Generalized Normal Ruled Surface of a Curve in the Euclidean 3-space

TL;DR

This paper introduces the generalized normal ruled surface of a curve in Euclidean 3-space, analyzing its geometric properties including curvature, minimality, and special curve conditions, with examples illustrating these concepts.

Contribution

It defines and studies the properties of generalized normal ruled surfaces, including curvature calculations and conditions for special curves, extending the understanding of ruled surface geometry.

Findings

Conditions for flat and minimal surfaces identified

Relations between the surface and helices/slant ruled surfaces established

Examples illustrating the theoretical results provided

Abstract

In this study, we define the generalized normal ruled surface of a curve in the Euclidean 3-space $E^3$. We study the geometry of such surfaces by calculating the Gaussian and mean curvatures to determine when the surface is flat or minimal (equivalently, helicoid). We examine the conditions for the curves lying on this surface to be asymptotic curves, geodesics or lines of curvature. Finally, we obtain the Frenet vectors of generalized normal ruled surface and get some relations with helices and slant ruled surfaces and we give some examples for the obtained results.