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A Simple Algorithm to Triangulate a Special Class of 3d Non-convex Polyhedra Without Steiner Points

Conference paper
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 131)

Abstract

We describe a simple algorithm to triangulate a special class of 3d non-convex polyhedra without Steiner points (vertices which are not the vertices of the given polyhedron). We prove sufficient conditions for the termination of this algorithm, and show that it runs in O(n3) time, where n is the number of input vertices.

Keywords

Weighted Delaunay triangulations Non-regular triangulations Lawson’s flip algorithm Directed flips Monotone sequence Flip graph Redundant interior vertices Schönhardt polyhedron Indecomposable polyhedra Steiner points 

References

  1. 1.
    Bagemihl, F.: On indecomposable polyhedra. Am. Math. Mon. 55(7), 411–413 (1948)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Bern, M.: Compatible tetrahedralizations. In: Proc. 9th Annual ACM Symposium on Computational Geometry, pp. 281–288 (1993)Google Scholar
  3. 3.
    Bezdek, A., Carrigan, B.: On nontriangulable polyhedra. Contrib. Algebra Geom. 57(1), 51–66 (2016)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Chazelle, B.: Convex partition of polyhedra: a lower bound and worst-case optimal algorithm. SIAM J. Comput. 13(3), 488–507 (1984)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Chazelle, B., Palios, L.: Triangulating a non-convex polytope. Discrete Comput. Geom. 5(3), 505–526 (1990)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Chazelle, B., Shouraboura, N.: Bounds on the size of tetrahedralizations. Discrete Comput. Geom. 14(3), 429–444 (1995)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Chew, P.L.: Constrained Delaunay triangulation. Algorithmica 4, 97–108 (1989)MathSciNetCrossRefGoogle Scholar
  8. 8.
    De Loera, J.A., Rambau, J., Santos, F.: Triangulations, Structures for Algorithms and Applications. Algorithms and Computation in Mathematics, vol. 25. Springer, Berlin (2010)Google Scholar
  9. 9.
    Edelman, P., Reiner, V.: The higher Stasheff-Tamari posets. Mathematika 43, 127–154 (1996)MathSciNetCrossRefGoogle Scholar
  10. 10.
    George, P.-L., Borouchaki, H., Saltel, E.: últimateŕobustness in meshing an arbitrary polyhedron. Int. J. Numer. Methods Eng. 58, 1061–1089 (2003)Google Scholar
  11. 11.
    George, P.-L., Hecht, F., Saltel, E.: Automatic mesh generator with specified boundary. Comput. Methods Appl. Mech. Eng. 92, 269–288 (1991)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Goodman, J., Pach, J.: Cell decomposition of polytopes by bending. Isr. J. Math. 64, 129–138 (1988)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Hershberger, J., Snoeyink, J.: Erased arrangements of lines and convex decompositions of polyhedra. Comput. Geom. Theory Appl. 9(3), 129–143 (1998)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Jessen, B.: Orthogonal icosahedra. Nordisk Mat. Tidskr 15, 90–96 (1967)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Lawson, C.L.: Software for c 1 surface interpolation. In: Mathematical Software III, pp. 164–191. Academic Press, New York (1977)Google Scholar
  16. 16.
    Lee, D.T., Lin, A.K.: Generalized Delaunay triangulations for planar graphs. Discrete Comput. Geom. 1, 201–217 (1986)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Lennes, N.J.: Theorems on the simple finite polygon and polyhedron. Am. J. Math. 33(1/4), 37–62 (1911)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Rambau, J.: Polyhedral subdivisions and projections of polytopes. PhD thesis, Fachbereich 3 Mathematik der Technischen Universität Berlin, Berlin (1996)Google Scholar
  19. 19.
    Rambau, J.: On a generalization of Schönhardt’s polyhedron. In: Goodman, J.E., Pach, J., Welzl, E. (eds.), Combinatorial and Computational Geometry, vol. 52, pp. 501–516. MSRI Publications, Berkeley (2005)Google Scholar
  20. 20.
    Ruppert, J., Seidel, R.: On the difficulty of triangulating three-dimensional nonconvex polyhedra. Discrete Comput. Geom. 7, 227–253 (1992)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Schönhardt, E.: Über die zerlegung von dreieckspolyedern in tetraeder. Math. Ann. 98, 309–312 (1928)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Si, H.: TetGen, a Delaunay-based quality tetrahedral mesh generator. ACM Trans. Math. Softw. 41(2), 11:1–11:36 (2015)Google Scholar
  23. 23.
    Si, H.: On monotone sequence of directed flips, triangulations of polyhedra, and the structural properties of a directed flip graph. arXiv:1809.09701 [cs.DM] (2018)Google Scholar
  24. 24.
    Si, H., Goerigk, N.: Generalised Bagemihl polyhedra and a tight bound on the number of interior Steiner points. Comput. Aided Des. 103, 92–102 (2018)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Sleator, D.D., Thurston, W.P., Tarjan, R.E.: Rotation distance, triangulations, and hyperbolic geometry. J. Amer. Math. Soc. 1, 647–682 (1988)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Toussaint, G.T., Verbrugge, C., Wang, C., Zhu, B.: Tetrahedralization of simple and non-simple polyhedra. In: Proc. 5th Canadian Conference on Computational Geometry, pp. 24–29 (1993)Google Scholar
  27. 27.
    Weatherill, N.P., Hassan, O.: Efficient three-dimensional Delaunay triangulation with automatic point creation and imposed boundary constraints. Int. J. Numer. Methods Eng. 37, 2005–2039 (1994)CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Weierstrass Institute for Applied Analysis and Stochastics (WIAS)BerlinGermany

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