Curves in ℤn

  • Grit Thürmer
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2301)


A new definition of closed curves in n-dimensional discrete space is proposed. This definition can be viewed as a generalization of closed quasi curves and is intended to overcome the limitations of known definitions for practical purposes. Following the proposed definition, a set of points forms a closed curve in discrete space if the set admits a parameterization, i.e. there exists a Hamiltonian cycle in the set. Adjacencies that do not indicate the parameterization are allowed only between points that are “close to each other” along the parameterization. Additionally, it is proven that discrete curves satisfying the new definition in two-dimensional discrete space have the Jordan property.


Travel Salesman Problem Closed Curve Hamiltonian Cycle Discrete Space Endoscopic Sinus Surgery 
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  1. 1.
    Cohen-Or, D., and Kaufman, A. Fundamentals of surface voxelization. Graphical Models and Image Processing 57, 6 (1995), 453–461.CrossRefGoogle Scholar
  2. 2.
    Kong, T. Y., and Rosenfeld, A. Digital topology: a comparison of the graphbased and topological approaches. In Topology and category theory in computer science, G. M. Reed, A. W. Roscoe, and R. F. Wachter, Eds. Oxford University Press, 1991, pp. 273–289.Google Scholar
  3. 3.
    Lawler (ed.), E. L., and Rinnooy-Kan, A. H. The Traveling Salesman problem: A guided tour of combinatorial optimization. Wiley-Interscience Series in Discrete Mathematics. John Wiley and Son, 1985.Google Scholar
  4. 4.
    Lebiedź, J. Discrete arcs and curves. Machine Graphics and Vision 9, 1/2(2000), 25–30.Google Scholar
  5. 5.
    Lincke, C., and Wüthrich, C. A. Morphologically closed surfaces and their digitization. submitted for publication.Google Scholar
  6. 6.
    Malgouyres, R. Phd thesis, Université d’Auvergne, Clermont-Ferrand, France, 1994.Google Scholar
  7. 7.
    Malgouyres, R. Graphs generalizing closed curves with linear construction of the Hamiltonian cyle — parameterization of discretized curves. Theoretical Computer Science 143 (1995), 189–249.zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Malgouyres, R. A definition of surfaces of Z3: A new 3D discrete Jordan theorem. Theoretical Computer Science 186 (1997), 1–41.zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Moise, E. E. Geometric topology in dimensions 2 and 3. Springer-Verlag, New York, 1977.zbMATHGoogle Scholar
  10. 10.
    Rosenfeld, A. Connectivity in digital pictures. Journal of the ACM 17, 1 (1970), 146–160.zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Rosenfeld, A. Arcs and curves in digital pictures. Journal of the ACM 20, 1 (1973), 81–87.zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Rosenfeld, A. A converse to the Jordan curve theorem for digital curves. Information and Control 29 (1975), 292–293.CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Rosenfeld, A. Three-dimensional digital topology. Information and Control 50 (1981), 119–127.zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Shareef, N., and Yagel, R. Rapid previewing via volume-based solid modeling. In Proc. of Solid Modeling’ 95 (1995), pp. 281–292.Google Scholar
  15. 15.
    Wang, S. W., and Kaufman, A. E. Volume-sampled 3D modeling. IEEE Computer Graphics and Applications 14, 5 (1994), 26–32.CrossRefGoogle Scholar
  16. 16.
    Wüthrich, C. A. A model for curve rasterization in n-dimensional space. Computers and Graphics 22, 2–3 (1998), 153–160.CrossRefGoogle Scholar
  17. 17.
    Yagel, R., Stredney, D., Wiet, G. J., Schmalbrock, P., Rosenberg, L., Sessanna, D. J., and Kurzion, Y. Building a virtual environment for endoscopic sinus surgery simulation. Computers and Graphics 20, 6 (1996), 813–823.CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • Grit Thürmer
    • 1
  1. 1.Computer Graphics, Visualization, Man-Machine Communication Group Faculty of MediaBauhaus-University WeimarWeimarGermany

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