In this paper, a developable surface with geodesic or line of curvature coordinates is constructed in the Euclidean 3-space. A developable surface is coordinated by two families of parametric curves, base curves (directrices) and lines (rulings). Since any part of a straight line on a developable surface is geodesic and line of curvature, we only need to show that the directrices curves are geodesics or lines of curvature to ensure that the developable surface is parameterized by geodesic or line of curvature coordinates. The necessary and sufficient conditions for the directrices curves to be geodesics or lines of curvature are studied. The main results of this paper show that the developable surface with geodesic coordinates is a generalized cylinder, and the developable surface with line of curvature coordinates is a tangent surface.

This paper aims to design a generalized cylinder with a geodesic base curve according to the Darboux frame in Euclidean 3-space. A generalized cylinder is a special ruled surface that is constructed by a continuous fixed motion of a generator line called the ruling along a given curve called the base curve. The necessary and sufficient conditions for the base curve to be geodesic are studied. The main results show that the generalized cylinder with a geodesic base curve is an osculating cylinder whose base curve is a helical geodesic, and the rulings are directed by the unit osculating Darboux vector.

Constructing a surface with geodesic or line of curvature parameterization is an important problem in many practical applications. The present paper aims to design a generalized cylinder that is parametrized along the geodesics and lines of curvature curves in Euclidean 3- space. The main results show that the generalized cylinder with geodesic or line of curvature parameterization is a rectifying cylinder or a right cylinder respectively.