## 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 language | English |
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Pages (from-to) | 283-310 |

Number of pages | 28 |

Journal | Algorithmica |

Volume | 55 |

Issue number | 2 |

DOIs | |

State | Published - 1 Oct 2009 |

Externally published | Yes |

## Keywords

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

## ASJC Scopus subject areas

- General Computer Science
- Computer Science Applications
- Applied Mathematics