Urn Models and Beta-Splines

Splines were originally studied in approximation theory where the focus is on approximating explicit functions of the form y = f(x) or z = f(x,y). These splines were later adopted by mathematicians and computer sicentists for use in computer-aided geometric design (CAGD) where the emphasis was shifted to parametric curves and surfaces. Initially the continuity conditions for splines developed in approximation theory were retained in CAGD, but it was soon realized that the old constraints were unnecessarily restrictive in this new context and that they could be relaxed without losing the essential property of smoothness. Beta-splines were developed to take advantage of this new freedom by introducing shape parameters into the constraint equations. These parameters could then be manipulated by a designer to change the shape of a curve of surface in an intuitively meaningful and useful way. Another seemingly unrelated context in which shape parameters appear is in blending functions constructed from discrete urn models. The purpose of this article is to begin to unify these two independent approaches to shape parameters, and in the process apply the techniques of urn models to gain some insight into the properties of Beta-splines.