On structuring a family of parametric and ruled surfaces with common line of curvatures according to Bishop frame

In this study, we expressed conditions for the unit speed curve given by the Bishop frame in 3-dimensional Euclidean space to be both line of curvature and parametric curve on the surface. We have shown with examples including known special surfaces that members of the surface family with the common curve can be obtained for each different choice of the marching scale functions in the parametric writing of the surface. In addition, in the case of a special type of surface, a ruled surface, conditions for the curve to serve as the line of curvature were expressed, and the potential developability of these surfaces was examined. Finally, various examples related to the subject are given.