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