Geometry of Solutions of the Quasi-Vortex Filament Equation in Euclidean 3-Space E3

Abstract

This work aims at investigating the geometry of surfaces corresponding to the geometry of solutions of the vortex filament equation in Euclidean 3-space E3 using the quasi-frame. In particular, we discuss some geometric properties and some characterizations of parameter curves of these surfaces in E3.

On Some Quasi-Curves in Galilean Three-Space

“…The vector that is perpendicular to both the tangent vector and the quasi-normal vector is referred to as the quasi-binormal vector. Much research on the quasi-frame has been conducted in a variety of Euclidean and Minkowski spaces and may be found in [7][8][9][10].…”

Geometry of solutions of the geometric curve flows in space

“…In recent years, the theory of surfaces with the connection of the geometric curve flows in different spaces and integrable non-linear equations is a subject of research attention (see e.g. [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16]) and finding geometric curve flows which generate various types of surfaces may expand our understanding of their geometric and topological properties, in order to a deep understanding of the physical world (see e.g. [17][18][19][20]).…”

“…Surfaces generated by the binormal flow, referred to as the Hasimoto surfaces, have been previously considered in [1,12]. For the Hasimoto surfaces, a lot of researches are done using the different frames (such as the Frenet frame [1], the Bishop frame [13], the Darboux frame [7,9], the quasi-frame [11], the modified orthogonal frame [8], the hybrid frame [16] and so on ) in Euclidean 3-space [1], Minkowski 3-space [5,14,15], Galilean 3space [6] and pseudo-Galilean 3-space [2]. Trajectory surfaces have been studied for the special case of inextensible flows in [21], curves flow of elastic rods in [22], and the curve shortening flow in [23][24][25].…”

Geometry of solutions of the geometric curve flows in space

“…In recent years, the theory of surfaces with the connection of the geometric curve flows in different spaces and integrable non-linear equations is a subject of research attention (see e.g. 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16 ) and finding geometric curves flows which generate various types of surfaces may expand our understanding of their geometric and topological properties, in order to a deep understanding of the physical world (see e.g. 17,18,19,20 ).…”

“…Surfaces generated by the binormal flow, referred to as Hasimoto surfaces, have been previously considered in 1,12 . For Hashimoto surfaces, A lot of researches were done using the different frames (such as the Frenet frame 1 , the Bishop frame 13 , the Darboux frame 7,9 , the quasi-frame 11 , the modified orthogonal frame 8 , the hybrid frame 16 and so on ) in Euclidean 3-space 1 , Minkowski 3-space 5,14,15 , Galilean 3-space 6 and pseudo-Galilean 3-space 2 . Trajectory surfaces have been studied for the special case of inextensible flows in 21 , curves flows of elastic rods in 22 , and the curve shortening flow in 23,24,25 .…”