The Generalized Taxicab Group

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.

Keywords

• Taxicab metric,generalized taxicab metric,distance,angle measure,reflection,rotation,isometry,the generalized taxicab group

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