Cyclic surfaces in E 5 generated by equiform motions

Abstract

In this paper, we study cyclic surfaces in E 5 generated by equiform motions of a circle. The properties of this cyclic surfaces up to the first order are discussed. We prove the following new result: A cyclic 2-surfaces in E 5 in general are contained in canal hypersurfaces. Finally we give an example. (2000): 53A05, 53A17. Mathematical Subject Classification

Local Study of Scalar Curvature of Cyclic Surfaces Obtained by Homothetic Motion of Lorentzian Circle

“…Abdel-All and Hamdoon studied a cyclic surface in 5  . In this sense, they proved that such surface in 5  is in general contained in a canal hypersurface [7]. Solouma ([8]- [10]) studied locally some geometric problems on surfaces obtained by the equiform motion up to the first order.…”

Three Dimensional Surfaces Obtained by the Equiform Motion of a Surface of Revolution

“…An equiform transformation in the 3-dimensional Euclidean space E 3 is an affine transformation whose linear part is composed of an orthogonal transformation and a homothetical transformation. This motion can be represented by a translation vector d and a rotation matrix A as the following.…”

“…Under the assumption of the constancy of the scalar curvature, kinematic surfaces obtained by the motion of a circle have been obtained in [3]. In a similar context, one can consider hypersurfaces in space forms generated by oneparameter family of spheres and having constant curvature see [4,5].…”

Spiral surfaces enveloped by one-parametric sets of spheres

“…In the case that the Gauss curvature vanishes on the surface, then the planes containing the circles must be parallel. In [1], we studied cyclic surfaces in E 5 generated by equiform motions of circles and we proved that: any 2-dimensional cyclic surface in E 5 is contained in canal hypersurface. In E 3 any cyclic surface can be generated by an equiform motion of a circle.…”