Drawing nice projections of objects in space

  • Prosenjit Bose
  • Pedro Ramos
  • Francisco Gomez
  • Godfried Toussaint
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1027)


Our results on regular and minimum-crossing projections of line segments have immediate corollaries for polygonal chains, polygons, trees and more general geometric graphs in 3-D since these are all special cases of sets of line segments. Our results also have application to graph drawing for knot-theorists. Let K be a knot with n vertices. To study the knot's combinatorial properties, knot theorists obtain a planar graph G called the diagram of K by a regular projection of K. Many of their algorithms are applied to G and therefore their time complexity depends on the space complexity of G. By combining our algorithms we can obtain regular projections with the minimum number of crossings thereby minimizing the time complexity of their algorithms.


  1. [AGR95]
    Amato, N. M., Goodrich, M. T. and Ramos, E. A., “Computing faces in segment and simplex arrangements,” Proc. Symp. on the Theory of Computing, 1995.Google Scholar
  2. [AW88]
    Avis, D., and Wenger, R., “Polyhedral line transversals in space,” Discrete and Computational Geometry, vol. 3, 1988, pp. 257–265.CrossRefGoogle Scholar
  3. [AW87]
    Avis, D., and Wenger, R., “Algorithms for line stabbers in space,” Proc. Third ACM Symp. on Computational Geometry, 1987, pp. 300–307.Google Scholar
  4. [Ba95]
    Balaban I. J., “An optimal algorithm for finding segments intersections,” Proc. ACM Symp. on Comp. Geom., Vancouver, Canada, June 1995, pp. 211–219.Google Scholar
  5. [BO79]
    Bentley, J.L. and Ottmann, T.A., “Algorithms for reporting and counting geometric intersections”, IEEE Trans. Comput. vol. 8, pp. 643–647, 1979.Google Scholar
  6. [BR94]
    Bhattacharya, P. and Rosenfeld, A., “Polygons in three dimensions,” J. of Visual Communication and Image Representation, vol. 5, June 1994, pp. 139–147.CrossRefGoogle Scholar
  7. [BM76]
    Bondy, J. and Murty, U. S. R., Graph Theory with Applications, Elsevier Science, New York, 1976.Google Scholar
  8. [BGK95]
    Burger, T., Gritzmann, P. and Klee, V., “Polytope projection and projection polytopes,” TR No. 95-14, Dept. Mathematics, Trier University.Google Scholar
  9. [CE92]
    Chazelle, B. and Edelsbrunner, H., “An optimal algorithm for intersecting line segments in the plane,” J. ACM, vol. 39, 1992, pp. 1–54.CrossRefGoogle Scholar
  10. [CN89]
    Chiang, K., Nahar, S., and Lo, C., “Time-efficient VLSI artwork analysis algorithms in GOALIE2”, IEE Trans. CAD, vol. 39, pp. 640–647, 1989.Google Scholar
  11. [Co90]
    Colin, C., “Automatic computation of a scene's good views,” MICAD'90, Paris, February, 1990.Google Scholar
  12. [DETT]
    Di Battista, G., Eades, P., Tamassia, R. and Tollis, I. G., “Algorithms for drawing graphs: an annotated bibliography,” Computational Geometry: Theory and Applications, vol. 4, 1994, pp. 235–282.MathSciNetGoogle Scholar
  13. [FDFH]
    Foley, J. D., van Dam, A., Feiner, S. K. and Hughes, J. F., Computer Graphics: Principles and Practice, Addison-Wesley, 1990.Google Scholar
  14. [Ga95]
    Gallagher, R. S., Ed., Computer Visualization: Graphic Techniques for Engineering and Scientific Analysis, IEEE Computer Society Press, 1995.Google Scholar
  15. [GJ83]
    Garey, M.R., and Johnson, D.S., “Crossing number is NP-complete,” SIAM J. Alg. Discrete Methods, vol. 4, 1983, pp. 312–316.Google Scholar
  16. [GT95]
    Garg, A. and Tamassia, R., “On the computational complexity of upward and rectilinear planarity testing,” eds., R. Tamassia and I. G. Tollis, Proc. Graph Drawing'94, LNCS 894, Springer-Verlag, 1995, 286–297.Google Scholar
  17. [HMTT]
    Hirata, T., Matousek, J., Tan, X.-H. and Tokuyama, T., “Complexity of projected images of convex subdivisions,” Comp. Geom., vol. 4., 1994, pp. 293–308.CrossRefGoogle Scholar
  18. [KK88]
    Kamada, T. and Kawai, S., “A simple method for computing general position in displaying three-dimensional objects,” Computer Vision, Graphics and Image Processing, vol. 41, 1988, pp. 43–56.Google Scholar
  19. [KK93]
    Keller, P. R. & Keller, M. M., Visual Cues: Practical Data Visualization, IEEE Computer Society Press, 1993.Google Scholar
  20. [Li93]
    Livingston, C, Knot Theory, The Carus Mathematical Monographs, vol. 24, The Mathematical Association of America, 1993.Google Scholar
  21. [Me83]
    Megiddo, N., “Linear-time algorithms for linear programming in R 3 and related problems,” SIAM Journal of Computing, vol. 12, 1983, pp. 759–776.CrossRefGoogle Scholar
  22. [MS85]
    McKenna, M. & Seidel, R., “Finding the optimal shadows of a convex polytope,” Proc. ACM Symp. on Comp. Geom., June 1985, pp. 24–28.Google Scholar
  23. [NP82]
    Nievergelt, J., Preparata, F., “Plane-sweep algorithms for intersecting geometric figures”, Communications of ACM vol.25, pp. 739–747, 1982.CrossRefGoogle Scholar
  24. [PS85]
    Preparata, F., and Shamos, M., Computational Geometry: An introduction, Springer-Verlag, New York, 1985.Google Scholar
  25. [PT92]
    Preparata, F. and Tamassia, R., “Efficient point location in a convex spatial cell-complex,” SIAM Journal of Computing, vol. 21, 1992, pp. 267–280.CrossRefGoogle Scholar
  26. [Re83]
    Reidemeister, R., Knotentheorie, Ergebnisse der Mathematic, Vol. 1, Springer-Verlag, Berlin, 1932; L. F. Boron, C. O. Christenson and B. A. Smith (English translation) Knot Theory, BSC Associates, Moscow, Idaho, USA, 1983.Google Scholar
  27. [SB92]
    Souvaine, D. and Bjorling-Sachs, I., “The contour problem for restricted-orientation polygons”, Proc. of the IEEE, vol. 80, pp. 1449–1470, 1992.CrossRefGoogle Scholar
  28. [SSV94]
    Shahrokhi, F., Szekely, L., and Vrt'o, I., “Crossing number of graphs, lower bound techniques and algorithms: A survey,” Lecture Notes in Comp. Science, vol. 894, Princeton, New Jersey, 1994, pp. 131–142.Google Scholar
  29. [St82]
    Strang, G., “The width of a chair,” The American Mathematical Monthly, vol. 89, No. 8, October 1982, pp. 529–534.Google Scholar
  30. [To85]
    Toussaint, G. T., “Movable separability of sets,” in Computational Geometry, G. T. Toussaint, Ed., Elsevier Science Publishers, 1985, pp. 335–375.Google Scholar

Copyright information

© Springer-Verlag 1996

Authors and Affiliations

  • Prosenjit Bose
    • 1
  • Pedro Ramos
    • 2
  • Francisco Gomez
    • 2
  • Godfried Toussaint
    • 3
  1. 1.University of British ColumbiaVancouverCanada
  2. 2.Universidad Politecnica de MadridMadridSpain
  3. 3.McGill UniversityMontrealCanada

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