The Delaunay constrained triangulation: the Delaunay stable algorithms

Abstract

Delaunay triangulation is well known for its use in geometric design. A derived version of this structure, the Delaunay constrained triangulation, takes into account the triangular mesh problem in presence of rectilinear constraints. The Delaunay constrained triangulation is very useful for CAD, topography and mapping and in finite element analysis. This technique is still developing. We present a taxonomy of this geometric structure. First we describe the different tools used to introduce the problem. Then we introduce the different approaches highlighting various points of view of the problem. We focus on the Delaunay stable methods. A Delaunay stable method preserves the Delaunay nature of the constrained triangulation. Each method is detailed by its algorithms, performances, and properties. For instance we show how these methods approximate the generalised Voronoi diagram of the configuration. The Delaunay stable algorithms are used for 2.5D DEM design. The aim of this work is to demonstrate that the use of topographic constraints in a regular DEM without adding new points preserves the terrain shape. So the resulting DEM can be more easily interpreted because its realism is preserved and the mesh still owns all the Delaunay triangulation properties.