The 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 elements 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 will focus on the Delaunay stable methods presenting our researches on the subject. 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 generalized Vorono? diagram of the configuration.Finally, we expose an application of our algorithms in maintaining at lowest cost the DEM realism during the resampling process. 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.