## Ray shooting amidst convex polyhedra and polyhedral terrains in three dimensions (1996)

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Venue: | SIAM Journal on Computing |

Citations: | 7 - 5 self |

### BibTeX

@ARTICLE{Agarwal96rayshooting,

author = {Pankaj K. Agarwal and Micha Sharir},

title = {Ray shooting amidst convex polyhedra and polyhedral terrains in three dimensions},

journal = {SIAM Journal on Computing},

year = {1996}

}

### Years of Citing Articles

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### Abstract

Abstract. 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):+) preprocessing time and storage, and can answer ray-shooting queries in O(log n) time. This is a significant improvement over previously known techniques (which require O(n4+) 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 TM) space and preprocessing, and answers queries in time O(ml/4nl/2+e), again a significant improvement over previous techniques when m << n.

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Citation Context ...ch triangle in F* has one vertical edge (i.e., parallel to the y-axis) and two nonvertical edges. Let I denote the set of the x-projections of triangles in F*. We construct a segment tree B on I; see =-=[30]-=- for details on segment trees. Every node v of B is associated with an interval 3v and stores a "canonical" subset I (v) of I, where each interval in I (v) contains v (but does not contain p(, where p... |

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Citation Context ...ject off intersected by a query ray. The ray-shooting problem has received much attention in the last few years because of its applications in computer graphics and other geometric problems [1], [3], =-=[4]-=-, [5], [6], [9], [10], [14], [17], [21], [28]. Most of the work to date studies the planar case, where F is a collection of line segments in 2. Chazelle and Guibas proposed an optimal algorithm for th... |

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Citation Context ...e that lUll, Itzxl _< st. We map the lines containing the edges of Uzx to their PRicker hyperplanes in 5 and preprocess their lower envelope for point-location queries, using an algorithm of Clarkson =-=[19]-=-. That is, we preprocess the hyperplanes into a data structure, so that we can quickly determine whether a query point lies below all hyperplanes. Similarly, we map the edges of L x to their PRicker h... |

110 | A deterministic view of random sampling and its use in geometry - Chazelle, Friedman - 1990 |

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86 |
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Citation Context ...le polygon [17]. Their algorithm answers a ray-shooting query in O (log n) time using O (n) space; simpler algorithms, with the same asymptotic performance bounds, were recently developed in [14] and =-=[22]-=-. If F is a collection of arbitrary segments in the plane, the best-known algorithm answers a ray-shooting query in time O( log (1) n) using O(S l+e) space and preprocessing [1], [6], [9], where s is ... |

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Citation Context ...nd [28] for more details. We remark that no nontrivial lower bounds are known for the three-dimensional problem as well, although such bounds are known for the related simplex range-searching problem =-=[12]-=-, which is used as a subprocedure in the solutions just mentioned. The performance of these algorithms is rather inefficient when n is large, so a natural objective is to find special cases where this... |

31 | Ray shooting and other applications of spanning trees with low stabbing number - Agarwal - 1992 |

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Citation Context ... by a query ray. The ray-shooting problem has received much attention in the last few years because of its applications in computer graphics and other geometric problems [1], [3], [4], [5], [6], [9], =-=[10]-=-, [14], [17], [21], [28]. Most of the work to date studies the planar case, where F is a collection of line segments in 2. Chazelle and Guibas proposed an optimal algorithm for the special case where ... |

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21 |
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Citation Context ...st object off intersected by a query ray. The ray-shooting problem has received much attention in the last few years because of its applications in computer graphics and other geometric problems [1], =-=[3]-=-, [4], [5], [6], [9], [10], [14], [17], [21], [28]. Most of the work to date studies the planar case, where F is a collection of line segments in 2. Chazelle and Guibas proposed an optimal algorithm f... |

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18 |
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Citation Context ...ime if r is constant [11], [23]. We triangulate the arrangement 4(R). Let E denote the simplices of the triangulation that intersect the PRicker surface. By a result of Aronov, Pellegrini, and Sharir =-=[8]-=-, the number of simplices in E is O(r 4 log 5 r). By construction, each simplex in ,E intersects at most t/r hyperplanes of H. For each A 6 F,, let H/ H denote the set of hyperplanes that intersect th... |

17 | Efficient partition trees, Discrete Comput - Matoušek - 1992 |

14 |
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Citation Context ...he ray-shooting problem has received much attention in the last few years because of its applications in computer graphics and other geometric problems [1], [3], [4], [5], [6], [9], [10], [14], [17], =-=[21]-=-, [28]. Most of the work to date studies the planar case, where F is a collection of line segments in 2. Chazelle and Guibas proposed an optimal algorithm for the special case where F is the boundary ... |

12 | Applications of a new space partitioning technique, Discrete Comput - AGARWAL, SHARIR - 1993 |

12 | Finding stabbing lines in 3-space
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Citation Context ...sions is the so-called stabbing problem, where one wants to determine whether a query line intersects all polyhedra. This problem seems to be easier than the ray-shooting problem: Pellegrini and Shor =-=[29]-=- have described a data structure of size O(n 2+e) that can answer a stabbing query in O (log n) time. We will first describe, in 2, the overall structure of the algorithm. We next present, in 3, an al... |

8 | Good splitters with applications to ray shooting - YEHUDA, FOGEL - 1990 |

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8 |
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Citation Context ...tersecting more than IHl/r hyperplanes of H intersects a hyperplane of R; it is well known that there exists such an R with the prescribed size.) R can be computed in O(t) time if r is constant [11], =-=[23]-=-. We triangulate the arrangement 4(R). Let E denote the simplices of the triangulation that intersect the PRicker surface. By a result of Aronov, Pellegrini, and Sharir [8], the number of simplices in... |

3 |
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Citation Context ... similarly modify the algorithm of Agarwal and Sharir [6], so that the overall query time of the first data structure, summed over all canonical subsets of E*, also reduces to O (log n); see [10] and =-=[27]-=- for details. A similar analysis shows that the time spent in querying 2(79) is O(log 2 n) if r is chosen to be a constant, and that it can be improved to O (log n) by choosing r n, modifying the stru... |

3 |
Ray shooting in triangles in 3-space
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Citation Context ...-shooting problem has received much attention in the last few years because of its applications in computer graphics and other geometric problems [1], [3], [4], [5], [6], [9], [10], [14], [17], [21], =-=[28]-=-. Most of the work to date studies the planar case, where F is a collection of line segments in 2. Chazelle and Guibas proposed an optimal algorithm for the special case where F is the boundary of a s... |

2 |
shooting amidst convex polygons in 2D
- Ray
- 1996
(Show Context)
Citation Context ...also present another algorithm that answers a query in time O (m 1/4n 1/2+) using O (n +) space and preprocessing, so it matches the bound of [5] when rn n, but is considerably faster when m << n. In =-=[7]-=- we have presented an algorithm to preprocess a collection of rn convex polygons in the plane, with a total of n vertices, into a data structure of size O(mn log m), so that a ray-shooting query can b... |

2 |
Variations on ray shooting
- YEHUDA, FOGEL
- 1994
(Show Context)
Citation Context ...ected by a query ray. The ray-shooting problem has received much attention in the last few years because of its applications in computer graphics and other geometric problems [1], [3], [4], [5], [6], =-=[9]-=-, [10], [14], [17], [21], [28]. Most of the work to date studies the planar case, where F is a collection of line segments in 2. Chazelle and Guibas proposed an optimal algorithm for the special case ... |

1 | Ray shooting amidst convex polygons - Agarwal, Sharir - 1993 |

1 | An optimal computing convex hull algorithm and new results on cuttings - Chazelle - 1991 |

1 |
Determining the separation ofpreprocessed polyhedra: A uni3ed approach
- DOBKIN, KIRKPATRICK
- 1991
(Show Context)
Citation Context ...ngles are the faces of a convex polyhedron, then an optimal algorithm, with O(log n) query time and linear space, can be obtained using the hierarchical decomposition scheme of Dobkin and Kirkpatrick =-=[20]-=-. If the triangles form a polyhedral terrain (a piecewise-linear surface intersecting every vertical line in ex*Received by the editors February 16, 1993; accepted for publication (in revised form) Ju... |

1 |
searching with efficient hierarchical cuttings, Discrete Comput
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(Show Context)
Citation Context .... The maximum number of simplices intersected by a hyperplane is called the crossing number of 1-I. Matougek has shown that there exists a simplicial r-partition with crossing number O(r 1-1/d) [25], =-=[26]-=-. Agarwal and Matougek [5] proved that if S lies on an algebraic surface0 PANKAJ K. AGARWAL AND MICHA SHARIR of some fixed degree, then the crossing number can be improved to O(r1-1 logd/(d-l r). Mor... |