Let $P$ be a simple polygon with $n$ vertices. We present a simple decomposition scheme that partitions the interior of $P$ into $O(n)$ so-called geodesic triangles, so that any line segment interior to $P$ crosses at most 2 log $n$ of these triangles. This decomposition can be used to preprocess $P$ in a very simple manner, so that any ray-shooting query can be answered in time $O(log^n)$. The data structure required $O(n)$ storage and $O(n$ log $n)$ preprocessing time. By using more sophisticated techniques, we can reduce the preprocessing time to $O(n)$. We also extend our general technique to the case of ray-shooting amidst $k$ polygonal obstacles with a total of $n$ edges, so that a query can be answered in $O( sqrt k$ log $n$) time.

This technical report has been published as

Ray Shooting in Polygons Using Geodesic Triangulations. Bernard Chazelle, Herbert Edelsbrunner, Michelangelo Grigni, Leonidas Guibas, John Hershberger, Micha Sharir, and Jack Snoeyink, Algorithmica, 12, 1994, pp. 54-68.