Dual Curves of Constant Breadth in Dual Euclidean Space D 3

In this paper, we study dual curves of constant breadth in dua l E clidean SpaceD3. We obtain the differential equations characterizing dual curves of constant breadth in D3 and we introduce some special cases for these dual curves.


Introduction
The properties of plane convex curves have been studied by many geometers so far. Two brief reviews of the most important publications on this subject have been published by Struik [18]. Also, a number of interesting properties of plane curves of constant breadth are included in the works of Euler [8], Ball [2], Barbier [3], Blaschke [4] and Mellish [15].
A space curve of constant breadth was defined by Fujivara [9]. He considered a closed curve whose normal plane at a point P has only one more point Q in common with the curve, and for which d(P, Q) is constant. For such curves PQ is also normal at Q. Furthermore, spherical curve of constant breadth was defined by Blaschke [5]. Köse presented some concept for space curves of constant breadth in Euclidean 3-space [12,13]. Furthermore, differential equations characterizing space curves of constant breadth were obtained by Sezer [17]. Similar characterization of space curves of constant breadth in Euclidean 4-space were given by Magden and Köse [14]. Also, the curves of constant breadth have been studied in Minkowski 3-space spacelike and timelike curves of constant breadth are normal curves, helices and spherical curves in some special cese. In [10], Kocayigit andÖnder showed that in Minkowski 3-space spacelike and timelike curves of constant breadth in are normal curves, helices and spherical curves in some special ceses. Moreover, Onder, Kocayigit and Candan studied differential equations characterizing timelike and spacelike curves of constant breadth in Minkowski 3-space E 3 1 and gave a criterion for a timelike or spacelike curve of to be the curve of constant breadth in E 3 1 [16]. Spacelike curves of constant breadth in Minkowski 4-space were given by Kazaz,Önder and Kocayigit [11]. Furthermore, Yılmaz have studied dual timelike curves of constant breadth in dual Lorentzian space [20].
In this paper we study dual curves of constant breadth in dual Lorentzian space D 3 . We obtain a third order differential equation which characterizes dual curves of constant breadth in D 3 and give some special cases.
Let D = R × R = {ā = (a, a * ) : a, a * ∈ R} be the set of the pairs (a, a * ).Forā = (a, a * ),b = (b, b * ) ∈ D the following operations are defined on D: The element ε = (0, 1) ∈ D satisfies the relationships Let consider the elementā ∈ D of the formā = (a, 0). Then the mapping f : D → R, f (a, 0) = a is a isomorphism. So, we can write a = (a, 0).By the multiplication rule we have that a = (a, a * ) = (a, 0) + (0, a * ) = (a, 0) + (0, 1)(a * , 0) = a + εa * Thenā = a + εa * is called dual number and ε is called dual unit. Thus the set of dual numbers is given by The set D forms a commutative group under addition [6,7]. The associative laws hold for multiplication. Dual numbers are distributive and form a ring over the real number field.
Dual function of dual number presents a mapping of a dual numbers space on itself. The general expression for dual analytic (differentiable) function as follows the set of all triples of dual numbers, i.e., Then the set D 3 is called dual space. The elements of D 3 are called dual vectors. Similar to the dual numbers, a dual vectorã may be expressed in the formã = a + εa * = (a, a * ), where a and a * are the vectors of R 3 . Then for any vectors a = a + εa * andb = b + εb * of D 3 , the scalar product and the vector product are defined by respectively, where < a, b > and a × b are the inner product and the vector product of the vectors a and b in R 3 , respectively.
The norm of a dual vectorã is given by ã = a +ε < a, a * > a , (a = 0).

Dual curves of constant breadth in D 3
In this section, we consider dual curves of constant breadth in dual Euclidean space D 3 . Let now C and C ζ be a pair of unit speed dual curves of class C 3 with nonzero dual curvature and dual torsion in D 3 and let those dual curves have parallel tangents in opposite directions at corresponding points. Then we may write the position vector of dual curve C ζ as followsζ =φ +γt +δñ +λb (2) whereγ,δ andλ are arbitrary functions ofs. By differentiating (2) with respect tos we get

Definition 1. Let C be a dual curve in D 3 with dual position vectorφ =φ (s). If C has parallel tangents in opposite directions at corresponding points and if the distance between these points is always constant then C is called a dual curve of constant breadth in
Since we havet ζ = −t from (3) where f (φ) =ρ +ρ ζ . Using the system of ordinary differential equations (5), we have the following dual third order differential equation with respect toγ d dφ Then we can give the following corollary:

Corollary 1. The dual differential equation given in (6) is a characterization for dual curve C ζ and the position vector of this dual curve can be determined by the solution of this equation.
Let now investigate the solution of equation (6) in a special case. Assume thatκ(s),τ(s) be constants i.e., κ, κ * , τ and τ * be constant. It means that dual curve C is a dual helix. In the case equation (6) has the form d 3γ Then we have the following corollary: Corollary 2. The dual differential equation characterizing dual helix curves of constant breadth in D 3 is given as follows Corollary 3. Let C be a dual helix. In this case equation (7) gives us followings where A and B are constants. Then we have the following theorem.
By differentiating (9) with respect toφ it follows By virtue of (5), the differential equation (10) yields From (11) the following cases are obtained.
(i)γ = 0. Then, from (5) the other components arē and we getζ (ii) dγ dφ =δ . That is f = 0. Then followings are obtained (a) κ = τ and κ * = τ * . Thus, the components are (b) κ τ = κ * τ * , τ > κ and 1 +τ In this study, some characterizations of space curves according to Bishop frame in Euclidean 3-space are given by using Laplacian operator and Levi-Civita connection. Furthermore, the general differential equations which characterize the space curves according to Bishop frame are given. It is obtained some useful results and given theorems. Similar characterizations can be applied to other curves.