Elliptic Inversions in Taxicab Geometry

Yıl 2024, Cilt: 24 Sayı: 2, 266 - 272, 29.04.2024

Zeynep Can

https://doi.org/10.35414/akufemubid.1353161

Öz

The goal of this research is to introduce inversion with respect to an ellipse which is a generalization of the classical circular inversion in taxicab plane and to investigate general properties and basic concepts of this transformation in taxicab geometry. The cross ratio is preserved under the elliptic inversion in taxicab plane though this transformation is not an isometry. Thus some properties such as cross ratio and harmonic conjugates of the elliptic inversions in R_T^2 are also studied.

Anahtar Kelimeler

Inversion, elliptic inversion, taxicab metric, cross ratio, harmonic conjugates

Taksi Geometride Eliptik İnversiyonlar

Öz

Bu çalışmanın amacı klasik çembersel inversiyonların bir genelleştirmesi olan elipse göre inversiyonları taksi düzleminde çalışmak ve bu dönüşümün genel özelliklerini ve temel yapılarını taksi geometride araştırmaktır. Bu dönüşüm bir izometri olmasa da çapraz oran taksi düzleminde eliptik inversiyonlar altında korunur. Bu yüzden bu araştırmada eliptik inversiyonların çapraz oran ve harmonik eşlenik gibi bazı özellikleri üzerine de R_T^2 de çalışılmıştır.

Anahtar Kelimeler

inversiyon, eliptik inversiyon, taksi metriği, çapraz oran, harmonik eşlenik

Kaynakça

• Akça, Z. and Kaya, R., 1997. On the Taxicab Trigonometry. Journal of Inst. of Math. & Comp. Sci., 10(3), 151-159. Bayar, A. and Ekmekçi, S., 2014. On Circular Inversions in Taxicab Plane. J. Adv. Res. Pure Math., 6, 33-39. https://doi.org/10.5373/jarpm.1934.013114
• Blair, D., 2000. Inversion Theory and Conformal Mapping, Student Mathematical Library, American Mathematical Society. Chen, G., 1992. Lines and Circles in Taxicab Geometry, M.S. thesis, Department of Mathematics and Computer Science, Centered Missouri State University. Childress, N., 1965. Inversion with Respect to the Central Conics. Math. Mag., 38(3). https://doi.org/10.1080/0025570X.1965.11975615
• Divjak, B., 2000. Notes on Taxicab Geometry. Scientific and Professional Information Journal of Croatian Society for Constructive Geometry and Computer Graphics (KoG), 5, 5-9. Gelişgen, Ö., 2007. On the Minkowski Geometries: A General Analysis About Taxicab, Chinese Checkers and -Geometries, Phd Thesis, Eskişehir Osmangazi University, 163. Gelişgen , Ö. and Ermiş, T., 2019. Some Properties of Inversions in the Alpha Plane. Forum Geometricorum, 19, 1-9. Ho, Y. P. and Lıu, Y., 1996. Parabolas in Taxicab Geometry. Missouri J. of Math. Sci., 8, 63-72. Kaya, R., 2004. Area Formula For Taxicab Triangles. Pi Mu Epsilon, 12(4), 219-220.
• Krause, E.F., 1975. Taxicab Geometry, Addision-Wesley.
• Laatsch, R., 1982. Pramidal Sections in Taxicab Geometry. Mitt. Math. Magazine, 55, 205-212.
• Menger, K., 1952. You Will Like Geometry, Guildbook of the Illinois Institute of Technology Geometry Exhibit, Museum of Science and Industry, Chicago, IL.
• Minkowski, H., 1967. Gesammelte Abhandlungen, Chelsa Publishing Co. New York.
• Nickel, J. A., 1995. A Budget of inversion. Math. Comput. Modelling, 21, 87-93.
• Özcan, M. and 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.
• Patterson, B.C., 1933. The origins of the Geometric Principle of Inversion. Isis, 19(1), 154 - 180.
• Ramirez, J., 2014. Inversions in an ellipse. Forum Geom., 14, 107-115.
• Ramirez, J. and Rubiano, G. 2014. A Geometrical Construction of Inverse Points with Respect to an Ellipse. Int. J. Math. Ed. Sci. Tech., 45(8), 1254-1259. https://doi.org/10.1080/0020739X.2014.914255
• Reynolds, B.E., 1980. Taxicab Geometry. Pi Mu Epsilon Journal, 7, 77-88. Schattschneider, D. J., 1984. The Taxicab group. Amer. Math. Monthly, 91, 423-428. https://doi.org/10.1080/00029890.1984.11971453
• So, S.S., 2002. Recent Developments in Taxicab Geometry. Cubo Mathematica Educational, 4(2), 79-96. Thompson, A. C., 1996. Minkowski Geometry, Cambridge University Press.
• Tian, S., So, S. S. and Chen, G., 1997. Concerning Circles in Taxicab Geometry. J. Math. Educ. Sci. Technol., 28, 727-733. https://doi.org/10.1080/0020739970280509