Ruled Surfaces of Adjoint Curve with the Modified Orthogonal Frame

This paper analyzes several specific ruled surfaces generated by the base curve $\alpha$ and its director curve or the $\alpha$'s adjoint curve $\beta$ and its director curve, where the director curves are frame vectors of the modified orthogonal frame in $E^{3}$. Furthermore, this paper studies the flat or minimal properties of the surfaces, as well as their asymptotic and geodesic curves. Afterward, it exemplifies the theoretical results herein. Finally, this paper discusses the need for further research.