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G 1 -smooth splines on quad meshes with 4-split macro-patch elements

Ahmed Blidia, Bernard Mourrain, Nelly Villamizar Orcid Logo

Computer Aided Geometric Design, Volume: 52-53, Pages: 106 - 125

Swansea University Author: Nelly Villamizar Orcid Logo

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DOI (Published version): 10.1016/j.cagd.2017.03.003

Abstract

We analyze the space of smooth spline functions on quad-meshes. These functions are composed of 4-split spline macro-patch elements on each quadrangular face. We describe explicit transition maps across shared edges so that the space of smooth splines is ample on a quad-mesh of arbitrary topology. T...

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Published in: Computer Aided Geometric Design
ISSN: 01678396
Published: 2017
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URI: https://cronfa.swan.ac.uk/Record/cronfa32862
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first_indexed 2017-03-30T18:46:33Z
last_indexed 2020-07-15T12:50:30Z
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spelling 2020-07-15T12:56:15.1326290 v2 32862 2017-03-30 G 1 -smooth splines on quad meshes with 4-split macro-patch elements 41572bcee47da6ba274ecd1828fbfef4 0000-0002-8741-7225 Nelly Villamizar Nelly Villamizar true false 2017-03-30 SMA We analyze the space of smooth spline functions on quad-meshes. These functions are composed of 4-split spline macro-patch elements on each quadrangular face. We describe explicit transition maps across shared edges so that the space of smooth splines is ample on a quad-mesh of arbitrary topology. These transition maps define a finite dimensional vector space of G1 spline functions on each quadrangular face of the mesh. We determine the dimension of this space of G1 spline functions for degree big enough and provide explicit constructions of basis functions attached respectively to vertices, edges and faces. This construction requires the analysis of the module of syzygies of univariate b-spline functions with b-spline function coefficients. We provide new results on their generators and the dimension of the graded pieces.Examples of bases of G1 splines of small degree for simple topological surfaces are detailed and illustrated by parametric surface constructions. Journal Article Computer Aided Geometric Design 52-53 106 125 01678396 Geometrically continuous splines, Multivariate spline approximation, B-splines, surfaces with arbitrary topology, quad meshes, macro patch elements 31 3 2017 2017-03-31 10.1016/j.cagd.2017.03.003 COLLEGE NANME Mathematics COLLEGE CODE SMA Swansea University 2020-07-15T12:56:15.1326290 2017-03-30T16:52:57.3235664 Faculty of Science and Engineering School of Mathematics and Computer Science - Mathematics Ahmed Blidia 1 Bernard Mourrain 2 Nelly Villamizar 0000-0002-8741-7225 3 0032862-30032017165600.pdf ABN-CAGD.pdf 2017-03-30T16:56:00.1430000 Output 208223 application/pdf Accepted Manuscript true 2018-03-21T00:00:00.0000000 true eng
title G 1 -smooth splines on quad meshes with 4-split macro-patch elements
spellingShingle G 1 -smooth splines on quad meshes with 4-split macro-patch elements
Nelly Villamizar
title_short G 1 -smooth splines on quad meshes with 4-split macro-patch elements
title_full G 1 -smooth splines on quad meshes with 4-split macro-patch elements
title_fullStr G 1 -smooth splines on quad meshes with 4-split macro-patch elements
title_full_unstemmed G 1 -smooth splines on quad meshes with 4-split macro-patch elements
title_sort G 1 -smooth splines on quad meshes with 4-split macro-patch elements
author_id_str_mv 41572bcee47da6ba274ecd1828fbfef4
author_id_fullname_str_mv 41572bcee47da6ba274ecd1828fbfef4_***_Nelly Villamizar
author Nelly Villamizar
author2 Ahmed Blidia
Bernard Mourrain
Nelly Villamizar
format Journal article
container_title Computer Aided Geometric Design
container_volume 52-53
container_start_page 106
publishDate 2017
institution Swansea University
issn 01678396
doi_str_mv 10.1016/j.cagd.2017.03.003
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 We analyze the space of smooth spline functions on quad-meshes. These functions are composed of 4-split spline macro-patch elements on each quadrangular face. We describe explicit transition maps across shared edges so that the space of smooth splines is ample on a quad-mesh of arbitrary topology. These transition maps define a finite dimensional vector space of G1 spline functions on each quadrangular face of the mesh. We determine the dimension of this space of G1 spline functions for degree big enough and provide explicit constructions of basis functions attached respectively to vertices, edges and faces. This construction requires the analysis of the module of syzygies of univariate b-spline functions with b-spline function coefficients. We provide new results on their generators and the dimension of the graded pieces.Examples of bases of G1 splines of small degree for simple topological surfaces are detailed and illustrated by parametric surface constructions.
published_date 2017-03-31T03:40:25Z
_version_ 1763751835068268544
score 11.013148

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