Finding a Covering Triangulation Whose Maximum Angle is Provably Small
Scott A. Mitchell · International Journal of Computational Geometry & Applications · 1997
We consider the following problem: given a planar straight-line graph, find a covering triangulation whose maximum angle is as small as possible. A covering triangulation is a triangulation whose vertex set contains the input vertex set and whose edge set contains the input edge set. The covering triangulation problem differs from the usual Steiner triangulation problem in that we may not add a vertex on any input edge. Covering triangulations provide a convenient method for triangulating multiple regions sharing a common boundary, as each region can be triangulated independently. We give an explicit lower bound γopt on the maximum angle in any covering triangulation of a particular input graph in terms of its local geometry. Our algorithm produces a covering triangulation whose maximum angle γ is provably close to γopt. Specifically, we show that [Formula: see text], i.e., our γ is not much closer to π than is γopt. To our knowledge, this result represents the first nontrivial bound on a covering triangulation's maximum angle. Our algorithm adds O(n) Steiner points and runs in time O(n log n), where n is the number of vertices of the input. We have implemented an O(n2) time version of our algorithm.