On the Inverses of Lines under the Inversion in a Generalized Taxicab Circle

Öz

In this study, the images of lines under the inversion in a generalized taxicab circle are examined. It is observed that the image of the line not passing through the inversion center is not a generalized taxicab circle, but the closed curve. The images of the lines under the inversion mapping are investigated depending on their positions and some features related to the images are presented. Furthermore, it is concluded that the inversion in a generalized taxicab circle maps the pencil of parallel lines (except the line passing through the center) to the set of the curves passing through the inversion center.

Anahtar Kelimeler

• Generalized taxicab distance
• Inversion in the generalized taxicab circle
• Image of the line under inversion

Kaynakça

1. Akça, Z., & Kaya R. (1997). On the taxicab trigonometry. Journal of Inst. Math. Comput. Sci. Math. Ser., 10(3), 151–159.
2. Akça, Z., & Kaya R. (2004). On the distance formulae in three dimensional taxicab space. Hadronic Journal, 27(5), 521-532.
3. Akça, Z., & Çalış, C. (2021). On the voronoi diagram and taxicab plane. Erzincan University Journal of Science and Technology, 14(1), 175-181.
4. Bayar A., & Ekmekçi, S. (2014). On circular inversions in taxicab plane. Journal of Advanced Research in Pure Mathematics, 6(4), 33-39.
5. Kaya, R., Akça, Z., Günaltılı, İ. & Özcan, M. (2000). General equation for Taxicab conics and their classification. Mitt. Math. Ges. Hamburg, 19(0), 136-148.
6. Krause, E. F. (1975). Taxicab geometry. Addison -Wesley Publishing Company, Menlo Park, California, USA.
7. Çolakoğlu, H. B. (2018). The generalized taxicab group. International Electronic Journal of Geometry, 11(2), 83-89.
8. Ekmekçi, S., Bayar, A., & Altıntaş, A. (2015). On the group of isometries of the generalized Taxicab metric. International Journal of Comtemporary Mathematical Sciences, 10(4), 159-166.

9. Ekmekçi, S., Akça, Z., & Altıntaş, A. (2015). On trigonometric functions and norm in the generalized Taxicab plane. Mathematical Sciences And Applications E-Notes, 3(2), 27-33.
10. Altıntaş, A. (2009). The application of some geometric problems on Euclidean plane using generalized taxicab metric. Master’s Thesis, Eskişehir Osmangazi University, Eskişehir.
11. Wallen, L. J. (1995). Kepler, the taxicab metric, and beyond: An isoperimetric primer. The College Mathematics Journal, 26(3), 178-190.
12. Blair, D. E. (2000). Inversion theory and conformal mapping. Student Mathematical Library, American Math. Society.
13. Gdawiec, K. (2014). Star-shaped set inversion fractals. Fractals, 22(4), 1450009-1-1450009-7.
14. Childress, N. (1965). Inversion with respect to the central conics. Mathematics Magazine, 38(3), 147-149.
15. Nickel, J. A. (1995). A budget of inversion. Math. Comput. Modelling, 21(6), 87-93.
16. Ramirez, J. L. (2014). Inversions in an ellipse. Forum Geometricorum, 14, 107-115.
17. 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.
18. 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.
19. Ekmekçi, S. (2023). A note on the maximum circle inverses of lines in the maximum plane. Ikonion Journal of Mathematics, 5(2), 1-9.
20. Gelişgen Ö., & Ermiş ̧ T. (2023). Inversions and fractal patterns in Alpha plane. Int. Electron. J. Geom., 16(1), 398-411.
21. Bilgin, Y. (2023). Inversions with respect to circles in the generalized Taxicab plane. Master’s Thesis, Eski s ̧ ehir Osmangazi University, Eskişehir.
22. Can, Z. (2022). On spherical inversions in three dimensional tetrakis hexahedron space. Erciyes Üniversitesi Fen Bilimleri Enstitüsü Dergisi, 38(1), 100-108.
23. 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.
24. Pekzorlu, A., & Bayar, A. (2020). Taxicab spherical invesions in Taxicab space. Journal of Mahani Mathematical Research Center, 9(1), 45-54.
25. 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.
26. Yüca, G., & Can, Z. (2020). On The circular inversion in maximum plane. Ikonion Journal of Mathematics, 2(2), 26-34.
27. Cırık, Y., & Ekmekçi, S. (2022). On the maksimum spherical inversions. Erzincan University, Journal of Science and Technology, 15(1), 360-371