Ruled Surfaces with〖 t〗_α ζ_1^α -Smarandache Base Curve Derived from the Bishop Frame

Abstract

In this study, a class of ruled surfaces defined by unit vector fields whose rulings consist of Smarandache-type curves has been investigated. The base curve of these surfaces is taken as the adjoint curve, which is obtained by integrating a Smarandache curve constructed through the combination of the tangent vector and the first Bishop vector defined in the context of the Bishop frame. Closed-form expressions for the Gauss and mean curvatures of the surfaces have been derived, and their fundamental geometric properties have been analyzed in detail. The study also examines the conditions under which these surfaces become developable, minimal, or singular. Moreover, the geodesic and asymptotic behaviors of certain parametric curves on the surface are evaluated, and the necessary and sufficient conditions for these properties are determined.