On The Inversion in the Generalized Taxicab Circle

Yıl 2023, Cilt: 5 Sayı: 2, 50 - 58, 30.12.2023

Süheyla Ekmekçi Yeliz Bilgin

Öz

In this study, inversions in generalized taxicab circles are defined and their properties are presented. The results obtained by examining the images of points under inversions in generalized taxicab circles are provided. Additionally, the concept of the directed generalized taxicab length of a line segment is introduced. Based on this concept, the definitions of the generalized taxicab cross-ratio and the generalized taxicab harmonic conjugate are given, along with some of their properties. The impact of inversion in the generalized taxicab circle on these concepts is investigated.

Anahtar Kelimeler

Generalized taxicab distance, Inversion in the generalized taxicab circle, Cross-ratio, Harmonic Conjugate

Kaynakça

• Akça, Z., & Kaya R. (2004). On the norm in higher dimensional taxicab spaces. Hadronic Journal Supplement, 19(5), 491-501.
• Akça, Z., & Kaya R. (2004). On the distance formulae in three dimensional taxicab space. Hadronic Journal, 27(5), 521-532.
• Akça, Z., & Nazlı, S. (2023). On the Thales theorem in the iso-taxicab plane. Hagia Sophia Journal of Geometry, 5(1), 15-20.
• Bayar A., & Ekmekçi S. (2014). On circular inversions in taxicab plane. Journal of Advanced Research in Pure Mathematics, 6(4), 33-39.
• Kaya, R., Akça, Z., Günaltılı, ̇İ, & Özcan, M. (2000). General equation for taxicab conics and their classification. Mitteilungen der Mathematische Gesellschaft in Hamburg, 19(0), 135-148.
• Krause, E. F. (1975). Taxicab geometry. Addison-Wesley Publishing Company, Menlo Park, California, USA.
• Altıntaş , A. (2009). The application of some geometric problems on Euclidean plane using generalized taxi metric (Manhattan metric). Master’s Thesis, Eskişehir Osmangazi University, Eskişehir.
• Colakoğlu, H. B. (2018). The generalized taxicab group. International Electronic Journal of Geometry, 11(2), 83-89.
• Ekmekçi, S., Bayar, A., & Altıntaş, A. (2015). On the group of isometries of the generalized taxicab plane. International Journal of Contemporary Mathematical Sciences, 10(4), 159-166.
• Ekmekçi, S., Akça, Z., & Altıntaş, K. (2015). On trigonometric functions and norm in the generalized taxicab metric. Mathematical Sciences and Applications E-Notes, 3(2), 27-33.
• Wallen, L.J. (1995). Kepler, the taxicab metric, and beyond: an isoperimetric primer. The College Mathematics Journal, 26(3), 178-190.
• Blair, D. E. (2000). Inversion theory and conformal mapping. Student Mathematical Library (Vol. 9), The American Mathematical Society.
• Childress, N. A. (1965). Inversion with respect to the central conics. Mathematics Magazine, 38(3), 147-149.
• Nickel, J. A. (1995). A budget of inversion. Mathematical and Computer Modelling, 21(6), 87-93.
• Ramírez, J. L. (2014). Inversions in an ellipse. Forum Geometricorum, 14, 107-115.
• Ramírez, 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.
• Ramírez, J. L., Rubiano, G. N., & Zlobec, B. J. (2015). Generating fractal patterns by using p-circle inversion. Fractals, 23(04), Article Number: 1550047.
• Ekmekçi, S. (2023). A note on the maximum circle inverses of lines in the maximum plane. Ikonion Journal of Mathematics, 5(2),1-9.
• Gelişgen, Ö., & Ermiş, T. (2019). Some properties of inversions in alpha plane. Forum Geometricorum, 19, 1-9.
• Bilgin, Y. (2023). Inversions with respect to circles in the generalized taxicab plane. Master’s Thesis, Eskişehir Osmangazi University, Eskişehir.
• Can, Z. (2022). On spherical inversions in three dimensional Tetrakis Hexahedron space. Erciyes Üniversitesi Fen Bilimleri Enstitüsü Dergisi, 38(1), 100-108.
• Gelişgen, Ö., & Ermiş, T. (2023). Inversions and fractal patterns in Alpha plane. International Electronic Journal of Geometry, 16(1), 398-411.
• 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.
• Yüca, G., & Can, Z. (2020). On the circular inversion in Maximum plane. Ikonion Journal of Mathematics, 2(2), 26-34.
• Cırık, Y., & Ekmekçi, S. (2022). On the maksimum spherical inversions. Erzincan University Journal of Science and Technology, 15(1), 360-371.
• 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.
• Pekzorlu, A., & Bayar, A. (2020). Taxicab spherical inversions in taxicab space. Journal of Mahani Mathematical Research Center, 9(1), 45-54