Ruled Surfaces with Bishop vectors via Smarandache geometry
Abstract
The paper studies new ruled surfaces with the vectors of Bishop frame via Smarandache geometry. The fundamental forms and the corresponding curvatures are provided for each ruled surface to draw characteristics of the surfaces such as developability and minimality. Moreover, the properties of the base curve and the corresponding striction curves of each surface are also discussed through asymptoticity, geodesicity and principal line. It is found that a ruled surface designed by Bishop vectors of a slant helix-like curve apart from other special kinds of curves has a direct effect on the characteristics of some constructed ruled surfaces.
Keywords
Bishop frame, Smarandache ruled surfaces, fundamental forms, mean and Gaussian curvatures, developable and minimal surfaces
Supporting Institution
None
Project Number
None
Ethical Statement
No need
Thanks
We would like to thank the editors of the journal for taking care of the manuscript.
References
- [1] P. Do-Carmo, “Differential geometry of curves and surfaces: revised and updated second edition”, Courier Dover Publications, 2016.
- [2] E. Abbena, S. Salamon and A. Gray, “Modern differential geometry of curves and surfaces with Mathematica”, Chapman and Hall/CRC, 2017.
- [3] H. H. Hacısaliho˘glu , “Differential geometry II”, AnkaraUniversity Press, 2000.
- [4] D. J. Struik, “Lectures on classical differential geometry”, Courier Corporation, 2012.
- [5] M. Juza, “Ligne de striction sur unegeneralisation a plusierurs dimensions d’une surface regle”, Czechoslovak Mathematical Journal 12(87) (1962), 243-250.
- [6] S. Ouarab and A. O. Chahdi, “Some characteristic properties of ruled surface with Frenet frame of an arbitrary non-cylindrical ruled surface in Euclidean 3-space”, International Journal of Applied Physics and Mathematics 10(1) (2020), 16-24.
- [7] R. L. Bishop, “There is more than one way to frame a curve”, The American Mathematical Monthly 82 (1975), 246-251.
- [8] M. Masal and A. Z. Azak, “Ruled surfaces according to Bishop frame in the Euclidean 3-space”, Proceedings of the National Academy of Sciences, India Section A: Physical Sciences 89(2) (2019), 415-424.
- [9] Y. Tunc¸er, “Ruled surfaces with the Bishop frame in Euclidean 3–space”, Gen. Math. Notes 26 (2015), 74-83.
- [10] S. Ouarab, A. O. Chahdi, M. Izid, “Ruled surfaces with alternative moving frame in Euclidean 3-space”, International Journal of Mathematical Sciences and Engineering Applications 12(2) (2018), 43-58.