On the Chinese Checkers Circular Inversions in the Chinese Checkers Plane

Year 2022, Volume: 4 Issue: 2, 28 - 34, 30.12.2022

Adnan Pekzorlu , Ayşe Bayar

Abstract

In present article, we introduce an inversion with respect to a Chinese Checkers circle in the Chinese Checkers plane, and prove several properties of this inversion. We also study cross ratio, harmonic conjugates and the images of lines, planes and Chinese Checkers circle in the Chinese Checkers plane.

Keywords

Chinese Checkers plane, Chinese Checkers circle, inversion, harmonic conjugates.

References

• Krause, E. F. (1975). Taxicab geometry, Addison-Wesley, Menlo Park, California.
• Chen, G. (1992). Lines and circles in Taxicab geometry. Master Thesis, Department of Mathematic and Computer Science, Central Missouri State Uni.
• Minkowski, H. (1967). Gasammelte abhandlungen, Chelsea Publishing Co., New York.
• Gelisgen, Ö., & Kaya, R. (2008). The Taxicab space group. Acta Mathematica Hungarica, 122(1), 187-200.
• Kaya, R., Akça, Z., Günaltılı, İ., & Özcan, M. (2000). General equation for Taxicab conics and their classification. Mitt.Math. Ges. Hamburg, 19, 135-148.
• Menger, K., & Sutton, R. M. (1952). You will like geometry. American Journal of Physics, 20(8), 521-521.
• Schattschneider, D. J. (1984). The Taxicab group. American Mathematical Monthly, 91, 423-428.
• Akça Z., & Kaya, R. (2004). On the distance formulae in three dimensional Taxicab space. Hadronic Journal, 27(5), 521-532.
• Blasjo,V. (2009). Jakob Steiner’s Systematische Entwickelung: The culmination of classical geometry. The Mathematical Intelligencer, 31(1), 21-29.
• Blair, D. (2000). Inversion theory and conformal mapping. Student Mathematical Library, American Mathematical Society, 9.
• Deza, E., & Deza, M. (2006). Dictionary of distances, Elsevier Science.
• Nickel, J. A. (1995). A budget of inversion, Math. Comput. Modelling, 21(6), 87-93.
• Childress, N. (1965). Inversion with respect to the central conics. Mathematics Magazine, 38(3), 147-149.
• Ramirez, J. L. (2013). An Introduction to inversion in an ellipse. arXiv preprint arXiv:1309.6378v1.
• Ramirez, J. L., & Rubiano, G. N. ( 2017). A generalization of the spherical inversion. International Journal of Mathematical Education in Science and Technology, 48(1), 132-149.
• Bayar, A., & Ekmekçi, S. (2014). On circular inversions in Taxicab plane. J. Adv. Res. Pure Math. 6(4), 33-39.
• Pekzorlu, A., & Bayar, A. (2020). Taxicab spherical inversions in Taxicab space. Journal of Mahani Math. Research Center, 9(1-2), 45-54.
• Gelisgen, Ö., & Ermiş, T. (2019). Some properties of inversions in alpha plane. Forum Geometricorum, 19, 1-9.
• Ramirez, J. L., Rubiano, G. N., & Zlobec, B. J. (2015). Generating fractal patterns by using p−circle inversion. Fractals, 23(4), 1-13.
• Pekzorlu, A., & Bayar, A. (2020). On the Chinese Checkers spherical inversions in three dimensional Chinese Checkers space. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, 69(2), 1498-1507.
• Gelisgen, Ö., Kaya, R., & Ozcan, M. (2006). Distance formulae in the Chinese Checker space. Int.J. Pure Appl. Math., 26(1), 35-44.
• Kaya, R., Gelisgen, Ö., Bayar, A., & Ekmekçi, S. (2006). Group of isometries of CC-plane. Missouri Journal of Mathematical Sciences, 18, 221-233.
• Özcan, M., & Kaya, R. (2002). On the ratio of directed lengths in the Taxicab plane and related properties. Missouri Journal of Mathematical Sciences, 14(2), 107-117