ON THE OFFSETS OF RULED SURFACES IN EUCLIDEAN SPACE

D.W. Yoon

doi:10.12732/ijpam.v108i4.22

Abstract

In the present paper, we study evolute offsets of a non-developable ruled surface in Euclidean 3-space E and classify the evolute offset with constant Gaussian curvature and constant mean curvature. In last section, we investigate linear Weingarten evolute offsets in E . A linear Weingarten surface is the surface having a linear equation between the Gaussian curvature and the mean curvature of a surface. AMS Subject Classification: 53C30, 53B25

Highlights

A surface M in Euclidean 3-space E3 is called a Weingarten surface if there exists a non-trivial differentiable function relating the Gaussian curvature K and the mean curvature H of a surface M
If a surface M satisfies a linear equation with respect to K and H, that is, aK + bH = c for some real numbers a, b, c not all zero, it is said to be a linear Weingarten surface
In [14], Lopez investigated rotational linear Weingarten surfaces of hyperbolic type in Euclidean 3-space E3 and he proved that the profile curves of these surfaces are periodic and have selfintersections

Summary

Introduction

A surface M in Euclidean 3-space E3 is called a Weingarten surface if there exists a non-trivial differentiable function relating the Gaussian curvature K and the mean curvature H of a surface M. The ruled Weingarten surfaces in E3 were classified by Kuhnel [11]. For the another space forms S3 and H3 Valerio was studied in [17], but a complete classification is not known yet. Following this line of reasoning, the classification of ruled Weingarten hypersurfaces in En+1, Sn+1 and Hn+1, n ≥ 3 is obtained in [1], [4] and [15], respectively. We classify a linear Weingarten evolute offset of ruled surfaces

Preliminaries

Offsets with Constant Gaussian Curvature and Mean Curvature

Linear Weingarten Offsets of Ruled Surfaces