Beihui Yuan

We derive an explicit formula, valid for all integers $r,d\ge 0$, for the dimension of the vector space $C^r_d(Δ)$ of piecewise polynomial functions continuously differentiable to order $r$ and whose constituents have degree at most $d$, where $Δ$ is a planar triangulation that has a single totally interior edge. This extends previous results of Tohǎneanu, Mináč, 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.

Geometrically continuous splines are piecewise polynomial functions defined on a collection of patches which are stitched together through transition maps. They are called $G^{r}$-splines if, after composition with the transition maps, they are continuously differentiable functions to order $r$ on each pair of patches with stitched boundaries. This type of splines has been used to represent smooth shapes with complex topology for which (parametric) spline functions on fixed partitions are not sufficient. In this article, we develop new algebraic tools to analyze $G^r$-spline spaces. We define $G^{r}$-domains and transition maps using an algebraic approach, and establish an algebraic criterion to determine whether a piecewise function is $G^r$-continuous on the given domain. In the proposed framework, we construct a chain complex whose top homology is isomorphic to the $G^{r}$-spline space. This complex generalizes Billera-Schenck-Stillman homological complex used to study parametric splines. Additionally, we show how previous constructions of $G^r$-splines fit into this new algebraic framework, and present an algorithm to construct a bases for $G^r$-spline spaces. We illustrate how our algebraic approach works with concrete examples, and prove a dimension formula for the $G^r$-spline space in terms of invariants to the chain complex. In some special cases, explicit dimension formulas in terms of the degree of splines are also given.