No one knows how to compute the shortest path between two points in a line arrangement in less than \theta(n^2). We first show that one can simplify this to finding a shortest path in a grid of quadrilaterals (which has the complication of having paths traveling on a boundary around the grid).
The only previously published approximation algorithm gives a factor of 2 approximation to the shortest path in O(n log n). We use the simpler grid structure to give a much better approximation algorithm.
This is a preliminary version of what will be a job talk. The associated paper is to be submitted for publication.