On Discrete Simplex Splines and Subdivision

A variety of techniques to facilitate fast graphical display of curves and surfaces for interactive CAGD purposes have been developed, in the past years. Many of these are commonly referred to as subdivision algorithms PI. In this paper we propose a method for subdividing simplex splines, i.e., splines that can be defined over non-regular grid partitions. There is a relatively exhaustive theory of multivariate simplex splints (or shortly B-splines) [6]. Due to some favorable properties, similar to those in the univariate case, the B-splines are suitable, in principle, for application, e.g., in the Finite Element Method or in Computer Aided Geometric Design. On the other hand, their computational properties seem to be very restrictive compared to other available methods. In spite of the existence of a number of recurrence relations to facilitate the numerical manipulation of B-splines, these are often computationally expensive in sharp contrast with the univariate case. Therefore, there naturally arises a need for a subdivisionlike algorithm for simplex B-splines. Some basic ideas on subdivision of B-splines have been described in [l, 12, 14, 151, all based on a geometric interpretation of B-splines. Here, we suggest a different approach. It is based on the notion of discrete simplex splines, which is proven to be an extension of the ideas underlying the definitions of dikcrete cube splines, also called box splines, and discrete cone splines [3, 71. The notion of the