Ruled Surfaces with $\{\overline{u}_1\,\overline{u}_3\}$-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 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

References

1. [1] H. Pottmann, M. Eigensatz, A. Vaxman, and J. Wallner, Architectural geometry, Computers and Graphics, 47 (2015), 145–164.
2. [2] J. Stillwell, Mathematics and Its History, Undergraduate texts in mathematics, Third Edition Springer (2010).
3. [3] D. J. Struik, Lectures on Classical Differential Geometry, Addison-Wesley Publishing Company, (1961).
4. [4] K. Akutagawa and S. Nishikawa, The Gauss Map and Space-like surfaces with prescribed mean curvature in Minkowski 3-space,Tohoku Mathematical Journal Second Series, 42 (1990), 67-82.
5. [5] L. Grill, S. S¸enyurt, and S. G. Mazlum, Gaussian curvatures of parallel ruled surfaces, Applied Mathematical Sciences, 14 (2020), 171-183.
6. [6] S. G. Mazlum, S. S¸enyurt, and L. Grill, The invariants of Dual Parallel equidistant ruled surfaces, Symmetry, 15 (2023), 206.
7. [7] S. S¸enyurt, D. Canlı, E. Can, and S. G. Mazlum, Some special smarandache ruled surfaces by Frenet frame in E3 􀀀II, Honam Mathematical Journal, 44 (2022), 594-617.
8. [8] S. S¸enyurt, D. Canlı, and E. Can, Some special Smarandache ruled surfaces by Frenet frame in E3 􀀀I, Turkısh Journal of Science, 7(1) (2022), 31-42.

9. [9] W. S. Massey, Surfaces of gaussian curvature zero in euclidean 3-space, Tohoku Mathematical Journal Second Series, 14 (1962), 73-79.
10. [10] A. Pressley, Elementary Differential Geometry, Springer Science and Business Media, (2010). [11] S. Ouarab, Smarandache ruled surfaces according to Frenet-serret frame of a regular curve in E3, Hindawi Abstract and Applied Analysis (2021), 8 pages.
11. [12] S. Ouarab, Smarandache ruled surfaces according to Darboux frame in E3, Hindawi Journal of Mathematics (2021), 10 pages.
12. [13] S. Ouarab, NC-smarandache ruled surface and NW-Smarandache ruled surface according to Alternative moving frame in E3, Hindawi Journal of Mathematics (2021), 6 pages.
13. [14] E. Kemal and S. S¸enyurt, On ruled surface with Sannia frame in Euclidean 3- Space, Kyunpook Math. J. 62 (2022), 509-531.
14. [15] S. S¸enyurt, K. H. Ayvacı, and D.Canlı, Some characterizations of spherical indicatrix curves generated by Sannia frame, Konuralp Journal of Mathematics, 9(2) (2021), 222-232.
15. [16] A. Menninger, Characterization of the slant helix as Successor curves of the general helix, International Electronic Journal of Geometry, 2 (2014), 84-91.
16. [17] M. Masal, Curves according to the Successor frame in Euclidean 3-space, Sakarya University Journal of Science, 6 (2018), 1868-1873.
17. [18] S. S¸enyurt and G. Kaya, Smarandache curves obtained from Frenet vectors of Successor curve (In Turkish), G. G¨urc¸ay (Ed.), Karadeniz 1. Uluslararası Multidisipliner C¸ alıs¸malar Kongresi, Giresun (2019), 318-324
18. [19] S. S¸enyurt and G. Kaya, Vector moment curves of Frenet vectors of Successor curve (In Turkish), G. G¨urc¸ay (Ed.), Karadeniz 1. Uluslararası Multidisipliner C¸ alıs¸malar Kongresi Giresun 2019 pp.325-330.
19. [20] T. Eris¸ir and H.K. O¨ ztas¸, Spinor equations of Successor curve, Universal of Mathematics and Applications, 5(1) (2022), 32–41.
20. [21] G. Kaya, Successor curves and equidistant ruled surfaces on the Dual, Doctoral Dissertation Ordu University, Ordu, (2023) .
21. [22] A. T. Ali, Special Smarandache curves in the Euclidean space, International Journal of Mathematical Combinatorics, 2 (2010), 30-36.
22. [23] K. Tas¸k¨opr¨u and M. Tosun, Smarandache curves on S2, Bol. Soc. Paran. Mat. 32(1) (2014), 51-59.
23. [24] S. S¸enyurt and S. Sivas, An application of Smarandache curve, Ordu Univ. J. Sci. Tech. 3 (2013), 46-60.
24. [25] N. Turgut and S. Yılmaz, Smarandache curves in Minkowski space-time, International Journal of Mathematical Combinatorics, 3 (2008), 51–55.
25. [26] J. Monterde, Salkowski curves revisited: a family of curves with constant curvature and con-constant Torsion, Computer Aided Geometric Design, 3 (2009), 271-278.