PARABOLAS IN GENERALIZED TAXICAB PLANE

Year 2026, Volume: 27 Issue: 1 , 99 - 110 , 27.03.2026

Abdilkadir Altıntaş , Süheyla Ekmekçi

https://izlik.org/JA58ZS58YN

Abstract

This study investigates parabolas in the generalized taxicab plane, a non-Euclidean geometry where distance is measured using weighted coordinate axes with positive parameters (a,b). Using the focus-directrix definition, it examines the structures of generalized taxicab parabolas (briefly, GTPs) with respect to the positions of their directrices. It is determined that generalized taxicab parabolas are simple rectilinear figures. It further provides a detailed analysis of GTPs, including their axes, vertices, latus rectum, and focal lengths. It reveals that the latus rectum length of a GTP is four times its focal length regardless of the directrix type. Also, the algorithm is presented to visualize GTPs for all types of directrices. Additionally, the study identifies degenerate cases in which the focus is on the directrix, and it is demonstrated that the obtained geometric structures reduce to single lines or unions of planar regions defined by vertical and horizontal lines through the focus.

Keywords

Generalized taxicab distance , Generalized taxicab parabola , Focus-directrix conics , Degenerate conics , non-Euclidean geometry

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, Çalış C. On the Voronoi diagram and taxicab plane. Erzincan University Journal of Science and Technology 2021;14(1):175–181.
• [3] Altıntaş A. The application of some geometric problems on Euclidean plane using generalized taxicab metric, MSc, Eskişehir Osmangazi University, Eskişehir, Turkey, 2009.
• [4] Bayar A, Ekmekçi S, Öztürk İ. On complex numbers and taxicab plane. Mathematical Sciences and Applications E-Notes 2015;3(1):58–64.
• [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, Özcan M. On trigonometric functions and cosine and sine rules in taxicab plane. International Electronic Journal of Geometry 2009;2(1):17–24.
• [7] Çolakoğlu BH. The generalized taxicab group. International Electronic Journal of Geometry 2018;11(2):83-89.
• [8] Ekmekçi S, Bayar A, Akça Z. On the plane geometry with generalized absolute value metric. Math Probl Eng 2008;1(0):1-8.
• [9] Ekmekçi S, Akça Z, Altıntaş A. On trigonometric functions and norm in the generalized taxicab metric. Mathematical Sciences and Applications E-Notes 2015;3(2):27-33.
• [10] Ekmekçi S, Bayar A, Altıntaş A. On the group of isometries of the generalized taxicab plane. International Journal of Contemporary Mathematical Sciences 2015;10(4):159–166.
• [11] Gelişgen Ö, Kaya R. The taxicab space group. Acta Mathematica Hungarica 2008;122(1):187-200.
• [12] 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.
• [13] Krause EF. Taxicab geometry. Menlo Park, CA:Addison –Wesley Publishing Company;1975.
• [14] Laatsch R. Pyramidal sections in taxicab geometry. Math Mag 1982;55:205–212.
• [15] Reynolds BE. Taxicab geometry. Pi Mu Epsilon J 1980; 7: 77–88.
• [16] Salihova S. On the geometry of maximum metric. Doctoral dissertation, Eskişehir Osmangazi University, Institute of Science and Technology, Turkey, 2006.
• [17] Turan M, Özcan M. General equation for Chinese checker conics and focus-directrix Chinese checker conics. International Journal of Pure and Applied Mathematics 2006;30(3):393-401.
• [18] Turan M, Özcan M. Two-foci Chinese checker ellipses. International Journal of Pure and Applied Mathematics 2004;16(1):119-127.
• [19] Turan M, Özcan M. Two-foci Chinese checker hyperbolas. International Journal of Pure and Applied Mathematics 2004;16(4):509-520.
• [20] Wallen LJ. Kepler, the taxicab metric, and beyond: An isoperimetric primer. The College Mathematics Journal 1995;26(3):178-190.