Polynomial splines of non-uniform degree on triangulations: Combinatorial bounds on the dimension

Abstract

For T a planar triangulation, let Rmr(T) denote the space of bivariate splines on T such that f∈Rmr(T) is Cr(τ) smooth across an interior edge τ and, for triangle σ in T, f|σ is a polynomial of total degree at most m(σ)∈Z≥0. The map m:σ↦Z≥0 is called a non-uniform degree distribution on the triangles in T, and we consider the problem of computing (or estimating) the dimension of Rmr(T) in this paper. Using homological techniques, developed in the context of splines by Billera (1988), we provide combinatorial lower and upper bounds on the dimension of Rmr(T). When all polynomial degrees are sufficiently large, m(σ)≫0, we prove that the number of splines in Rmr(T) can be determined exactly. In the special case of a constant map m, the lower and upper bounds are equal to those provided by Mourrain and Villamizar (2013).