An algebraic framework for geometrically continuous splines

Mantzaflaris, Angelos; Mourrain, Bernard; Villamizar, Nelly; Yuan, Beihui

doi:10.1090/mcom/4068

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An algebraic framework for geometrically continuous splines

Angelos Mantzaflaris, Bernard Mourrain, Nelly Villamizar

, 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...

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Published in:	Mathematics of Computation
ISSN:	0025-5718 1088-6842
Published:	American Mathematical Society (AMS) 2025
Online Access:	Check full text
URI:	https://cronfa.swan.ac.uk/Record/cronfa69106

first_indexed	2025-03-15T21:26:12Z
last_indexed	2025-05-15T10:49:17Z
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spelling	2025-05-14T15:31:25.1634107 v2 69106 2025-03-15 An algebraic framework for geometrically continuous splines 41572bcee47da6ba274ecd1828fbfef4 0000-0002-8741-7225 Nelly Villamizar Nelly Villamizar true false 2025-03-15 MACS 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. Journal Article Mathematics of Computation American Mathematical Society (AMS) 0025-5718 1088-6842 13 3 2025 2025-03-13 10.1090/mcom/4068 COLLEGE NANME Mathematics and Computer Science School COLLEGE CODE MACS Swansea University 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. 2025-05-14T15:31:25.1634107 2025-03-15T21:16:41.1729128 Faculty of Science and Engineering School of Mathematics and Computer Science - Mathematics Angelos Mantzaflaris 1 Bernard Mourrain 2 Nelly Villamizar 0000-0002-8741-7225 3 Beihui Yuan 4 69106__33822__38cda733ea4e4f1ba2dbd4e92f876b44.pdf 230606-Villamizar.pdf 2025-03-15T21:29:21.3787286 Output 426913 application/pdf Accepted Manuscript true Author accepted manuscript document released under the terms of a Creative Commons CC-BY licence using the Swansea University Research Publications Policy (rights retention). true eng https://creativecommons.org/licenses/by/4.0/deed.en
title	An algebraic framework for geometrically continuous splines
spellingShingle	An algebraic framework for geometrically continuous splines Nelly Villamizar
title_short	An algebraic framework for geometrically continuous splines
title_full	An algebraic framework for geometrically continuous splines
title_fullStr	An algebraic framework for geometrically continuous splines
title_full_unstemmed	An algebraic framework for geometrically continuous splines
title_sort	An algebraic framework for geometrically continuous splines
author_id_str_mv	41572bcee47da6ba274ecd1828fbfef4
author_id_fullname_str_mv	41572bcee47da6ba274ecd1828fbfef4_***_Nelly Villamizar
author	Nelly Villamizar
author2	Angelos Mantzaflaris Bernard Mourrain Nelly Villamizar Beihui Yuan
format	Journal article
container_title	Mathematics of Computation
publishDate	2025
institution	Swansea University
issn	0025-5718 1088-6842
doi_str_mv	10.1090/mcom/4068
publisher	American Mathematical Society (AMS)
college_str	Faculty of Science and Engineering
hierarchytype
hierarchy_top_id	facultyofscienceandengineering
hierarchy_top_title	Faculty of Science and Engineering
hierarchy_parent_id	facultyofscienceandengineering
hierarchy_parent_title	Faculty of Science and Engineering
department_str	School of Mathematics and Computer Science - Mathematics{{{_:::_}}}Faculty of Science and Engineering{{{_:::_}}}School of Mathematics and Computer Science - Mathematics
document_store_str	1
active_str	0
description	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.
published_date	2025-03-13T18:30:50Z
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score	11.059359

An algebraic framework for geometrically continuous splines

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