Linear data structures for fast ray-shooting amidst convex polyhedra

We consider the problem of ray shooting in a three-dimensional scene consisting of k (possibly intersecting) convex polyhedra with a total of n facets. That is, 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 describe data structures that require tilde{O}(n\cdot \mathrm{poly}(k)) preprocessing time and storage (where the tilde{O}(\cdot) notation hides polylogarithmic factors), and have polylogarithmic query time, for several special instances of the problem. These include the case when the ray origins are restricted to lie on a fixed line ℓ ₀, but the directions of the rays are arbitrary, the more general case when the supporting lines of the rays pass through ℓ ₀, and the case of rays orthogonal to some fixed line with arbitrary origins and orientations. We also present a simpler solution for the case of vertical ray-shooting with arbitrary origins. In all cases, this is a significant improvement over previously known techniques (which require Ω(n ²) storage, even when k ∼ n).