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Degree Aware Triangulation of Annular Regions

  • Laxmi P. Gewali
  • Bhaikaji Gurung
Conference paper
Part of the Advances in Intelligent Systems and Computing book series (AISC, volume 738)

Abstract

Generating constrained triangulation of point sites distributed in the plane is an important problem in computational geometry. We present theoretical and experimental investigation results for generating triangulations for polygons and point sites that address node degree constraints. We characterize point sites that have almost all vertices of odd degree. We present experimental results on the node degree distribution of Delaunay triangulation of point sites generated randomly. Additionally, we present a heuristic algorithm for triangulating a given normal annular region with increased number of even degree vertices.

Keywords

Triangulation Degree-constrained triangulation Annular region triangulation 

References

  1. 1.
    M. de Berg, M. van Kreveld, M. Overmars, O. Schwarzkopf, Computational Geometry: Algorithms and Applications (Springer-Verlag New York, Inc., Secaucus, NJ, 1997)CrossRefGoogle Scholar
  2. 2.
    J. O’Rourke, Art Gallery Theorems and Algorithms (Oxford University Press Inc, New York, NY, 1987)zbMATHGoogle Scholar
  3. 3.
    T. Auer, M. Held, Heuristics for the generation of random polygons, in Proceedings of the 28th Canadian Conference on Computational Geometry, CCCG 2016, (Carleton University, Ottawa, 1996), pp. 38–43Google Scholar
  4. 4.
    W. Mulzer, G. Rote, Minimum-weight triangulation is np-hard. J. ACM 55(2), 11:1–11:29 (2008)MathSciNetCrossRefGoogle Scholar
  5. 5.
    C. Pelaez, A. Ramrez-Vigueras, and J. Urrutia. Triangulations with many points of even degree, in Proceedings of the 22nd Annual Canadian Conference on Computational Geometry (2010), pp. 103–106Google Scholar
  6. 6.
    O. Aichholzer, T. Hackl, M. Homann, A. Pilz, G. Rote, B. Speckmann, and B. Vogtenhuber. Plane graphs with parity constraints, in Algorithms and Data Structures, 11th International Symposium, WADS 2009, Ban, Canada, August 21–23, 2009. Proceedings (2009), pp. 13–24Google Scholar
  7. 7.
    F. Homann, K. Kriegel, A graph-coloring result and its consequences for polygon-guarding problems. SIAM J. Discret. Math. 9(2), 210–224 (May 1996)MathSciNetCrossRefGoogle Scholar
  8. 8.
    R. Gyawali, Degree Constrained Triangulation. Master’s thesis (University of Nevada, Las Vegas, 2012)Google Scholar
  9. 9.
    M.R. Garey, D.S. Johnson, Computers and Intractability: A Guide to the Theory of NP-Completeness (W. H. Freeman & Co., New York, NY, 1979)zbMATHGoogle Scholar
  10. 10.
    E.M. Arkin, M. Held, J.S.B. Mitchell, S.S. Skiena, Hamiltonian triangulations for fast rendering. Vis. Comput. 12(9), 429–444 (1996)CrossRefGoogle Scholar
  11. 11.
    M. Bern, H. Edelsbrunner, D. Eppstein, S. Mitchell, T.S. Tan, Edge insertion for optimal triangulations. Discrete Comput. Geom. 10(1), 47–65 (1993)MathSciNetCrossRefGoogle Scholar
  12. 12.
    T.C. Biedl, A. Lubiw, S. Mehrabi, S. Verdonschot, Rectangle of- inuence triangulations, in Proceedings of the 28th Canadian Conference on Com-putational Geometry, CCCG 2016, August 3–5, 2016, (Simon Fraser University, Vancouver, British Columbia, 2016), pp. 237–243Google Scholar
  13. 13.
    C. Burnikel, K. Mehlhorn, S. Schirra, On degeneracy in geometric computations, in Proceedings of the Fifth Annual ACM-SIAM Symposium on Dis-crete Algorithms, SODA '94, (Society for Industrial and Applied Mathematics, Philadelphia, PA, 1994), pp. 16–23zbMATHGoogle Scholar
  14. 14.
    B. Chazelle, Triangulating a simple polygon in linear time. Discrete Comput. Geom. 6(3), 485–524 (1991)MathSciNetCrossRefGoogle Scholar
  15. 15.
    S. Fortune, A sweepline algorithm for voronoi diagrams. Algorithmica 2, 153–174 (1987)MathSciNetCrossRefGoogle Scholar
  16. 16.
    L.J. Guibas, D.E. Knuth, M. Sharir, Randomized incremental construction of delaunay and voronoi diagrams. Algorithmica 7(4), 381–413 (1992)MathSciNetCrossRefGoogle Scholar
  17. 17.
    S.K. Ghosh, D.M. Mount, An output-sensitive algorithm for computing visibility graphs. SIAM J. Comput. 20(5), 888–910 (1991)MathSciNetCrossRefGoogle Scholar
  18. 18.
    L. Guibas, J. Stol, Primitives for the manipulation of general subdivisions and the computations of voronoi diagrams. ACM Trans. Graph 4, 74–123 (1985)CrossRefGoogle Scholar
  19. 19.
    S. Hertel, K. Mehlhorn, Fast triangulation of simple polygons, in Proceedings of the 1983 International FCT-Conference on Fundamentals of Computation Theory, (Springer-Verlag, London, UK, 1983), pp. 207–218Google Scholar
  20. 20.
    F. Hurtado, M. Noy, The graph of triangulations of a convex polygon, in Proceedings of the Twelfth Annual Symposium on Computational Geometry, SCG ‘96, (ACM, New York, NY, 1996), pp. 407–408CrossRefGoogle Scholar
  21. 21.
    F. Hurtado, M. Noy, J. Urrutia, Flipping edges in triangulations, in Proceedings of the Twelfth Annual Symposium on Computational Geometry, SCG '96, (ACM, New York, NY, 1996), pp. 214–223CrossRefGoogle Scholar
  22. 22.
    D.V. Hutton, Fundamentals of finite element analysis, in Mcgraw-Hill Series in Mechanical Engineering, (McGraw-Hill, New York, 2004)Google Scholar
  23. 23.
    C.L. Lawson. Software for c1 surface interpolation, in Mathematical Software III (1977), pp. 161–194Google Scholar
  24. 24.
    G.H. Meisters, Polygon have ears. Am. Math. Mon. 82, 648–651 (1975)MathSciNetCrossRefGoogle Scholar
  25. 25.
    D.E. Muller, F.P. Preparata, Finding the intersection of two convex polyhedra. Theor. Comput. Sci. 7, 217–236 (1978)MathSciNetCrossRefGoogle Scholar
  26. 26.
    E. Osherovich, A.M. Bruckstein, All triangulations are reachable via sequences of edge-ips: an elementary proof. Comput. Aided Geom. Des. 25, 157–161 (2008)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Computer ScienceUniversity of NevadaLas VegasUSA

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