On polar relative normalizations of ruled surfaces

TL;DR

This paper investigates skew ruled surfaces in Euclidean space with polar normalizations, analyzing their invariants, vector fields, and special cases where the relative image reduces to a curve.

Contribution

It introduces the concept of polar normalizations for ruled surfaces and explores their geometric properties and invariants, extending the understanding of relative normalizations.

Findings

Determined invariants of polar normalized ruled surfaces

Analyzed properties of Tchebychev and support vector fields

Studied cases where the relative image degenerates into a curve

Abstract

This paper deals with skew ruled surfaces in the Euclidean space $\mathbb{E}^{3}$ which are equipped with polar normalizations, that is, relative normalizations such that the relative normal at each point of the ruled surface lies on the corresponding polar plane. We determine the invariants of a such normalized ruled surface and we study some properties of the Tchebychev vector field and the support vector field of a polar normalization. Furthermore, we study a special polar normalization, the relative image of which degenerates into a curve.