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A Lower Bound for Splines on Tetrahedral Vertex Stars

Michael DiPasquale, Nelly Villamizar Orcid Logo

SIAM Journal on Applied Algebra and Geometry, Volume: 5, Issue: 2, Pages: 250 - 277

Swansea University Author: Nelly Villamizar Orcid Logo

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DOI (Published version): 10.1137/20m1341118

Abstract

A tetrahedral complex all of whose tetrahedra meet at a common vertex is called a vertex star. Vertex stars are a natural generalization of planar triangulations, and understanding splines on vertex stars is a crucial step to analyzing trivariate splines. It is particularly diffcult to compute the d...

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Published in: SIAM Journal on Applied Algebra and Geometry
ISSN: 2470-6566
Published: Society for Industrial & Applied Mathematics (SIAM) 2021
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URI: https://cronfa.swan.ac.uk/Record/cronfa56990
first_indexed 2021-05-31T22:10:27Z
last_indexed 2023-01-11T14:36:35Z
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spelling 2022-08-15T10:21:32.7456448 v2 56990 2021-05-31 A Lower Bound for Splines on Tetrahedral Vertex Stars 41572bcee47da6ba274ecd1828fbfef4 0000-0002-8741-7225 Nelly Villamizar Nelly Villamizar true false 2021-05-31 MACS A tetrahedral complex all of whose tetrahedra meet at a common vertex is called a vertex star. Vertex stars are a natural generalization of planar triangulations, and understanding splines on vertex stars is a crucial step to analyzing trivariate splines. It is particularly diffcult to compute the dimension of splines on vertex stars in which the vertex is completely surrounded by tetrahedra|we call theseclosed vertex stars. A formula due to Alfeld, Neamtu, and Schumaker gives the dimension of splines whose derivatives up to order r are continuous on closed vertex stars of degree at least 3r + 2. We show that this formula is a lower bound on the dimension of splines of degree at least (3r + 2)/2. Our proof uses apolarity and thevso-called Waldschmidt constant of the set of points dual to the interior faces of the vertex star. We furthermore observe that arguments of Alfeld, Schumaker, and Whiteley imply that the only splines of degree at most (3r + 1)/2 on a generic closed vertex star are global polynomials. Journal Article SIAM Journal on Applied Algebra and Geometry 5 2 250 277 Society for Industrial & Applied Mathematics (SIAM) 2470-6566 spline functions, apolarity, fat point ideals, Waldschmidt constant 16 6 2021 2021-06-16 10.1137/20m1341118 COLLEGE NANME Mathematics and Computer Science School COLLEGE CODE MACS Swansea University EP/V012835/1 2022-08-15T10:21:32.7456448 2021-05-31T22:34:30.6740893 Faculty of Science and Engineering School of Mathematics and Computer Science - Mathematics Michael DiPasquale 1 Nelly Villamizar 0000-0002-8741-7225 2 56990__20033__6f4fed675d084e139e5e6e0365bb8ff9.pdf DiPasquale-Villamizar_2021.pdf 2021-05-31T23:09:27.3045834 Output 502625 application/pdf Accepted Manuscript true true eng
title A Lower Bound for Splines on Tetrahedral Vertex Stars
spellingShingle A Lower Bound for Splines on Tetrahedral Vertex Stars
Nelly Villamizar
title_short A Lower Bound for Splines on Tetrahedral Vertex Stars
title_full A Lower Bound for Splines on Tetrahedral Vertex Stars
title_fullStr A Lower Bound for Splines on Tetrahedral Vertex Stars
title_full_unstemmed A Lower Bound for Splines on Tetrahedral Vertex Stars
title_sort A Lower Bound for Splines on Tetrahedral Vertex Stars
author_id_str_mv 41572bcee47da6ba274ecd1828fbfef4
author_id_fullname_str_mv 41572bcee47da6ba274ecd1828fbfef4_***_Nelly Villamizar
author Nelly Villamizar
author2 Michael DiPasquale
Nelly Villamizar
format Journal article
container_title SIAM Journal on Applied Algebra and Geometry
container_volume 5
container_issue 2
container_start_page 250
publishDate 2021
institution Swansea University
issn 2470-6566
doi_str_mv 10.1137/20m1341118
publisher Society for Industrial & Applied Mathematics (SIAM)
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 A tetrahedral complex all of whose tetrahedra meet at a common vertex is called a vertex star. Vertex stars are a natural generalization of planar triangulations, and understanding splines on vertex stars is a crucial step to analyzing trivariate splines. It is particularly diffcult to compute the dimension of splines on vertex stars in which the vertex is completely surrounded by tetrahedra|we call theseclosed vertex stars. A formula due to Alfeld, Neamtu, and Schumaker gives the dimension of splines whose derivatives up to order r are continuous on closed vertex stars of degree at least 3r + 2. We show that this formula is a lower bound on the dimension of splines of degree at least (3r + 2)/2. Our proof uses apolarity and thevso-called Waldschmidt constant of the set of points dual to the interior faces of the vertex star. We furthermore observe that arguments of Alfeld, Schumaker, and Whiteley imply that the only splines of degree at most (3r + 1)/2 on a generic closed vertex star are global polynomials.
published_date 2021-06-16T05:05:58Z
_version_ 1821380663248420864
score 11.04748

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