Ray shooting amidst convex polytopes in three dimensions

We consider the problem of ray shooting in a 3-dimensional scene consisting of m (possibly intersecting) convex polyhedra 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)^2+ε) 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(n^4+ε) space and preprocessing) if m is much smaller than n, which is often the case in practice. We also present a variant of the technique that requires O(n^1+ε) space and preprocessing, and answers queries in time O(m^1/4n^1/2+ε), again a significant improvement over previous techniques when m≪n. Our algorithms also work when the given polyhedra intersect, and also apply to a collection of (possibly intersecting) polyhedral terrains.