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

In this work, we consider the Darboux frame <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" id="M1"> <a:mfenced open="(" close=")"> <a:mrow> <a:mi>T</a:mi> <a:mo>,</a:mo> <a:mi>V</a:mi> <a:mo>,</a:mo> <a:mi>U</a:mi> </a:mrow> </a:mfenced> </a:math> of a curve lying on an arbitrary regular surface and we construct ruled surfaces having a base curve which is a <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" id="M2"> <e:mi>V</e:mi> </e:math> -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.