The transformation of the involute curves using by lifts on $R^{3}$ to tangent space $TR^{3}$

Abstract

"How we can speak about the features of involute curve on space $TR^{3}$ by looking at the characteristics of the first curve $\alpha $?" In this paper, we investigate the answer of this question using by lifts. In this direction firstly, we define the involute curve of any curve with respect to the vertical, complete and horizontal lifts on space $R^{3}$ to its tangent space $TR^{3}=R^{6}.$ Secondly, we examine the Frenet-Serret aparatus $\left\{ T^{\ast }(s),N^{\ast }(s),B^{\ast }(s),\kappa ^{\ast },\tau ^{\ast }\right\} $ and the unit Darboux vector $\tilde{D}^{\ast }$ of the involute curve $\alpha ^{\ast }$ according to the vertical, complete and horizontal lifts on $TR^{3}$ depending on the lifting of Frenet-Serret aparatus $\left\{ T(s),N(s),B(s),\kappa ,\tau \right\} $ of the first curve $\alpha $ on space $R^{3}.$ In addition, we include all special cases the curvature $\kappa ^{\ast }(s)$ and torsion $\tau ^{\ast }(s)$ of the Frenet-Serret aparatus of the involute curve $\alpha ^{\ast }$ with respect to lifts on space $R^{3}$ to its tangent space $TR^{3}.$ As a result of this transformation on space $R^{3}$ to its tangent space $TR^{3}$, we could have some information about the features of involute curve of any curve on space $TR^{3}$ by looking at the characteristics of the first curve $\alpha$. Moreover, we get the transformation of the involute curves using by lifts on $R^{3}$ to tangent space $TR^{3}$. Finally, some examples are given for each curve transformation to validated our theorical claims.