Evolutions of the Ruled Surfaces via the Evolution of Their Directrix Using Quasi Frame along a Space Curve

In this paper, evolutions of ruled surfaces generated by the quasi normal and quasi binormal vector fields of space curve are presented. These evolutions of the ruled surfaces depend on the evolutions of their directrix using quasi frame along a space curve.


Introduction
Recently, the study of the motion of inelastic plane curves has arisen in a number of diverse engineering applications. Chirikjian and Burdick [1] describe the motion of a planar hyper redundant (or snake-like) robot as the flow of a plane curve, while Brockett [2] explicitly proposes the idea of an inelastic string machine as a robotic device. Jagadeesan Jayender and Kirby G. Vosburgh [3] showed that the colonoscope can be modeled as a set of infinitesimal rigid links along a backbone curve defined in terms of the Frenet-Serret frame. They used evolution of space curve theory in their model. Kalantar, et al. [4] they considered a collection of robots which can act as markers on an imaginary curve moving according to the local curvature and the external environmental force projected unto the normal to the curve.
The geometric link between integrable equation and the motion of curves may be said to have its origin in an analysis by Da Rios [5] in 1906. He obtained partial differential equation that governing the moving curves. Takao [6], showed that the Da Rios equations may be mapped to produce the celebrated nonlinear Schrodinger equation. Lamb [7], later in 1977, linked the motion of curves with  [8], derived the Heisenberg spin chain equation via the spatial motion of a space curve.
In recent times, Santini and Doliwa [9] linked the motion of inextensible curves with solitonic systems. [10] studied evolution of the translation surfaces and their generating curves in E 3 and obtained the evolution equations of the fundamental quantities and the Christoffel symbols for the translation surfaces.
[11] studied generated surfaces via inextensible flows of curves in R 3 . They constructed and plotted the surfaces generated from the motion inextensible curves in R 3 . D. Y. Kwona and F. C. Park [12] [13] studied evolution of inelastic plane curves and inextensible flows of curves and developable ruled surfaces. They get partial differential equation that governing the flow of curves and the flow of In this paper, we introduce a different approach to this problem. The evolution of curves is represented by two sets of quasi Serret-Frenet equations for tangent, quasi normal and quasi binormal vectors to the curve. By applying compatibility condition on these vectors, three partial differential equations for the curvatures 1 2 3 , , κ κ κ are derived. We derive system of partial differen- The article is organized as follows. In Section 2, we introduce differential geometry of curves focusing on Serret-Frenet frame and quasi frame along a space curve. In Section 3, the evolution of curves is represented by two sets of quasi Serret-Frenet equations for tangent, quasi normal and quasi binormal vectors to the curve. By applying compatibility condition on these vectors, three partial differential equations for the curvatures 1 2 3 , , κ κ κ are derived. In Section 4, Ruled surface is constructed on the evolving curve where the generator is quasi normal and quasi binormal vectors to the curve. The coefficients of the first, second fundamental forms, Gaussian curvatures, mean curvatures are obtained.

Frenet Frame and Quasi Frame along a Space Curve
There is a more moving frame that can be associated to a space curve in space Journal of Applied Mathematics and Physics such as Frenet frame [17], Bishop frame [18] [19] [20], Kepler frame [21]. In this section we define a new frame along a space curve as an alternant to Frenet frame which called quasi Frame [22]. Also geometric proprieties for the Frenet frame and quasi frame along a space curve are presented.
be a vector valued function of a regular space curve represented with its arc-length s , the vectors associated to the curve are where T is the tangent vector, N is the normal vector, B is the binomial vector.
The curvature 1 κ and the torsion 2 κ are given by The Frenet frame ( ) , , T N B vary along r according to the well-known Serret-Frenet relations [17] The quasi frame of a regular space curve where k is the projection vector can be chosen as ( ) Let θ is the angel between the normal N and quasi normal q N . Then, the relation between two frames is given by Thus,

) Journal of Applied Mathematics and Physics
A short calculation using Equations (3), (5) and (6) shows that the variation of quasi frame is given by where the quasi curvatures are

Evolution of a Space Curve with Time by Quasi Frame
In this section we study the evolution of a regular space curve using quasi frame.
We can write the Serret-Frenet equations and the equation of the evolution in the matrix form as follows. Defining a short calculation using Equations (9), (10) and (11) Thus the compatibility conditions becomes The set of Equation (16)

Evolution of Ruled Surfaces Depends on Their Directrix by Quasi Frame of a Space Curve
We provide a general scheme for studying evolution of ruled surfaces using an approach different from the one proposed by [13] [15]. We apply our method by using quasi frame along a space curve. Evolutions of ruled surfaces generated by the quasi normal and quasi binormal vector fields of space curve are presented.
These evolutions of the ruled surfaces depend on the evolutions of their directrix using quasi frame along a space curve.

Evolution of Quasi Normal Ruled Surface
The equation of surfaces generated by quasi normal is [26] ( The tangent space to the surface ψ is, The second derivative is calculated and given by

Evolution of Quasi Binormal Ruled Surface
The equation of surfaces generated by quasi normal is The tangent space to the surface ψ is, ( ) If we compute components of the second fundamental form, we have