On generalized taxicab metric in three dimensional space

In this paper, we define the generalized taxicab distance function in three dimensional space, which includes the taxicab distance function as a special case, and we show that three dimensional generalized taxicab distance function determines a metric. Then we give some properties of three dimensional generalized taxicab metric, and determine Euclidean isometries of the space preserving the generalized taxicab metric.

___

  • Akça, Z. and Kaya, R., On the distance formulae in three dimensional taxicab space, Hadronic Journal, 27, No.5 (2004), 521-532.
  • Akça, Z. and Kaya, R., On the taxicab trigonometry, Jour. of Inst. of Math. & Comp. Sci. (Math. Ser.), 10, No.3 (1997), 151-159.
  • Çolakoğlu H. B., Concerning the alpha distance, Algebras Groups Geom. 8(2011), 1-14.
  • Ekici, C., Kocayusufoğlu, İ. and Akça, Z., The norm in taxicab geometry, Tr. J. of Mathematics, 22 (1998), 295-307.
  • Ekmekçi, E., Bayar, A. and Altıntaş, A. K., On the group of isometries of the generalized taxicab plane, International Journal of Contemporary Mathematical Sciences, 10, No.4 (2015), 159-166.
  • Ekmekçi, S., Akça, Z. and Altıntaş, A. K., On trigonometric functions and norm in the generalized taxicab metric, Mathematical Sciences And Applications E-Notes, 3, No.2 (2015), 27-33.
  • Gelişgen, Ö. and Kaya, R., The taxicab space group, Acta Math. Hungar., 122, Issue 1â€"2 (2009), 187â€"200.
  • Gelişgen, Ö. and Kaya, R., On α-distance in three dimensional space, Applied Science, 8 (2006), 65-69.
  • Gelişgen, Ö. and Kaya, R., Generalization of α-distance to n-dimensional Space, KoG, 10 (2006), 33-35.
  • Kaya, R., Akça, Z., Günaltılı, İ. and Ö zcan, M., General equation for taxicab conics and their classification, Mitt. Math. Ges. Hamburg, 19 (2000), 135-148.
  • Kaya, R. and Çolakoğlu, H. B., Taxicab versions of some Euclidean theorems, International Journal of Pure And Applied Mathematics, 26, No.1 (2006), 69-81.
  • Kocayusufoglu, İ. and Özdamar, E., Isometries of taxicab geometry, Commum. Fac. Sci. Univ. Ank. Series A1, 47 (1998), 73-83.
  • Krause, E. F., Taxicab Geometry, Addison-Wesley, Menlo Park, California, 1975.
  • Martin, G. E., Transformation Geometry, Springer-Verlag, New York Inc., 1997, pp.182.
  • Menger, K., You Will Like Geometry, Guidebook of Illinois Institute of Technology Geometry Exhibit, Museum of Science and Industry, Chicago, Illinois, 1952.
  • Schattschneider, D. J., The taxicab group, American Mathematical Monthly, 91, No.7 (1984), 423-428.
  • Thompson, K. P., The nature of length, area, and volume in taxicab geometry, International Electronic Journal of Geometry, 4, No.2 (2011), 193-207.
  • Wallen, L. J., Kepler, the taxicab metric, and beyond: An isoperimetric primer, The College Mathematics Journal, 26, No.3 (1995), 178-190.