2023/05/16 by Angelos Mantzaflaris, Mantzaflaris, Angelos, Bernard Mourrain +5
Engineering · #41A15(Primary) 13D02 #65D07(Secondary) #Advanced Numerical Analysis Techniques #Commutative Algebra (math.AC) #FOS: Mathematics #Numerical Analysis (math.NA)
paper · pdf · doi:10.48550/arxiv.2305.09096
openalex publication_date 2023/05/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Geometrically continuous splines are piecewise polynomial functions defined on a collection of patches which are stitched together through transition maps. They are called Gr-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 Gr-spline spaces. We define Gr-domains and transition maps using an algebraic approach, and establish an algebraic criterion to determine whether a piecewise function is Gr-continuous on the given domain. In the proposed framework, we construct a chain complex whose top homology is isomorphic to the Gr-spline space. This complex generalizes Billera-Schenck-Stillman homological complex used to study parametric splines. Additionally, we show how previous constructions of Gr-splines fit into this new algebraic framework, and present an algorithm to construct a bases for Gr-spline spaces. We illustrate how our algebraic approach works with concrete examples, and prove a dimension formula for the Gr-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.