Abstract
Coxeter triangulations are triangulations of Euclidean space based on a single simplex. By this we mean that given an individual simplex we can recover the entire triangulation of Euclidean space by inductively reflecting in the faces of the simplex. In this paper we establish that the quality of the simplices in all Coxeter triangulations is O(1/sqrt{d}) of the quality of regular simplex. We further investigate the Delaunay property for these triangulations. Moreover, we consider an extension of the Delaunay property, namely protection, which is a measure of non-degeneracy of a Delaunay triangulation. In particular, one family of Coxeter triangulations achieves the protection O(1/d^2). We conjecture that both bounds are optimal for triangulations in Euclidean space.
Highlights
1.1 Objectives and MotivationWell-shaped simplices are of importance for various fields of application such as finite element methods and manifold meshing [2,9,10,18,21,29]
Even in Euclidean space, of which all simplices have good quality is a non-trivial exercise in arbitrary dimension
The notion of Coxeter triangulations was introduced to the computational geometry community by Dobkin, Wilks, Levy and Thurston in [14], where they tackled the problem of contour-tracing in Rd
Summary
Well-shaped simplices are of importance for various fields of application such as finite element methods and manifold meshing [2,9,10,18,21,29]. Even in Euclidean space, of which all simplices have good quality is a non-trivial exercise in arbitrary dimension. All dihedral angles of simplices in Coxeter triangulations are 45◦, 60◦ or 90◦, with the exception of the G 2 triangulation of the plane where we can find an angle of 30◦. This is a clear sign of the exceptional quality of the simplices involved. Take the d-dimensional Coxeter triangulation Ad. As we will see in the following, this highly-structured triangulation is Delaunay with protection. This protection value is the greatest in a general d-dimensional Delaunay triangulation we know
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