Self-Delaunay meshes for surfaces

In the Euclidean plane, a Delaunay triangulation can be characterized by the requirement that the circumcircle of each triangle be empty of vertices of all other triangles. For triangulating a surface S in R^3, the Delaunay paradigm has typically been employed in the form of the restricted Delaunay triangulation, where the empty circumcircle property is defined by using the Euclidean metric in R^3 to measure distances on the surface. More recently, the intrinsic (geodesic) metric of S has also been employed to define the Delaunay condition. In either case the resulting mesh M is known to approximate S with increasing accuracy as the density of the sample points increases. However, the use of the reference surface S to define the Delaunay criterion is a serious limitation. In particular, in the absence of the original reference surface, there is no way of verifying if a given mesh meets the criterion. We define a self-Delaunay mesh as a triangle mesh that is a Delaunay triangulation of its vertex set with respect to the intrinsic metric of the mesh itself. This yields a discrete surface representation criterion that can be validated by the properties of the mesh alone, independent of any reference surface the mesh is supposed to represent. The intrinsic Delaunay triangulation that characterizes self-Delaunay meshes makes them a natural domain for discrete differential geometry, and the discrete exterior calculus in particular. We examine self-Delaunay meshes and their relationship with other Delaunay structures for surface representation. We study sampling conditions relevant to the intrinsic approach, and compare these with traditional sampling conditions which are based on extrinsic quantities and distances in the ambient Euclidean space. We also provide practical and provably correct algorithms for constructing self-Delaunay meshes. Of particular interest in this context is the extrinsic edge flipping algorithm which extends the familiar algorithm for producing planar Delaunay triangulations.