Approximating the maxmin-angle covering triangulation

Abstract

Given a planar straight-line graph or polygon with holes, we seek a covering triangulation whose minimum angle is as large as possible. A covering triangulation is a Steiner triangulation with the following restriction: no Steiner vertices may be added on an input edge. We give an explicit upper bound on the largest possible minimum angle in any covering triangulation of a given input. This upper bound depends on local geometric features of the input. We then show that our covering triangulation has minimum angle at least a constant factor times this upper bound. This is the first known algorithm for generating a covering triangulation of an arbitrary input with a provable bound on triangle shape. Covering triangulations can be used to triangulate intersecting regions independently, and so solve several subproblems of mesh generation.