Abstract
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.
This work has been supported by a postdoctoral grant of the DFG (Deutsche Forschungsgemeinschaft).
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References
Cohen-Or, D., and Kaufman, A. Fundamentals of surface voxelization. Graphical Models and Image Processing 57, 6 (1995), 453–461.
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.
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.
Lebiedź, J. Discrete arcs and curves. Machine Graphics and Vision 9, 1/2(2000), 25–30.
Lincke, C., and Wüthrich, C. A. Morphologically closed surfaces and their digitization. submitted for publication.
Malgouyres, R. Phd thesis, Université d’Auvergne, Clermont-Ferrand, France, 1994.
Malgouyres, R. Graphs generalizing closed curves with linear construction of the Hamiltonian cyle — parameterization of discretized curves. Theoretical Computer Science 143 (1995), 189–249.
Malgouyres, R. A definition of surfaces of Z3: A new 3D discrete Jordan theorem. Theoretical Computer Science 186 (1997), 1–41.
Moise, E. E. Geometric topology in dimensions 2 and 3. Springer-Verlag, New York, 1977.
Rosenfeld, A. Connectivity in digital pictures. Journal of the ACM 17, 1 (1970), 146–160.
Rosenfeld, A. Arcs and curves in digital pictures. Journal of the ACM 20, 1 (1973), 81–87.
Rosenfeld, A. A converse to the Jordan curve theorem for digital curves. Information and Control 29 (1975), 292–293.
Rosenfeld, A. Three-dimensional digital topology. Information and Control 50 (1981), 119–127.
Shareef, N., and Yagel, R. Rapid previewing via volume-based solid modeling. In Proc. of Solid Modeling’ 95 (1995), pp. 281–292.
Wang, S. W., and Kaufman, A. E. Volume-sampled 3D modeling. IEEE Computer Graphics and Applications 14, 5 (1994), 26–32.
Wüthrich, C. A. A model for curve rasterization in n-dimensional space. Computers and Graphics 22, 2–3 (1998), 153–160.
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.
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© 2002 Springer-Verlag Berlin Heidelberg
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Thürmer, G. (2002). Curves in ℤn . In: Braquelaire, A., Lachaud, JO., Vialard, A. (eds) Discrete Geometry for Computer Imagery. DGCI 2002. Lecture Notes in Computer Science, vol 2301. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45986-3_3
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DOI: https://doi.org/10.1007/3-540-45986-3_3
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