Skip to main content
SpringerLink
Log in

Lines in space: Combinatorics and algorithms

  • Published:
Algorithmica Aims and scope Submit manuscript

Abstract

Questions about lines in space arise frequently as subproblems in three-dimensional computational geometry. In this paper we study a number of fundamental combinatorial and algorithmic problems involving arrangements ofn lines in three-dimensional space. Our main results include:

  1. 1.

    A tight Θ(n 2) bound on the maximum combinatorial description complexity of the set of all oriented lines that have specified orientations relative to then given lines.

  2. 2.

    A similar bound of Θ(n 3) for the complexity of the set of all lines passing above then given lines.

  3. 3.

    A preprocessing procedure usingO(n 2+ɛ) time and storage, for anyε>0, that builds a structure supportingO(logn)-time queries for testing if a line lies above all the given lines.

  4. 4.

    An algorithm that tests the “towering property” inO(n 2+ɛ) time, for anyε>0; don given red lines lie all aboven given blue lines?

The tools used to obtain these and other results include Plücker coordinates for lines in space andε-nets for various geometric range spaces.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. P. Agarwal, Geometric partitioning and its applications, inDiscrete and Computational Geometry: Papers from the DIMACS Special Year, American Mathematical Society, Providence, RI, 1991.

    Google Scholar 

  2. B. Aronov, M. Pellegrini, and M. Sharir, On the zone of a surface in a hyperplane arrangement, In preparation. See alsoProc. 2nd WADS Workshop, 1991, Lecture Notes in Computer Science, Vol. 519, Springer-Verlag, Berlin, pp. 13–19, for an early version by two of the authors.

    Google Scholar 

  3. O. Bottema and B. Roth,Theoretical Kinematics, Dover, New York, 1990.

    Google Scholar 

  4. B. Chazelle, H. Edelsbrunner, L. Guibas, R. Pollack, R. Seidel, M. Sharir, and J. Snoeyink, Counting and cutting cycles of lines and line segments in space,Proc. 31st IEEE Symp. on Foundations of Computer Science, 1990, pp. 242–251.

  5. B. Chazelle, H. Edelsbrunner, L. Guibas, and M. Sharir, Algorithms for bichromatic line segment problems and polyhedral terrains, In preparation. An earlier version of this paper, combined with the current paper, has appeared inProc. 21st ACM Symp. on Theory of Computing, 1989, pp. 382–393.

  6. B. Chazelle, H. Edelsbrunner, L. Guibas, and M. Sharir, A singly-exponential stratification scheme for real semi-algebraic varieties and its applications,Proc. 16th ICALP Colloq., Lectures Notes in Computer Science, Vol. 372, Springer-Verlag, Berlin, 1989, pp. 179–193.

    Google Scholar 

  7. K. Clarkson, H. Edelsbrunner, L. Guibas, M. Sharir, and E. Welzl, Combinatorial complexity bounds for arrangements of curves and surfaces,Proc. 29th IEEE Symp. on Foundations of Computer Science, 1988, pp. 568–579.

  8. B. Chazelle and J. Friedman, A deterministic view of random sampling and its use in geometry,Combinatorica,10 (1990), 229–249.

    Google Scholar 

  9. B. Chazelle, An optimal convex hull algorithm and new results on cuttings,Proc. 32nd Symp. on Foundations of Computer Science, 1991, pp. 29–38.

  10. K. Clarkson, New applications of random sampling in computational geometry,Discrete Comput. Geom.,2 (1987), 195–222.

    Google Scholar 

  11. K. Clarkson, A probabilistic algorithm for the post office problem,Proc. 17th ACM Symp. on Theory of Computing, 1985, pp. 175–184.

  12. K. Clarkson, Applications of random sampling in computational geometry,Discrete Comput. Geom.,4 (1989), 387–421.

    Google Scholar 

  13. H. Edelsbrunner,Algorithms in Combinatorial Geometry, Springer-Verlag, Heidelberg, 1987.

    Google Scholar 

  14. H. Edelsbrunner, L. Guibas, and M. Sharir, The complexity of many faces in arrangements of lines and of segments,Proc. 4th ACM Symp. on Computational Geometry, 1988, pp. 44–55.

  15. H. Edelsbrunner, L. Guibas, and M. Sharir, The complexity of many faces in arrangements of planes and related problems,Discrete Comput. Geom.,5 (1990), 197–216.

    Google Scholar 

  16. G. Fano and A. Terracini,Lezioni di Geometria Analitica e Proiettiva, Paravia, 1939.

  17. Z. Gigus, J. Canny, and R. Seidel, Efficiently computing and representing aspect graphs of polyhedral objects, Technical Report UCB/CSD 88/432, Univesrity of California at Berkeley, August 1988.

    Google Scholar 

  18. W. Hodge and D. Pedoe,Methods of Algebraic Geometry, Cambridge University Press, Cambridge, 1952.

    Google Scholar 

  19. K. Hunt,Kinematic Geometry of Mechanisms, Oxford, 1990.

  20. D. Haussler and E. Welzl, Epsilon nets and simplex range searching,Discrete Comput. Geom.,2 (1987), 127–151.

    Google Scholar 

  21. A. Inselberg,N-dimensional graphics. Part I —lines and hyperplanes. Scientific Center Report G320-2711, IBM, Los Angeles, CA, 1981.

    Google Scholar 

  22. C. M. Jessop,A Treatise on the Line Complex, Chelsea, New York, 1969.

    Google Scholar 

  23. J. Matoušek, Construction ofε-nets,Discrete Comput. Geom.,5 (1990), 427–448.

    Google Scholar 

  24. J. Matoušek, Cutting hyperplane arrangements,Discrete Comput. Geom.,6 (1991), 385–406.

    Google Scholar 

  25. J. Matoušek, Approximations and optimal geometric divide-and-conquer,Proc. 23rd Annual ACM Symp. on Theory of Computing, 1991, pp. 506–511.

  26. M. McKenna, Arrangements of lines in 3-space: a data structure with applications, Ph.D. Dissertation, Johns Hopkins University, 1989.

  27. J. Milnor. On the Betti numbers of real varieties,Proc. Amer. Math. Soc.,15 (1964), 275–280.

    Google Scholar 

  28. M. McKenna and J. O'Rourke, Arrangements of lines in 3-space: a data structure with applications,Proc. 4th ACM Symp. on Computational Geometry, 1988, pp. 371–380.

  29. M. Paterson, Personal communication.

  30. W. H. Plantinga and C. R. Dyer, An algorithm for constructing the aspect graph,Proc. 27th IEEE Symp. on Foundations of Computer Science, 1986, pp. 123–131.

  31. J. Plücker, On a new geometry of space,Philos. Trans. Roy. Soc. London,155 (1865), 725–791.

    Google Scholar 

  32. J. Pach, R. Pollack, and E. Welzl, Weaving patterns of lines and line segments,Proc. 1st SIGAL Symp., Lecture Notes in Computer Science, Vol. 450, Springer-Verlag, Berlin, 1990, pp. 437–446.

    Google Scholar 

  33. R. Seidel, Constructing higher dimensional convex hulls at logarithmic cost per face,Proc. 18th ACM Symp. on Theory of Computing, 1986, pp. 404–413.

  34. M. Sharir, Onk-sets in arrangements of curves and surfaces,Discrete Comput. Geom.,6 (1991), 593–613.

    Google Scholar 

  35. M. Sharir, On joints in arrangements of lines in space and related problems, In preparation.

  36. D. M. Y. Sommerville,Analytical Geometry of Three Dimensions, Springer-Verlag, Heidelberg, 1987.

    Google Scholar 

  37. J. Stolfi,Oriented Projective Geometry, Academic Press, New York, 1991.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Computer Science, Princeton University, 08544, Princeton, NJ, USA

    B. Chazelle

  2. Department of Computer Science, University of Illinois at Urbana-Champaign, 61801, Urbana, IL, USA

    H. Edelsbrunner

  3. Stanford University, 94305, Stanford, CA, USA

    L. J. Guibas

  4. DEC Systems Research Center, 94301, Palo Alto, CA, USA

    L. J. Guibas & J. Stolfi

  5. Tel Aviv University, 69 978, Ramat Aviv, Israel

    M. Sharir

  6. New York University, 10012, New York, NY, USA

    M. Sharir

Authors
  1. B. Chazelle
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. H. Edelsbrunner
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. L. J. Guibas
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. M. Sharir
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. J. Stolfi
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Communicated by D. P. Dobkin.

Work on this paper by Bernard Chazelle has been supported by NSF Grant CCR-87-00917. Work on this paper by Herbert Edelsbrunner has been supported by NSF Grant CCR-87-14565. Work on this paper by Leonidas Guibas has been supported by grants from the Mitsubishi and Toshiba Corporations. Work on this paper by Micha Sharir has been supported by ONR Grant N00014-87-K-0129, by NSF Grants DCR-83-20085 and CCR-89-01484, and by grants from the U.S.-Israeli Binational Science Foundation, the NCRD — the Israeli National Council for Research and Development, and the EMET Fund of the Israeli Academy of Sciences.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Chazelle, B., Edelsbrunner, H., Guibas, L.J. et al. Lines in space: Combinatorics and algorithms. Algorithmica 15, 428–447 (1996). https://doi.org/10.1007/BF01955043

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01955043

Key words

Search

Navigation