Continuous Maps on Digital Simple Closed Curves

ABSTRACT

We give digital analogues of classical theorems of topology for continuous functions defined on spheres, for digital simple closed curves. In particular, we show the following. ? A digital simple closed curve of more than 4 points is not contractible, i.e., its identity map is not nullhomotopic in . ? Let and be digital simple closed curves, each symmetric with respect to the origin, such that (where is the number of points in ). Let be a digitally continuous antipodal map. Then is not nullho- motopic in . ? Let be a digital simple closed curve that is symmetric with respect to the origin. Let be a digitally continuous map. Then there is a pair of antipodes such that .

We give digital analogues of classical theorems of topology for continuous functions defined on spheres, for digital simple closed curves. In particular, we show the following. ? A digital simple closed curve of more than 4 points is not contractible, i.e., its identity map is not nullhomotopic in . ? Let and be digital simple closed curves, each symmetric with respect to the origin, such that (where is the number of points in ). Let be a digitally continuous antipodal map. Then is not nullho- motopic in . ? Let be a digital simple closed curve that is symmetric with respect to the origin. Let be a digitally continuous map. Then there is a pair of antipodes such that .

Cite this paper

nullL. Boxer, "Continuous Maps on Digital Simple Closed Curves,"*Applied Mathematics*, Vol. 1 No. 5, 2010, pp. 377-386. doi: 10.4236/am.2010.15050.

nullL. Boxer, "Continuous Maps on Digital Simple Closed Curves,"

References

[1] A. Rosenfeld, “Digital Topology,” American Mathe- matical Monthly, Vol. 86, 1979, pp. 76-87.

[2] A. Rosenfeld, “Continuous Functions on Digital Pic- tures,” Pattern Recognition Letters, Vol. 4, No. 3, 1986, pp. 177-184.

[3] A. Rosenfeld, “Directions in Digital Topology,” 11th Summer Conference on General Topology and Applica- tions, 1995. http://atlas-conferences.com/cgi-bin/abstract/ caaf-71

[4] Q. F. Stout, “Topological Matching,” Proceedings 15th Annual Symposium on Theory of Computing, Boston, 1983, pp. 24-31.

[5] T. Y. Kong, “A Digital Fundamental Group,” Computers and Graphics, Vol. 13, No. 1, 1989, pp. 159-166.

[6] T. Y. Kong, A. W. Roscoe and A. Rosenfeld, “Concepts of Digital Topology,” Topology and Its Applications, Vol. 46, No. 3, 1992, pp. 219-262.

[7] L. Boxer, “Digitally Continuous Functions,” Pattern Recognition Letters, Vol. 15, No. 8, 1994, pp. 833-839.

[8] L. Boxer, “A Classical Construction for the Digital Fundamental Group,” Journal of Mathematical Imaging and Vision, Vol. 10, No. 1, 1999, pp. 51-62.

[9] T. Y. Kong and A. Rosenfeld, “Topological Algorithms for Digital Image Processing,” Elsevier, New York, 1996.

[10] L. Boxer, “Homotopy Properties of Sphere-Like Digital Images,” Journal of Mathematical Imaging and Vision, Vol. 24, No. 2, 2006, pp. 167-175.

[11] G. T. Herman, “Oriented Surfaces in Digital Spaces,” CVGIP: Graphical Models and Image Processing, Vol. 55, No. 1, 1993, pp. 381-396.

[12] L. Chen, “Gradually Varied Surfaces and Its Optimal Uniform Approximation,” SPIE Proceedings, Bellingham, Vol. 2182 1994, pp. 300-307.

[13] L. Chen, “Discrete Surfaces and Manifolds,” Scientific Practical Computing, Rockville, 2004.

[14] L. Boxer, “Digital Products, Wedges, and Covering Spaces,” Journal of Mathematical Imaging and Vision, Vol. 25, 2006, pp. 159-171.

[15] U. Eckhardt and L. Latecki, “Digital Topology,” In: Current Topics in Pattern Recognition Research, Research Trends, Council of Scientific Information, 1994. http://cosmic.rrz.uni-hamburg.de/webcat/mathematik/eckhardt/eck00001/eck00001.pdf

[16] R. O. Duda, P. E. Hart and J. H. Munson, “Graphical Data Processing Research Study and Experimental Investigation,” Descriptive Note: Quarterly Report No. 7, March 1967, pp. 28-30.

[17] E. Khalimsky, “Motion, Deformation, and Homotopy in Finite Spaces,” Proceedings IEEE International Confer- ence on Systems, Man, and Cybernetics, Boston, 1987, pp. 227-234.

[18] L. Boxer, “Properties of Digital Homotopy,” Journal of Mathematical Imaging and Vision, Vol. 22, No. 1, 2005, pp. 19-26.

[19] J. Dugundji, “Topology,” Allyn and Bacon, Inc., Boston, 1966.

[20] S. E. Han, “Non-Product Property of the Digital Funda- mental Group,” Information Sciences, Vol. 171, No. 1-3, 2005, pp. 73-91.

[21] S. E. Han, “Digital Coverings and Their Applications,” Journal of Applied Mathematics and Computing, Vol. 18, No. 1-2, 2005, pp. 487-495.

[1] A. Rosenfeld, “Digital Topology,” American Mathe- matical Monthly, Vol. 86, 1979, pp. 76-87.

[2] A. Rosenfeld, “Continuous Functions on Digital Pic- tures,” Pattern Recognition Letters, Vol. 4, No. 3, 1986, pp. 177-184.

[3] A. Rosenfeld, “Directions in Digital Topology,” 11th Summer Conference on General Topology and Applica- tions, 1995. http://atlas-conferences.com/cgi-bin/abstract/ caaf-71

[4] Q. F. Stout, “Topological Matching,” Proceedings 15th Annual Symposium on Theory of Computing, Boston, 1983, pp. 24-31.

[5] T. Y. Kong, “A Digital Fundamental Group,” Computers and Graphics, Vol. 13, No. 1, 1989, pp. 159-166.

[6] T. Y. Kong, A. W. Roscoe and A. Rosenfeld, “Concepts of Digital Topology,” Topology and Its Applications, Vol. 46, No. 3, 1992, pp. 219-262.

[7] L. Boxer, “Digitally Continuous Functions,” Pattern Recognition Letters, Vol. 15, No. 8, 1994, pp. 833-839.

[8] L. Boxer, “A Classical Construction for the Digital Fundamental Group,” Journal of Mathematical Imaging and Vision, Vol. 10, No. 1, 1999, pp. 51-62.

[9] T. Y. Kong and A. Rosenfeld, “Topological Algorithms for Digital Image Processing,” Elsevier, New York, 1996.

[10] L. Boxer, “Homotopy Properties of Sphere-Like Digital Images,” Journal of Mathematical Imaging and Vision, Vol. 24, No. 2, 2006, pp. 167-175.

[11] G. T. Herman, “Oriented Surfaces in Digital Spaces,” CVGIP: Graphical Models and Image Processing, Vol. 55, No. 1, 1993, pp. 381-396.

[12] L. Chen, “Gradually Varied Surfaces and Its Optimal Uniform Approximation,” SPIE Proceedings, Bellingham, Vol. 2182 1994, pp. 300-307.

[13] L. Chen, “Discrete Surfaces and Manifolds,” Scientific Practical Computing, Rockville, 2004.

[14] L. Boxer, “Digital Products, Wedges, and Covering Spaces,” Journal of Mathematical Imaging and Vision, Vol. 25, 2006, pp. 159-171.

[15] U. Eckhardt and L. Latecki, “Digital Topology,” In: Current Topics in Pattern Recognition Research, Research Trends, Council of Scientific Information, 1994. http://cosmic.rrz.uni-hamburg.de/webcat/mathematik/eckhardt/eck00001/eck00001.pdf

[16] R. O. Duda, P. E. Hart and J. H. Munson, “Graphical Data Processing Research Study and Experimental Investigation,” Descriptive Note: Quarterly Report No. 7, March 1967, pp. 28-30.

[17] E. Khalimsky, “Motion, Deformation, and Homotopy in Finite Spaces,” Proceedings IEEE International Confer- ence on Systems, Man, and Cybernetics, Boston, 1987, pp. 227-234.

[18] L. Boxer, “Properties of Digital Homotopy,” Journal of Mathematical Imaging and Vision, Vol. 22, No. 1, 2005, pp. 19-26.

[19] J. Dugundji, “Topology,” Allyn and Bacon, Inc., Boston, 1966.

[20] S. E. Han, “Non-Product Property of the Digital Funda- mental Group,” Information Sciences, Vol. 171, No. 1-3, 2005, pp. 73-91.

[21] S. E. Han, “Digital Coverings and Their Applications,” Journal of Applied Mathematics and Computing, Vol. 18, No. 1-2, 2005, pp. 487-495.