Triangulations of Line Segment Sets in the Plane

Given a set S of line segments in the plane, we introduce a new family of partitions of the convex hull of S called segment triangulations of S. The set of faces of such a triangulation is a maximal set of disjoint triangles that cut S at, and only at, their vertices. Surprisingly, several properties of point set triangulations extend to segment triangulations. Thus, the number of their faces is an invariant of S. In the same way, if S is in general position, there exists a unique segment triangulation of S whose faces are inscribable in circles whose interiors do not intersect S. This triangulation, called segment Delaunay triangulation, is dual to the segment Voronoi diagram. The main result of this paper is that the local optimality which characterizes point set Delaunay triangulations [10] extends to segment Delaunay triangulations. A similar result holds for segment triangulations with same topology as the Delaunay one.