The Transformation of the Evolute Curves Using Lifts on R^3 to Tangent Space Tr^3

Abstract

In this paper, firstly, we define the evolute curve of any curve concerning the vertical, complete, and horizontal lifts on space R^3 to its tangent space TR^3=R^6. Secondly, we examine the Frenet-Serret apparatus {T^* (s),N^* (s),B^* (s),κ^*,τ^* } and the Darboux vector W^* of the evolute curve α^* according to the vertical, complete and horizontal lifts on TR^3 by depend on the lifting of Frenet-Serret aparatus {T(s),N(s),B(s),κ,τ} of the first curve α on space R^3. In addition, we include all special cases the curvature κ^* (s) and torsion τ^* (s) of the Frenet-Serret aparatus {T^* (s),N^* (s),B^* (s),κ^*,τ^* } of the evolute curve α^* with respect concerning complete and horizontal 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 the volute curve of any curve on space TR^3 by looking at the characteristics of the first curve α. Moreover, we get the transformation of the evolute curves using bifts on R^3 to tangent space TR^3. Finally, some examples are given for each curve transformation to validate our theoretical claims.