Parallel-Redrawing Mechanisms, Pseudo-Triangulations and Kinetic Planar Graphs

  • Ileana Streinu
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3843)

Abstract

We study parallel redrawing graphs: graphs embedded on moving point sets in such a way that edges maintain their slopes all throughout the motion.

The configuration space of such a graph is of an oriented-projective nature, and its combinatorial structure relates to rigidity theoretic parameters of the graph. For an appropriate parametrization the points move with constant speeds on linear trajectories. A special type of kinetic structure emerges, whose events can be analyzed combinatorially. They correspond to collisions of subsets of points, and are in one-to-one correspondence with contractions of the underlying graph on rigid components. We show how to process them algorithmically via a parallel redrawing sweep.

Of particular interest are those planar graphs which maintain non-crossing edges throughout the motion. Our main result is that they are (essentially) pseudo-triangulation mechanisms: pointed pseudo-trian- gulations with a convex hull edge removed. These kinetic graph structures have potential applications in morphing of more complex shapes than just simple polygons.

References

  1. 1.
    Agarwal, P., Basch, J., Guibas, L., Hershberger, J., Zhang, L.: Deformable free space tilings for kinetic collision detection. International Journal of Robotics Research 21, 179–197 (2003); Preliminary version appeared in Proc. 4th International Workshop on Algorithmic Foundations of Robotics (WAFR) (2000)Google Scholar
  2. 2.
    Aichholzer, O., Rote, G., Speckmann, B., Streinu, I.: The zig-zag path of a pseudo-triangulation. In: Dehne, F., Sack, J.-R., Smid, M. (eds.) WADS 2003. LNCS, vol. 2748, pp. 377–388. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  3. 3.
    Bespamyatnikh, S.: Enumerating pseudo-triangulations in the plane. Comput. Geom. Theory Appl. 30(3), 207–222 (2005)MathSciNetGoogle Scholar
  4. 4.
    Bokowski, J., Mock, S., Streinu, I.: The folkman-lawrence topological representation theorem for oriented matroids - an elementary proof in rank 3. European Journal of Combinatorics 22, 601–615 (2001)MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Gotsman, C., Surazhsky, V.: Guaranteed intersection-free polygon morphing. Computers and Graphics 25(1), 67–75 (2001)CrossRefGoogle Scholar
  6. 6.
    Graver, J., Servatius, B., Servatius, H.: Combinatorial Rigidity. Graduate Studies in Mathematics, vol. 2. American Mathematical Society (1993)Google Scholar
  7. 7.
    Guibas, L., Hershberger, J., Suri, S.: Morphing simple polygons. Discrete and Computational Geometry 24, 1–34 (2000)MATHMathSciNetGoogle Scholar
  8. 8.
    Haas, R., Orden, D., Rote, G., Santos, F., Servatius, B., Servatius, H., Souvaine, D., Streinu, I., Whiteley, W.: Planar minimally rigid graphs and pseudo-triangulations. Computational Geometry: Theory and Applications, 31–61 (May 2005)Google Scholar
  9. 9.
    Lee, A., Streinu, I., Theran, L.: Finding and maintaining rigid components. In: Proc. Canad. Conf. Comp. Geom., Windsor, Canada (August 2005)Google Scholar
  10. 10.
    Pocchiola, M., Vegter, G.: Topologically sweeping visibility complexes via pseudo-triangulations. Discrete & Computational Geometry 16(4), 419–453 (1996)MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Rote, G., Santos, F., Streinu, I.: Expansive motions and the polytope of pointed pseudo-triangulations. In: Aronov, J.P.B., Basu, S., Sharir, M. (eds.) Discrete and Computational Geometry - The Goodman-Pollack Festschrift, Algorithms and Combinatorics, pp. 699–736. Springer, Berlin (2003)Google Scholar
  12. 12.
    Speckmann, B., Tóth, C.: Allocating vertex π-guards in simple polygons via pseudo-triangulations. In: Proc. ACM-SIAM Symp. Discrete Algorithms (SODA), pp. 109–118 (2003); To appear in Discrete and Computational Geometry (2004)Google Scholar
  13. 13.
    Stolfi, J.: Oriented Projective Geometry: A Framework for Geometric Computations. Academic Press, New York (1991)MATHGoogle Scholar
  14. 14.
    Streinu, I.: A combinatorial approach to planar non-colliding robot arm motion planning. In: IEEE Symposium on Foundations of Computer Science, pp. 443–453 (2000)Google Scholar
  15. 15.
    Streinu, I.: Pseudo-triangulations, rigidity and motion planning. Discrete and Computational Geometry (2005) (to appear); A preliminary version appeared in [14]Google Scholar
  16. 16.
    Whiteley, W.: Some matroids from discrete applied geometry. In: Oxley, J., Bonin, J., Servatius, B. (eds.) Matroid Theory. Contemporary Mathematics, vol. 197, pp. 171–311. American Mathematical Society (1996)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Ileana Streinu
    • 1
  1. 1.Computer Science DepartmentSmith CollegeNorthampton

Personalised recommendations