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Dimension and bases for geometrically continuous splines on surfaces of arbitrary topology

Bernard Mourrain, Raimundas Vidunas, Nelly Villamizar Orcid Logo

Computer Aided Geometric Design, Volume: 45, Pages: 108 - 133

Swansea University Author: Nelly Villamizar Orcid Logo

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DOI (Published version): 10.1016/j.cagd.2016.03.003

Abstract

We analyze the space of geometrically continuous piecewise polynomial functions, or splines, for rectangular and triangular patches with arbitrary topology and general rational transition maps. To define these spaces of G 1 spline functions, we introduce the concept of topological surface with gluin...

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Published in: Computer Aided Geometric Design
ISSN: 01678396
Published: 2016
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URI: https://cronfa.swan.ac.uk/Record/cronfa32863
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spelling 2017-05-08T12:22:30.0267331 v2 32863 2017-03-30 Dimension and bases for geometrically continuous splines on surfaces of arbitrary topology 41572bcee47da6ba274ecd1828fbfef4 0000-0002-8741-7225 Nelly Villamizar Nelly Villamizar true false 2017-03-30 SMA We analyze the space of geometrically continuous piecewise polynomial functions, or splines, for rectangular and triangular patches with arbitrary topology and general rational transition maps. To define these spaces of G 1 spline functions, we introduce the concept of topological surface with gluing data attached to the edges shared by faces. The framework does not require manifold constructions and is general enough to allow non-orientable surfaces. We describe compatibility conditions on the transition maps so that the space of differentiable functions is ample and show that these conditions are necessary and sufficient to construct ample spline spaces. We determine the dimension of the space of G1 spline functions which are of degree less than or equal to k on triangular pieces and of bi-degree less than or equal to (k, k) on rectangular pieces, for k big enough. A separability property on the edges is involved to obtain the dimension formula. An explicit construction of basis functions attached resspectively to vertices, edges and faces is proposed; examples of bases of G1 splines of small degree for topological surfaces with boundary and without boundary are detailed. Journal Article Computer Aided Geometric Design 45 108 133 01678396 geometrically continuous splines, dimension and bases of spline spaces, gluing data, polygonal patches, surfaces of arbitrary topology 31 7 2016 2016-07-31 10.1016/j.cagd.2016.03.003 http://www.sciencedirect.com/science/article/pii/S0167839616300309 COLLEGE NANME Mathematics COLLEGE CODE SMA Swansea University 2017-05-08T12:22:30.0267331 2017-03-30T17:01:36.2943668 Faculty of Science and Engineering School of Mathematics and Computer Science - Mathematics Bernard Mourrain 1 Raimundas Vidunas 2 Nelly Villamizar 0000-0002-8741-7225 3 0032863-30032017173026.pdf cagd_paper.pdf 2017-03-30T17:30:26.5400000 Output 589630 application/pdf Accepted Manuscript true 2017-03-30T00:00:00.0000000 true eng
title Dimension and bases for geometrically continuous splines on surfaces of arbitrary topology
spellingShingle Dimension and bases for geometrically continuous splines on surfaces of arbitrary topology
Nelly Villamizar
title_short Dimension and bases for geometrically continuous splines on surfaces of arbitrary topology
title_full Dimension and bases for geometrically continuous splines on surfaces of arbitrary topology
title_fullStr Dimension and bases for geometrically continuous splines on surfaces of arbitrary topology
title_full_unstemmed Dimension and bases for geometrically continuous splines on surfaces of arbitrary topology
title_sort Dimension and bases for geometrically continuous splines on surfaces of arbitrary topology
author_id_str_mv 41572bcee47da6ba274ecd1828fbfef4
author_id_fullname_str_mv 41572bcee47da6ba274ecd1828fbfef4_***_Nelly Villamizar
author Nelly Villamizar
author2 Bernard Mourrain
Raimundas Vidunas
Nelly Villamizar
format Journal article
container_title Computer Aided Geometric Design
container_volume 45
container_start_page 108
publishDate 2016
institution Swansea University
issn 01678396
doi_str_mv 10.1016/j.cagd.2016.03.003
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
url http://www.sciencedirect.com/science/article/pii/S0167839616300309
document_store_str 1
active_str 0
description We analyze the space of geometrically continuous piecewise polynomial functions, or splines, for rectangular and triangular patches with arbitrary topology and general rational transition maps. To define these spaces of G 1 spline functions, we introduce the concept of topological surface with gluing data attached to the edges shared by faces. The framework does not require manifold constructions and is general enough to allow non-orientable surfaces. We describe compatibility conditions on the transition maps so that the space of differentiable functions is ample and show that these conditions are necessary and sufficient to construct ample spline spaces. We determine the dimension of the space of G1 spline functions which are of degree less than or equal to k on triangular pieces and of bi-degree less than or equal to (k, k) on rectangular pieces, for k big enough. A separability property on the edges is involved to obtain the dimension formula. An explicit construction of basis functions attached resspectively to vertices, edges and faces is proposed; examples of bases of G1 splines of small degree for topological surfaces with boundary and without boundary are detailed.
published_date 2016-07-31T03:40:25Z
_version_ 1763751835192000512
score 11.012678

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