# Ruled surfaces right normalized

## TL;DR

This paper studies a special class of skew ruled surfaces in Euclidean space that are equipped with a particular type of relative normalization, analyzing their geometric properties and associated vector fields.

## Contribution

It introduces and investigates right normalized skew ruled surfaces with a specific support function form, exploring their geometric properties and related vector fields.

## Key findings

- Characterization of right normalized ruled surfaces with the given support function
- Analysis of the Tchebychev and support vector fields on these surfaces
- Conditions under which the relative image is a curve or a ruled surface

## Abstract

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