Constrained Construction of Planar Delaunay Triangulations without Flipping

The construction of Voronoi diagrams and Delaunay triangulations finds wide application in many branches of science. Delaunay triangulations have properties which make them more desirable than other triangulations for the same node set. Delaunay has characterized his triangulations by the empty circle property. The partitioning and flipping methods which have been developed for digital construction of Voronoi diagrams and Delaunay triangulations only make indirect use of this property. A novel method of construction is proposed, which is based directly on the empty circle property of Delaunay. The geometry of the steps of the algorithm is simple and can be grasped intuitively. The method can be applied to constrained triangulations, in which a triangulation domain and some of the edges are prescribed. A data structure for triangulations of concave and multiply-connected domains is presented which permits convenient specification of the constraints and the triangulation. The method is readily implemented, efficient and robust.