On ruled surfaces by Smarandache geometry in $\mathbf{E^3}$
Keywords:
Smarandache geometry, Ruled surfaces, Fundamental forms, Principal curvatures, Developable and minimal surfaces, Geodesic, asymptotic and curvature linesAbstract
The paper introduces a series of new ruled surfaces by following the idea of Smarandache geometry according to Frenet frame by taking into account all the possible linear combinations of the frame vectors. The metric properties of each defined ruled surface is examined by computing the $1^{st}$ and $2^{nd}$ fundamental forms as well as the curvatures of Gaussian and the mean expressed by the harmonic curvature function. Therefore, the conditions for each surface to be minimal or developable are provided. Moreover, the constraints for the characteristics of the base curve are discussed whether it is geodesic, asymptotic or a curvature line on the generated ruled surface. Finally, the graphical illustrations are presented for each ruled surface with a given appropriate example.
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