Some Characterizations of Normal Curves Under Isometry Using Darboux Frame

Abstract

This paper deals with the study of characterizations of a normal curve in 3-dimensional Euclidean space and presents a sufficient condition for the invariance of a normal curve on regular surfaces under isometric transformations using the Darboux frame $\{T_1,P_1,U_1\}$. Furthermore, we compute the deviations of the position vector of the normal curve on regular surfaces along the unit tangent vector $T_{1}$, the unit normal $U_{1}$  perpendicular to the oriented surface, and the cross product of $U_{1}$ and $T_{1}$ due to isometric transformations.