A Novel Approach to Ruled Surfaces Using Adjoint Curve

In this study, ruled surfaces are examined where the direction vectors are unit vectors derived from Smarandache curves, and the base curve is taken as an adjoint curve constructed using the integral curve of a Smarandache-type curve generated from the first and second Bishop normal vectors. The newly generated ruled surfaces will be referred to as Bishop adjoint ruled surfaces. Explicit expressions for the Gaussian and mean curvatures of these surfaces have been obtained, and their fundamental geometric properties have been analyzed in detail. Additionally, the conditions for developability, minimality, and singularities have been investigated. The asymptotic and geodesic behaviors of parametric curves have been examined, and the necessary and sufficient conditions for their characterization have been derived. Furthermore, the geometric properties of the surface generated by the Bishop adjoint curve and its relationship with the choice of the original curve have been established. The constructed ruled surfaces exhibit a notable degree of geometric regularity and symmetry, which naturally arise from the structural behavior of the associated adjoint curves and direction fields. This underlying symmetry plays a central role in their formulation and classification within the broader context of differential geometry. Finally, the obtained surfaces are illustrated with figures.