On Isometries of Rpn2

Abstract

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 D2n and the translation group T (2). 

Keywords

• Isometry,Distance function,Metric,2000 AMS Classification: 51 F 15

References

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 Trans- actions on Image Processing, 7 (1998), 10.
3. H. B. C¸ olako˘glu, ¨O. Geli¸sgen and R. Kaya, Pythagorean Theorem in Alpha Plane, Mathe- matical Comminations, 14, 2, (2009), 211-221.
4. ¨O. 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, ¨O. 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,
13. Kepler, the taxicab metric, and beyond : An isoperimetric Primer, The
14. College Mathematics Journal, 26 (1995), 3.
15. M. J. Willard, Symmetry Groups and their Applications, Academic Press, New York, 190 (1972), 16-23.