Linear data structures for fast ray-shooting amidst convex polyhedra
ESA'07 Proceedings of the 15th annual European conference on Algorithms
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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 preprocessing time and storage (where the $ 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).