an optimal algorithm for computing angle-constrained spanners

Let S be a set of n points in R^d and let t>1 be a real number. A graph G=(S,E) is called a t-spanner for S, if for any two points p and q in S, the shortest-path distance in G between p andq is at most t|pq|, where |pq| denotes the Euclidean distance between p and q. The graph G is called θ-angle-constrained, if any two distinct edges sharing an endpoint make an angle of at least θ. It is shown that, for any θ with 0<θ<π/3, a θ-angle-constrained t-spanner can be computed in O(nlog n) time, where t depends only on θ. For values of θ approaching 0, we havet=1 + O(θ).