Approximating Uniform Triangular Meshes in Polygons

Given a convex polygon P in the plane and a positive integer n, we consider the problem of generating a length-uniform triangular mesh for the interior of P using n Steiner points. More specifically, we want to find both a set Sn of n points inside P , and a triangulation of P using Sn, with respect to the following minimization criteria: (1) ratio of the maximum edge length to the minimum one, (2) maximum edge length, and (3) maximum triangle perimeter. These problems can be formalized as follows: Let V be the set of vertices of P . For an n-point set Sn interior to P let T (Sn) denote the set of all possible triangulations of Sn ∪ V . Further, let l(e) denote the (Euclidean) length of edge e, and let peri(∆) be the perimeter of triangle ∆.