Constrained Delaunay Triangulation Using Delaunay Visibility

  • Yi-Jun Yang
  • Hui Zhang
  • Jun-Hai Yong
  • Wei Zeng
  • Jean-Claude Paul
  • Jiaguang Sun
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4291)


An algorithm for constructing constrained Delaunay triangulation (CDT) of a planar straight-line graph (PSLG) is presented. Although the uniform grid method can reduce the time cost of visibility determinations, the time needed to construct the CDT is still long. The algorithm proposed in this paper decreases the number of edges involved in the computation of visibility by replacing traditional visibility with Delaunay visibility. With Delaunay visibility introduced, all strongly Delaunay edges are excluded from the computation of visibility. Furthermore, a sufficient condition for DT (CDT whose triangles are all Delaunay) existence is presented to decrease the times of visibility determinations. The mesh generator is robust and exhibits a linear time complexity for randomly generated PSLGs.


Delaunay Triangulation Steiner Point Polygonal Domain Active Edge Visibility Determination 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    Ronfard, R.P., Rossignac, J.R.: Triangulating multiply-connected polygons: A simple, yet efficient algorithm. Computer Graphics Forum 13(3), 281–292 (1994)CrossRefGoogle Scholar
  2. 2.
    Rockwood, A., Heaton, K., Davis, T.: Real-time rendering of trimmed surfaces. Computers & Graphics 23(3), 107–116 (1989)CrossRefGoogle Scholar
  3. 3.
    Sheng, X., Hirsch, B.E.: Triangulation of trimmed surfaces in parametric space. Computer-Aided Design 24(8), 437–444 (1992)MATHCrossRefGoogle Scholar
  4. 4.
    Obabe, H., Imaoka, H., Tomiha, T., Niwaya, H.: Three dimensional apparel CAD system. Computer & Graphics 26(2), 105–110 (1992)CrossRefGoogle Scholar
  5. 5.
    Zeng, W., Yang, C.L., Meng, X.X., Yang, Y.J., Yang, X.K.: Fast algorithms of constrained Delaunay triangulation and skeletonization for band-images. In: SPIE Defense and Security Symposium, vol. 5403, pp. 337–348 (2004)Google Scholar
  6. 6.
    Gopi, M., Krishnan, S., Silva, C.T.: Surface reconstruction based on lower dimensional localized Delaunay triangulation. Computer Graphics Forum 19(3), 467–478 (2000)CrossRefGoogle Scholar
  7. 7.
    Nonato, L.G., Minghim, R., Oliveira, M.C.F., Tavares, G.: A novel approach for Delaunay 3D reconstruction with a comparative analysis in the light of applications. Computer Graphics Forum 20(2), 161–174 (2001)MATHCrossRefGoogle Scholar
  8. 8.
    Marco, A., Michela, S.: Automatic surface reconstruction from point sets in space. Computer Graphics Forum 19(3), 457–465 (2000)CrossRefGoogle Scholar
  9. 9.
    Ho-Le, K.: Finite element mesh generation methods: a review and classification. Computer-Aided Design 20(1), 27–38 (1988)MATHCrossRefGoogle Scholar
  10. 10.
    Lee, D.T., Lin, A.K.: Generalized Delaunay triangulations. Discrete & Computational Geometry 1, 201–217 (1986)MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Yang, Y.J., Yong, J.H., Sun, J.G.: An algorithm for tetrahedral mesh generation based on conforming constrained Delaunay tetrahedralization. Computers & Graphics 29(4), 606–615 (2005)CrossRefGoogle Scholar
  12. 12.
    Shewchuk, J.R.: Constrained Delaunay tetrahedralizations and provably good boundary recovery. In: Eleventh International Meshing Roundtable, pp. 193–204 (2002)Google Scholar
  13. 13.
    Cignoni, P., Montani, C., Perego, R., Scopigno, R.: Parallel 3D Delaunay triangulation. Computer Graphics Forum 12(3), 129–142 (1993)CrossRefGoogle Scholar
  14. 14.
    Edelsbrunner, H., Tan, T.S.: An Upper Bound for Conforming Delaunay Triangulations. Discrete & Computational Geometry 10(2), 197–213 (1993)MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Nackman, L.R., Srinivasan, V.: Point placement for Delaunay triangulation of polygonal domains. In: Proceeding of Third Canadian Conference Computational Geometry, pp. 37–40 (1991)Google Scholar
  16. 16.
    Saalfeld, A.: Delaunay edge refinements. In: Proceeding of Third Canadian Conference on Computational Geometry, pp. 33–36 (1991)Google Scholar
  17. 17.
    Chew, L.P.: Constrained Delaunay triangulations. Algorithmica 4, 97–108 (1989)MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    George, P.L., Borouchaki, H.: Delaunay Triangulation and Meshing: Application to Finite Elements. Hermes, Paris (1998)MATHGoogle Scholar
  19. 19.
    Klein, R.: Construction of the constrained Delaunay triangulation of a polygonal domain. In: CAD Systems Development, pp. 313–326 (1995)Google Scholar
  20. 20.
    Piegl, L.A., Richard, A.M.: Algorithm and data structure for triangulating multiply connected polygonal domains. Computer & Graphics 17(5), 563–574 (1993)CrossRefGoogle Scholar
  21. 21.
    Bentley, J.L., Weide, B.W., Yao, A.C.: Optimal expected-time algorithms for closest point problems. ACM Transactions on Mathematical Software 6(4), 563–580 (1980)MATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Devillers, O., Estkowski, R., Gandoin, P.M., Hurtado, F., Ramos, P., Sacristan, V.: Minimal Set of Constraints for 2D Constrained Delaunay Reconstruction. International Journal of Computational Geometry and Applications 13(5), 391–398 (2003)MATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Fang, T.P., Piegl, L.: Delaunay triangulation using a uniform grid. IEEE Computer Graphics and Applications 13(3), 36–47 (1993)CrossRefGoogle Scholar
  24. 24.
    Edelsbrunner, H., Mcke, E.P.: Simulation of simplicity: a technique to cope with degenerate cases in geometric algorithms. ACM Transactions on Graphics 9(1), 66–104 (1990)MATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Yi-Jun Yang
    • 1
    • 2
  • Hui Zhang
    • 1
  • Jun-Hai Yong
    • 1
  • Wei Zeng
    • 3
  • Jean-Claude Paul
    • 1
  • Jiaguang Sun
    • 1
    • 2
  1. 1.School of SoftwareTsinghua UniversityBeijingChina
  2. 2.Department of Computer Science and Tech.Tsinghua UniversityBeijingChina
  3. 3.Institute of Computing TechnologyChinese Academy of SciencesBeijingChina

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