Construction of the segment Delaunay triangulation by a flip algorithm (Construction de la triangulation de Delaunay de segments par un algorithme de flip)

Given a set S of points in the plane, a triangulation of S is a partition of the convex hull of S into triangles whose vertices are the points of S. A triangulation of S is said to be Delaunay if no point of S lies inside the triangles’ circumcircles. In this thesis, we study a generalization of these notions to a set S of disjoint segments in the plane. At first, we define a new family of diagrams, called segment triangulations. We study their geometric and topologic properties and we give an algorithm that efficiently computes such a triangulation. Then, we generalize the notion of Delaunay triangulation to segment triangulations and we point out that it is dual to the segment Voronoi diagram. We also extend the notion of edge legality to segment triangulations. On the one hand, we define the geometric legality, which characterizes the segment Delaunay triangulation among the set of all possible segment triangulations. On the other hand, we introduce a topologic legality, which characterizes the segment triangulations that have the same topology as the Delaunay one. Finally, we give a « flip » algorithm that transforms any segment triangulation in a triangulation that has the same topology as the Delaunay one. By using locally convex functions, we show that the sequence of triangulations computed by this algorithm converges to the segment Delaunay triangulation. Moreover, we prove that a segment triangulation with the same topology as the Delaunay one is achieved after a finite number of steps.