The transformation of the involute curves using by lifts on R3 to tangent space TR3

H. Çayir

"How we can speak about the features of involute curve on space TR3 by looking at the characteristics of the first curve α?" 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 R3 to its tangent space TR3=R6. Secondly, we examine the Frenet-Serret aparatus {T*(s), N*(s), B*(s), κ*, τ*} and the unit Darboux vector D* of the involute curve α* according to the vertical, complete and horizontal lifts on TR3 depending on the lifting of Frenet-Serret aparatus {T(s), N(s), B(s), κ, τ} of the first curve α on space R3. In addition, we include all special cases the curvature κ*(s) and torsion τ*(s) of the Frenet-Serret aparatus of the involute curve α* with respect to lifts on space R3 to its tangent space TR3. As a result of this transformation on space R3 to its tangent space TR3, we could have some information about the features of involute curve of any curve on space TR3 by looking at the characteristics of the first curve α. Moreover, we get the transformation of the involute curves using by lifts on R3 to tangent space TR3. Finally, some examples are given for each curve transformation to validated our theorical claims. 

Tbilisi Mathematical Journal, Vol. 14(1) (2021), pp. 119-134