Curves and ruled surfaces according to alternative frame in dual space

In this paper, the vectorial moments of the alternative vectors are  expressed in terms of alternative frame. According to the new versions of these  vectorial moments, the parametric equations of the closed ruled surfaces corresponding  to the (^N); (^C); (^W) dual curves are given. The integral invariants of  the these surfaces are computed and illustrated by presenting with examples.


Introduction
There are many studies on the classical di¤erential geometry of curve and surface theories and are still being studied. A ruled surface in IR 3 is a surface which contains at least one 1-parameter family of straight lines. Thus a ruled surface has a parametrization in the form' where we call the anchor curve and the generator vector x as ruled surface. When the above ruled surface satis…es '(s + 2 ; v) = '(s; v) it is called closed ruled surface. The properties of the ruled surface obtained according to the condition of the anchor curve or the generator vector are available in the books of di¤erential geometry, [1,2,13]. Bertrand o¤sets, Mannheim o¤sets and involute-evolute o¤sets are obtained when special curves such as Bertrand, Mannheim and involute-evolute are taken as base curves. The geometric properties of these curves and surfaces are available in some references, [8,9,10,12,14,15,16].
If the vectorial moment of the x vector is denoted by x , then x = ^x. If X has the norm kXk = 1, then it is dual point on the dual unit sphere. According to E.Study theorem, there exists a one-to-one transformation between the dual points on the unit dual sphere and the oriented lines in IR 3 . A one-parameter set of points (a dual curve) on dual unit sphere corresponds to a one-parameter family of oriented lines in E 3 , which de…nes a ruled surface. This dual curve is called the dual spherical image of the ruled surface, [5,7].
The dual expression of a ruled surface in (1) is where thex(s)^x (s) is the anchor curve. s is not the arc-parameter of this curve [5,7]. The dual angle of pitch of the closed ruled surface in (2) is de…ned by [5] Figure 1. The dual expression of a ruled surface.
Here, x and L x are real integral invariants [5]. Osman Gürsoy's study showed that the dual integral invariant of a closed ruled surface, the dual angle of pitch, corresponds to the dual spherical surface area described by the dual spherical indicatrix of the closed ruled surface. Further, geometric interpretations of the real angle of pitch and the real pitch of a closed ruled surface were given [6]. In [4], the pitch, the angle of pitch and the dual angle of pitch of closed ruled surface corresponding to a closed curve on dual unit sphere were investigated. In [17], a di¤erential equation characterizing the dual spherical curves and an explicit solution of this di¤erential equation was given. By investigating one parameter spherical motion in with two di¤erent kinds of dual indicatrice curves, Yayl¬and Saraço¼ glu obtained the ruled surfaces that correspond to tangent, principal normal and binormal indicatrices of the dual curve were developable, [20].

Preliminaries
In E 3 , standard inner product is given by Here curvature and torsion of the curve (s) are de…ned with [2] (s) = k 00 (s)k; (s) = h 0 (s)^ 00 (s); 000 (s)i k 0 (s)^ 002 The vector W is called unit Darboux vector and de…ned by [3] It is obvious that the Darboux vector is perpendicular to the principal normal vector …eld N . If C is taken as C = W^N , then fN; C; W g are another orthonormal moving frame along the curve . This frame is called an alternative frame. The derivative formulae of the alternative frame is given by 2 where principal normal vector N is same in both frames, = and = , [11,19]. Let f (s), g(s) and h(s) be al least C 3 functions. (s) can be written in the form of (s) = f (s)T (s) + g(s)N (s) + h(s)B(s) (10) as a linear combination of the Frenet vectors fT; N; Bg, [18]. By di¤erentiating both side of (10), it is obtained [18] f 0 (s) g(s) (s) = 1; h 0 (s) + g(s) (s) = 0; g 0 (s) + f (s) (s) h(s) (s) = 0: (11)

Curves and Ruled Surfaces According to Alternative Frame in Dual Space
The geometric location of b N = N + "N , b C = C + "C and c W = W + "W vectors draws closed curves on the dual sphere. These closed curves are shown as respectively. According to Study's theorem these closed curves correspond to closed ruled surfaces. The dual expressions of the closed ruled sur- It is known that the curve is written in the form of Frenet vectors. Using the equations (9) and (10), we can write as the linear combination of alternative vectors as follows: Considering the above equation, vectorial moments of N; C; W are given respectively Using (14) in (12), The dual expressions of the closed ruled surfaces corresponding Theorem 1. Distribution parameters of the closed ruled surfaces corresponding to the ( b N ); ( b C); ( c W ) dual curves are given by Proof. We know that the distribution parameter of the closed ruled surface corresponding to the ( b N ) dual curve is calculated by It can be written that If this value is substituted into (17), the following result is obtained Similarly, with the values of distribution parameters of the closed ruled surfaces corresponding to the ( b C) and ( c W ) dual curves are Theorem 2. Gauss curvatures of closed ruled surfaces corresponding to ( b K c W (P ) = 0: Proof. For the closed ruled surface b N (s; v), the partial derivative is taken according to s and v, it is found Taking into account that inner product, we compute Using the Gram-Schmidt process, it can be seen that For a closed ruled surface with parametrization b N (s; v), the normal vector is given by On the other hand, we compute where = hy1;x2i hy1;y1i : Since shape operator is self-adjoint, we can write hS(E 2 ); E 1 i = hS(E 1 ); E 2 i: If the main direction of the surface is the asymptotic direction, the shape operator is hS(E 1 ); E 1 i = 0; [1]. Then, Gauss curvature of closed ruled Likewise, Gauss curvatures of closed ruled surfaces b C (s; v) and c W (s; v) are Vectorial moment of Darboux vector is given by

If this statement is substituted in (24), the instantaneous dual Pfa¢ an vector is
Also, by taking de…nition of dual Steiner vector, [5], we can write Theorem 4. The dual angles of pitch of closed ruled surfaces corresponding to (27) Here T and B are the angles of pitch of closed ruled surfaces drawn by T and B, respectively.
Proof. If take into account equations (3) and (23), the dual angle of pitch of closed ruled surface corresponding to the Similarly, the dual angles of pitch of closed ruled surfaces corresponding to the ( b C) and  Considering equation (2), we obtain closed ruled surfaces corresponding to the Using the equation (11), the solutions of f (s); g(s); h(s) are given by By taking into account the above equation and the equation (27), the dual angles of pitch of closed ruled surfaces corresponding to Here L T is the pitch of closed ruled surface drawn by the T and c is an arbitrary constant known as the integration constant. where c is an arbitrary constant known as the integration constant. Example 3. The Viviani's curve is formed by the intersection of a cylinder and a sphere. It is parametrized by (t) = a(1 + cos t); a sin t; 2a sin t 2 : Here, 2a is radius of sphere. The expression for the alternative invariants of Viviani's curve are given by N (s) = 3 12 cos t cos 2t p 88 cos t + 162 + 6 cos 2t ; 12 sin t sin 2t p 88 cos t + 162 + 6 cos 2t ; ; (s) = 6 cos t 2 3a cos t + 13a Considering the equation (15), closed ruled surfaces b N (s; v); b C (s; v) and c W (s; v) corresponding to ( b N ); ( b C) and ( c W ) dual curves is plotted by using Maple program (Fig. 4). Herein, associated calculations of these surfaces are computed by Maple program.

Conclusion
In this study, the vectorial moments of the alternative vectors are written using the data in equation (13). The dual expressions of the closed ruled surfaces which corresponds to the dual curves drawn by the b N ; b C and c W on the dual sphere are expressed in terms of alternative vectors. The distribution parameters and Gauss curvatures of closed ruled surfaces b N (s; v), b C (s; v) and c W (s; v), which are obtained using the equation (13), are calculated. Applying (13), likewise, it is shown that the closed ruled surface corresponding to the ( c W ) is developable. The dual angles of pitch of these surfaces obtained using the equation (13) are expressed in terms of the angles of pitch of closed ruled surfaces drawn by T and B. Upon inspection of helix and Viviani's curves, the related ruled surfaces are generated.