Notes on relative normalizations of ruled surfaces in the three-dimensional Euclidean space

TL;DR

This paper explores the properties of relative normalizations of skew ruled surfaces in three-dimensional Euclidean space, deriving new formulas and characterizations for special classes like relative spheres and centrally normalized surfaces.

Contribution

It introduces new formulae for invariants of ruled surfaces and classifies surfaces with specific normalizations, including relative spheres and centrally normalized surfaces.

Findings

Derived new formulas for Pick invariant, relative curvature, and mean curvature.

Characterized surfaces that are relative spheres under various normalizations.

Analyzed properties of the Tchebychev vector field and central images.

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 $\varPhi$.