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

Evan VanderZee, Anil N. Hirani, Damrong Guoy, Vadim Zharnitsky, Edgar A. Ramos

Research output: Contribution to journalArticlepeer-review

Abstract

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.

Original languageEnglish (US)
Pages (from-to)700-724
Number of pages25
JournalComputational Geometry: Theory and Applications
Volume46
Issue number6
DOIs
StatePublished - Aug 1 2013

Keywords

  • Acute triangulations
  • Circumcentric dual
  • Discrete exterior calculus
  • Finite element method
  • Mesh generation

ASJC Scopus subject areas

  • Computer Science Applications
  • Geometry and Topology
  • Control and Optimization
  • Computational Theory and Mathematics
  • Computational Mathematics

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