Maximizing maximal angles for plane straight-line graphs

Let G = (S, E) be a plane straight-line graph on a finite point set S subset of R-2 in general position. The incident angles of a point p is an element of S in G are the angles between any two edges of G that appear consecutively in the circular order of the edges incident to p. A plane straight-line graph is called phi-open if each vertex has an incident angle of size at least phi. In this paper we study the following type of question: What is the maximum angle phi such that for any finite set S subset of R-2 of points in general position we can find a graph from a certain class of graphs on S that is phi-open? In particular, we consider the classes of triangulations, spanning trees, and paths on S and give tight bounds in most cases.