On isometries of $Bbb{R}^2_{pi dot n}$

On isometries of $Bbb{R}^2_{pi dot n}$

In this work, we introduce a family of distance functions and show that the group of isometries of the plane associated with the induced metrics is the semi-direct product of the Dihedral group $D_{2n}$ and the translation group T (2).

___

  • [1] A.Bayar,S.Ekmek¸ci and Z. Ak¸ca, On the plane geometry with generalized absolute value metric, Mathematical problems in Engineering, (2008),673275, 8 pages.
  • [2] M. A. Butt, P. Marogos, Optimum Design of Chamfer Distance Transforms, IEEE Transactions on Image Processing, 7 (1998), 10.
  • [3] H. B. C¸ olako˘glu, Ö. Geli¸sgen and R. Kaya, Pythagorean Theorem in Alpha Plane, Mathematical Comminations, 14, 2, (2009), 211-221.
  • [4] Ö. Geli¸sgen and R.Kaya, The taxicab space group, Acta Math. Hungar, 122 (2009), 187-200.
  • [5] S.-M. Jung, Mappings preserving some geometric figures, Acta Math. Hungar., 100 (2003),167-175.
  • [6] S.-M. Jung, On mappings preserving pentagons, Acta Math. Hungar., 110 (2006), 261-266.
  • [7] R. Kaya, Ö. Geli¸sgen, S. Ekmek¸ci and A. Bayar, Group of Isometries of CC-plane, Missouri J. of Math. Sci., 3 (2006), 3.
  • [8] E. F. Krause, Taxicab Geometry, Addison - Wesley Publishing Company, (Menlo Park, CA 1975).
  • [9] E. W. Miller, Revisiting The Geometry of Ternary Diagram with the Half-Taxi Metric,Mathematical Geology, 34, (2002), 3.
  • [10] D. J. Schattschneider, The taxicab group, Amer. Math. Monthly, 91 (1984), 423-428.
  • [11] K. O. Sowell, Taxicab geometry-A new slant, Mathematics Magazine, 62 (1989), 4.
  • [12] A. C. Thompson, Minkowski Geometry, Cambridge University Press (1996).
  • [13] L. J. Wallen, Kepler, the taxicab metric, and beyond : An isoperimetric Primer, The College Mathematics Journal , 26 (1995), 3.
  • [14] M. J. Willard, Symmetry Groups and their Applications, Academic Press, New York, 190 (1972), 16-23.