Delaunay Edge Flips in Dense Surface Triangulations

Delaunay flip is an elegant, simple tool to convert a triangulation of a point set to its Delaunay triangulation. The technique has been researched extensively for full dimensional triangulations of point sets. However, an important case of triangulations which are not full dimensional is surface triangulations in three dimensions. In this paper we address the question of converting a surface triangulation to a subcomplex of the Delaunay triangulation with edge flips. We show that the surface triangulations which closely approximate a smooth surface with uniform density can be transformed to a Delaunay triangulation with a simple edge flip algorithm. The condition on uniformity becomes less stringent with increasing density of the triangulation. If the condition is dropped completely, the flip algorithm still terminates although the output surface triangulation becomes “almost Delaunay” instead of exactly Delaunay. Research supported by NSF grants CCF-0430735 and CCF-0635008 and RGC, Hong Kong, China (HKUST 6181/04E). Department of Computer Science and Engineering, HKUST, Clear Water Bay, Hong Kong. Email: [email protected] Department of Computer Science and Engineering, The Ohio State University, Columbus, OH 43210, USA. Email: [email protected]