On the dimension of spline spaces on planar T-meshes

TL;DR

This paper provides new combinatorial bounds for the dimension of bivariate spline spaces on planar T-meshes, using homological techniques and relating them to the mesh's structure, with applications to hierarchical T-meshes.

Contribution

It introduces new bounds for spline space dimensions on T-meshes and relates them to the mesh's combinatorial structure, especially for hierarchical and regular meshes.

Findings

Lower and upper bounds coincide for high degrees or regular hierarchical T-meshes.

A subdivision rule is provided to construct T-meshes with matching bounds.

Results are illustrated with small degree and smoothness spline spaces.

Abstract

We analyze the space of bivariate functions that are piecewise polynomial of bi-degree \textless{}= (m, m') and of smoothness r along the interior edges of a planar T-mesh. We give new combinatorial lower and upper bounds for the dimension of this space by exploiting homological techniques. We relate this dimension to the weight of the maximal interior segments of the T-mesh, defined for an ordering of these maximal interior segments. We show that the lower and upper bounds coincide, for high enough degrees or for hierarchical T-meshes which are enough regular. We give a rule of subdivision to construct hierarchical T-meshes for which these lower and upper bounds coincide. Finally, we illustrate these results by analyzing spline spaces of small degrees and smoothness.