Linear data structures for fast ray-shooting amidst convex polyhedra

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Abstract

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).

Original languageEnglish
Pages (from-to)283-310
Number of pages28
JournalAlgorithmica
Volume55
Issue number2
DOIs
StatePublished - Oct 2009

Keywords

  • Computational geometry
  • Convex polyhedra
  • Lines in space
  • Ray-shooting

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