On ruled surfaces relatively normalized

Abstract

This paper deals with skew ruled surfaces Φ in the Euclidean space E 3 which are right normalized, that is they are equipped with relative normalizations, whose support function is of the form q, where w 2 (u, v) is the discriminant of the first fundamental form of Φ. This class of relatively normalized ruled surfaces contains surfaces such that their relative image Φ * is either a curve or it is as well as Φ a ruled surface whose generators are, additionally, parallel to those of Φ. Moreover we investigate various properties concerning the Tchebychev vector field and the support vector field of such ruled surfaces.

On polar relative normalizations of ruled surfaces

“…We concentrate now on the main topic of this paper, namely the polar normalizations of a skew ruled surface Φ, i.e., relative normalizations such that the relative normal at each point P of Φ lies on the corresponding polar plane {P ; n, z}. In [7] it was shown that the support function of y is of the form…”

“…In [5] it was shown that the coordinate functions of the Tchebychev vector T (u, v) of (Φ, y), which is defined by T := T m x /m , where T m := 1 2…”

“…The divergence div I T of T with respect to the first fundamental form I of Φ, which initially reads (see [5])…”

“…G. Stamou and A. Magkos in [9] and G. Stamou, St. Stamatakis and I. Delivos in [10] studied ruled surfaces in the Euclidean space E 3 which are equipped with Manhart's normalizations. Later, S. Stamatakis and I. Kaffas studied in [5] the asymptotic relative normalizations of a ruled surface Φ, that is, relative normalizations such that the relative normals at each point P of Φ lie on the corresponding asymptotic plane of Φ. Following this idea the authors introduced in [7] three special relative normalizations:…”

“…Later, S. Stamatakis and I. Kaffas studied in [5] the asymptotic relative normalizations of a ruled surface Φ, that is, relative normalizations such that the relative normals at each point P of Φ lie on the corresponding asymptotic plane of Φ. Following this idea the authors introduced in [7] three special relative normalizations:…”