Research Article
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Year 2018, Volume: 11 Issue: 2, 83 - 89, 30.11.2018
https://doi.org/10.36890/iejg.545135

Abstract

References

  • [1] Akça, Z. and Kaya, R., On the taxicab trigonometry. Jour. of Inst. of Math. Comp. Sci. (Math. Ser.) 10 (1997), no.3, 151-159.
  • [2] Çolako˘ glu, H.B. and Kaya, R., A generalization of some well-known distances and related isometries. Math. Commun. 16 (2011), 21-35.
  • [3] Ekmekçi, E., Bayar, A. and Altınta¸s, A.K., On the group of isometries of the generalized taxicab plane. International Journal of Contemporary Mathematical Sciences 10 (2015), no.4, 159-166.
  • [4] Ekmekçi, S., Akça, Z. and Altınta¸s, A.K., On trigonometric functions and norm in the generalized taxicab metric. Mathematical Sciences And Applications E-Notes 3 (2015), no.2, 27-33.
  • [5] Geli¸sgen, Ö. and Kaya, R., The taxicab space group. Acta Math. Hung. 122 (2009), no.1-2, 187-200.
  • [6] Kaya, R., Akça, Z., Günaltılı, ˙I. and Ö zcan, M., General equation for taxicab conics and their classification. Mitt. Math. Ges. Hamburg 19 (2000), 135-148.
  • [7] Kocayusufoglu, ˙I. and Özdamar, E., Isometries of taxicab geometry. Commum. Fac. Sci. Univ. Ank. Series A1 47 (1998), 73-83. [8] Krause, E.F., Taxicab Geometry. Dover, New York, 1986.
  • [9] Martin, G.E., Transformation Geometry. Springer-Verlag, New York Inc., 1997.
  • [10] Menger, K., You Will Like Geometry. Guidebook of Illinois Institute of Technology Geometry Exhibit, Museum of Science and Industry, Chicago, Illinois, 1952.
  • [11] Richard, S. M. and George, D.P., Geometry, A Metric Approach With Models. Springer-Verlag, New York, 1981.
  • [12] Schattschneider, D.J., The taxicab group. American Mathematical Monthly 91 (1984), no.7, 423-428.
  • [13] Thompson, K.P. and Dray T., Taxicab Angles and Trigonometry. The Pi Mu Epsilon Journal 11 (2000), no.2, 87-96.
  • [14] Thompson, K.P., The nature of length, area, and volume in taxicab geometry. International Electronic Journal of Geometry 4 (2011), no.2, 193-207.
  • [15] Wallen, L.J., Kepler, the taxicab metric, and beyond: An isoperimetric primer. The College Mathematics Journal 26 (1995), no.3, 178-190.

The Generalized Taxicab Group

Year 2018, Volume: 11 Issue: 2, 83 - 89, 30.11.2018
https://doi.org/10.36890/iejg.545135

Abstract

In this study, we determine the generalized taxicab group consisting all isometries of the real plane
endowed with the generalized taxicab metric. First we develop natural analogues of Euclidean
reflection and rotation notions, and then determine all isometries in the generalized taxicab plane.
Finally, we show that the generalized taxicab group is semidirect product of the translation group
and the generalized taxicab symmetry group of the unit generalized taxicab circle, as Euclidean
group. We also see that there are transformations of the real plane onto itself which preserve the
generalized taxicab distance, but not preserve the Euclidean distance.

References

  • [1] Akça, Z. and Kaya, R., On the taxicab trigonometry. Jour. of Inst. of Math. Comp. Sci. (Math. Ser.) 10 (1997), no.3, 151-159.
  • [2] Çolako˘ glu, H.B. and Kaya, R., A generalization of some well-known distances and related isometries. Math. Commun. 16 (2011), 21-35.
  • [3] Ekmekçi, E., Bayar, A. and Altınta¸s, A.K., On the group of isometries of the generalized taxicab plane. International Journal of Contemporary Mathematical Sciences 10 (2015), no.4, 159-166.
  • [4] Ekmekçi, S., Akça, Z. and Altınta¸s, A.K., On trigonometric functions and norm in the generalized taxicab metric. Mathematical Sciences And Applications E-Notes 3 (2015), no.2, 27-33.
  • [5] Geli¸sgen, Ö. and Kaya, R., The taxicab space group. Acta Math. Hung. 122 (2009), no.1-2, 187-200.
  • [6] Kaya, R., Akça, Z., Günaltılı, ˙I. and Ö zcan, M., General equation for taxicab conics and their classification. Mitt. Math. Ges. Hamburg 19 (2000), 135-148.
  • [7] Kocayusufoglu, ˙I. and Özdamar, E., Isometries of taxicab geometry. Commum. Fac. Sci. Univ. Ank. Series A1 47 (1998), 73-83. [8] Krause, E.F., Taxicab Geometry. Dover, New York, 1986.
  • [9] Martin, G.E., Transformation Geometry. Springer-Verlag, New York Inc., 1997.
  • [10] Menger, K., You Will Like Geometry. Guidebook of Illinois Institute of Technology Geometry Exhibit, Museum of Science and Industry, Chicago, Illinois, 1952.
  • [11] Richard, S. M. and George, D.P., Geometry, A Metric Approach With Models. Springer-Verlag, New York, 1981.
  • [12] Schattschneider, D.J., The taxicab group. American Mathematical Monthly 91 (1984), no.7, 423-428.
  • [13] Thompson, K.P. and Dray T., Taxicab Angles and Trigonometry. The Pi Mu Epsilon Journal 11 (2000), no.2, 87-96.
  • [14] Thompson, K.P., The nature of length, area, and volume in taxicab geometry. International Electronic Journal of Geometry 4 (2011), no.2, 193-207.
  • [15] Wallen, L.J., Kepler, the taxicab metric, and beyond: An isoperimetric primer. The College Mathematics Journal 26 (1995), no.3, 178-190.
There are 14 citations in total.

Details

Primary Language English
Journal Section Research Article
Authors

Harun Barış Çolakoğlu

Publication Date November 30, 2018
Published in Issue Year 2018 Volume: 11 Issue: 2

Cite

APA Çolakoğlu, H. B. (2018). The Generalized Taxicab Group. International Electronic Journal of Geometry, 11(2), 83-89. https://doi.org/10.36890/iejg.545135
AMA Çolakoğlu HB. The Generalized Taxicab Group. Int. Electron. J. Geom. November 2018;11(2):83-89. doi:10.36890/iejg.545135
Chicago Çolakoğlu, Harun Barış. “The Generalized Taxicab Group”. International Electronic Journal of Geometry 11, no. 2 (November 2018): 83-89. https://doi.org/10.36890/iejg.545135.
EndNote Çolakoğlu HB (November 1, 2018) The Generalized Taxicab Group. International Electronic Journal of Geometry 11 2 83–89.
IEEE H. B. Çolakoğlu, “The Generalized Taxicab Group”, Int. Electron. J. Geom., vol. 11, no. 2, pp. 83–89, 2018, doi: 10.36890/iejg.545135.
ISNAD Çolakoğlu, Harun Barış. “The Generalized Taxicab Group”. International Electronic Journal of Geometry 11/2 (November 2018), 83-89. https://doi.org/10.36890/iejg.545135.
JAMA Çolakoğlu HB. The Generalized Taxicab Group. Int. Electron. J. Geom. 2018;11:83–89.
MLA Çolakoğlu, Harun Barış. “The Generalized Taxicab Group”. International Electronic Journal of Geometry, vol. 11, no. 2, 2018, pp. 83-89, doi:10.36890/iejg.545135.
Vancouver Çolakoğlu HB. The Generalized Taxicab Group. Int. Electron. J. Geom. 2018;11(2):83-9.