Towards a definition of higher order constrained Delaunay triangulations

Abstract

When a triangulation of a set of points and edges is required, the constrained Delaunay triangulation is often the preferred choice because of its well-shaped triangles. However, in applications like terrain modeling, it is sometimes necessary to have flexibility to optimize some other aspect of the triangulation, while still having nicely-shaped triangles and including a set of constraints. Higher order Delaunay triangulations were introduced to provide a class of well-shaped triangulations, flexible enough to allow the optimization of some extra criterion. But they are not able to handle constraints: a single constraining edge may cause that all triangulations with that edge have high order, allowing ill-shaped triangles at any part of the triangulation. In this paper we generalize the concept of the constrained Delaunay triangulation to higher order constrained Delaunay triangulations. We study several possible definitions that assure that an order-k constrained Delaunay triangulation exists for any k ≥ 0, while maintaining the character of higher order Delaunay triangulations of point sets. Several properties of these definitions are studied, and efficient algorithms to support computations with order-k constrained Delaunay triangulations are also discussed. For the special case of k = 1, we show that many measures can be optimized efficiently in the presence of constraints.

Optimal higher order Delaunay triangulations of polygons

“…The algorithms presented in this paper are general, in the sense that they were designed to be able to work for a large number of measures. Even though any of the definitions proposed in [22] can easily be plugged in into the algorithms, many of the improvements on the running time achieved here would be lost. It may be interesting to try to adapt the polygon triangulation algorithms to some of the definitions proposed in [22].…”

“…As mentioned in the introduction, there seems to be no single definition of the constrained order of a triangle that generalizes the concept of constrained Delaunay triangulation to higher orders in a completely natural way. The authors concluded in [22] that the choice of such a definition will depend on the application in mind. The algorithms presented in this paper are general, in the sense that they were designed to be able to work for a large number of measures.…”

“…A first one is studying what other measures, relevant to specific applications, can be optimized with a similar approach. A second interesting topic is combining the algorithms presented here with some definition of higher order constrained Delaunay triangulations [22]. We would like to have a definition of the order of a triangle that is influenced by the polygon edges, allowing us to always obtain some triangulation that is close to the constrained Delaunay triangulation.…”

“…In such a case, it is always possible to increase k until a triangulation is found. The authors recently studied possible definitions of the order of a triangle that take a set of constraining edges into account [22], which guarantee that a polygon can always be triangulated for any k. However, although seven different definitions for this notion of constrained order were proposed in [22], no single definition can be regarded as the natural or right one.…”

Generalized Delaunay Triangulations:Graph-Theoretic Properties and Algorithms

“…In this chapter we study the flip graph of higher order Delaunay triangulations. Most previous work on such triangulations focused on algorithmic questions related to finding order-k triangulations that are optimal with respect to extra criteria [104,97,98], or on evaluating their effectiveness in practical settings [20,45]. One of the few theoretical aspects studied is the asymptotic number of order-k triangulations [82].…”