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Dimension and bases for geometrically continuous splines on surfaces of arbitrary topology / Bernard Mourrain; Raimundas Vidunas; Nelly Villamizar

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

Swansea University Author: Villamizar, Nelly

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
Online Access: Check full text

URI: https://cronfa.swan.ac.uk/Record/cronfa32863
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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 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.
Keywords: geometrically continuous splines, dimension and bases of spline spaces, gluing data, polygonal patches, surfaces of arbitrary topology
College: College of Science
Start Page: 108
End Page: 133