Planar splines on a triangulation with a single totally interior edge

TL;DR

This paper provides a comprehensive explicit formula for the dimension of spline spaces on a planar triangulation with a single interior edge, extending classical results and utilizing commutative algebra methods.

Contribution

It introduces a general dimension formula for $C^r_d$ splines on such triangulations, covering all degrees and smoothness levels, building upon and extending previous work.

Findings

Derived an explicit dimension formula valid for all degrees and smoothness levels.

Extended classical spline dimension results to a broader class of triangulations.

Utilized commutative algebra techniques for the derivation.

Abstract

We derive an explicit formula, valid for all integers $r,d\ge 0$, for the dimension of the vector space $C^r_d(\Delta)$ of piecewise polynomial functions continuously differentiable to order $r$ and whose constituents have degree at most $d$, where $\Delta$ is a planar triangulation that has a single totally interior edge. This extends previous results of Toh\v{a}neanu, Min\'{a}\v{c}, and Sorokina. Our result is a natural successor of Schumaker's 1979 dimension formula for splines on a planar vertex star. Indeed, there has not been a dimension formula in this level of generality (valid for all integers $d,r\ge 0$ and any vertex coordinates) since Schumaker's result. We derive our results using commutative algebra.