Variational Subdivision for Natural Cubic Splines

This paper explores the intrinsic link between natural cubic splines and subdivision. Natural cubic splines are deened via the varia-tional problem of minimizing a simple approximation of bending energy. A subdivision scheme is derived which converges to the minimizer of this particular variational problem. x1. Introduction Geometric design is the study of the representation of shapes with mathematical models. Today, curved shapes are most commonly described using a parametric representation. These are based on the weighting of some number of control points with appropriate parametric basis functions. The most successful family of such basis functions is without any doubt the B-spline basis 7,5]. More recently subdivision has evolved as a novel approach for representing shape 2,3]. In this framework smooth shapes are represented as the limit of a repeated weighted averaging process of control points. This paper exposes the intrinsic link between the two concepts: Starting with the variational deenition of natural cubic spline curves, a representation by means of a subdivision process is derived. The principles underlying this derivation are applicable to other variational problems in perfect analogy. Thus, the method presented in this paper can be used to derive subdivision schemes which produce the minimizers of variational problems. Historically, a spline was a thin, exible piece of wood used in drafting. The draftsman attached the spline to a sequence of anchor points on a drafting table. The spline was then allowed to slide through the anchor points and assume a smooth, minimum energy shape. All rights of reproduction in any form reserved.