The new study of some characterization of canal surfaces with Weingarten and linear Weingarten types according to Bishop frame

M A Soliman,E M Solouma,W M Mahmoud,M Bary

doi:10.1186/s42787-019-0032-y

M A Soliman, E M Solouma + Show 2 more

Open Access

https://doi.org/10.1186/s42787-019-0032-y

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Abstract

In this paper, we have a tendency to investigate a particular Weingarten and linear Weingarten varieties of canal surfaces according to Bishop frame in Euclidean 3-space E3 satisfying some fascinating and necessary equations in terms of the Gaussian curvature, the mean curvature, and therefore the second Gaussian curvature. On the premise of those equations, some canal surfaces are introduced.

Highlights

Theorem 1 The mean curvature H and the second Gaussian curvature KII of nondevelopable canal surface Υ = f (u, v) according to Bishop frame in E3 satisfy
A surface Υ = f (u, v) in Euclidean 3-space E3 is said to be a (x, y)-Weingarten surface if for a pair (x, y), x = y of the K Gaussian curvature, H mean curvature, and KII second Gaussian curvature of a surface Υ satisfies ψ(x, y) = 0, where ψ is the Jacobi function defined by ψ = xy − yx
The “Canal surface according to Bishop frame in E3” section is precise to prepare some fundamental facts about the first, second, and third fundamental forms, the Gaussian curvature, the mean curvature, and the second Gaussian curvature, in the “(x, y)-Weingarten canal surface according to Bishop frame in E3” and “(x, y)-linear Weingarten canal surface according to Bishop frame in E3” sections, the (x, y)-Weingarten and (x, y)-linear Weingarten canal surfaces are discussed

Summary

Introduction

Theorem 1 The mean curvature H and the second Gaussian curvature KII of nondevelopable canal surface Υ = f (u, v) according to Bishop frame in E3 satisfy Theorem 6 A non-developable canal surface Υ = f (u, v) according to Bishop frame in E3 with vanishing second Gaussian curvature KII = 0 is a surface of revolution which satisfies

Results

Conclusion