Constructing The Explicit Expression Of Bivariate Simplex Splines And Applications

A spline function is a piecewise polynomial, which have certain smooth properties. Multivariate splines are widely used in function approximations, computations of science and engineering, CAGD and wavelets. Moreover, multivariate splines also have certain relations with pure mathematics, such as, abstract algebra, algebraic geometry combinatorics and so forth. Further, as Multivariate splines depend severely on the geometric properties of the defining region partition, so there are several complicated situations. But for m generic points in R², connecting any tow of them, we can get a complete graph, which forms the support partition of S_m-3^m-4 bivariate simplex splines.In this paper, we use the informal explicit expressions of higher dimensional B-splines which are deduced by homogenizing and symmetrizing lower dimensional B-splines. For the bivariate simplex splines, there two ways. One gives an algorithm to traverse the complete graph which can get the explicit expression on every cell. The other is by the combination of bivariate cone splines. Then homogenizing the bivariate simplex splines, we can smooth polyhedral angles. In the end, using the combination of bivariate cone splines, we prove the S₄³ splines which have minimal support can be formed by convolution of two S₁⁰ simplex splines.