Ruled Surfaces with $T_1N_1$-Smarandache Base Curve Obtained from the Successor Frame

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 principal normal vectors of the successor curve of a curve were defined. Then, the Gaussian and mean curvatures of each ruled surface were calculated. It has been shown that the ruled surface formed by the tangent vector of the successor curve moving along the Smarandache curve is a developable ruled surface. In addition, it was found that the surface formed by the principal normal vector of the successor curve along the Smarandache curve is a minimal developable ruled surface if the principal curve is planar. Conditions are given for other surfaces to be developable or minimal surfaces. Finally, the examples of these surfaces were provided and their shapes were drawn.

Keywords

• Smarandache ruled surfaces
• mean curvature
• Gaussian curvature
• successor curve

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