Coxeter Triangulations Have Good Quality

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/ √ 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 nondegeneracy 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.

Triangulating Submanifolds: An Elementary and Quantified Version of Whitney’s Method

“…Proof Choudhary et al [25,App. B] provide explicit values of all the quantities mentioned in Corollary 4.4 for a Coxeter triangulation of typeÃ, with the exception of μ, which can be easily derived from a more general result.…”

“…They are also attractive from the geometrical perspective, because they provide simplices with very good quality and some particular Coxeter triangulations are Delaunay protected and thus very stable Delaunay triangulations. We will now very briefly introduce both the concepts of Coxeter triangulations and Delaunay protection, but refer to [25] for more details on Coxeter triangulations and to [13,14] for Delaunay protection. This definition imposes very strong constraints on the geometry of the simplices, implying that there are only a small number of such triangulations in each dimension.…”

“…II.16]. The argument for stability of triangulations forà type Coxeter triangulations is much simpler, because it is a Delaunay triangulation and is δ-protected, see [25]. Before we can recall this result we need to introduce some notation and a definition:…”

“…B] provide explicit values of all the quantities mentioned in Corollary 4.4 for a Coxeter triangulation of typeÃ, with the exception of μ, which can be easily derived from a more general result. If we fix the scale (which in [25] we did by a convenient choice of coordinates for the vertices), we have…”

Local Conditions for Triangulating Submanifolds of Euclidean Space

“…The altitude of a vertex in a simplex is the distance from the vertex to the affine hull of the opposite face. Observe that t(σ ) ≤ 1/m and we have conjectured in [9] that the largest thickness that can be achieved is in fact O(m −3/2 ). We set t(σ ) = 1 if σ has dimension 0.…”