Shooting amidst Convex Polyhedra and Polyhedral Terrains in Three Dimensions

We consider the problem of ray shooting in a three-dimensional scene consisting of m (possibly intersecting) convex polyhedra or polyhedral terrains with a total of n faces, i.e., we want to preprocess them into a data structure, so that the first intersection point of a query ray and the given polyhedra can be determined quickly. We present a technique that requires O ((mn):+) preprocessing time and storage, and can answer ray-shooting queries in O(log n) time. This is a significant improvement over previously known techniques (which require O(n4+) space and preprocessing) if m is much smaller than n, which is often the case in practice. Next, we present a variant of the technique that requires O(n TM) space and preprocessing, and answers queries in time O(ml/4nl/2+e), again a significant improvement over previous techniques when m << n. Key words, arrangements, ray shooting, range searching, data structures, parametric search, random sampling AMS subject classifications. 52B 11, 68P05, 68Q20, 68Q25