The Generalized Taxicab Group

ÇOLAKOĞLU H. B.

INTERNATIONAL ELECTRONIC JOURNAL OF GEOMETRY, vol.11, no.2, pp.83-89, 2018 (ESCI) identifier

  • Publication Type: Article / Article
  • Volume: 11 Issue: 2
  • Publication Date: 2018
  • Journal Name: INTERNATIONAL ELECTRONIC JOURNAL OF GEOMETRY
  • Journal Indexes: Emerging Sources Citation Index (ESCI), TR DİZİN (ULAKBİM)
  • Page Numbers: pp.83-89
  • Keywords: Taxicab metric, generalized taxicab metric, distance, angle measure, reflection, rotation, isometry, the generalized taxicab group
  • Akdeniz University Affiliated: Yes

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.