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arXiv · 2605.14577

Dimension Calculation for Spline Spaces over Rectilinear Partitions via Smoothing Cofactor Method

Bingru Huang·Falai Chen
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Abstract

This paper presents a general framework for calculating the dimension of spline spaces over arbitrary rectilinear partitions using the smoothing cofactor method. The approach extends existing dimension theory for polynomial splines over T-meshes by introducing the concept of TE-connected components, reducing the problem to the rank computation of explicitly constructible conformality matrices. Furthermore, a new class of rectilinear partitions, termed partitions with disjoint truncated l-edges, is introduced. It is proven that under specific conditions, the dimension of the corresponding spline space attains Schumaker's lower bound. This shows that the lower bound is attainable for arbitrary degree d and smoothness order mu in certain partition configurations. Numerical examples, including the Morgan-Scott and Yuan-Stillman partitions, validate the effectiveness and generality of the framework for both triangular and non-triangular rectilinear partitions.

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Bingru Huang, Falai Chen. 2026-05-14. Dimension Calculation for Spline Spaces over Rectilinear Partitions via Smoothing Cofactor Method. https://arxiv.org/abs/2605.14577

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