Article

Free access

Incidence and nearest-neighbor problems for lines in 3-space

Author:

Marco PellegriniAuthors Info & Claims

SCG '92: Proceedings of the eighth annual symposium on Computational geometry

Pages 130 - 137

https://doi.org/10.1145/142675.142703

Published: 01 July 1992 Publication History

Abstract

In the first part of the paper we solve the problem of detecting efficiently if a query simplex is collison-free among polyhedral obstacles. In order to solve this problem we develop new on-line data structures to detect intersections of query halfplanes with sets of lines and segments.

In the second part we consider the nearest-neighbor problems. Given a set of n lines in 3-space, the shortest vertical segment between any pair of lines is found in randomized expected time O(n^8/5+ε) for every ε > 0. The longest connecting vertical segment is found in time O(n^4/3+ε). The shortest connecting segment is found in time O(n^5/3+ε).

References

[1]

P.K. Agarwal, B. Aronov, M. Sharir, and S. Suri. Selecting distances in the plane. In Proceedings of the 6th A CM Symposium on Computational Geomelry, pages 321-331, 1990.

Digital Library

[2]

P.K. Agarwal. Improved ray-shooting trade-off. Personal communication, 1991.

[3]

P.K. Agarwal and J. Matou~ek. Ray shooting and parametric search. Manuscript. To appear in STOC '92, June 1991.

Digital Library

[4]

B. Aronov and M. Sharir. Triangles in space or building (and analyzing) castles in the air. Combinaiorica, 10(2):137-173, 1990.

[5]

P.K. Agarwal and M. Sharir. Applications of a new space partitioning technique. In Proceedings of the 1991 Workshop on Algorithms and Data Structures, number 519 in Lecture Notes in Computer Science, pages 379-391. Springer Verlag, 1991.

[6]

B. Aronov and M. Sharir. On the complexity of a single cell in an arrangement of (d- 1)-simplices in d-space and its applications. Manuscript, 1991.

[7]

B. Aronov and M. Sharir. On the zone of a surface in a hyperplane arrangement. In Proceedings of the 1991 Workshop on Algorithms and Data Structures, number 519 in Lecture Notes in Computer Science, pages 13-19. Springer Verlag, 1991.

[8]

B. Chazelle, H. Edelsbrunher, L. Guibas, R. Pollack, R. Seidel, M. Sharir, and J. Snoeyink. Counting and cutting circles of lines and rods in space. In Proceedings of the 31st Annual Symposium on Foundations of Computer Science, 1990.

Digital Library

[9]

B. Chazelle, H. Edelsbrunnet, L. Guibas, and M. Sharir. Lines in space: combinatorics, algorithms and applications. In Proc. of the 21st Symposium on Theory of Computing, pages 382-393, 1989.

Digital Library

[10]

B. Chazelle, H. Edelsbrunner, L. Guibas, and M. Sharir. A singly exponential stratification scheme for real semi-algebraic varieties and its applications. In Proceedings of the 16th International Colloquium on Automata, languages and Programming, number 372 in Lecture Notes in Computer Science, pages 179-193. Springer Verlag, 1989.

Digital Library

[11]

B. Chazelle, H. Edelsbrunner, L. Guibas, and M. Sharir. Diameter, width, closest line pair and parametric search. In Proceedings of the 8th ACM Symposium on Computational Geometry, 1992.

Digital Library

[12]

L.P. Chew and K. Kedem. Placing the largest similar copy of a convex polygon among polygonal obstacles. In Proceedings of the 5th A CM Symposium on Computational Geometry, pages 167-174, 1989.

Digital Library

[13]

R. Cole. Slowing down sorting networks to obtain faster sorting algorithms. J. of A CM, 34:200-208, 1987.

Digital Library

[14]

B. Chazelle and M. Sharir. An algorithm for generalized point location and its applications. Technical Report 153, Courant Institute of Mathematical Sciences, Robotics Lab., May 1988.

[15]

B. Chazelle, M. Sharir, and E. Welzl. Quasi-optimal upper bounds for simplex range searching and new zone theorems. In Proceedings of the 6th ACM Symposium on Computational Geometry, pages 23-33, 1990.

Digital Library

[16]

M. de Berg, D. Halperlin, M. Overmars, J. Snoeyink, and M. van Kreveld. Efficient ray-shooting and hidden surface removal, in Proceedings of the 71h A CM Symposium on Computational Geometry, 1991.

Digital Library

[17]

H. Edelsbrunner, L. Guibas, and M. Sharir. The complexity and construction of many faces in arrangements of lines and segments. Discrete Computational Geometry, 5:161- 196, 1990.

Digital Library

[18]

J. Friedman, j. Hershberger, and J. Snoeyink. Compliant motion in a simple polygon. In Proceedings of the 5th A CM Symposium on Computational Geometry, pages 175-186, 1989.

Digital Library

[19]

L.J. Guibas, M. Sharir, and S. Sifrony. On the general motion planning problem with two degrees of freedom. In Proceedings of the jib A CM Symposium on Computational Geometry, pages 289-298, 1988.

Digital Library

[20]

L. Guibas. Personal communicationion. November 1991.

[21]

D. Halperlin. On the complexity of a single cell in certain arrangements of surfaces in 3-space. In Proceedings of the 7th A CM Symposium on Computational Geometry, pages 314-323, 1991.

Digital Library

[22]

D. Halperlin and M. Overmars. Efficient Motion Planning of an L-she.ped Object. In Proceedings of the 5th A CM Symposium on Computational Gee, metry, pages 156-166, 1989.

Digital Library

[23]

K. Kedem and M. Sharir. An automatic motion planning system for a convex polygonal mobile robot in 2- d polygonal space. In Proceedin~'s of the Jth A CM Symposium on Computational Geometry, pages 329-340, 1988.

Digital Library

[24]

D. Leven and M. Sharir. An efficient simple motion planning algorithm for a ladder moving in two-dimensional space amidst polygonal barriers. J. of Algorithms, 8:192-215, 1987.

Digital Library

[25]

D. Leven and M. Sharir. Planning a purely translational motion for a convex object in two-dimensional space using generalized voronoi diagrams. Discrete ~4 Computational Geometry, 2:9-31, 1987.

[26]

J. Matou~ek. Efficient partition trees. In Proceedings of the 7th A CM c posture on Computational Geometry, pages 1-9, 1991.

Digital Library

[27]

N. Megiddo. Applying parallel c. omputation algorithms in the design of sequential algorithms. J. of A CM, 30:852-865, 1983.

Digital Library

[28]

M. Pellegrini. Stabbing and ray shooting in 3-dimensional space. In Proceedings of the 6th A CM Symposium on Computational Geometry, pages 177- 186, 1990.

Digital Library

[29]

M.Pellegrini. Ray shooting and isotopy classes of lines in 3-dimensional space. In Proceedings of the 1991 Structures, number 519 in Lecture Notes in Computer Science, pages 20- 31. Springer Verlag, 1991.

Digital Library

[30]

M. Pellegrini and P. Shor. Finding stabbing lines in 3-space. Discrete Computational Geometry, 6~, 1992. To appear. Preliminary version in Proc. of ACM-SIAM Symp. on Discrete Algorithms 1991.

Digital Library

[31]

J.T. Schwartz and M. SharJir. A survey of motion planning and related geometric algorithms. In D. Kapur and J.L. Mundy, editors, Geometric reasoning, pages 157-169. The MIT press, 1989.

Digital Library

[32]

S. Toledo. Extremal polygon containment problems. In Proceedings of the 7th A CM Symposium on Computational Geometry, pages 176-185, 1991.

Digital Library

Cited By

Gupta PJanardan RSmid M(2006)Fast algorithms for collision and proximity problems involving moving geometric objectsAlgorithms — ESA '9410.1007/BFb0049415(278-289)Online publication date: 23-Feb-2006
https://doi.org/10.1007/BFb0049415
Agarwal PSharir M(2005)Algorithmic techniques for geometric optimizationComputer Science Today10.1007/BFb0015247(234-253)Online publication date: 9-Jun-2005
https://doi.org/10.1007/BFb0015247
Erickson J(1996)New lower bounds for Hopcroft's problemDiscrete & Computational Geometry10.1007/BF0271287516:4(389-418)Online publication date: 1-Apr-1996
https://dl.acm.org/doi/10.1007/BF02712875
Show More Cited By

Index Terms

Incidence and nearest-neighbor problems for lines in 3-space

Recommendations

Lines Pinning Lines

A line ℓ is a transversal to a family F of convex polytopes in ź3 if it intersects every member of F. If, in addition, ℓ is an isolated point of the space of line transversals to F, we say that F is a pinning of ℓ. We show that any minimal pinning of a ...
On lines missing polyhedral sets in 3-space

We show some combinatorial and algorithmic results concerning finite sets of lines and terrains in 3-space. Our main results include:(1)An $$O(n^3 2^{c\sqrt {\log n} } )$$ upper bound on the worst-case complexity of the set of lines that can be ...
Lines in 3-space isotopy, chirality and weavings

Comments

Information & Contributors

Information

Published In

cover image ACM Conferences

SCG '92: Proceedings of the eighth annual symposium on Computational geometry

July 1992

384 pages

ISBN:0897915178

DOI:10.1145/142675

Chairman:
David Avis

Copyright © 1992 ACM.

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

Sponsors

Publisher

Association for Computing Machinery

New York, NY, United States

Publication History

Published: 01 July 1992

Permissions

Request permissions for this article.

Request Permissions

Check for updates

Qualifiers

Article

Conference

SOCG92

Sponsor:

SOCG92: 8th Annual Symposium on Computational Geometry 1992

June 10 - 12, 1992

Berlin, Germany

Acceptance Rates

Overall Acceptance Rate 625 of 1,685 submissions, 37%

Contributors

Other Metrics

View Article Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

5
Total Citations
View Citations
297
Total Downloads

Downloads (Last 12 months)29
Downloads (Last 6 weeks)5

Reflects downloads up to 10 Nov 2024

Other Metrics

View Author Metrics

Citations

Cited By

Gupta PJanardan RSmid M(2006)Fast algorithms for collision and proximity problems involving moving geometric objectsAlgorithms — ESA '9410.1007/BFb0049415(278-289)Online publication date: 23-Feb-2006
https://doi.org/10.1007/BFb0049415
Agarwal PSharir M(2005)Algorithmic techniques for geometric optimizationComputer Science Today10.1007/BFb0015247(234-253)Online publication date: 9-Jun-2005
https://doi.org/10.1007/BFb0015247
Erickson J(1996)New lower bounds for Hopcroft's problemDiscrete & Computational Geometry10.1007/BF0271287516:4(389-418)Online publication date: 1-Apr-1996
https://dl.acm.org/doi/10.1007/BF02712875
Schömer EThiel CSnoeyink J(1995)Efficient collision detection for moving polyhedraProceedings of the eleventh annual symposium on Computational geometry10.1145/220279.220285(51-60)Online publication date: 1-Sep-1995
https://dl.acm.org/doi/10.1145/220279.220285
Pellegrini MYap C(1993)On lines missing polyhedral sets in 3-spaceProceedings of the ninth annual symposium on Computational geometry10.1145/160985.160990(19-28)Online publication date: 1-Jul-1993
https://dl.acm.org/doi/10.1145/160985.160990

View Options

View options

PDF

View or Download as a PDF file.

eReader

View online with eReader.

Get Access

Login options

Check if you have access through your login credentials or your institution to get full access on this article.

Full Access

Get this Publication

Media

Figures

Other

Tables

View Table of Contents