Ray Shooting Amidst Convex Polyhedra and PolyhedralTerrains in Three Dimensions

• Authors: Pankaj K. Agarwal;Micha Sharir
• Venue: SIAM Journal on Computing
• Year: 1996

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 $O((mn)^{2+\eps})$ preprocessing time and storage, and can answer ray-shooting queries in $O(\log^2 n)$ time. This is a significant improvement over previously known techniques (which require $O(n^{4+\eps})$ 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 $O(n^{1+\eps})$ space and preprocessing, and answers queries in time $O(m^{1/4}n^{1/2+\eps})$, again a significant improvement over previous techniques when $m \ll n$.