ON THE MAXIMUM CIRCULAR INVERSES OF MAXIMUM CIRCLES

Year 2023, Volume: 24 Issue: 4 - Eskişehir Technical University Journal of Science and Technology A - Applied Sciences and Engineering, 324 - 335, 27.12.2023

Süheyla Ekmekçi

https://doi.org/10.18038/estubtda.1370728

Abstract

In this study, the inverses of maximum circles under the maximum circular inversion are examined. The maximum circular inversion is observed to transform maximum circles to curves consisting of parabolic arcs, line segments, and sometimes rays. It is seen that the image curve occurs in different shapes depending on the relative positions of the inversion circle and the maximum circle. While the images of maximum circles not passing through the inversion center are closed curves, the images of those passing through the inversion center are not closed. Inverse curves have been studied by considering the radii of the maximum circles and the positions of the lines joining their centers and the center of maximum inversion. Classifications of the inverses of maximum circles are presented.

Keywords

Maximum metric, Inversion in maximum circle, Maximum circular inversion, Inverse of maximum circle

References

• [1] Akça Z, Kaya R. On the taxicab trigonometry. Journal of Inst. Math. Comput. Sci. Math. Ser., 1997;10(3): 151–159.
• [2] Akça Z, Kaya R. On the distance formulae in three dimensional taxicab space. Hadronic Journal, 2004; 27(5): 521–532.
• [3] Akça Z, Nazlı S. On the versions in the plane R_π3^2 of some Euclidean theorems. New Trends in Mathematical Sciences, 2022; 10(1): 20–27.
• [4] Akça Z, Çalış C. On the voronoi diagram and taxicab plane. Erzincan Üniversity Journal of Science and Technology, 2021; 14(1): 175–181.
• [5] Bayar A, Ekmekçi S. On circular inversions in taxicab plane. Journal of Advanced Research in Pure Mathematics, 2014; 6(4): 33-39.
• [6] Bayar A, Ekmekçi S, Öztürk İ. On complex numbers and taxicab plane. Mathematical Sciences and Applications E-Notes, 2015; 3(1): 58–64.
• [7] Blair DE. Inversion Theory and Conformal Mapping. USA, American Math. Society, 2000.
• [8] Childress N. Inversion with respect to the central conics. Mathematics Magazine, 1965; 38(3): 147-149.
• [9] Cırık Y. The inversions in the plane and in the space donated with the maximum metric. MSc, Eskişehir Osmangazi University, Eskişehir, Turkey, 2022.
• [10] Cırık Y, Ekmekçi S. On the maksimum spherical ınversions. Erzincan University, Journal of Science and Technology, 2022; 15(1): 360-371.
• [11] Ekmekçi S. A note on the maximum circle inverses of lines in the maximum plane. Ikonion Journal of Mathematics, 2023; 5(2): 1-9.
• [12] Gdawiec K. Star-shaped set inversion fractals. Fractals, 2014; 22(4): 1450009-1-1450009-17.
• [13] Kaya R, Akça Z, Özcan M, Günaltılı İ. General equation for taxicab conics and their classification. Mitt. Math. Ges. Hamburg, 2000; 19(0): 135–148.
• [14] Krause EF. Taxicab Geometry. Menlo Park, California, USA, Addison –Wesley Publishing Company, 1975.
• [15] Nickel JA. A budget of inversion. Math. Comput. Modelling, 1995; 21(6): 87-93.
• [16] Pekzorlu A, Bayar A. On the Chinese checkers spherical invesions in three dimensional Chinese checker space. Communications Faculty of Sciences University of Ankara Series A1: Mathematics and Statistics, 2020; 69(2): 1498-1507.
• [17] Pekzorlu A, Bayar A. Taxicab spherical invesions in taxicab space. Journal of Mahani Mathematical Research Center, 2020; 9(1): 45-54.
• [18] Pekzorlu A, Bayar A. On the Chinese checkers circular inversions in the Chinese checkers plane. Hagia Sophia Journal of Geometry, 2022; 4(2): 28–34.
• [19] Ramirez JL. Inversions in an ellipse. Forum Geometricorum, 2014; 14: 107–115.
• [20] Ramirez JL, Rubiano GN. A geometrical construction of inverse points with respect to an ellipse. International Journal of Mathematical Education in Science and Technology, 2014; 45(8): 1254-1259.
• [21] Ramirez JL, Rubiano GN, Zlobec BJ. A generating fractal patterns by using p-circle inversion. Fractals, 2015; 23(4):1550047-1-1550047-13.
• [22] Salihova S. On the geometry of maximum metric. Ph.D, Eskişehir Osmangazi University, Eskişehir, Turkey, 2006.
• [23] Yüca G, Can Z. On the circular inversion in maximum plane. Ikonion Journal of Mathematics, 2020; 2(2): 26-34.