Homological techniques for the analysis of the dimension of triangular spline spaces

TL;DR

This paper uses homological algebra techniques to derive more accurate bounds and exact values for the dimension of bivariate spline spaces over triangulated regions, improving upon previous methods.

Contribution

It introduces a homological approach to determine the dimension of spline spaces on triangulations, providing more precise bounds and exact formulas in certain cases.

Findings

More accurate upper bounds for spline space dimensions.

Exact dimension formulas for specific cases when k ≥ 4r+1.

Short proof of the dimension formula using homological methods.

Abstract

The spline space $C_k^r(\Delta)$ attached to a subdivided domain $\Delta$ of $\R^{d} $ is the vector space of functions of class $C^{r}$ which are polynomials of degree $\le k$ on each piece of this subdivision. Classical splines on planar rectangular grids play an important role in Computer Aided Geometric Design, and spline spaces over arbitrary subdivisions of planar domains are now considered for isogeometric analysis applications. We address the problem of determining the dimension of the space of bivariate splines $C_k^r(\Delta)$ for a triangulated region $\Delta$ in the plane. Using the homological introduced by Billera (1988), we number the vertices and establish a formula for an upper bound on the dimension. There is no restriction on the ordering and we obtain more accurate approximations to the dimension than previous methods and furthermore, in certain cases even an exact value can be found. The construction makes also possible to get a short proof for the dimension formula when $k\ge 4r+1$, and the same method we use in this proof yields the dimension straightaway for many other cases.