Invariant Properties of Curves in the Taxicab Geometry

Abstract

Let $E^{2}_{T}$ be the group of all isometries of the $2$-dimensional taxicab space $R^{2}_{T}$. For the taxicab group $E^{2}_{T}$, the taxicab type of curves is introduced. All possible taxicab types are found. For every taxicab type, an invariant parametrization of a curve is described. The $E^{2}_{T}$-equivalence of curves is reduced to the problem of the $E^{2}_{T}$-equivalence of paths.