Geometric and combinatorial properties of well-centered triangulations in three and higher dimensions

An n-simplex is said to be n-well-centered if its circumcenter lies in its interior. We introduce several other geometric conditions and an algebraic condition that can be used to determine whether a simplex is n-well-centered. These conditions, together with some other observations, are used to des...

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Bibliographic Details
Published in:Computational geometry : theory and applications 2013-08, Vol.46 (6), p.700-724
Main Authors: VanderZee, Evan, Hirani, Anil N., Guoy, Damrong, Zharnitsky, Vadim, Ramos, Edgar A.
Format: Article
Language:English
Subjects:
Acute triangulations
Algebra
Circumcentric dual
Combinatorial analysis
Computational geometry
Constrictions
Discrete exterior calculus
Finite element method
Links
Mesh generation
Triangles
Triangulation
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Summary:An n-simplex is said to be n-well-centered if its circumcenter lies in its interior. We introduce several other geometric conditions and an algebraic condition that can be used to determine whether a simplex is n-well-centered. These conditions, together with some other observations, are used to describe restrictions on the local combinatorial structure of simplicial meshes in which every simplex is well-centered. In particular, it is shown that in a 3-well-centered (2-well-centered) tetrahedral mesh there are at least 7 (9) edges incident to each interior vertex, and these bounds are sharp. Moreover, it is shown that, in stark contrast to the 2-dimensional analog, where there are exactly two vertex links that prevent a well-centered triangle mesh in R2, there are infinitely many vertex links that prohibit a well-centered tetrahedral mesh in R3.
ISSN:0925-7721
DOI:10.1016/j.comgeo.2012.11.003