On geometry of focal surfaces due to Flc frame in Euclidean 3-space

Abstract

In this study, firstly, tube surfaces given with Flc frame are introduced. The Gaussian and mean curvature of these surfaces were found and their singular points were determined. By investigating the conditions for parameter curves on the surface to be asymptotic, geodesic and curvature lines, it has been shown that θ parameter curves of the surface can never be asymptotic, but always geodesic. In particular, it is concluded that the parameter curves t and θ are curvature lines if they are planar curves. Afterwards, the focal surfaces of the tube surfaces given with the Flc frame are defined. By calculating the Gaussian and mean curvatures of these focal surfaces, it was found that they are always developable and never minimal. Afterwards, the conditions for parameter curves on the focal surface to be asymptotic, geodesic and curvature lines were investigated, and it was seen that the t parameter curves of the surface could never be asymptotic and the θ parameter curves were always asymptotic curves and curvature lines. Finally, examples of these surfaces are given using the Maple program.