An urnful of blending functions

The author explores the link between probability and geometry. In the process, he shows 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. He also uses this probabilistic approach to introduce many new types of curves and surfaces into computer-aided geometric design, and demonstrates how probability theory can be used to simplify, unify, and generalize many well-known results. He concludes that urn models are a powerful tool for generating discrete probability distributions, and built into these special distributions are many propitious properties essential to the blending functions of computer-aided geometric design. This fact allows mathematicians to use probabilistic arguments to simplify, unify, and generalize many geometric results. He believes that this link between probability and geometry will ultimately prove beneficial to both disciplines, and expects that it will continue to be a productive area for future inspiration and research.