A relative-geometric treatment of ruled surfaces

TL;DR

This paper investigates relative normalizations of ruled surfaces in Euclidean space, characterizing them via support functions based on Gaussian curvature, and explores properties of their relative normals, invariants, and affine normal images.

Contribution

It introduces a new class of relative normalizations for ruled surfaces using support functions tied to Gaussian curvature and analyzes their geometric properties.

Findings

Characterization of ruled surfaces with specific properties of relative normals

Analysis of the Pick invariant and Tchebychev vector field for these surfaces

Study of the affine normal image of non-conoidal ruled surfaces

Abstract

We consider relative normalizations of ruled surfaces with non-vanishing Gaussian curvature $K$ in the Euclidean space $\mathbb{R} ^{3}$, which are characterized by the support functions $^{\left( \alpha \right) }q=\left \vert K\right \vert ^{\alpha}$ for $\alpha \in \mathbb{R}$ (Manhart's relative normalizations). All ruled surfaces for which the relative normals, the Pick invariant or the Tchebychev vector field have some specific properties are determined. We conclude the paper by the study of the affine normal image of a non-conoidal ruled surface.