VDM Vassilev, T: Optimal Area Triangulation

Triangulations of point sets play an important role in Computational Geometry and have been studied extensively in the last decades. The results on optimizing angles and edge lengths are classical in the field. Here we present a study on optimizing the area in two ways: minimizing the maximum area of a triangle, and maximizing the minimum area of a triangle. In the case of a point set in convex position we present nearly quadratic algorithms for both problems. The geometric properties of these two optimal triangulations are derived and extensively discussed. We strongly believe that both problems admit no worse than quadratic solution. Such will be based on a refinement of the geometric properties. Furthermore, the properties and the methods described here can serve as a starting point to obtaining efficient optimal triangulation algorithms for other quality measures such as maximizing inradius or aspect ratio of a triangle. In the case of a point set in general position, we present a polynomial time approximation algorithm. The algorithm is based on the matching properties of triangulations and further geometric considerations.