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arXiv · 2306.16825

Planar splines on a triangulation with a single totally interior edge

Michael DiPasquale·Beihui Yuan
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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.

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Michael DiPasquale, Beihui Yuan. 2023-06-29. Planar splines on a triangulation with a single totally interior edge. https://arxiv.org/abs/2306.16825

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