ODF Using a Beta1-Spline

Abstract

A 5-point B-spline is cast into the form of two Bézier segments with internal constraints on the parameters. A new parameter, β ₁, is introduced. This parameter breaks $\mathcal {C}^1$ continuity and replaces it with $\mathcal {G}^1$ continuity at the breakpoint. It defines an asymmetric change in the Bézier arm lengths at the splice. We develop a relationship between the dimensionless β ₁ and the Bézier arm lengths. Nonlinear coupling introduced by β ₁ is defined. Discontinuities in the response functions, introduced by β ₁, are described and shown to have no net effect. ODF results from fitting this spline to a hypoTrochoid shape are presented and compared to the original 5-point B-spline. The new parameter β ₁ is shown to be quite advantageous when fitting an asymmetric shape, but leads to very little improvement for a symmetric shape. It also introduces a considerable amount of complexity to the solution set, including a number of new causes of abnormal termination of the solution due to numerical convergence problems. These are classified by type.