Received 1 June 2014; revised 1 July 2014; accepted 13 July 2014

ABSTRACT

In this paper, we define dual curvature motion on the dual hyperbolic unit sphere in the dual Lorentzian space with dual signature. We carry the obtained results to the Lorentzian line space by means of Study mapping. Then we make an analysis of the orbits during the dual hyperbolic spherical curvature motion. Finally, we find some line congruences, the family of ruled surfaces and ruled surfaces in.

Keywords:

Dual Curvature Motion, Dual Lorentzian Space, Study Mapping

1. Introduction

Dual numbers had been introduced by W.K. Clifford (1845-1849) as a tool for his geometrical investigations. After him, E. Study (1860-1930) used dual numbers and dual vectors in his research on the geometry of lines and kinematics [1] . 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 and the directed lines of space of lines. Hence, a differentiable curve on the sphere corresponds to a ruled surface in the line space [2] - [4] . Ruled surfaces have been widely applied in surface design, manufacturing technology and simulation of rigid bodies [5] .

E. Study’s mapping plays a fundamental role in the real and dual Lorentzian spaces [6] . By this mapping, a curve on a dual hyperbolic unit sphere corresponds to a timelike ruled surface in the Lorentzian line space, in other words, there exists a one-to-one correspondence between the geometry of curves on and the geometry of timelike ruled surfaces in. Similarly, a timelike (spacelike) curve on a dual Lorentzian unit sphere corresponds to a spacelike (timelike) ruled surface in the Lorentzian line space, this means that, there exists a one-to-one correspondence between the geometry of timelike (spacelike) curves on and the geometry of spacelike (timelike) ruled surfaces in [7] . Since the dual Lorentzian metric is indefinite, the angle concept in this space is very interesting. For instance, the dual hyperbolic angle between two dual timelike unit vectors is a dual value formed with the (real) hyperbolic angle between corresponding two directed timelike lines in the Lorentzian line space and the shortest Lorentzian distance between these directed timelike lines.

Real spherical curvature motion had been introduced by A. Karger and J. Novak [8] . Also, a dual spherical curvature motion has been defined by Z. Yapar [9] . In recent years, study about the real spherical motion has been generalized to the Lorentz spherical motion [6] [7] [10] [11] . In this work, we consider the curvature motion on the dual hyperbolic unit sphere of the dual Lorentzian space and the results are carried over to the Lorentzian line space by the E. Study’s mapping.

2. Preliminaries and Definitions

In this section, we give a brief summary of the theory of dual numbers, dual Lorentzian vectors and Study’s mapping.

Let be the 3-dimensional Minkowski space over the field of real numbers with the Lorentzian inner product given by

where and.

A vector of is said to be timelikeif, spacelike if or, and lighlike (null) if and.

The norm of a vector is defined by. Let and be two vec-

tors in, then the Lorentzian cross product of and is given by

If and are real numbers and, the combination is called a dual number, where is a dual unit.

The set of all dual numbers forms a commutative ring over the real numbers field and is denoted by 𝔻. Then the set

is a module over the ring 𝔻 which is called a 𝔻-module or dual space. The elements of are called dual vectors. Thus a dual vector can be written as

where and are real vectors at.

If and with then the division is given by

Let be a differentiable function with dual variable. Then the Maclaurin series generated by is

where is the derivative of. Then it is easy to see that

The norm of a dual number is defined by

Then we can write

The Lorentzian inner product of two dual vectors, is defined by

where is the Lorentzian inner product of the vectors and in the Minkowski 3-space.

A dual vector is said to be timelike if, spacelike if and lightlike (or null) if

,

where is a Lorentzian inner product with signature.

The set of all dual Lorentzian vectors is called dual Lorentzian space and it is denoted by

The Lorentzian cross product of dual vectors and is defined by

where is the Lorentzian cross product in.

Lemma 2.1. Let. Then [12]

1)

2) ; and,

3) ,

4) .

Let. Then is said to be dual timelike unit vector (resp., dual spacelike unit vector) if the vectors and satisfy the following properties:

, (resp.,),

The set of all dual timelike unit vectors (resp., all dual spacelike unit vectors) is called the dual hyperbolic unit sphere (resp., dual Lorentzian unit sphere) and is denoted by (resp.,) [6] . (See [13] - [16] for Lorentzian basic concepts.)

Theorem 2.2. (E. Study Map) [6] There exists one-to-one correspondence between directed timelike (resp., spacelike) lines of and an ordered pair of vectors such that (resp.,) and

.

Definition 2.1. A directed timelike line in may be given by two points on it, and. If is any non-zero constant, the parametric equation of the line is. In this case, the vector given by

is called the moment of the vector with respect to the origin 0.

This means that the direction vector of the timelike line and its moment vector are independent of the choice of the points on the line. However the vector and are not independent of one another. Also, they satisfy the following properties:

Let, 0 and denote the dual hyperbolic unit sphere, the center of and the dual orthonormal system at 0, respectively where we have

and

[7] . In this case the orthonormal system is the system of the space of lines.

A ruled surface is a surface generated by the motion of a straight line in. This line the generator of the surface. This follows the following definition.

Definition 2.2. A ruled surface is said to be timelike if the normal of surface at every point is spacelike, and spacelike if the normal of surface at every point is timelike [7] .

Let and denote two different points at and denote the dual hyperbolic angle. The hyperbolic angle has a value which is a dual number, where and are the hyperbolic angle and the minimal Lorentzian distance between directed lines and, respectively.

3. Dual Curvature Motion on the Dual Hyperbolic Unit Sphere

Let us consider a fixed dual orthonormal frame and represent this frame by the dual hyperbolic unit sphere. Consider the dual hyperbolic spherical motion of a hyperbolic spherical segment

of constant such that its endpoints move along circles, one of them being a great circle which lying on the plane on. Let a circle with radius which is perpendicular to the great circle

be given in a plane which is parallel to the plane. Its center is on the vector and with distance

from the plane. The segment moves so that. The position vectors of the

endpoints of segment are chosen as the vectors (timelike) and (spacelike) of the moving frame. The vector is then defined by the relation. As the parameter of motion we choose the dual hyperbolic angle of the timelike vectors and. Let us denote the dual hyperbolic angle of the vectors and by. Then

where the vector is spacelike. Further, we have

(timelike).

where is timelike. It must be

, i.e.,

i.e. or, where. Then. Thus we obtain

(1)

Thus, we have the orthonormal dual frame. Let this system be represented by moving hyperbolic sphere. Then, a dual hyperbolic curvature motion takes place. This motion will be called a dual hyperbolic curvature motion. Let be a fixed point on the arc. During the dual hyperbolic curvature motion, the point draws an orbit on the fixed hyperbolic sphere.

Denote the dual hyperbolic angles of by and respectively. Then it can be written

(2)

where [5] . From Equation (2), making the necessary calculations for, we have

(3)

(4)

where and are the real and dual parts of, also. Since, Ө₁ and Ө₂ are constant (i.e., are all constants), is constant. Equations (3) and (4) depend only two parameters and. Thus, Equations (3) and (4) represent a timelike congruence in (for more details on congruences, see [10] [12] ).

Let denote the position vector of an arbitrary point of a directed timelike line of this timelike line congruence in. Then we have

(5)

Since are the coordinates of, making the necessary calculations, we obtain

(6)

(7)

and

(8)

If, (i.e.) then, i.e.. In this case, from Equation (2) we have. Thus, from Equations (6)-(8) we obtain

(9)

From Equation (9) we have

(10)

which represents a line congruence. Thus, we have the following theorem.

Theorem 3.1. During the dual hyperbolic spherical curvature motion in the case of (hence) in Equation (2), the Study map of the orbit which is drawn on the by is the congruence in

(11)

If we take in the Equation (9), then we have

(12)

Thus, we have the following theorem.

Theorem 3.2. During the dual hyperbolic spherical curvature motion in the case of, the Study map in of the orbit drawn on the by is the cone which is given by

In addition, if we take (c = constant) then we have

which represents a right helicoid.

If, i.e., then, i.e. In this case , from Equation (2) we have. Thus, from Equations (6)-(8) we obtain

(13)

If we put, from Equation (13) we have

(14)

From Equation (14) we have

(15)

which represents a cone whose axis is the vector. Thus, we have the following theorem.

Theorem 3.3. During the dual hyperbolic spherical curvature motion, the orbit drawn on by (if) represents a cone in the, whose axis is the vector.

If we put, from Equation (13) we have

(16)

From Equation (16) we have

(17)

which represents a cone whose axis is the vector. Thus, we have the following theorem.

Theorem 3.4. During the dual hyperbolic spherical curvature motion, the orbit drawn on by (if) represents a cone in the, whose axis is the vector.

4. Analysis of the Orbit of during the Dual Hyperbolic Spherical Curvature Motion

Seperating real and dual parts of, from Equation (1) we have

(18)

(19)

Equations (18) and (19) have only two parameters and. Hence Equations (18) and (19) represent a line congruence in. Let denote the position vector of an arbitrary point of an oriented line of this congruence in, then considering Equation (5) we have

(20)

Since are the coordinates of, making the necessary calculations, we obtain

(21)

In the case of from Equation (21) we have

(22)

From Equation (22) we obtain

(23)

which represents an one-parameter family of cone in.

If we put in the Equation (23), then we have

(24)

which represents an elliptic cone, whose axis is the vector. Thus, we have the following theorem.

Theorem 4.1. During the dual hyperbolic spherical curvature motion, the orbit drawn on by (if) represents an elliptic cone, whose axis is the vector in the.

In addition, putting various values of parameters in the Equations (21) or (22) we have different line congruences or ruled surfaces in.

5. Conclusion

This paper presents the curvature motion on the dual hyperbolic unit sphere. We define the curvature motion on the dual hyperbolic unit sphere of the dual Lorentzian space and the results are carried over to the Lorentzian line space by the E. Study mapping. The orbits drawn on the fixed dual hyperbolic unit sphere by unit dual vectors of an orthonormal base are obtained. During this carrying, we do an analysis of orbits they drawn by the vectors of dual hyperbolic unit sphere and then we get some line congruences, the families of ruled surfaces and ruled surfaces in according to variables of parameters. Moreover we find equations of these line congruences, the families of ruled surfaces and ruled surfaces. This motion and its results may give a way to define new motions and contribute to the study of surface design, manufacturing technology, robotic research and special and general theory of relativity, and many other areas in 3-dimensional Lorentzian space.

References

NOTES

^*Corresponding author.