Efficient Computation of 3D Clipped Voronoi Diagram

  • Dong-Ming Yan
  • Wenping Wang
  • Bruno Lévy
  • Yang Liu
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6130)


The Voronoi diagram is a fundamental geometry structure widely used in various fields, especially in computer graphics and geometry computing. For a set of points in a compact 3D domain (i.e. a finite 3D volume), some Voronoi cells of their Voronoi diagram are infinite, but in practice only the parts of the cells inside the domain are needed, as when computing the centroidal Voronoi tessellation. Such a Voronoi diagram confined to a compact domain is called a clipped Voronoi diagram. We present an efficient algorithm for computing the clipped Voronoi diagram for a set of sites with respect to a compact 3D volume, assuming that the volume is represented as a tetrahedral mesh. We also describe an application of the proposed method to implementing a fast method for optimal tetrahedral mesh generation based on the centroidal Voronoi tessellation.


Voronoi diagram Delaunay triangulation centroidal Voronoi tessellation tetrahedral meshing 


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  1. 1.
    CGAL, Computational Geometry Algorithms Library,
  2. 2.
    Alliez, P., Cohen-Steiner, D., Yvinec, M., Desbrun, M.: Variational tetrahedral meshing. ACM Transactions on Graphics (Proceedings of ACM SIGGRAPH 2005) 24(3), 617–625 (2005)Google Scholar
  3. 3.
    Alliez, P., Ucelli, G., Gotsman, C., Attene, M.: Recent advances in remeshing of surfaces. Shape Analysis and Structuring, 53–82 (2008)Google Scholar
  4. 4.
    Aurenhammer, F.: Voronoi diagrams: a survey of a fundamental geometric data structure. ACM Computing Surveys 23(3), 345–405 (1991)CrossRefGoogle Scholar
  5. 5.
    Balzer, M., Deussen, O.: Voronoi treemaps. In: Proceedings of the 2005 ACM Symposium on Software Visualization, pp. 165–172 (2005)Google Scholar
  6. 6.
    Barber, C.B., Dobkin, D.P., Huhdanpaa, H.: The quickhull algorithm for convex hulls. ACM Trans. Math. Software 22, 469–483 (1996)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Du, Q., Faber, V., Gunzburger, M.: Centroidal Voronoi tessellations: applications and algorithms. SIAM Review 41(4), 637–676 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Du, Q., Gunzburger, M., Ju, L.: Advances in studies and applications of centroidal Voronoi tessellations. Numer. Math. Theor. Meth. Appl. (to appear 2010)Google Scholar
  9. 9.
    Edelsbrunner, H., Shah, N.R.: Triangulating topological spaces. Int. J. Comput. Geometry Appl. 7(4), 365–378 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Fortune, S.: Voronoi diagrams and Delaunay triangulations. In: Computing in Euclidean Geometry, pp. 193–233 (1992)Google Scholar
  11. 11.
    Gold, C.M.: What is GIS and what is not? Transactions in GIS 10(4), 505–519 (2006)CrossRefGoogle Scholar
  12. 12.
    Hoff III, K.E., Keyser, J., Lin, M.C., Manocha, D.: Fast computation of generalized Voronoi diagrams using graphics hardware. In: Proceedings of ACM SIGGRAPH 1999, pp. 277–286 (1999)Google Scholar
  13. 13.
    Iri, M., Murota, K., Ohya, T.: A fast Voronoi diagram algorithm with applications to geographical optimization problems. In: Proceedings of the 11th IFIP Conference on System Modelling and Optimization, pp. 273–288 (1984)Google Scholar
  14. 14.
    Liu, Y., Wang, W., Lévy, B., Sun, F., Yan, D.-M., Lu, L., Yang, C.: On centroidal Voronoi tessellation: Energy smoothness and fast computation. ACM Transactions on Graphics 28(4), Article No. 101 (2009)Google Scholar
  15. 15.
    Okabe, A., Boots, B., Sugihara, K., Chiu, S.N.: Spatial Tessellations: Concepts and Applications of Voronoi Diagrams, 2nd edn. Wiley, Chichester (2000)zbMATHGoogle Scholar
  16. 16.
    Poupon, A.: Voronoi and Voronoi-related tessellations in studies of protein structure and interaction. Current Opinion in Structural Biology 14(2), 233–241 (2004)CrossRefGoogle Scholar
  17. 17.
    Si, H.: TetGen: A quality tetrahedral mesh generator and three-dimensional Delaunay triangulator,
  18. 18.
    Sud, A., Andersen, E., Curtis, S., Lin, M.C., Manocha, D.: Real-time path planning in dynamic virtual environments using multiagent navigation graphs. IEEE Transactions on Visualization and Computer Graphics 14(3), 526–538 (2008)CrossRefGoogle Scholar
  19. 19.
    Sud, A., Govindaraju, N.K., Gayle, R., Kabul, I., Manocha, D.: Fast proximity computation among deformable models using discrete Voronoi diagrams. ACM Transactions on Graphics (Proceedings of ACM SIGGRAPH 2006) 25(3), 1144–1153 (2006)Google Scholar
  20. 20.
    Sutherland, I.E., Hodgman, G.W.: Reentrant polygon clipping. Communications of the ACM 17(1), 32–42 (1974)zbMATHCrossRefGoogle Scholar
  21. 21.
    Yan, D.-M.: Variational Shape Segmentation and Mesh Generation. Phd dissertation, The University of Hong Kong (2010)Google Scholar
  22. 22.
    Yan, D.-M., Lévy, B., Liu, Y., Sun, F., Wang, W.: Isotropic remeshing with fast and exact computation of restricted Voronoi diagram. Computer Graphics Forum (Proceedings of SGP 2009) 28(5), 1445–1454 (2009)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Dong-Ming Yan
    • 1
    • 2
  • Wenping Wang
    • 1
    • 2
  • Bruno Lévy
    • 1
    • 2
  • Yang Liu
    • 1
    • 2
  1. 1.The University of Hong KongHong KongChina
  2. 2.Project Alice, INRIANancyFrance

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