OPTIMAL TRIANGULATIONS OF POINTS AND SEGMENTS WITH STEINER POINTS

Consider a set X of points in the plane and a set E of non-crossing segments with endpoints in X. One can efficiently compute the triangulation of the convex hull of the points, which uses X as the vertex set, respects E, and maximizes the minimum internal angle of a triangle. In this paper we consider a natural extension of this problem: Given in addition a Steiner point p, determine the optimal location of p and a triangulation of X boolean OR {p} respecting E, which is best among all triangulations and placements of p in terms of maximizing the minimum internal angle of a triangle. We present a polynomial- time algorithm for this problem and then extend our solution to handle any constant number of Steiner points.