Discrete & Computational Geometry

, Volume 7, Issue 3, pp 227–253 | Cite as

On the difficulty of triangulating three-dimensional Nonconvex Polyhedra

  • Jim Ruppert
  • Raimund Seidel


A number of different polyhedraldecomposition problems have previously been studied, most notably the problem of triangulating a simple polygon. We are concerned with thepolyhedron triangulation problem: decomposing a three-dimensional polyhedron into a set of nonoverlapping tetrahedra whose vertices must be vertices of the polyhedron. It has previously been shown that some polyhedra cannot be triangulated in this fashion. We show that the problem of deciding whether a given polyhedron can be triangulated is NP-complete, and hence likely to be computationally intractable. The problem remains NP-complete when restricted to the case of star-shaped polyhedra. Various versions of the question of how many Steiner points are needed to triangulate a polyhedron also turn out to be NP-hard.


Conjunctive Normal Form Steiner Point Simple Polygon Satisfying Assignment Variable Niche 
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  1. 1.
    D. Avis and H. ElGindy. Triangulating simplicial point sets in space.Discrete Comput. Geom., 2:99–111, 1987.MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    C. L. Bajaj and T. K. Dey. Convex Decompositions of Simple Polyhedra. Technical Report CSD-TR-833, Dept. Comput. Sci., Purdue University, 1989.Google Scholar
  3. 3.
    B. Chazelle. Computational Geometry and Convexity. Ph.D. thesis, Yale University, 1980. Also available as Technical Report CMU-CS-80-150, Dept. Comput. Sci., Carnegie Mellon University, July 1980.Google Scholar
  4. 4.
    B. Chazelle. Triangulating a simple polygon in linear time. InProceedings of the 31st Annual Symposium on Foundations of Computer Science, IEEE, 1990, pages 220–230.Google Scholar
  5. 5.
    B. Chazelle and L. Palios. Triangulating a nonconvex polytope.Discrete Comput. Geom., 5:505–526, 1990.MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    J. Culberson and R. Reckhow. Covering polygons is hard. InProceedings of the 29th Annual Symposium on Foundations of Computer Science, IEEE, 1988, pages 601–611.Google Scholar
  7. 7.
    H. Edelsbrunner, F. P. Preparata, and D. B. West. Tetrahedrizing point sets in three dimensions. Technical Report UIUCDCS-R-86-1310, Dept. Comput. Sci., University of Illinois, 1986.Google Scholar
  8. 8.
    M. R. Garey and D. S. Johnson.Computers and Intractability: A Guide to the Theory of NP-Completeness. Freeman, San Francisco, 1979.MATHGoogle Scholar
  9. 9.
    J. O'Rourke.Art Gallery Theorems and Algorithms. Oxford University Press, New York, 1987.MATHGoogle Scholar
  10. 10.
    J. O'Rourke and K. Supowit. Some NP-hard polygon decomposition problems.IEEE Trans. Inform. Theory, 29:181–190, 1983.MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    F. P. Preparata and M. I. Shamos.Computational Geometry—An Introduction. Springer-Verlag, New York, 1985.Google Scholar
  12. 12.
    E. Schönhardt. Über die Zerlegung von Dreieckspolyedern in Tetraeder.Math. Ann., 98:309–312, 1928.MathSciNetCrossRefGoogle Scholar
  13. 13.
    B. Von Hohenbalken. Finding simplicial subdivisions of polytopes.Math. Programming, 21:233–234, 1981.MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag New York Inc. 1992

Authors and Affiliations

  • Jim Ruppert
    • 1
  • Raimund Seidel
    • 1
  1. 1.Computer Science DivisionUniversity of California at BerkeleyBerkeleyUSA

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