Polynomial splines of non-uniform degree on triangulations

Abstract

For T a planar triangulation, let R_m ^r(T) denote the space of bivariate splines on T such that f∈R_m ^r(T) is C^r(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 R_m ^r(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 R_m ^r(T). When all polynomial degrees are sufficiently large, m(σ)≫0, we prove that the number of splines in R_m ^r(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).