Skip to main content
SpringerLink
Log in

Triangulation in 2D and 3D space

  • Conference paper
  • First Online:
Geometry and Robotics (GeoRob 1988)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 391))

Included in the following conference series:

Abstract

This paper is a review of some problems related to the triangulation of polygons or point sets in 2D and 3D space. It includes in particular a proof of the equiangularity properties of the Delaunay triangulation in 2D space and a short review on the different algorithms for the triangulation of polygons. The (still open) question of the intrinsic complexity of the triangulation problem for a simple polygon is also raised. As far as 3D space is concerned, the combinatory relations which provide bounds on the number of tetrahedra which appear in the triangulation of a set of points are given. A divide and conquer algorithm for triangulating arbitrary set of points is also presented. This algorithm is based on a splitting theorem which has been proved independently by Avis and ElGindy on one side and Edelsbrunner, Preparata and West on the other side.

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. D. Avis and H. ElGindy. Triangulating point sets in space. Discrete Comput. Geom., 2, 1987.

    Google Scholar 

  2. J. D. Boissonnat. Geometric structures for three-dimensional shape representation. ACM Trans. on Graphics, 3, 1984.

    Google Scholar 

  3. J. D. Boissonnat. Shape Reconstruction from Planair Cross-sections. Technical Report 546, INRIA, Rocquencourt, France, 1986.

    Google Scholar 

  4. J. D. Boissonnat and M. Teillaud. An hierarchical representation of objects: The delaunay tree. In Proc. of the 2d annual Symposium of Computational Geometry, pages 260–268, Yorktown Heights, New York, June 86, 1986.

    Google Scholar 

  5. B. Chazelle. A theorem on polygon cutting with applications. In Proc. of the 23rd IEEE Annual Symp. on Found. of Comp. Sci., IEEE, New York, Chicago, Nov. 3–5 1982, 1982.

    Google Scholar 

  6. B. Chazelle and J. Incerpi. Triangulation and shape complexity. ACM Transactions on Graphics, 3(2), 1984.

    Google Scholar 

  7. H. Edelsbrunner. Algorithms in Combinatorial Geometry. EATCS Monographs on Theoretical Computer Science, Springer-Verlag, 1987.

    Google Scholar 

  8. H. Edelsbrunner, F. P. Preparata, and D. B. West. Tetrahedrizing point sets in three dimension. Technical Report UIUC DCS-R-86-1310, Department of Computer Sciences, Univ. of Illinois, Urbana, Champaign, 1986.

    Google Scholar 

  9. A. Fournier and D. Y. Montuno. Triangulating simple polygons and equivalent problems. ACM Transactions on Graphics, 3(2), 1984.

    Google Scholar 

  10. L. Guibas, J. Hershberger, D. Leven, M. Sharir, and R. E. Tarjan. Linear-time algorithm for visibility and shortest path problems inside triangulated polygons. Algorithmica, 2, 1987.

    Google Scholar 

  11. A. R. Garey, D. S. Johnson, F. P. Preparata, and R. E. Tarjan. Triangulating a simple polygon. Inform. Process. Lett., 7, 1978.

    Google Scholar 

  12. S. Hertel and K. Mehlhorn. Fast triangulation of simple polygons. In Proc. of the Conference on Foundations of Computing Theory, Springer-Verlag, Borgholm, Sweden, 1983.

    Google Scholar 

  13. D. G. Kirkpatrick. Optimal search in planar subdivision. SIAM J. Comput., 12, 1983.

    Google Scholar 

  14. K. Mehlhorn. Data Structures and Algorithms 3: Multidimensional Searching and Computational Geometry. McGraw-Hill, 1985.

    Google Scholar 

  15. F. P. Preparata and S. J. Hong. Convex hulls of finite sets in two and three dimensions. Comm. ACM, 20, 1977.

    Google Scholar 

  16. F. P. Preparata and M. I. Shamos. Computational Geometry. Springer-Verlag, New York, 1985.

    Google Scholar 

  17. D. F. Rogers. Procedural Elements for Computer Graphics. EATCS Monographs on Theoretical Computer Science, Springer-Verlag, 1985.

    Google Scholar 

  18. R. Sibson. Locally equiangular triangulations. Comput. J., 21, 1978.

    Google Scholar 

  19. A. A. Schoone and J. Van Leeuwen. Triangulating a star shaped polygon. Technical Report RUV-CS-80-3, Univ. of Utrecht, 1980.

    Google Scholar 

  20. G. T. Toussaint and D. Avis. On a convex hull algorithm for polygons and its application to triangulation problems. Pattern Recogn., 15, 1982.

    Google Scholar 

  21. R. E. Tarjan and C. J. Vam Wyk. An O(nloglogn) Time Algorithm for Triangulating Simple Polygons. Technical Report CS-TR-52-6, Dept. Comput. Sci., Princeton Univ., NJ, 1986.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Liens _ URA Cnrs 1327 Ecole Normale Supérieure, 45 rue d'Ulm, 75230, Paris, France

    M. Yvinec

Authors
  1. M. Yvinec
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

J. -D. Boissonnat J. -P. Laumond

Rights and permissions

Reprints and permissions

Copyright information

© 1989 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Yvinec, M. (1989). Triangulation in 2D and 3D space. In: Boissonnat, J.D., Laumond, J.P. (eds) Geometry and Robotics. GeoRob 1988. Lecture Notes in Computer Science, vol 391. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-51683-2_35

Download citation

  • DOI: https://doi.org/10.1007/3-540-51683-2_35

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-51683-5

  • Online ISBN: 978-3-540-46748-9

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation