Generalized Rectifying Ruled Surfaces of Special Singular Curves

In this study, generalized rectifying ruled surfaces of Frenet-type framed base curves in the three-dimensional Euclidean space are introduced. These surfaces are a generalization of not only the tangent and binormal surfaces of Frenet-type framed base curves, but also the tangent and binormal surfaces of regular curves. Additionally, we present some geometric characterizations and properties of these surfaces. Then, the singular point classes of the surface are scrutinized and the conditions for being a cross-cap surface are stated. Moreover, generalized rectifying surfaces are examined as framed surfaces by using the framed surface theory, and we investigate the basic invariants and curvatures of them. Then, several illustrative examples with figures are given to support the theorems and results.