Coxeter triangulations have good quality

Aruni Choudhary (1) , Siargey Kachanovich (2) , Mathijs Wintraecken (2)

1 Max Planck Institute for Informatics [Saarbrücken]
2 DATASHAPE - Understanding the Shape of Data

Coxeter triangulations are triangulations of Euclidean space based on a simple 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 (and an extension thereof) for these triangulations. 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.

Mots clés: Coxeter triangulation; Delaunay triangulation; Euclidean geometry; Simplex quality; Delaunay protection

Domaines: Géométrie algorithmique [cs.CG]

Aruni Choudhary, Siargey Kachanovich, Mathijs Wintraecken. Coxeter triangulations have good quality. 2017. ⟨hal-01667404⟩