## An optimal algorithm for intersecting three-dimensional convex polyhedra (1992)

Venue: | SIAM J. Comput |

Citations: | 66 - 5 self |

### BibTeX

@ARTICLE{Chazelle92anoptimal,

author = {Bernard Chazelle},

title = {An optimal algorithm for intersecting three-dimensional convex polyhedra},

journal = {SIAM J. Comput},

year = {1992},

volume = {21},

pages = {586--591}

}

### Years of Citing Articles

### OpenURL

### Abstract

Abstract. This paper describes a linear-time algorithm for computing the intersection of two convex polyhedra in 3-space. Applications of this result to computing intersections, convex hulls, and Voronoi diagrams are also given. Key words, computational geometry, convex polyhedra AMS(MOS) subject classifications. 68Q25, 68H05 1. Introduction. Giventwo

### Citations

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Algorithms in Combinatorial Geometry
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Citation Context ...ons for intersecting convex polygons are given in Shamos and Hoey [27] and O’Rourke et al. [23]. For additional background material on polyhedral intersections, the reader should consult Edelsbrunner =-=[11]-=-, Mehlhorn [20], and Preparata and Shamos [25]. Our main result is an algorithm for constructing the intersection between two convex polyhedra in linear time. The algorithm does not use any complicate... |

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Introduction to piecewise-linear topology, Springer-Verlag Ed
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Citation Context ...h face is the interior of a simplex (in the relative topology of its affine closure). We take a rather general view of a polyhedron as any subset of 3-space that is locally a cone with a compact base =-=[26]-=-. Given a pointp 3 and a subset Cc3, we say that the points ap+(1-a)q, for all q C and 0-< a-< 1, form a cone pC if for each point ofpC distinct from p, the choice of q is unique. A subset P of3 is ca... |

205 |
Linear-time algorithms for linear programming in R 3 and related problems
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Citation Context ...convex polyhedra intersect can be done in a linear number of operations. By stating the problem as a linear program over three variables, other linear-time algorithms originate in the works ofMegiddo =-=[19]-=- and Dyer 10]. Previous results also include an efficient algorithm for intersecting two polyhedra, one ofwhich is convex (Mehlhorn and Simon [21]). Optimal solutions for intersecting convex polygons ... |

132 |
Data Structures and Algorithms 3: Multidimensional Searching and Computational Geometry, volume 3
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(Show Context)
Citation Context ...cting convex polygons are given in Shamos and Hoey [27] and O’Rourke et al. [23]. For additional background material on polyhedral intersections, the reader should consult Edelsbrunner [11], Mehlhorn =-=[20]-=-, and Preparata and Shamos [25]. Our main result is an algorithm for constructing the intersection between two convex polyhedra in linear time. The algorithm does not use any complicated data structur... |

89 |
Primitives for the Manipulation of Three-Dimensional Subdivisions
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Citation Context ... all coplanar. For storing two- and three-dimensional cell complexes we shall assume the representations of Baumgart [3], Muller and Preparata [22], or Guibas and Stolfi [15] and of Dobkin and Laszlo =-=[8]-=-, respectively, or any other data structures that allow us to navigate at ease between adjacent cells. Such representations will be called standard. To conclude this laundry list of assumptions and de... |

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Finding the intersection of two convex polyhedra
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Citation Context ...onvex polyhedra AMS(MOS) subject classifications. 68Q25, 68H05 1. Introduction. Giventwo convex polyhedra in 3-space,how fast canwe compute their intersection? Over a decade ago, Muller and Preparata =-=[22]-=- gave the first efficient solution to this problem by reducing it to a combination of intersection detection and convex hull computation. Another route was followed in 1984 by Hertel et al., who solve... |

72 |
A geometric consistency theorem for a symbolic perturbation scheme
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Citation Context ... always to choose the triangle that is (locally) highest (or leftmost if there are several highest ones). We can also submit the ray to a symbolic perturbationsee Edelsbrunner and Mficke [13] and Yap =-=[28]-=-. Note that we can easily generalize the mode of traversal to polygonal lines embedded in 3-space. The following summarizes our discussion. LEMMA 2.2. The complexity oftraversing theprimal (respective... |

44 |
A linear time algorithm for computing the voronoi diagram of a convex polygon
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(Show Context)
Citation Context ...ecomes a special case of intersecting two convex polyhedra. Applications include computing the Voronoi diagram of a polygon (Kirkpatrick [17]) and of the vertices of a convex polygon (Aggarwal et al. =-=[1]-=-). 5. Conclusions. Our main result is a linear-time algorithm for intersecting two convex polyhedra in 3-space. Whether the algorithm lends itselfto efficient and robust implementations remains to be ... |

40 |
New linear algorithm for intersecting convex polygons
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Citation Context ...ent algorithm for intersecting two polyhedra, one ofwhich is convex (Mehlhorn and Simon [21]). Optimal solutions for intersecting convex polygons are given in Shamos and Hoey [27] and O’Rourke et al. =-=[23]-=-. For additional background material on polyhedral intersections, the reader should consult Edelsbrunner [11], Mehlhorn [20], and Preparata and Shamos [25]. Our main result is an algorithm for constru... |

18 |
Efficient uses of the past
- Dobkin, Munro
- 1985
(Show Context)
Citation Context ...lem, however, remained elusive. The different but related problem of detecting whether two convex polyhedra intersect, by using preprocessing, was studied by Chazelle and Dobkin [5], Dobkin and Munro =-=[9]-=-, and Dobkin and Kirkpatrick [6]. More germane to our concerns here is the off-line version ofthe detection problem. Dobkin and Kirkpatrick [7] have shown that detecting whether two convex polyhedra i... |

13 |
Triangles in space or building (and analyzing) castles in the air
- Aronov, Sharir
- 1988
(Show Context)
Citation Context ...An outstanding open problem is that of intersecting two nonconvex polyhedra efficiently. The problem of intersecting arbitrarily placed triangles in 3-space has been investigated by Aronov and Sharir =-=[2]-=-. How much we can gain by having collections of faces structured into the boundaries of simple polyhedra is an intriguing open question. Acknowledgments. I wish to thank Herbert Edelsbrunner, David Ki... |

11 |
Intersecting two polyhedra one of which is convex
- Mehlhorn, Simon
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(Show Context)
Citation Context ...near-time algorithms originate in the works ofMegiddo [19] and Dyer 10]. Previous results also include an efficient algorithm for intersecting two polyhedra, one ofwhich is convex (Mehlhorn and Simon =-=[21]-=-). Optimal solutions for intersecting convex polygons are given in Shamos and Hoey [27] and O’Rourke et al. [23]. For additional background material on polyhedral intersections, the reader should cons... |

10 |
Finding Extreme Points in Three Dimensions and Solving the Post-Office Problem in the Plane
- Edelsbrunner, Maurer
- 1984
(Show Context)
Citation Context ...00917. Department of Computer Science, Princeton University, Princeton, New Jersey 08544. 671s672 BERNARD CHAZELLE polyhedron. Further applications of that versatile data structure have been given in =-=[12]-=-, [21]. The representation can be seen as a specialization of Kirkpatrick’s point location structure [18]. A convex polyhedron P of n vertices is made the first element of a descending chain of O(log ... |

7 |
Space Sweep Solves Intersection of Convex Polyhedra
- Hertel, Mantyla, et al.
- 1984
(Show Context)
Citation Context ...to this problem by reducing it to a combination of intersection detection and convex hull computation. Another route was followed in 1984 by Hertel et al., who solved the problem by using space sweep =-=[16]-=-. In both cases, a running time of (R)(n log n) was achieved, where n is the combined number of vertices in the two polyhedra. Resolving the true complexity of the problem, however, remained elusive. ... |

5 | Voronoi diagrams and arrangements, Discrete Comput - Edelsbrunner, Seidel - 1986 |

2 |
Intersection ofconvex objects in two and three dimensions
- CHAZELLE, DOBKIN
- 1987
(Show Context)
Citation Context ...complexity of the problem, however, remained elusive. The different but related problem of detecting whether two convex polyhedra intersect, by using preprocessing, was studied by Chazelle and Dobkin =-=[5]-=-, Dobkin and Munro [9], and Dobkin and Kirkpatrick [6]. More germane to our concerns here is the off-line version ofthe detection problem. Dobkin and Kirkpatrick [7] have shown that detecting whether ... |

1 |
BAUMGART,A polyhedron representationfor computer vision
- G
- 1975
(Show Context)
Citation Context ...date for practical implementation. As is customary, our result assumes that the input conforms with any one of the standard (linear-time equivalent) polyhedral representations given in the literature =-=[3]-=-, [15], [22]. This is not a minor point, because nonstandard representations can easily make the problem more difficult. (For example, think how much more difficult the problem would be if we were giv... |

1 |
Divide and conquerfor linear expected time
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- 1978
(Show Context)
Citation Context ...rged inM (-,,.-."-,,-k) ways, wherem rn. The lower bound follows from the fact that logM l)(n log k) and by-now standard algebraic decision-tree arguments [25]. B. CONVEX I-ItJLLS. Bentley and Shamos =-=[4]-=- have shown how to take advantage of certain point distributions to obtain linear expected-time algorithms for computing convex hulls. The idea is to use divide-and-conquer by splitting the input set ... |

1 |
Fast detection ofpolyhedral intersection, Theoret
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(Show Context)
Citation Context ...The different but related problem of detecting whether two convex polyhedra intersect, by using preprocessing, was studied by Chazelle and Dobkin [5], Dobkin and Munro [9], and Dobkin and Kirkpatrick =-=[6]-=-. More germane to our concerns here is the off-line version ofthe detection problem. Dobkin and Kirkpatrick [7] have shown that detecting whether two convex polyhedra intersect can be done in a linear... |

1 |
Linear time algorithmsfor two and three-variable linear programs
- DYER
- 1984
(Show Context)
Citation Context ... brief discussion of what makes the problem not so easy. We can assume that we have a point O inside both convex polytopes P and Q, since such a witness (if any) can be discovered in linear time [7], =-=[10]-=-, [19]. What remains to be done is in some sense merging P and Q. Imagine a sphere centered at O, on which OP and oQ are centrally projected. This gives us two spherical subdivisions Sp and S o. Mergi... |

1 |
Simulation ofsimplicity: a technique to cope with degenerate cases in geometric algorithms
- EDELSBRUNNER, MCKE
- 1988
(Show Context)
Citation Context ...e might agree always to choose the triangle that is (locally) highest (or leftmost if there are several highest ones). We can also submit the ray to a symbolic perturbationsee Edelsbrunner and Mficke =-=[13]-=- and Yap [28]. Note that we can easily generalize the mode of traversal to polygonal lines embedded in 3-space. The following summarizes our discussion. LEMMA 2.2. The complexity oftraversing theprima... |

1 |
Convex hulls offinite sets ofpoints in two and three dimensions
- PREPARATA, HONG
- 1977
(Show Context)
Citation Context ...ding to a parenthesis system. A traversal of such a curve is FIG. 3.5. B may come in and out off repeatedly. A similar situation occurs in the merge step of Preparata and Hong’s convex hull algorithm =-=[24]-=-, which is also discussed in detail in Edelsbrunner [11].s684 BERNARD CHAZELLE FIG. 3.6. A case in which b consists ofa single vertex. shown in Fig. 3.7: The curve can be obtained from a circle by pin... |

1 |
Geometric intersectionproblems
- SHAMOS, HOEY
- 1976
(Show Context)
Citation Context ...ts also include an efficient algorithm for intersecting two polyhedra, one ofwhich is convex (Mehlhorn and Simon [21]). Optimal solutions for intersecting convex polygons are given in Shamos and Hoey =-=[27]-=- and O’Rourke et al. [23]. For additional background material on polyhedral intersections, the reader should consult Edelsbrunner [11], Mehlhorn [20], and Preparata and Shamos [25]. Our main result is... |