Generalized delaunay triangulation for planar graphs

Abstract

We introduce the notion of generalized Delaunay triangulation of a planar straight-line graph G = (V, E) in the Euclidean plane and present some characterizations of the triangulation. It is shown that the generalized Delaunay triangulation has the property that the minimum angle of the triangles in the triangulation is maximum among all possible triangulations of the graph. A general algorithm that runs in O(I VI 2) time for computing the generalized Delaunay triangulation is presented. When the underlying graph is a simple polygon, a divide-andconquer algorithm based on the polygon cutting theorem of Chazelle is given that runs in O(I V I logl VI) time.

General-Dimensional Constrained Delaunay and Constrained Regular Triangulations, I: Combinatorial Properties

“…A twodimensional CDT shares these same optimality properties, if it is compared with every other constrained triangulation of the same PSLG [4,29].…”

“…The second alternative is to form a constrained Delaunay triangulation (CDT) [9,29,43], illustrated in Fig. 2(d).…”

“…Nevertheless, CDTs retain many of the desirable properties of Delaunay triangulations. For instance, a two-dimensional CDT maximizes the minimum angle in the triangulation, compared with all other constrained triangulations of X [29].…”

General-Dimensional Constrained Delaunay and Constrained Regular Triangulations, I: Combinatorial Properties

“…The second alternative is to form a constrained Delaunay triangulation (CDT) [29,9,43], illustrated at far right in Figure 2. A CDT of X has no vertices not in X, and every segment in X is a single edge of the CDT.…”

“…Nevertheless, CDTs retain many of the desirable properties of Delaunay triangulations. For instance, a two-dimensional CDT maximizes the minimum angle in the triangulation, compared with all other constrained triangulations of X [29]. Prior to the present work (in its first incarnation [45]), CDTs had not been generalized to dimensions higher than two.…”