A Note on the Maximum Circle Inverses of Lines in the Maximum Plane

Year 2023, Volume: 5 Issue: 2, 1 - 9, 20.11.2023

Süheyla Ekmekçi

https://doi.org/10.54286/ikjm.1319223

Cited By: 1

Abstract

In this study, the images of lines under inversion in maximum circle are analytically examined. It is observed that the image of line not passing through the center of the inversion is not a maximum circle, but the closed curve containing at least one parabola arc. Some properties regarding the images of lines are introduced. Then, the images of line segments are examined according to the positions of their end points . In addition, it is seen that the inversion in maximum circle transforms the pencil of parallel lines (except line passing the center) to the set of the closed curves passing the center of inversion. Also, the images of the concurrent lines under inversion with respect to a maximum circle are considered and the results are presented.

Keywords

maximum distance, chebyshev distance, inversion

References

• Akça, Z., Kaya, R. (1997) On the taxicab trigonometry. Jour. of Inst. of Math. & Comp. Sci.(Math. Ser.), 10(3), 151-159.
• Akça, Z., Kaya R. (2004) On the distance formulae in three dimensional taxicab space. Hadronic J., 27(5): 521–532.
• Akça, Z., Nazli, S. (2022) On the versions in the Plane \mathbb{R}_{\pi3}^2 of some Euclidean theorems. New Trends in Mathematical Sciences, 10(1), 20-27.
• Akça, Z., Çalış, C. (2021) On the Voronoi Diagram and Taxicab Plane. Erzincan University Journal of Science and Technology, 14(1), 175-181.
• Bayar, A., Ekmekçi, S. (2014) On circular inversions in taxicab plane. J. Adv. Res. Pure Math, 6(4), 33-39.
• Bayar, A., Ekmekçi, S., Öztürk, İ. (2015) On Complex Numbers and Taxicab Plane. Mathematical Sciences and Applications E-Notes, 3(1), 58-64.
• Blair, D. E. (2000) Inversion Theory and Conformal Mapping. Student Mathematical Library, American Math. Society, 9.
• Childress, N. (1965) Inversion with Respect to the Central Conics. Mathematics Magazine, 38(3):147-149.
• Cırık, Y. (2022) Maksimum Metrik ile Donatılmış Düzlemde ve Uzayda İnversiyonlar. MSc thesis, Eskişehir Osmangazi University, Institute of Science and Technology, Turkey.
• Cırık, Y., Ekmekçi, S. (2022) On the Maksimum Spherical Inversions. Erzincan University, Journal of Science and Technology, 15(1):360-371.
• Gdawiec, K. (2014) Star-shaped set inversion fractals. Fractals, 22(4): 1450009-1-1450009-17.
• Kaya R., Akça Z., Özcan M., Günaltılı, İ. (2000) General equation for taxicab conics and their classification. Mitt. Math. Ges. Hamburg, 19(0): 135–148.
• Krause, E. F. (1975) Taxicab Geometry. Addison –Wesley Publishing Company, Menlo Park, California.
• Nickel, J.A. (1995) A Budget of Inversion. Mathematical and Computer Modelling, 21(6): 87-93.
• Pekzorlu, A., Bayar, A. (2020) On The Chinese Checkers Spherical Invesions in Three Dimensional Chinese Checker Space. Communications Faculty of Sciences University of Ankara Series A1: Mathematics and Statistics, 69(2): 1498-1507.
• Pekzorlu, A., Bayar, A. (2020) Taxicab Spherical Invesions in Taxicab Space. Journal of Mahani Mathematical Research Center, 9(1): 45-54.
• Pekzorlu, A., Bayar, A. (2022) On the Chinese Checkers Circular Inversions in the Chinese Checkers Plane. Hagia Sophia Journal of Geometry, 4(2): 28–34.
• Ramirez, J.L. (2014) Inversions in an Ellipse. Forum Geometricorum,14:107–115.
• Ramirez, J.L., Rubiano, G.N. (2014) A Geometrical Construction of Inverse Points with Respect to An Ellipse. International Journal of Mathematical Education in Science and Technology, 45(8): 1254-1259.
• Ramirez, J.L., Rubiano, G.N., Zlobec, B. J. (2015) A Generating fractal patterns by using p-circle inversion. Fractals, 23(4):1550047-1-1550047-13.
• Salihova, S. (2006) Maksimum Metriğinin Geometrisi Üzerine. Doctoral dissertation, Eskişehir Osmangazi University, Institute of Science and Technology, Turkey.
• Yüca, G., Can, Z. (2020) On the Circular Inversion in Maximum Plane. Ikonion Journal Of Mathematics, 2(2):26-34.