Finding a Covering Triangulation Whose Maximum Angle is Provably Small

We consider the following problem given a pla nar straight line graph nd a covering triangulation whose maximum angle is as small as possible A cov ering triangulation is a triangulation whose vertex set contains the input vertex set and whose edge set con tains the input edge set The covering triangulation problem di ers 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 opt on the maxi mum angle in any covering triangulation of a particu lar input graph in terms of its local geometry Our al gorithm produces a covering triangulation whose max imum angle is provably close to opt Bounding by a constant times opt is trivial We prove something signi cantly stronger Speci cally we show that