Skip to main content
SpringerLink
Log in
Book cover

Workshop on Algorithms and Data Structures

WADS 1997: Algorithms and Data Structures pp 186–199Cite as

Checking the convexity of polytopes and the planarity of subdivisions (extended abstract)

  • Session 7A: Invited Lecture
  • Conference paper
  • First Online:

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 1272))

Abstract

This paper studies the problem of verifying the correctness of geometric structures. We design optimal checkers for convex polytopes in two and higher dimensions, and for various types of planar subdivisions, such as triangulations, Delaunay triangulations, and convex subdivisions. Our checkers are simpler and more general than the ones previously described in the literature. Their performance is studied also in terms of the degree, which characterizes the arithmetic precision required.

Research supported in part by the U.S. Army Research Office under grant DAAH04-96-1-0013, by the National Science Foundation under grant CCR-9423847, and by the EC ESPRIT Long Term Research Project ALCOM-IT under contract 20244.

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

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. N. Amenta. Directory of computational geometry software. http://www.geom.umn.edu/software/cglist/.

    Google Scholar 

  2. G. Barequet, S. S. Bridgeman, C. A. Duncan, M. T. Goodrich, and R. Tamassia Classic computational geometry with GeomNet. In Proc. ACM Symp. Computational Geometry, 1997.

    Google Scholar 

  3. J. L. Bentley and T. A. Ottmann. Algorithms for reporting and counting geometric intersections. IEEE Trans. Comput., C-28:643–647, 1979.

    Google Scholar 

  4. M. Blum and S. Kannan. Designing programs that check their work. J. ACM, 42(1):269–291, Jan. 1995.

    Google Scholar 

  5. C. Burnikel. Exact Computation of Voronoi Diagrams and Line Segment Intersections. Ph.D thesis, Universitat des Saarlandes, Mar. 1996.

    Google Scholar 

  6. B. Chazelle. Triangulating a simple polygon in linear time. Discrete Comput. Geom., 6:485–524, 1991.

    Google Scholar 

  7. G. Di Battista, R. Tamassia, and L. Vismara. On-line convex planarity testing. In Graph-Theoretic Concepts in Computer Science (Proc. WG 94), Lecture Notes in Computer Science. Springer-Verlag, 1994.

    Google Scholar 

  8. G. Di Battista, R. Tamassia, and L. Vismara. On-line convex planarity testing. Technical Report CS-95-26, Dept. Computer Science, Brown Univ., 1995. ftp://ftp.cs.brown.edu/pub/techreports/95/cs95-26.ps.Z.

    Google Scholar 

  9. G. Di Battista and L. Vismara. Angles of planar triangular graphs. SIAM J. Discrete Math., 9(3):349–359, 1996.

    Google Scholar 

  10. I. Fary. On straight lines representation of planar graphs. Acta Sci. Math. Szeged., 11:229–233, 1948.

    Google Scholar 

  11. S. Gottschalk, M. C. Lin, and D. Manocha. OBB-tree: A hierarchical structure for rapid interference detection. Comput. Graph., 1996. Proc. SIGGRAPH '96.

    Google Scholar 

  12. F. Harary. Graph Theory. Addison-Wesley, Reading, MA, 1972.

    Google Scholar 

  13. J. Hopcroft and R. E. Tarjan. Efficient planarity testing. J. ACM, 21(4):549–568, 1974.

    Google Scholar 

  14. D. G. Kirkpatrick. Establishing order in planar subdivisions. Discrete Comput. Geom., 3:267–280, 1988.

    Google Scholar 

  15. C. Levcopoulos and A. Lingas. On approximation behavior of the greedy triangulation for convex polygons. Algorithmica, 2:175–193, 1987.

    Google Scholar 

  16. G. Liotta, F. P. Preparata, and R. Tamassia. Robust proximity queries: an illustration of degree-driven algorithm design. SIAM J. Computing. to appear.

    Google Scholar 

  17. G. Liotta, F. P. Preparata, and R. Tamassia. Robust proximity queries: an illustration of degree-driven algorithm design. In Proc. ACM Symp. Computational Geometry, 1997.

    Google Scholar 

  18. K. Mehlhorn and S. Näher. Checking Geometric Structures, December 1996. Manual.

    Google Scholar 

  19. K. Mehlhorn, S. Näher, T. Schilz, S. Schirra, M. Seel, R. Seidel, and C. Uhrig. Checking geometric programs or verification of geometric structures. In Proc. 12th Annu. ACM Sympos. Comput. Geom., pages 159–165, 1996.

    Google Scholar 

  20. K. Mehlhorn, S. Näher, T. Schilz, S. Schirra, M. Seel, R. Seidel, and C. Uhrig. Checking geometric programs or verification of geometric structures. Manuscript, 1997.

    Google Scholar 

  21. G. F. Sullivan, D. S. Wilson, and G. M. Masson. Certification of computational results. IEEE Transactions on Computers, 44(7):833–847, 1995.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. INRIA, BP 93, 06902, Sophia Antipolis Cedex, France

    O. Devillers & R. Tamassia

  2. Dipartimento di Informatica e Sistemistica, Universitá di Roma “La Sapienza”, via Salaria 113, 00198, Rome, Italy

    G. Liotta

  3. Center for Geometric Computing, Department of Computer Science, Brown University, 02912-1910, Providence, RI, USA

    F. P. Preparata

Authors
  1. O. Devillers
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. G. Liotta
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. F. P. Preparata
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. R. Tamassia
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Frank Dehne Andrew Rau-Chaplin Jörg-Rüdiger Sack Roberto Tamassia

Rights and permissions

Reprints and permissions

Copyright information

© 1997 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Devillers, O., Liotta, G., Preparata, F.P., Tamassia, R. (1997). Checking the convexity of polytopes and the planarity of subdivisions (extended abstract). In: Dehne, F., Rau-Chaplin, A., Sack, JR., Tamassia, R. (eds) Algorithms and Data Structures. WADS 1997. Lecture Notes in Computer Science, vol 1272. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-63307-3_59

Download citation

  • DOI: https://doi.org/10.1007/3-540-63307-3_59

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-63307-5

  • Online ISBN: 978-3-540-69422-9

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation