Slant Helices, Darboux Helices and Similar Curves in Dual Space D

In this paper, we give definitions and characterizations of slant helices, normalized Darboux helices and similar curves in dual space D. First, we define dual slant helices and dual normalized Darboux helices and show that dual slant helices are also dual normalized Darboux helices. Then, we introduce the concept of dual similar curves and obtain that the family of dual slant helices forms a family of dual similar curves.


Introduction
Helix is one of the most interesting curves in science and nature. We can encounter helices in biology, fractal geometry, computer aided design, computer graphics, etc [1]. In differential geometry, a general helix is defined by the property that the tangent line of the curve makes a constant angle with a fixed straight line [14]. In 1802, Lancret stated that "a curve is a general helix if and only if the ratio of curvature to torsion is constant" [14].
There are many papers about helices and some of them include different types of helices and their properties. Barros proved the Lancret theorem for general helices in a space form by using killing vector fields along curves [2]. Izumiya and Takeuchi defined slant helices and conical geodesic curves [8]. Kula and Yaylı obtained that the indicatrices of a slant helix are spherical helices [9]. Moreover, they showed that a curve of constant precession is a slant helix. Kula et. al. investigated the relationships between slant helices and general helices in R 3 [10]. They obtained differential equations which are characterizations of a slant helix. Lee, Choi and Jin studied dual slant helix and Mannheim partner curves in the dual space D 3 [11]. Zıplar, Şenol and Yaylı introduced Darboux helices in R 3 and gave the relations between Darboux helices and slant helices [18].
On the other hand, in local differential geometry, associated curves such as Bertrand curves, Mannheim curves and involute-evolute curves are very fascinating research area. Recently, El-Sabbagh and Ali added a new one to these associated curves [6]. They defined a new family of curves and called a family of similar curves with variable transformation. Also, they introduced relationships between some special curves and similar curves.
This study consists of two original sections, Sections 3 and 4. In Section 3, the definitions and characterizations of slant helices and Darboux helices are introduced in dual space D 3 . It is shown that dual slant helices are also dual Darboux helices. In Section 4, dual similar curves are defined in D 3 and it is obtained that the family of dual slant helices and of course in a special case dual Darboux helices forms a family of dual similar curves.

Preliminaries
A dual number, as introduced by W. Clifford, can be defined as an ordered pair of real numbers (a, a * ) where a is called the real part and a * is called the dual part of the dual number. If both parts are nonzero, the dual number is said to be proper; if the real part is zero, it is called a pure dual number; and if the dual part is zero, it reduces to a real number. Dual numbers may be formally expressed asā = a + εa * where ε is the dual unit which is subjected to the following rules [15]: We denote the set of dual numbers by D, i.e., D = ā = a + εa * : a, a * ∈ R, ε 2 = 0 .
A dual numberā = a + εa * divided by a dual numberb = b + εb * , with b = 0, can be defined asā ( [3,4]). We can define the function of a dual number f (ā) by expanding it formally in a Maclaurin series with ε as variable. Since ε n = 0 for n > 1, we obtain where f (a) is derivative of f (a) with respect to a [5].
In analogy with dual numbers, a dual vector referred to an arbitrarily chosen origin can be defined as an ordered pair of vectors (a, a * ), where a, a * ∈ R 3 [16]. Also dual vectors can be expressed asã= a+εa * , where a, a * ∈ R 3 and ε 2 = 0. We denote the set of dual vectors by D 3 , i.e., D 3 is a module over the ring D and it is called dual space. For any dual vectorsã= a+εa * andb= b+εb * in D 3 , the scalar product and the vector product are defined by , respectively [4,7].
The norm of a dual vectorã is given by (See [7]). A dual vectorã with norm 1 + ε0 is called dual unit vector. The set of dual unit vectors is denoted bỹ and called dual unit sphere [3,7]. A dual angle, subtended by two oriented lines in space as introduced by Study in 1903, is defined asθ = θ + ε θ * , where θ is the projected angle and θ * is the shortest distance between the two lines [13].
Letα(s) be a dual space curve with dual arc length parameters. Dual unit tangent vector ofα is defined by 92 Slant Helices, Darboux Helices and Similar Curves in Dual Space D 3 DifferentiatingT with respect to dual arc length parameters we have whereκ =κ(s) is called dual curvature. We restrict thatκ(s) is never pure dual number. The dual unit vectorÑ = (1/κ)T is called dual unit principal normal vector ofα. The dual unit vectorB defined byB =T ×Ñ is called dual unit binormal vector ofα. The dual frame T (s),Ñ (s),B (s) is called moving dual Frenet frame along the dual space curveα(s) in D 3 . For the curveα, the dual Frenet derivative formulae can be given in matrix form as whereτ =τ (s) is called the dual torsion ofα [17]. Then the dual Darboux vector ofα(s) is defined byW =τT +κB which gives the derivative formulae (4) as follows The unit dual Darboux vector ofα(s) is defined byW 0 =τT +κB √τ 2 +κ 2 .
Now we give the following theorem which we will use in the following sections.
Theorem 2.1. Letα(s) be a dual curve parametrized by dual arclengths. Supposeα =α(θ) is another parametric representation of this dual curve by the parameterθ = κ(s) ds. Then the dual unit tangent vectorT satisfies the following differential equation: and prime shows the derivative with respect toθ.
Proof. Letα(s) be a unit speed dual curve. We can write this curve in another parametric representationα =α(θ), whereθ = κ(s) ds, and we have new dual Frenet equations as follows: κ(θ) . If we use the first and second equation of the new Frenet formulae we have (7) dÑ dθ = d 2T dθ 2 = −T +fB and we get the dual unit binormal vector as Differentiating the equation (8) and using the third equation of equation (6), we obtain the dual vector differential equation as desired.

Slant Helices and Darboux Helices in Dual Space
In this section, we will give the definitions and characterizations of dual slant helix and dual Darboux helix. After these definitions, we can give the following characterizations: Proof. Letd be a fixed dual unit vector which makes a constant dual anglē φ = ± arccos(n) with the dual unit principal normal vectorÑ , i.e., wheren ∈ D is a constant. If we differentiate equation (10) with respect tō θ = κ(s) ds and use the new dual Frenet equation (6), we get Therefore, We can put B ,d =b and write (13)d =fbT +nÑ +bB .
Since the dual vectord is a dual unit vector, we getb = ± 1−n 2 1+f 2 . Hence, dual unit vectord can be written as If we differentiate equation (11) again and use equation (6), we have Substituting equation (14) into equation (15), we obtain following differential equation wherem =n √ 1−n 2 . Integrating above equation, we get wherec 1 is a dual integration constant. We can use a parameter changē θ →θ −c 1 to eliminate the integration constant. Then,f can be found from equation (17) as (18)  The equation (20) shows that dual vectord is a dual constant vector and Ñ ,d =n is constant. This finishes the proof of Theorem 3.1. Sinced is a constant dual vector, we getd = 0. Then if we take the derivative of (22) and use the dual Frenet formulae, we obtain where the prime shows the derivative with respect tos. Since the dual Frenet frame T ,Ñ ,B is linearly independent, we have   ā 1 −κ cosφ = 0, a 1κ −ā 3τ = 0, a 3 +τ cosφ = 0.
From the second equation of the system (24) we obtain By taking derivative of (28) and using the third equation of system (24) we get which is desired function.
Conversely, let us assume that the function (29) is satisfied. We define the dual vector whereφ is a dual constant angle between dual vectorsd andÑ . If we take derivative of equation (30) and use dual Frenet formulae (4) and (29), we can easily see thatd = 0, which gives thatd = constant. On the other hand, Ñ ,d = cosφ is constant and which means that the curveα is a dual slant helix.

Similar Curves in Dual Space
In this section, we will give the definition of dual similar curves with variable transformation. Then we will present some theorems concerning the relations between dual Frenet elements of dual similar curves.  Proof. Letα(s α ) andβ(s β ) be two regular similar curves with variable transformations in D 3 . If we differentiate the equation (38) with respect tos β we have From the equation (41), we obtain the two equations (39) and (40). Conversely, letα(s α ) andβ(s β ) be two regular curves in D 3 satisfying two equations (39) and (40). Multiplying equation (39) by κ β (s β ) and integrating the result with respect tos β we get Using equations (39) and (40), the equation (42) becomes If we use the first equation of Frenet formulae (4) and integrate the result, we obtain the equation (38) which completes proof.
Conversely, letα(s α ) andβ(s β ) be two regular curves in D 3 with the same dual unit binormal vector under the particular transformationλ β α = ds β ds α = τ ᾱ τ β of the dual arclengths. Differentiating the equation (44) with respect tō The equation (47) leads to the following two equations .
From the hypothesis and equation (48) we have which completes the proof.
From the equation (51) we havef β (θ β ) =f α (θ α ) under the variable transformationθ β =θ α . Under the equation (51) and the transformation (52), two equations (57) and (58) are the same. So the solutions are the same, i.e., the dual unit tangent vectors are the same. This means that the dual curvesα(s α ) andβ(s β ) are dual similar curves with variable transformation. This completes the proof.
After these characterizations we can give the following special cases: Case 1. If the dual curveα is a general dual helix, i.e.,τ ᾱ κα = cotφ =m is a constant andφ is dual constant angle between dual unit tangent vector and a fixed dual unit vector, then from Theorem 4.3, any dual similar curvẽ β of this dual helix has the propertyτ β κ β =m. Thus we have the following corollary: Corollary 4.1. The family of general dual helices with fixed dual angleφ between a fixed dual unit vector and dual unit tangent vector forms a family of dual similar curves with variable transformation.
Case 2. Letα andβ be two dual slant helices such that the transformation (52) is satisfied. If we use the relation (9) and (52), it is easy to prove that: wherem = cotφ is a constant andφ is the angle between the dual unit principal normal vector ofα and a fixed dual unit vector. Thus we have the following corollary:

Conclusion
The characterizations of special curves are important and fascinating problem of differential geometry. These curves are characterized by relationships between the curvatures and torsions of curves and well-known examples of such curves are helices and slant helices which have been studied in different spaces such as Euclidean space and Minkowski space. But there are no many studies on these curves in dual space which is a more general space than the others. In this space, a dual curve consists of two real curves. So, the characterizations of dual special curves include the characterizations of real space curves. This paper gives some new characterizations of dual slant helix. Moreover, the dual normalized Darboux helix and dual similar curves are introduced and the relationships between these special dual curves are obtained.