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An algebraic framework for geometrically continuous splines
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Beihui Yuan
Mathematics of Computation
Swansea University Author:
Nelly Villamizar
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DOI (Published version): 10.1090/mcom/4068
Abstract
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 pai...
| Published in: | Mathematics of Computation |
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| ISSN: | 0025-5718 1088-6842 |
| Published: |
American Mathematical Society (AMS)
2025
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| Online Access: |
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| URI: | https://cronfa.swan.ac.uk/Record/cronfa69106 |
| Abstract: |
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 spline 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. |
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| College: |
Faculty of Science and Engineering |
| Funders: |
The first and third authors were supported by The Alliance Hubert Curien Programme, project number: 515492678. The third and fourth authors were supported by the UK Engineering and Physical Sciences Research Council (EPSRC) New Investigator Award EP/V012835/1. |