Abnormal Truncation of Solutions......Page 59 8.5 Topological Comparison of Béziers and B-Splines......Page 60 References......Page 61 Beta2-Spline......Page 62 9.1 Rendering a Beta2-Spline......Page 63 9.2 Nonlinear Effects Due to ψ......Page 66 9.4 ODF Results for a Beta2-Spline......Page 67 References......Page 70 10.1 Rendering a Beta1-Spline......Page 71 10.2 Nonlinear Effects Due to β1......Page 72 10.3 Continuity of the Beta1-Spline Integrands......Page 73 10.4 ODF Results for a Beta1-Spline......Page 75 10.5 Numerical Problems with the Beta1-Spline......Page 77 Conclusions......Page 80