A survey of Delaunay structures for surface representation

The Delaunay triangulation characterizes a natural neighbour relation amongst points distributed in a Euclidean space. In this survey we examine extensions of the Delaunay paradigm that have been used to define triangle meshes for representing smooth surfaces embedded in three dimensional Euclidean space. Progress in this area has stemmed primarily from work done in surface reconstruction and surface meshing. However, our focus is not on the surface reconstruction or meshing algorithms themselves, but rather on the structures that they aim to produce. In particular we concentrate on three distinct Delaunay structures which differ according to the metric involved in their definition: the metric of the ambient Euclidean space; the intrinsic metric of the original surface; or the intrinsic metric of the mesh itself. We study the similarities and distinctions between these objects; their strengths and weaknesses both as theoretical tools and as practical data structures. The topic of this survey lies within the realm of geometry processing, a field of study generally associated with computer graphics. However, it is hoped that this survey will be of interest not just to those who study computer graphics, but to anybody whose research touches on a need to represent non-Euclidean geometry with Euclidean