Advertisement

Ray-shooting and isotopy classes of lines in 3-dimensional space

  • Marco Pellegrini
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 519)

Abstract

A uniform approach to problems involving lines in 3-dimensional space is presented. This approach is based on mapping lines in R3 into points and hyperplanes in 5-dimensional projective space (Plücker space). We improve previously known results on the following problems:
  1. 1.

    Preprocess n triangles so as to efficiently answer the query: “Given a ray, which is the first triangle hit?” (Ray-shooting problem). We discuss the ray-shooting problem for both disjoint and non-disjoint triangles and several space/time trade offs.

     
  2. 2.

    Efficiently detect the first face hit by any ray in a set of axis-oriented polyhedra.

     
  3. 3.

    Preprocess n lines (segments) so as to efficiently answer the query “Given 2 lines, is it possible to move one into the other without crossing any of the initial lines (segments)?” (Isotopy problem). If the movement is possible produce an explicit representation of it.

     
  4. 4.

    Construct the arrangement generated by n intersecting triangles in 3-space in an output-sensitive way, with a subquadratic overhead term.

     
  5. 5.

    Count the number of pairs of intersecting lines in a set of n lines in space.

     

Keywords

Computational Geometry Query Point Query Time Hyperplane Arrangement Isotopy Class 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [AMS91]
    B. Aronov, J. Matoušek, and M. Sharir. On the sum of squares of cell complexities in hyperplane arrangements. To appear in the Proceedings of the 7th ACM Symposium on Computational Geometry, 1991.Google Scholar
  2. [AS91a]
    P. K. Agarwal and M. Sharir. Applications of a new space partitioning technique. To appear in the Proceedings of the 1991 Workshop on Algorithms and Data Structures, 1991.Google Scholar
  3. [AS91b]
    B. Aronov and M. Sharir. On the zone of a surface in an hyperplane arrangement. To appear in the Proceedings of the 1991 Workshop on Algorithms and Data Structures, 1991.Google Scholar
  4. [CE88]
    B. Chazelle and H. Edelsbrunner. An optimal algorithm for intersecting line segments in the plane. In Proc. of the 29th Ann. Symp. on Foundations of Computer Science, 1988.Google Scholar
  5. [CEGS89a]
    B. Chazelle, H. Edelsbrunner, 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.Google Scholar
  6. [CEGS89b]
    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, pages 179–193, 1989. In Springer Verlag LNCS 372.Google Scholar
  7. [Cha85]
    B. Chazelle. Fast searching in a real algebraic manifold with applications to geometric complexity, volume 185 of Lecture notes in computer science, pages 145–156. Springer Verlag, 1985.Google Scholar
  8. [Cla87]
    K.L Clarkson. New applications of random sampling in computational geometry. Discrete Computational Geometry, (2):195–222, 1987.Google Scholar
  9. [CSW90]
    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.Google Scholar
  10. [dBHO+91]
    [dBHO+91] M. de Berg, D. Halperlin, M. Overmars, J. Snoeyink, and M. van Kreveld. Efficient ray-shooting and hidden surface removal. To appear in Procedings of the 7th Annual Symposium on Computational Geometry, 1991.Google Scholar
  11. [DK90]
    D. Dobkin and D. Kirkpatrick. Determining the separation of preprocessed polyhedra: a unified approach. In Proc. of the 17th International Colloqium on Automata, Languages and Programming, pages 400–413, 1990.Google Scholar
  12. [EGS88]
    H. Edelsbrunner, L. Guibas, and M. Sharir. The complexity of many faces in arrangements of lines and segments. In Proceedings of the 4th ACM Symposium on Computational Geometry, pages 44–55, 1988.Google Scholar
  13. [EMP+82]
    H. Edelsbrunner, H. Mauer, F. Preparata, E. Welzl, and D. Wood. Stabbing line segments. BIT, (22):274–281, 1982.Google Scholar
  14. [GOS89]
    L. Guibas, M. Overmars, and M. Sharir. Ray shooting, implicit point location and related queries in arrangements of segments. Technical Report TR433, Courant Institute, 1989.Google Scholar
  15. [Mat90]
    J. Matouěk. Cutting hyperplane arrangements. In Proceedings of the 6th ACM Symposium on Computational Geometry, pages 1–9, 1990.Google Scholar
  16. [MO88]
    M. McKenna and J. O'Rourke. Arrangements of lines in 3-space: A data structure with applications. In Proceedings of the 4th annual Symposium on Computational Geometry, pages 371–380, 1988.Google Scholar
  17. [Pel90]
    M. Pellegrini. Stabbing and ray shooting in 3 dimensional space. In Proceedings of the 6th ACM Symposium on Computational Geometry, pages 177–186, 1990.Google Scholar
  18. [Pel91]
    M. Pellegrini. Combinatorial and Algorithmic Analysis of Stabbing and Visibility Problems in 3-Dimensional Space. PhD thesis, New York University-Courant Institute of Mathematical Sciences, February 1991.Google Scholar
  19. [PS91]
    M. Pellegrini and P. Shor. Finding stabbing lines in 3-dimensional space. In Proceedings of the Second SIAM-ACM Symposium on Discrete Algorithms, 1991.Google Scholar
  20. [Som51]
    D. M. H. Sommerville. Analytical geometry of three dimensions. Cambridge, 1951.Google Scholar
  21. [SS83]
    J.T. Schwartz and M. Sharir. On the piano mover's problem: II. General techniques for computing topological properties of real algebraic manifolds. Adv. in Appl. Math, 4:298–351, 1983.Google Scholar
  22. [Sto89]
    J. Stolfi. Primitives for computational geometry. Technical Report 36, Digital SRC, 1989.Google Scholar
  23. [Tar83]
    R.E. Tarjan. Data Structures and Network Algorithms, volume 44 of CBMS-NSF Regional Conference Series in Applied Mathematics. SIAM, 1983.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1991

Authors and Affiliations

  • Marco Pellegrini
    • 1
  1. 1.Courant InstituteNew York UniversityN.Y.

Personalised recommendations