Fitting Splines to a Parametric Function

This Brief investigates the intersections that occur between three different areas of study that normally would not touch each other: ODF, spline theory, and topology. The Least Squares Orthogonal Distance Fitting (ODF) method has become the standard technique used to develop mathematical models of the physical shapes of objects, due to the fact that it produces a fitted result that is invariant with respect to the size and orientation of the object. It is normally used to produce a single optimum fit to a specific object; this work focuses instead on the issue of whether the fit responds continuously as the shape of the object changes. The theory of splines develops user-friendly ways of manipulating six different splines to fit the shape of a simple family of epiTrochoid curves: two types of Bézier curve, two uniform B-splines, and two Beta-splines. This work will focus on issues that arise when mathematically optimizing the fit. There are typically multiple solutions to the ODF method, and the number of solutions can often change as the object changes shape, so two topological questions immediately arise: are there rules that can be applied concerning the relative number of local minima and saddle points, and are there different mechanisms available by which solutions can either merge and disappear, or cross over each other and interchange roles. The author proposes some simple rules which can be used to determine if a given set of solutions is internally consistent in the sense that it has the appropriate number of each type of solution. 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