Goldman, R. N. | Engineering Papers

NASA NTRS1985

Urn Models and Beta-splines

Some insight into the properties of beta-splines is gained by applying the techniques of urn models. Urn models are used to construct beta-spline basis functions and to derive the basic properties of these blending functions and the corresponding beta-spline curves. Only the simple notion of linear geometric continuity and with the most elementary beta parameter are outlined. Non-linear geometric continuity leads to additional beta parameters and to more complicated basis functions. Whether urn models can give us any insight into these higher order concepts still remains to be investigated.

Goldman, R. N.↗

NASA NTRS1984

An urnful of blinding functions

There is a fundamental connection between the mathematical theory of discrete probability distributions and the parametric curves and surfaces of computer-aided geometric design. It is no accident that the blending functions of Bezier curves and surfaces have an obvious probabilistic interpretation, nor is it a coincidence that the normalized uniform B-spline basis functions also model a simple stochastic process. The link between probability and geometry, and how to exploit simple probabilistic arguments to derive many of the classical geometric properties of the parametric curves and surfaces currently in vogue in computer-aided geometric design is discussed. This probabilistic approach is also used to introduce many new types of curves and surfaces, and it is demonstrated how probability theory can be used to simplify, unify, and generalize many well-known results.

Goldman, R. N.↗