Directional Evolution of the Ruled Surfaces via the Evolution of Their Directrix Using q-frame along a Timelike Space Curve Timelike Uzay Eğrisi Boyunca q-çatı Kullanılarak Doğrultmanların Gelişimine Göre Regle Yüzeylerin Yönlü Gelişimleri

Directional Evolution of the Ruled Surfaces via the Evolution of Their Directrix Using q-frame along a Timelike Space Curve. European Journal of Science and Technology , (20), 392-396. Abstract In this study, the ruled surfaces obtained by normal and binormal vectors along a timelike space curve by using q-frame are investigated in 3 dimensional Minkowski space. Directional evolutions of both quasi normal and quasi binormal ruled surfaces are studied by using their directrices. Then, we work on some geometric properties such as inextensibilty, developability and minimality of these ruled surfaces.


Introduction
The time evolution of a curve or a surface is generated by inextensible flows of a curve or a surface. The flow of a curve or a surface is said to be inextensible if its arclength is preserved or the intrinsic curvature is preserved, respectively. Physically, the inextensible curve flows lead to motions in which no strain energy is induced. Also, the evolutions of curves have many important applications of physics as magnetic spin chains and vortex filaments [3,12,17].
In recent times, the motion of inelastic plane curves has been studied by many authors. After Da Rios in 1906 found out the geometric relation between the motion of curves and the differential equation, Doliwa in 1994 [9] characterized the integrable motions of a curve. While Kwon 2005 [16] and Körpinar 2011 [15] worked on inextensible flows of curves in Euclidean space, Gurbuz 2009 [11] and Yüzbaş 2018 [22] studied these curves in Minkowski space. Abd. Ellah 2015, Hussein 2016, D.W. Yoon 2019, Soliman 2018 studied the evolutions of the ruled surfaces via the evolution of their directrix [1, 13,20,21]. After the quasi-normal vector of a curve was introduced by Coquillart [5], Dede et.al. [6] found quasi frame in 2015 and Soliman also used this frame to work on this subject in 2018 [20].
For a space curve () t  , quasi frame consists of three orthonormal vectors, the unit tangent vector t , the quasi-normal q n and the quasi-binormal vector q b . The quasi frame as where k is the projection vector [6]. The q-frame has many advantages versus other frames (Frenet, Bishop). For instance the q-frame can be defined even along a line ( 0)   and the construction of the q-frame doesn't change if the space curve has unit speed or not and the q-frame can also be calculated easily [6].
The projection vector k is a unit vector along , xy  and z  axes. We choose the projection vector (0,0,1)  k without loss of generality. A quasi frame along a space curve is shown in Figure 1.   (2) where the quasi curvatures are given as follows , .
A ruled surface is a surface that can be sweept out by moving a line in space. Therefore, it has a parametrization of the form where  is called the directrix and  is the director curve.
In Minkowski 3-space ℝ 1 3 , the inner product of two vectors w w w  w is defined as    e e e respectively [2]. If u and w are timelike vectors then  uw is a spacelike vector [19].
The norm of the vector w is given by We say that a Lorentzian vector w is spacelike, lightlike respectively. In particular, the vector 0  w is spacelike [18,19]. Let () t  be a timelike space curve with a non-vanishing second derivative.
Then Frenet formulas of timelike curve may be written as 00 where . v  t The curvature and torsion of timelike curve respectively [2,19].
Let  be a surface in Euclidean 3-space, the first The second fundamental form II of  is given by 22 In this paper, we give another approach to evolutions of the ruled surfaces depend on a timelike space curve by q-frame used in [7,8,10,14]. Using q-frame, we present two sets of quasi frame equations with respect to arc-length s and time t . We obtain three differential equations depend on q-curvatures for the qframe vectors of the timelike space curve. Calculating first and second fundamental forms of this ruled surface, we get geometric properties such as curvatures, flatness, inextensibility and minimality of the ruled surface.

q-frame Along a Timelike Space Curve
As an alternative frame, quasi frame as called q-frame in both Euclidean and Minkowski space is defined by Dede and Ekici et al. [6,10].  (11) Since the derivation formula for the q-frame for the timelike curve in Minkowski space does not depend on projection vector being timelike or spacelike, we work on spacelike projection vector without loss of generality.

Evolution of Timelike Space Curve with Time by q-frame
In this section, we obtain time evolution equations depending on q-curvatures of the evolving curve in order to obtain space curve with q-frame. That is, integrating time evolution equations for given , , ,    one can find qcurvatures. Using eq. (6), we get evolving curve.
Theorem 1. The evolution equations for the quasi curvatures of the evolving curve are given by  (16) where the quasi formula with respect to time t is in the form 0 0. 0 Proof. Using equation (11), and defining 12 13 23 00 , 0 , 0 , 00  (12) and (14), the matrix of evolution equations is obtained by
With the help of (10), if the quasi normal surface is inextensible, then one can derive this differential equation