Some Special Ruled Surfaces Generated by a Direction Curve according to the Darboux Frame and their Characterizations

Abstract

In this work, we consider the Darboux frame $\left(T,V,U\right)$ of a curve lying on an arbitrary regular surface and we construct ruled surfaces having a base curve which is a $V$-direction curve. Subsequently, a detailed study of these surfaces is made in the case where the directing vector of their generatrices is a vector of the Darboux frame, a Darboux vector field. Finally, we give some examples for special curves such as the asymptotic line, geodesic curve, and principal line, with illustrations of the different cases studied.