# Optimal triangulations by retriangulating

## Abstract

Given a set *S* of *n* points in the plane, a triangulations that minimizes the maximum angle can be contracted in polynomial time by repeatedly deleting all edges that intersect some line segment and retriangulating the thus created polygonal regions. The same method can be used to compute a triangulation that maximizes the minimum triangle height. The currently most efficient implementation of this paradigm runs in time *O(n*^{2}log *n*) and storage *O(n)*.

While the above method fails when the objective is to minimize the length of the longest edge, such an optimal triangulation can be computed as follows. First, construct the relative neighborhood graph of the points and then optimally triangulate the thus obtained polygonal regions. This idea leads to an algorithm that runs in *O(n*^{2}) time and storage.

In the first case (minmax angle and maxmin height) the retriangulation is done algorithmically, and in the second case (minmax length) it is a proof technique. In particular, various retriangulation methods are used to prove that, indeed, there is a minmax length triangulation that contains the relative neighborhood graph as a subgraph and to show several additional properties necessary for the final algorithm.