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

• Published in: Computational geometry : theory and applications Vol. 46; no. 6; pp. 700 - 724
• Main Authors: VanderZee, Evan, Hirani, Anil N., Guoy, Damrong, Zharnitsky, Vadim, Ramos, Edgar A.
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 01.08.2013
• Subjects: Acute triangulations; Algebra; Circumcentric dual; Combinatorial analysis; Computational geometry; Constrictions; Discrete exterior calculus; Finite element method; Links; Mesh generation; Triangles; Triangulation; Finite element method; Acute triangulations; Circumcentric dual; Discrete exterior calculus; Mesh generation
• ISSN: 0925-7721
• DOI: 10.1016/j.comgeo.2012.11.003

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.