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On the problem of instability in the dimension of a spline space over a T-mesh

Authors
Journal
Computers & Graphics
0097-8493
Publisher
Elsevier
Volume
36
Issue
5
Identifiers
DOI: 10.1016/j.cag.2012.03.005
Keywords
  • Spline Space
  • T-Mesh
  • Dimension
Disciplines
  • Mathematics

Abstract

Graphical Abstract For certain types of T-meshes, we prove that the dimension of spaces S(4,4,2,2)(T), S(5,5,3,3)(T) depends on the coordinates of vertices of T-mesh T. We show, in particular case, that the bound m≥2r+1 and m′≥2r′+1 are optimal. Motivation: Splines over T-mesh could be a useful tool in many areas such as surface modeling and finite element analysis. The instability in the dimension of a space S(m,m′,r,r′) is an important issue to investigate, because it provides a better understanding of the main problem, namely, how the basis of a space S(m,m′,r,r′) could be described. Highlights ► We examine the dimension of a bivariate spline space over T-mesh. ► We treat the case of polynomial degree 5, 5 and order of smoothness 3, 3. ► And we treat the case of polynomial degree 4, 4 and order of smoothness 2, 2. ► We provide the examples of T-meshes such the dimension depends not only on topology. ► But it also depends on the geometry of a given T-meshes.

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