Mannheim Offsets of Ruled Surfaces under the 1-Parameter Motions

abstract: In this study, the situation of Mannheim offsets of ruled surfaces under the 1-parameter motions is investigated. Firstly, relationships between geodesic Frenet trihedrons of Mannheim offsets of ruled surfaces are obtained and the relationship between the curvatures of the surface pairs is examined. Also, change of integral invariants the surface pairs under the 1-parameter motions is studied. Finally, the relevant example is given for every Mannheim offsets of ruled surfaces.


Introduction
Classical differential geometry is divided into two main sections: Curves and Surfaces. Properties of curves and surfaces are examined over a point in the neighborhood of this curve and surface components [3,6,11]. There are some special pairs of curves and pairs of surfaces. For example; pair of involuteevolute curves which is tangent vectors perpendicular to each other in the corresponding points [7], pair of Bertrand curves which is the principal normal vectors common [2] and pair of Mannheim curves which is the principal normal vector corresponds to binormal vector in the corresponding points [12,13,20] are pairs of curves recently studied. A pair of surfaces is defined using these pairs of curves [9,14,15].
Geometry is the theory of do not change under a transformation group. Kinematics is the geometry of transformation protecting distances. It is obtained a curve with the continuous movement of a point on a surface and a surface with a continuous movement in space of a curve. If a curve especially is a line, in this case, the obtained surface is called Ruled surface. Examine the properties of the surface to a period under a motion has attracted the attention of mathematicians. For example; the conoidal ruled surface which is a special surface was examined under the helical and homothetic motions [1,5].
Movement on each of the curves or surfaces are not of concern to the geometry only. It is used in physics, engineering, computer-aided design and manufacturing, robotics, and in many industrial applications. Wang, Liu, and Xiao studied kinematic differential geometry of a rigid body in spatial motion in three consecutive articles [17,18,19]. In these articles, the geometrical properties of a point trajectory and a line trajectory in spatial motion are searched by means of a new adjoint approach. The invariant of axoids are derived and their kinematic meanings are revealed. This new adjoint approach a curve adjoint to a ruled surface is expanded for another ruled surface adjoint to a ruled surface. Based on the results obtained from them, the kinematics conditions for the moving body to be connected with a binary link by a kinematic pair are discussed and described by the geometrical properties of the line trajectory and the constraint ruled surface. To explain the formation of a double curve, kinematic decomposition of the coupler plane is done by Lan, Huijun, and Liuming [10]. The analysis of coupler limit positions in a four-bar linkage, the formation of coupler curves are studied and the coupler plane is divided into four zones. Shapes and features of coupler curves in different zones are explained. Also, 2 K. Orbay and T. Şahin recently, Cui and Dai extend the adjoint approach to a general surface and investigate the kinematics of the relative motion of two rigid objects that maintain sliding-rolling contact [4].
In this study, the situation of Mannheim offsets of ruled surfaces under the 1-parameter motions is investigated. Relationships between geodesic Frenet trihedrons of Mannheim offsets of ruled surfaces are obtained and the relationship between the curvatures of the surface pairs is examined. Also, change of integral invariants the surface pairs under the 1-parameter motions is studied.

Preliminaries
It has been defined as a ruled surface as a surface generated by the motion of a straight line, its generating line, generator, or ruling. There are ∞ 1 generators on a ruled surface. Let e = e(s) be the unit vector in the direction of the generating line passing through a point of an arbitrary nonisotropic curve α, which is called the base curve, lying on the surface, of which the equation is α = α(s). The ruled surface is given by a parametric representation the vector e = e(s), drawn through the center 0 of the unit sphere, describe the director cone of the surface. When α is a constant the surface is a cone, when e is a constant, the surface is a cylinder [16].
The vector e traces a curve on the surface of unit sphere S 2 called spherical indicatrix of the ruled surface [15]. The orthonormal system {e, t, g} is called the geodesic Frenet trihedron of the ruled surface φ such that t = es es and g = e×es es are the central normal and the asymptotic normal direction of φ, respectively. For the geodesic Frenet vectors e, t and g, we can write where q and γ are the arc-length of spherical indicatrix (e) and the geodesic curvature of (e) with respect to S 2 , respectively [15]. The striction point on a ruled surface φ is the foot of the common normal between two consecutive generators. The set of striction points defines the striction curve given by If consecutive generators of a ruled surface intersect, then the surface is said to be developable. The spherical indicatrix e, of a developable surface is tangent of its striction curve [15].
The distribution parameter of the ruled surface φ is defined by The ruled surface is developable if and only if P e = 0 [15]. In this paper, the striction curve of the ruled surface φ will be taken as the base curve. In this case, for the parametric equation of φ, we can write The ruled surface φ * is said to be Mannheim offset of the ruled surface φ if there exists a one to one correspondence between their generators such that the asymptotic normal of φ is the central normal of φ * . In this case, (φ, φ * ) is called a pair of Mannheim ruled surface [14]. Let {e, t, g } and {e * , t * , g * } be two geodesic Frenet triples at the point c(s) and c * (s) of the striction curves (c) and (c * ) of the ruled surface φ and φ * , respectively. Let parametric representations of the ruled surface φ and φ * be given by between the geodesic Frenet triples {e, t, g} and {e * , t * , g * }, where θ is the angle between corresponding generators [14].

Mannheim Offsets under the 1-Parameter Motions
Let the ruled surfaceφ roll on the ruled surface φ along curves (c) and (c) whose arc-length parameter s ands under 1-parameter spatial motion, respectively. While (φ, φ * ) is a pair of Mannheim ruled surface, the pair of ruled surface φ ,φ * also continues to be a pair of Mannheim ruled surface during movement.   Proof. Reorganizing the equations first and third of (3.4), we get respectively. Considering together of (3.6) and (3.7), we obtain   Proof. Because (c) is a geodesic, using (3.5), we obtain γ = tan β After that, considering (3.7), we reachγ 2 = (qq) 2 − 1.
So, (c) is a geodesic ⇔γ = 0 Theorem 3.5. Let the ruled surfaceφ rolls on the ruled surface ϕ under 1-parameter spatial motion. There is the following relation between distribution parameters P e andPē of the Mannheim ruled surfaces ϕ andφ where R andR are distance between corresponding striction points of the pair of Mannheim ruled surface (ϕ, ϕ * ) and (φ,φ * ), respectively.
Example 3.6. The ruled surface is a Mannheim offset of the ruled surface  We can see the ruled surfaces ϕ(s, v), ϕ * (s, v),φ(s, v) andφ * (s, v) as follows:

Conclusions
In this study, it is investigated that relationship between the geodesic curvatures of these curves when the couple of Mannheim ruled surfaces roll on along the striction curves under the 1-parameter motions. If one of the curves is a geodesic, the other would be a circle. In order that while one of the curves is geodesic, the other of the curve to be the geodesic too, the absolute value of the arc-lengths of spherical indicatrix should be equal. Moreover, if the ratio of distribution parameters of the pair the Mannheim ruled surfaces and the ratio of the distance between corresponding striction points of the pair of Mannheim ruled surface is proportional, it must be the arc-length of spherical indicatrix. With these data, considering the couple of Mannheim ruled surfaces and using the relations obtained under the 1-parameter motions, new surfaces can be formed. The physical relationships of these surfaces can be examined. Studies on industrial use can be made. It can be used to obtain artistic products and architectural designs.