  We describe an algorithm that takes as input n points in the plane and a
parameter {\epsilon}, and produces as output an embedded planar graph having
the given points as a subset of its vertices in which the graph distances are a
(1 + {\epsilon})-approximation to the geometric distances between the given
points. For point sets in which the Delaunay triangulation has bounded sharpest
angle, our algorithm's output has O(n) vertices, its weight is O(1) times the
minimum spanning tree weight, and the algorithm's running time is bounded by
O(n \sqrt{log log n}). We use this result in a similarly fast deterministic
approximation scheme for the traveling salesperson problem.
