Skip to main content
SpringerLink
Log in

Short and smooth polygonal paths

  • Conference paper
  • First Online:
LATIN'98: Theoretical Informatics (LATIN 1998)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 1380))

Included in the following conference series:

Abstract

Automatic graph drawers need to compute paths among vertices of a simple polygon which besides remaining in the interior need to exhibit certain aesthetic properties. Some of these require the incorporation of some information about the polygonal shape without being too far from the actual shortest path. We present an algorithm to compute a locally convex region that “contains” the shortest Euclidean path among two vertices of a simple polygon. The region has a boundary shape that “follows” the shortest path shape. A cubic Bezier spline in the region interior provides a “short and smooth” collision free curve between the two given vertices. The obtained results appear to be aesthetically pleasant and the methods used may be of independent interest. They are elementary and implementable. Figure 7 is a sample output produced by our current implementation.

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

Access this chapter

Log in via an institution

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. J. Abello, O. Egecioglu, K. Kumar, “Visibility Graphs of Staircase Polygons and the Weak Bruhat Order I: Prom Polygons to Maximal Chains,” Discrete and Computational Geometry, Vol. 14, No. 3, 1995, pp. 331–358.

    MathSciNet  Google Scholar 

  2. J. Abello, K. Kumar, “Visibility Graphs and Oriented Matroids,” In R. Tamassia, I. Tollis, editors, Symposium on Graph Drawing GD'94, Princeton, Lecture Notes in Computer Science, Vol. 894, 1994, pp. 147–158.

    Google Scholar 

  3. R. Bartels, J. Beatty, B. Barsky, An Introduction to Splines for Use in Computer Graphics and Geometric Modeling, Morgan Kaufman, Los Altos, California, 1987.

    Google Scholar 

  4. D. P. Dobkin, E. Gansner, E. Koutsofios, S. C. North, “Implementing a general-purpose edge router,” To appear in, Proceedings of the Symposium on Graph Drawing GD'97, Rome, Sept. 1997.

    Google Scholar 

  5. L. E. Dubis. “On Curves of minimal length with a constraint on average curvature and with prescribed initial and terminal positions and tangents,” Amer. J. Math., 79:497–516, 1957.

    MathSciNet  Google Scholar 

  6. S. Fortune and G. Wilfong, “Planning constraint motion,” In Proc. 20th Annu. ACM Sympos. Theory Comput., pp. 445–459, 1988.

    Google Scholar 

  7. E. R. Gansner, E. Koutsofios, S.C. North and K.P. Vo, “A technique for drawing directed graphs,” IEEE Transactions on Software Engineering, March 1993.

    Google Scholar 

  8. S. K. Ghosh and D. M. Mount, “An output-sensitive algorithm for computing visibility graphs”, Siam J. Computing, 20(5):888–910, 1991.

    Article  MathSciNet  Google Scholar 

  9. J. Hershberger and S. Suri, “Efficient computation of Euclidean shortest paths in the plane,” In Proc. 34th Annu. IEEE Sympos. Found. Comput. Sci., pp. 508–517, 1993.

    Google Scholar 

  10. Y. Kanayama and B. I. Hartman, “Smooth local path planning for autonomous vehicles,” In Proc. IEEE Intl. Conf. on Robotics and Automation, Vol. 3, pp. 1265–1270, 1989.

    Google Scholar 

  11. J.C. Latombe, Robot Motion Planning, Kluwer Academic Publishers, Boston, 1991.

    Google Scholar 

  12. J. P. Laumond, “Finding collision free smooth trajectories for a non-holonomic mobile robot,” In Proc. IEEE Intl. Join Conf. on Artificial Intelligence, pp. 1120–1123, 1987.

    Google Scholar 

  13. J. O'Rourke, “The Computational Geometry Column,” SIGACT News, 1992.

    Google Scholar 

  14. J. O'Rourke, I. Streinu, “Pseudo Visibility Graphs,” Proc. ACM Symposium on Computational Geometry, June 1997, France.

    Google Scholar 

  15. F. Preparata, M. Shamos, Computational Geometry: An Introduction, Springer Verlag, NY, 1995.

    Google Scholar 

  16. J. A. Reeds and L. A. Shepp, “Optimal paths for a car that goes both forward and backwards,” Pacific Journal of Mathematics, 145(2), 1990.

    Google Scholar 

  17. G. Sander, M, Alt, A, Ferdinand, and R. Wilhelm, “Clax, a visualized Compiler,” In F. J. Brandenburg, editor, Symposium on Graph Drawing GD'95, Vol. 1027 of Lecture Notes in Computer Science, pp. 459–462, 1996.

    Google Scholar 

  18. J. T. Schwartz and M. Sharir, “Algorithmic motion planning in robotics,” In J. van Leeuwen, editor, Algorithms and Complexity, Vol A of Handbook of Theoretical Computer Science, pp. 391–430. Elsevier, Amsterdam, 1990.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Communication Information Systems Research, AT&T Labs-Research, USA

    James Abello

  2. Information Analysis and Display Research, AT&T Labs-Research, USA

    Emden Gansner

Authors
  1. James Abello
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Emden Gansner
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Cláudio L. Lucchesi Arnaldo V. Moura

Rights and permissions

Reprints and permissions

Copyright information

© 1998 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Abello, J., Gansner, E. (1998). Short and smooth polygonal paths. In: Lucchesi, C.L., Moura, A.V. (eds) LATIN'98: Theoretical Informatics. LATIN 1998. Lecture Notes in Computer Science, vol 1380. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0054318

Download citation

  • DOI: https://doi.org/10.1007/BFb0054318

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-64275-6

  • Online ISBN: 978-3-540-69715-2

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation