A generalization of the Minkowski distance and new definitions of the central conics

In this paper, we give a generalization of the well-known Minkowski distance family in the n-dimensional Cartesian coordinate space. Then we consider three special cases of this family, which are also generalizations of the taxicab, Euclidean, and maximum metrics, respectively, and we determine some circle properties of them in the real plane. While we determine some properties of circles of these generalized distances, we discover a new definition of ellipses, and then we also determine a similar definition of hyperbolas, which will be new members among different metrical definitions of central conics in the Euclidean plane.

Keywords:

Minkowski distance, lp-norm, lp-metric, taxicab distance, Manhattan distance, Euclidean distance, maximum distance, Chebyshev distance, ellipse, hyperbola, central conics, asymptote, eccentrix eccentricity,

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[1] Bhatia R. Matrix Analysis. New York, NY, USA: Springer-Verlag, 1997.
[2] Çolakoğlu HB. A generalization of the taxicab metric and related isometries. Konuralp Journal of Mathematics 2018; 6 (1): 158-162.
[3] Çolakoğlu HB. The generalized taxicab group. International Electronic Journal of Geometry 2018; 11 (2): 83-89.
[4] Çolakoğlu HB. On generalized taxicab metric in three dimensional space. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 2019; 68 (2): 1359-1369.
[5] Çolakoğlu HB. On the distance formulae in the generalized taxicab geometry. Turkish Journal of Mathematics 2019; 43 (3): 1578-1594.
[6] Deza MM, Deza E. Encyclopedia of Distances. Berlin, German: Springer, 2009.
[7] Giaquinta M, Hildebrandt S. Calculus of Variations II. Berlin, Germany: Springer-Verlag, 1996.
[8] Horvath AG, Martini H. Conics in normed planes. Extracta Mathematicae 2011; 26 (1): 29-43.
[9] Horvath AG, Prok I. On the constructibility of the axes of an ellipsoid: the construction of Chasles in practice. Journal of Geometry 2018; 109 (29). doi: 10.1007/s00022-018-0437-z
[10] Kaya R, Akça Z, Günaltılı İ, Özcan M. General equation for taxicab conics and their classification. Mitteilungen der Mathematischen Gesellschaft in Hamburg 2000; 19: 135-148.
[11] Krause EF. Taxicab Geometry: An Adventure in Non-Euclidean Geometry. New York, NY, USA: Dover Publications, 1986.
[12] McCartin BJ. A matrix analytic approach to conjugate diameters of an ellipse. Applied Mathematical Sciences 2013; 7 (36): 1797-1810.
[13] McCartin BJ. On Euler’s synthetic demonstration of Pappus’ construction of an ellipse from a pair of conjugate diameters. International Mathematical Forum 2013; 8 (22): 1049-1056.
[14] Mitchell DW. A property of hyperbolas and their asymptotes. The Mathematical Gazette 2012; 96 (536): 299-301.
[15] Salihova S. On the geometry of the maximum metric. PhD, Eskişehir Osmangazi University, Eskişehir, Turkey, 2006.
[16] Senlin W, Donghai J, Alonso J. Metric ellipses in Minkowski planes. Extracta Mathematicae 2005; 20 (3): 273-280.
[17] Thompson AC. Minkowski Geometry, Encyclopedia of Mathematics and Its Applications 63. Cambridge, UK: Cambridge University Press, 1996.
[18] Venit S. The asymptotes of an oblique hyperbola. Pi Mu Epsilon Journal 1980; 7 (2): 89-92.
[19] Wallen LJ. Kepler, the taxicab metric, and beyond: an isoperimetric primer. The College Mathematics Journal 1995; 26 (3): 178-190.
[20] Zwillinger D (editor). CRC Standard Mathematical Tables and Formulas. 33rd Edition. New York, NY, USA: CRC Press, Taylor & Francis Group, 2018.