Some new results for Hasimoto surfaces
Computer Science & Engineering
Proceedings of the 8th International Youth Science Forum "Litteris et Artibus", November 22–24, 2018, Lviv: Lviv Polytechnic National University, 2018, pp. 22–25
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First published on this website: 26.10.2018 - 14:33
Abstract
Let σ=σ(s,t) be the position vector of a curve Γ moving on surface M in E3 such that σ=σ(s,t) is a unit speed curve for all t. If the surface M is a Hasimoto surface, then, the position vector σ satisfy the following condition
σt = σs ⋀ σss
also called as smoke ring equation or vortex filament [1]. In that work, we investigate the geometric properties according to Bishop frame of Hasimoto surfaces in Euclidean 3-space. Also, we give some characterization of parameter curves given according to Bishop frame of Hasimoto surfaces.
References
[1] C. Rogers and W.K. Schief, Backlund and Darboux Transformations, Geometry of Modern Applications in Soliton Theory. Cambridge University Press, 2002.
[2] H. Hasimoto, “A Soliton on a vortex filament,” J. Fluid. Mech, Vol. 51, 1972, pp. 477-485.
[3] M. Erdogdu and M. Ozdemir, “Geometry of Hasimoto Surfaces in Minkowski 3-Space,” Math. Phys. Anal. Geom., Vol. 17, 2014, pp. 169-181.
[4] L.R. Bishop, "There is more than one way to frame a curve," Amer. Math. Monthly, Volume 82, Issue 3, 1975, pp. 246-251.
[5] B. Bukcu and M.K. Karacan, “The Slant Helices According to Bishop Frame,” World Academy of Science, Engineering and Technology, Vol.3, 2009, pp. 11-20.
[6] S. Yılmaz and M. Turgut, “A new version of Bishop frame and an application to spherical images,” Journal of Mathematical Analysis and Applications, Vol. 371, 2010, pp. 764-776.
[7] L.P. Eisenhart, A Treatise On The Differential Geometry Of Curves And Surfaces, 1909.
[8] L.S. Da Rios, “On the motions of an unbounded fluid with a vortex filament of any shape,” (in Italian). Rend. Circ. Mat. Palermo, Vol. 22, Iss. 17, 1906.
First published Full Text (26.10.2018 - 14:33)
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