Title: Notes on relative normalizations of ruled surfaces in the three-dimensional Euclidean space,
Abstract: This paper deals with relative normalizations of skew ruled surfaces in the
Euclidean space $\mathbb{E}^{3}$. In section 2 we investigate some new formulae
concerning the Pick invariant, the relative curvature, the relative mean
curvature and the curvature of the relative metric of a relatively normalized
ruled surface $\varPhi$ and in section 3 we introduce some special
normalizations of it. All ruled surfaces and their corresponding normalizations
that make $\varPhi$ an improper or a proper relative sphere are determined in
section 4. In the last section we study ruled surfaces, which are
\emph{centrally} normalized, i.e., their relative normals at each point lie on
the corresponding central plane. Especially we study various properties of the
Tchebychev vector field. We conclude the paper by the study of the central
image of $\varPhi$.