Some Results and Characterizations for Mannheim Offsets of the Ruled Surfaces

In this study, we give the dual characterizations of Mannheim offsets of the ruled surface in terms of their integral invariants and the new characterization of the Mannheim offsets of developable surface. Furthermore, we obtain the relationships between the area of projections of spherical images for Mannheim offsets of ruled surfaces and their integral invariants.


Introduction
Ruled surfaces are the surfaces which are generated by moving a straight line continuously in the space and are one of the most important topics of differential geometry. A ruled surface can always be described (at least locally) as the set of points swept by a moving straight line. These surfaces are used in many areas of sciences such as Computer Aided Geometric Design (CAGD), mathematical physics, moving geometry, kinematics for modeling the problems and model-based manufacturing of mechanical products. Especially, the offsets of ruled surface have an important role in (CAGD). Some studies dealing with offsets of the surfaces have been given in [2,8,10,12,13,14]. In [14], Ravani and Ku have defined and given a generalization of the theory of Bertrand curves for Bertrand trajectory ruled surfaces on the line geometry. Küçük and Gürsoy have studied the integral invariants of Bertrand trajectory ruled surfaces in dual space and given the relations between the invariants [8].
Furthermore, similar to the Bertrand curves, in [9] a new definition of special curve pair has been given by Liu and Wang: Let C and C * be two space curves. C is said to be a Mannheim partner curve of C * if there exists a one to one correspondence between their points such that the binormal vector of C is the principal normal vector of C * . Orbay and et.al. have given a generalization of the theory of Mannheim curves for ruled surfaces and called Mannheim offsets [10].
In this paper, we examine the Mannheim offsets of trajectory ruled surfaces in view of their integral invariants. Using the dual representations of the ruled surfaces, we give the results obtained in [10] and some new results in short forms. Moreover, we show that if the Mannheim offsets of trajectory ruled surfaces are developable then their striction lines are Mannheim partner curves. Furthermore, we give some characterizations of Mannheim offsets of trajectory ruled surfaces in terms of integral invariants (such as the angle of pitch and the pitch) of closed trajectory ruled surfaces. Finally, we obtain the relationship between the area of projections of spherical images of Mannheim offsets of trajectory ruled surfaces and their integral invariants. 3 E Let I be an open interval in the real line IR . Let ( ) k k s = be a curve in 3 E defined on I and ( ) q q s = be a unit direction vector of an oriented line in 3 IR . Then we have the following parametrization for a ruled surface

Differential Geometry of Ruled Surfaces in
(1) The parametric s -curve of this surface is a straight line of the surface which is called ruling. For 0 v = , the parametric v -curve of this surface is ( ) k k s = which is called base curve or generating curve of the surface. In particular, if q is constant, the ruled surface is said to be cylindrical, and non-cylindrical otherwise [7].
The striction point on a ruled surface is the foot of the common normal between two consecutive rulings. The set of the striction points constitute a curve ( ) c c s = lying on the ruled surface and is called striction curve. The parametrization of the striction curve ( ) c c s = on a ruled surface is given by , So that, the base curve of the ruled surface is its striction curve if and only if , The distribution parameter (or drall) of the ruled surface in (1) is given as If 0 q δ = , then the normal vectors of the ruled surface are collinear at all points of the same ruling and at the nonsingular points of the ruled surface the tangent planes are identical. We then say that the tangent plane contacts the surface along a ruling. Such a ruling is called a torsal ruling. If 0 q δ ≠ , then the tangent planes are distinct at all points of the same ruling which is called nontorsal.
A ruled surface whose all rulings are torsal is called a developable ruled surface. The remaining ruled surfaces are called skew ruled surfaces. Thus, from (3) a ruled surface is developable if and only if at all its points the distribution parameter 0 q δ = [7,14].
 be a moving othonormal trihedron making a spatial motion along a closed space curve ( ) k s , s ∈ » , in 3 E . In this motion, an oriented line fixed in the moving system generates a closed ruled surface called closed trajectory ruled surface (CTRS) in 3 E [8]. A parametric equation of a closed trajectory ruled surface generated by qaxis is Consider the moving orthonormal system { } where ( ) b b s = is the striction line of q ϕ -CTRS and the differential forms 1 2 , k k and σ are the natural curvature, the natural torsion and the striction of q ϕ -CTRS, respectively [7,8]. Here, the striction is restricted as / 2 / 2 π σ π − < < fort he orientation on q ϕ -CTRS and s is the length of the striction line.
The pole vector and the Steiner vector of the motion are given by respectively, where 2 1 k q k a ψ = + is the instantaneous Pfaffian vector of the motion.
The pitch of q The angle of pitch of q ϕ -CTRS is given one of the followings where q a and q g are the measure of the spherical surface area bounded by the spherical image of q ϕ -CTRS and the geodesic curvature of this image, respectively. The pitch and the angle of pitch are well-known real integral invariants of closed trajectory ruled surface [3][4][5][6].
The area vector of a x -closed space curve in 3 E is given by and the area of projection of a x -closed space curve in direction of the generator of a y -CTRS is , 2 , x y x f v y = .

Dual Numbers and Dual Vectors
Dual numbers had been introduced by W. K. Clifford (1845-1879). A dual number has the form ( , ) a a a a a ε * * = = + where a and * a are real numbers and (0,1) ε = is dual unit with 2 0 ε = . The product of dual numbers ( , ) a a a a a ε * * We denote the set of dual numbers by D: Clifford showed that dual numbers form algebra, but not a field. The pure dual numbers a ε * are zero divisors, ( )( ) 0 a b ε ε * * = . However, the other laws of the algebra of dual numbers are the same as the laws of algebra of complex numbers. This means that dual numbers form a ring over the real number field. Now let f be a differentiable function with dual variable x x x ε * = + . Then the Maclaurine series generated by f is given by D be the set of all triples of dual numbers, i.e., ε * = + in 3 D , the scalar product and the vector product are defined by which is called dual unit sphere.
E. Study used dual numbers and dual vectors in his research on the geometry of lines and kinematics. He devoted special attention to the representation of directed lines by dual unit vectors and defined the mapping that is known by his name: There exists one-to-one correspondence between the vectors of dual unit sphere 2 S and the directed lines of space of lines 3 IR . By the aid of this correspondence, the properties of the spatial motion of a line can be derived. Hence, the geometry of ruled surface is represented by the geometry of dual curves on the dual unit sphere in 3 D (Fig. 1).
By considering The E. Study Mapping, the geometric interpretation of dual angle is that θ is the real angle between lines 1 2 , L L corresponding to the dual unit vectors , a b , respectively, and * θ is the shortest distance between those lines [1,6].
Let K be a moving dual unit sphere generated by a dual orthonormal system , , , , , dq q h a q h q q q h h h a a a dq and K′ be a fixed dual unit sphere with the same center. Then, the derivative equations of the dual spherical closed motion of K with respect to K′ are

Mannheim Offsets of Trajectory Ruled Surfaces
where θ θ εθ * = + , (0 , ) θ π θ * ≤ ≤ ∈ » is the dual angle between the generators q and 1 q of Mannheim trajectory ruled surface q ϕ and 1 q ϕ . The angle θ is called the offset angle and θ * is called the offset distance (Fig. 2).   (22) and (28) Separating (32) into real and dual parts we have Then, we may give the following results On the other hand from (5) and (28)  , , T N B . Thus, the following theorem may be given.

Conclusion
In this paper, we give the characterizations of Mannheim offsets of ruled surfaces. We find new relations between the invariants of Mannheim offsets of ruled surfaces. Furthermore, we show that the striction lines of the Mannheim offsets of the developable ruled surface are Mannheim partner curves. By using the similar methods given for Bertrand offsets, Mannheim offsets of the ruled surfaces can be used in CAGD.