The Properties of Bertrand Curves In Dual Space

In this study, we investigate Bertrand curves in three dimensional dual space D3 and we obtain the characterizations of these curves in dual space D3. Also we show that involutes of a curve constitute Bertrand pair curves.


Introduction
In the study of differential geometry, the characterizations of the curves and the corrsponding relations between the curves are significant problem. It is well known that many important results in the theory of the curves in E 3 were given by G. Monge and then G. Darboux detected the idea of moving frame. After this Frenet defined moving frame and special equations which are used in mechanics, kinematics and physics.
A set of orthogonal unit vectors can be built, if a curve is differentiable in an open interval, at each point. These unit vectors are called Frenet frame. The Frenet vectors along the curve, define curvature and torsion of the curve. The frame vectors, curvature and torsion of a curve constitute Frenet apparatus of the curve.
It is certainly well known that a curve can be explained by its curvature and torsion except as to its position in space. The curvature (κ) and torsion (τ ) of a regular curve help us to specify the shape and size of the curve. Such as; If κ = τ = 0, then the curve is a geodesic. If κ = 0 (constant) and τ = 0, then the curve is a circle with radius 1 κ . If κ = 0 (constant) and τ = 0 (constant), then the curve is a helix.
Bertrand curves can be given as another example of that relation. Bertrand curves are discovered in 1850, by J. Bertrand who is known for his applications of differential equations to physics, especially thermodynamics. A Bertrand curve in E 3 is a curve such that its principal normal vectors are the principal normal vectors of an other curve. It is proved in most studies on the subject that the characteristic property of a Bertrand curve is the existence of a linear relation between its curvature and torsion as: with constants λ, µ where λ = 0 (see [6]).
Dual numbers were defined by W.K.Clifford (1849-1879). After him E. Sudy used dual numbers and dual vectors to clarify a mapping from dual unit sphere to three dimensional Euclidean space E 3 . This mapping is called Study mapping. Study mapping corresponds the dual points of a dual unit sphere to the oriented lines in E 3 . So the set of oriented lines in Euclidean space E 3 is one to one correspondence with the points of dual space in D 3 .
In this paper, we study Bertrand curves in dual space D 3 .

Preliminaries
We now recall some basic notions about dual space and apparatus of curves. The set D is called the dual number system and the elements of this set are in type of a = a + εa * . Here a and a * are real numbers and ε 2 = 0 which is called a dual unit. The elements of the set D are called dual numbers. The set D is given by D = { a = a + εa * | a, a * ǫ R} . For the dual number a = a + εa * , a ∈ R is called the real part of a and a * ∈ R is called the dual part of a.
Two inner operations and equality on D are defined for a = a+εa * and b = b+εb * , as ; is called the addition in D.
is called the multiplication in D.
3) a = b if and only if a = b and a * = b * .(see [5,8]) Also the set D = { a = a + εa * | a, a * ǫ R} forms a commutative ring with the following operations The division of two dual numbers a = a + εa * and The set is a module on the ring D which is called D-Module and the elements are dual vectors consisting of two real vectors.
The inner product and vector product of Now we will give dual Frenet vectors of the dual curve with the dual arc-length parameter s. Then is called the unit tangent vector of − → α (s). The norm of the vector d − → T d s which is given by Here κ : I −→ D is nowhere pure-dual. Then the unit principal normal vector of − → α (s) is defined as which are called Frenet formulas [5]. The function τ :

4İLKAY ARSLAN GÜVEN ANDİPEK AGAOGLU
For a general parameter t of a dual space curve − → α , the curvature and torsion of − → α can be calculated as ;

Bertrand curves in D 3
In this section, we define Bertrand curves in dual space D 3 and give characterizations and theorems for these curves. where s ∈ I ⊂ D and c ∈ D(constant). Namely we mean that the distance between the corresponding points of the dual Bertrand curves is constant. If the dual angle between the tangent vectors − → T (s) and If we differentiate the equation (3.3), we get If we take the inner product of the equation (3.4) with − → T (s) both sides and we use the Frenet equations, we have This completes the proof.
then from the previous proof we have From the equation (3.2) we write In above equations, if we take into account d s ds = a (constant) then we get We can write Finally, from the equations (3.8) and (3.9), we find λ. κ(s) + µ. τ (s) = 1. It can be written from [4] (3.