Towards a Definition of Higher Order Constrained Delaunay Triangulations

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.