On Computing New Classes of Optimal Triangulations with Angular Constraints

Abstract

Given a planar point set S, a triangulation of S is a maximal set of non-intersecting edges connecting points in S. Triangulating a point set has many applications in computational geometry and other related fields. Specifically, in numerical solutions for scientific and engineering applications, poorly shaped triangles can cause serious difficulty. Traditionally, triangulations which minimize the maximum angle, maximize the minimum angle, minimize the maximum edge length, and maximize the minimum hight are considered. For example, if angles of triangles become too large, the discretization error in the finite element solution is increased and, if the angles become too small, the condition number of the element matrix is increased [1, 10, 11]. Polynomial time algorithms have been developed in determining those triangulations [2, 7, 8, 15]. In computational geometry another important research object is to compute the minimum weight triangulation. The weight of a triangulation is defined to be the sum of the Euclidean lengths of the edges in the triangulation. Despite the intensive study made during the lase two decades, it remains unknown that whether the minimum weight triangulation problem is NP-complete or polynomially solvable.

This research is supported by the Grant-in-Aid of Ministry of Science, Culture and Education of Japan.