I.5. - Incremental Delaunay Triangulation

Abstract

Dani Lischinski 580 ETC Building Cornell University Ithaca, NY 14850, USA danix@graphics.cornell.edu } Introduction } This gem gives a simple algorithm for the incremental construction of the Delaunay triangulation (DT) and the Voronoi diagram (VD) of a set of points in the plane. A triangulation is called Delaunay if it satis es the empty circumcircle property: the circumcircle of a triangle in the triangulation does not contain any input points in its interior. DT is the straight-line dual of the Voronoi diagram of a point set, which is a partition of the plane into polygonal cells, one for each point in the set, so that the cell for point p consists of the region of the plane closer to p than to any other input point (Preparata and Shamos 1985,Fortune 1992). Delaunay triangulations and Voronoi diagrams, which can be constructed from them, are a useful tool for e ciently solving many problems in computational geometry (Preparata and Shamos 1985). DT is optimal in several respects. For example, it maximizes the minimum angle and minimizes the maximum circumcircle over all possible triangulations of the same point set (Fortune 1992). Thus, DT is an important tool for high quality mesh generation for nite elements (Bern and Eppstein 1992). It should be noted, however, that standard DT doesn't allow edges that must appear in the triangulation to be speci ed in the input. Thus, in order to mesh general polygonal regions the more complicated constrained DT should be used (Bern and Eppstein 1992). The incremental DT algorithm given in this gem was originally presented by Green and Sibson (Green and Sibson 1978), but the implementation is based entirely on the quad-edge data structure and the pseudocode from the excellent paper by Guibas and Stol (Guibas and Stol 1985). I will brie y describe the data structures and the algorithm, but the reader is referred to Guibas and Stol for more details.