Algorithms pp 241-250 | Cite as

Spatial point location and its applications

  • Xue-Hou Tan
  • Tomio Hirata
  • Yasuyoshi Inagaki
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 450)


This paper considers the problem of locating a point in a polyhedral subdivision of the space defined by planar polygonal faces. A persistent form of binary-binary search tree is presented so that the point location problem can be solved in O(log N) query time and O(N+K) space, where N is the total number of edges and K the edge intersections in the image plane. The persistent structure also gives new better solutions for many other geometric problems.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    J.L. Bentley and D. Wood, An optimal worst-case algorithm for reporting intersections of rectangles, IEEE Trans. on Comput. C-29(1980), 571–577.Google Scholar
  2. [2]
    B. Chazelle, How to search in history, Inform. Control 64(1985), 77–99.CrossRefGoogle Scholar
  3. [3]
    B.Chazelle and H.Edelsbrunner, An optimal algorithm for intersecting line segments in the plane, in Proceedings, 29th Annu. IEEE Symp. Found. of Comput. Sci.(1988), pp. 590–600.Google Scholar
  4. [4]
    B.Chazelle and J.Friedman, A deterministic view of random sampling and its use in geometry, in Proceedings, 29th Annu. IEEE Symp. Found. of Comput. Sci.(1988), pp. 539–548.Google Scholar
  5. [5]
    K.L. Clarkson, New applications of random sampling in computational geometry, Discrete Comput. Geometry 2(1987), 195–222.Google Scholar
  6. [6]
    R. Cole, Searching and storing similar lists, J. Algorithms 7(1986), 202–220.CrossRefGoogle Scholar
  7. [7]
    P.Dietz and D.D.Sleator, Two algorithms for maintaining order in a list, in Proceedings, 19th Annu. ACM Symp. Theory of Computing(1987), pp. 365–372.Google Scholar
  8. [8]
    D. Dobkin and R.J. Lipton, Multidimensional search problems, SIAM J. Comput. 5(1976), 181–186.CrossRefGoogle Scholar
  9. [9]
    J.R. Driscoll, N. Sarnak, D.D. Sleator and R.T. Tarjan, Making data structures persistent, J. Comput. Sys. Sci. 38(1989), 86–124.CrossRefGoogle Scholar
  10. [10]
    H.Edelsbrunner, Algorithms in Combinatorial Geometry, Springer-Verlag, 1987.Google Scholar
  11. [11]
    H. Edelsbrunner, H.A. Maurer and D.G. Kirkpatrick, Polygonal intersection searching, Inform. Process. Lett. 14(1982), 74–77.CrossRefGoogle Scholar
  12. [12]
    H. Edelsbrunner, M.H. Overmars and R. Seidel, Some methods of computational geometry applied to computer graphics, Comput. Vision Graphics Image Process. 28(1984), 92–108.CrossRefGoogle Scholar
  13. [13]
    L.J.Guibas and R.Sedgewick, A dichromatic framework for balanced trees, in Proceedings, 19th Annu. IEEE Symp. Found. of Comput. Sci.(1978), pp. 8–21.Google Scholar
  14. [14]
    M. McKenna, Worst case optimal hidden surface removal, ACM Trans. Graphics 6, 1987, 19–28.CrossRefGoogle Scholar
  15. [15]
    J. Nivergelt and F.P. Preparata, Plane sweep algorithms for intersecting geometric figures, Comm. ACM 25(1982), 739–747.CrossRefGoogle Scholar
  16. [16]
    O. Nurmi, A fast line-sweep algorithm for hidden line elimination, BIT 25(1985), 466–472.MathSciNetGoogle Scholar
  17. [17]
    O. Nurmi, On translating a set of objects in 2-and 3-dimensional space, Comput. Vision Graphics Image Process. 36(1986), 42–52.Google Scholar
  18. [18]
    F.P.Preparata and M.I.Shamos, Computational Geometry, Springer-Verlag, 1985.Google Scholar
  19. [19]
    F.P.Preparata and R.Tamassia, Fully dynamic techniques for point location and transitive closure in planar structures, in Proceedings, 29th Annu. IEEE Symp. Found. of Comput. Sci.(1988), pp. 558–567.Google Scholar
  20. [20]
    F.P. Preparata and R. Tamassia, Fully dynamic point location in a monotone subdivision, SIAM J. Comput. 18(1989), 811–830.CrossRefGoogle Scholar
  21. [21]
    F.P. Preparata and R. Tamassia, Efficient spatial point location, in Algorithms and Data Structures (WADS'89), Lect. Notes in Comput. Sci. 382, Springer-Verlag, 1989, pp. 3–11.Google Scholar
  22. [22]
    N. Sarnak and R.E. Tarjan, Planar point location using persistent search trees, Comm. ACM 29(1986), 669–679.MathSciNetGoogle Scholar
  23. [23]
    A.Schmitt, H.Müller and W.Leister, Ray tracing algorithms — theory and practice, Proc. NATO Advanced Study Inst. Theoret. Found. Comput. Graphics and CAD, Springer-Verlag, 1987, pp. 997–1029.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1990

Authors and Affiliations

  • Xue-Hou Tan
    • 1
  • Tomio Hirata
    • 1
  • Yasuyoshi Inagaki
    • 1
  1. 1.Faculty of EngineeringNagoya UniversityNagoyaJapan

Personalised recommendations