Geometric and Combinatorial Properties of Well-Centered Triangulations in Three and Higher Dimensions

TL;DR

This paper explores geometric and combinatorial properties of well-centered simplices in higher dimensions, establishing conditions and restrictions for such meshes, with specific results for 3D tetrahedral meshes.

Contribution

It introduces new geometric and algebraic criteria for n-well-centered simplices and characterizes the local combinatorial restrictions in well-centered tetrahedral meshes.

Findings

In 3D, each interior vertex in a well-centered mesh has at least 7 incident edges.

Sharp bounds are established for the number of incident edges in 3D well-centered meshes.

Infinitely many vertex links prevent well-centered tetrahedral meshes, unlike in 2D.

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 R^2, there are infinitely many vertex links that prohibit a well-centered tetrahedral mesh in R^3.