Ruled Surfaces with $\{\overline{u}_1\,\overline{u}_3\}$-Smarandache Base Curve Obtained From the Successor Frame

Year 2024, Volume: 12 Issue: 1, 28 - 45, 30.04.2024

Gülşah Uzun , Süleyman Şenyurt , Kübra Akdağ

Abstract

In this study, ruled surfaces formed by the movement of the Frenet vectors of the Successor curve along the Smarandache curve obtained from the tangent and binormal vectors of the Successor curve of a curve are defined. Then, the Gaussian and mean curvatures of each ruled surface are calculated. It is shown that the ruled surface formed by the movement of the tangent vector of the Successor curve along the $\{\overline{u}_1\,\overline{u}_3\}$ curve is a developable minimal surface and the ruled surface formed by the movement of the binormal vector is only a developable surface. It is also stated that if the principal curve is a planar curve, the ruled surface formed by the principal normal vector of the Successor curve along the $\{\overline{u}_1\,\overline{u}_3\}$ curve is also a developable minimal surface. Conditions for other surfaces to be developable or minimal surfaces are given.

Keywords

Ruled surfaces, Gaussian curvature, Successor curve

Ethical Statement

Not necessary

Supporting Institution

None

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