On Classes of Surfaces with a Common Special Curve

Abstract

In recent years, many researchers have studied surfaces with common geodesic or common line of curvature in different
spaces. In this paper we give necessary and sufficient conditions for some classes of surfaces in the three dimensional Euclidean space
E^3 to have a curve as common geodesic or common line of curvature. Precisely, for a given curve, we suggest a family of surfaces
which have it as their common line of curvature or common geodesic. Moreover, we illustrate the results by examples.