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 $\varPhi$.