Dual Smarandache Curves and Smarandache Ruled Surfaces

In this paper, by considering dual geodesic trihedron (dual Darboux frame) we define dual Smarandache curves lying fully on dual unit sphere S^2 and corresponding to ruled surfaces. We obtain the relationships between the elements of curvature of dual spherical curve (ruled surface) x(s) and its dual Smarandache curve (Smarandache ruled surface) x1(s) and we give an example for dual Smarandache curves of a dual spherical curve.


Introduction
In the Euclidean space 3  E , an oriented line L can be determined by a point p L ∈ and a normalized direction vector a of L , i.e. 1 a = .The components of L are obtained by the moment vector a p a * = × with respect to the origin in 3  E .The two vectors a and a * are not independent of one another; they satisfy the relationships , 1, , 0 a a a a * = = .The pair ( , ) a a * of the vectors a and a * , which satisfies those relationships, is called dual unit vector [2].The most important properties of real vector analysis are valid for the dual vectors.Since each dual unit vector corresponds to a line of 3  E , there is a one-to-one correspondence between the points of a dual unit sphere 2  S and the oriented lines of 3  E .This correspondence is known as E. Study Mapping [2].As a sequence of that, a differentiable curve lying fully on dual unit sphere in dual space 3  D represents a ruled surface which is a surface generated by moving of a line L along a curve ( ) s α in 3 E and has the parametrization ( , ) ( ) ( ) r s u s u l s α = + , where ( ) s α is called generating curve and ( ) l s , the direction of the line L , is called ruling.
In the study of the fundamental theory and the characterizations of space curves, the special curves are very interesting and an important problem.The most mathematicians studied the special curves such as Mannheim curves and Bertrand curves.Recently, a new special curve which is called Smarandache curve is defined by Turgut and Yılmaz in Minkowski space-time [6].Then Ali have studied Smarandache curves in the Euclidean 3space 3  E [1].Moreover, Önder has studied the Bertrand offsets of ruled surface according to the dual geodesic trihedron(Darboux frame) and given the relationships between the dual and real curvatures of a ruled surface and its offset surface [5].
In this paper, we give Darboux approximation for dual Smarandache curves on dual unit sphere 2  S .Firstly, we define the four types of dual Smarandache curves (Smarandache ruled surfaces) of a dual spherical curve(ruled surface).Then, we obtain the relationships between the dual curvatures of dual spherical curve ( ) s α and its dual Smarandache curves.Furthermore, we show that dual Smarandache eg -curve of a dual curve is always its Bertrand offset.Finally, we give an example for Smarandache curves of an arbitrary curve on dual unit sphere 2  S .

Dual Numbers and Dual Vectors Let
(1) Let consider the element a D ∈ of the form ( , 0) a a = .Then the mapping : , ( , 0) = is a isomorphism.So, we can write ( , 0) a a = .By the multiplication rule we have that ( , ) Then a a a is called dual number and ε is called dual unit.Thus the set of all dual numbers is given by The set D forms a commutative group under addition.The associative laws hold for multiplication.Dual numbers are distributive and form a ring over the real number field [2,4].Dual function of dual number presents a mapping of a dual numbers space on itself.Properties of dual functions were thoroughly investigated by Dimentberg [3].He derived the general expression for dual analytic (differentiable) function as follows ( ) Then the set 3 D is module together with addition and multiplication operations on the ring D and called dual space.The elements of 3  D are called dual vectors.Similar to the dual numbers, a dual vector a may be expressed in the form respectively, where , a b and a b × are the inner product and the cross product of the vectors a and a * in 3 IR , respectively.The norm of a dual vector a is given by , , ( 0) a a a a a a A dual vector a with norm 1 0 ε + is called unit dual vector.The set of all dual unit vectors is given by and called dual unit sphere [2,4].E. Study used dual numbers and dual vectors in his research on the geometry of lines and kinematics.He devoted special attention to the representation of directed lines by dual unit vectors and defined the mapping that is known by his name:

Theorem2.1.(E. Study Mapping):
There exists one-to-one correspondence between the vectors of dual unit sphere 2  S and the directed lines of space of lines 3 » [2,4].
By the aid of this correspondence, the properties of the spatial motion of a line can be derived.Hence, the geometry of the ruled surface is represented by the geometry of dual curves lying fully on the dual unit sphere in 3  D .The angle By considering The E. Study Mapping, the geometric interpretation of dual angle is that θ is the real angle between lines 1 2 , L L corresponding to the dual unit vectors , a b respectively, and * θ is the shortest distance between those lines [2,4].

Dual Representation of Ruled Surfaces
In this section, we introduce dual representation of a ruled surface which is given by Veldkamp in [7]  Therefore c e t ′× = ∆ and we obtain in view of (11): ( ) Let t be dual unit vector with the same sense as e′ ; then we find as a consequence of (12): (1 ) e t ε ′ = + ∆ .This leads in view of (13) to: (15) Introducing the dual unit vector e t g g g we observe e t g × = ; hence by means of (10) and ( 14): Then the dual frame { } , , e t g is called dual geodesic trihedron( or dual Darboux frame) of the ruled surface corresponding to dual curve e .Thus, the derivative formulae of this frame are given as follows, , , de dt dg t g e t ds ds ds where γ is called dual spherical curvature and given by ( ) ( ) The unit vector 0 d with the same sense as the Darboux vector d e g γ = + is given by The dual angle ρ between 0 d and e satisfies therefore: The point M on the dual unit sphere indicated by 0 d is called the dual spherical centre of curvature of k at the point Q given by the parameter value s , whereas ρ is the dual spherical radius of curvature [7].

Dual Smarandache curves and Smarandache Ruled Surfaces
From E. Study Mapping, it is well-known that dual curves lying on dual unit sphere correspond to ruled surfaces of the line space 3 IR .Thus, by defining the dual smarandache curves lying fully on dual unit sphere, we also define the smarandache ruled surfaces.Then, the differential geometry of smarandache ruled surfaces can be investigated by considering the corresponding dual smarandache curves on dual unit sphere.
In this section, we first define the four different types of the dual smarandache curves on dual unit sphere.Then by the aid of dual geodesic trihedron(Dual Darboux frame), we give the characterizations of these dual curves(or ruled surfaces).

Dual Smarandache et -curve of a unit dual spherical curve(ruled surface)
In this section, we define the first type of dual Smarandache curves as dual Smarandache et -curve.Then, we give the relationships between the dual curve and its dual Smarandache et -curve.Using the found results and relationships we study the developable of the corresponding ruled surface and its Smaranadache ruled surface.α are given by where γ is as given in (18).
Proof.Let  If we represent the dual Darboux frames of α and 1 α by the dual matrixes E and 1 E , respectively, then (24) can be written as follows It is easily seen that det( ) 1 A = and where I is the 3 3 × unitary matrix.It means that A is a dual orthogonal matrix.Then we can give the following corollary.

Corollary 5.1. The relationship between Darboux frames of the dual curves(ruled surfaces) α and 1
α is given by a dual orthogonal matrix defined in (28).

Theorem 5.2. The relationship between the dual Darboux formulae of dual Smarandache etcurve 1
α and dual Darboux frame of α is as follows de ds e dt t ds g dg ds given by ( ) Proof.Since α is zero.
Corollary 5.3.The Darboux instantaneous vector of dual Smarandache et -curve is given by ( ) Proof: It is known that the dual Darboux instantaneous vector of dual Smarandache et -curve is  ( ) − ∆ Then substituting these equalities into to equation (30), we have ( ) ( ) .
Since the ruled surface corresponding to dual curve α is developable, 0 ∆ = .Hence, ( ) ( ) Proof.From (20), ( ) . Then from (30), the radius of dual curvature is In the following sections we define dual Smarandache eg , tg and etg curves.The proofs of the theorems and corollaries of these sections can be given by using the similar way used in previous section.

Dual Smarandache eg -curve of a unit dual spherical curve(ruled surface)
In this section, we define the second type of dual Smarandache curves as dual Smarandache eg -curve.Then, we give the relationships between the dual curve and its dual Smarandache eg -curve.Using obtained results and relationships we study the developable of the corresponding ruled surface and its Smarandache ruled surface., , e t g be its moving Darboux frame.The dual curve 2 α defined by ( ) is called the dual Smarandache eg -curve of α and fully lies on 2 S .Then the ruled surface corresponding to 2 α is called the Smarandache eg -ruled surface of the surface corresponding to dual curve α .Now we can give the relationships between α and its dual smarandache eg -curve 2 α as follows.
Theorem 5.7.Let ( ) s α α = be a unit speed regular dual curve lying on dual unit sphere 2 S .Then the relationships between the dual Darboux frames of α and its dual Smarandache eg - curve 2 α are given by α is a Bertrand offset of α [5].
In [5], Önder has given the relationship between the geodesic frames of Bertrand surface offsets as follows where θ θ εθ * = + , (0 , )   = be a unit speed regular curve on dual unit sphere.Then the relationship between the radius of dual spherical curvature of dual Smarandache eg -curve 2 α and the elements of dual curvature of α is, ( ) 3. Dual Smarandache tg -curve of a unit dual spherical curve(ruled surface) In this section, we define the second type of dual Smarandache curves as dual Smarandache tg -curve.Then, we give the relationships between the dual curve and its dual smarandache tg -curve.Using the found results and relationships we study the developable of the corresponding ruled surface and its Smaranadache ruled surface., , e t g be its moving Darboux frame.The dual curve 3 α defined by ( ) is called the dual Smarandache tg -curve of α and fully lies on 2 S .Then the ruled surface corresponding to 3 α is called the Smarandache tg -ruled surface of the surface corresponding to dual curve α .Now we can give the relationships between α and its dual Smarandache tg -curve 3 α as follows.
Theorem 5.13.Let ( ) s α α = be a unit speed regular dual curve lying on dual unit sphere 2 S .Then the relationships between the dual Darboux frames of α and its dual Smarandache tg - curve 3 α are given by If we represent the dual darboux frames of α and 3 α by the dual matrixes E and 1 E , respectively, then (37) can be written as follows It is easily seen that det( ) 1 A = and α is given by a dual orthogonal matrix defined by (38).
Theorem 5.14.The relationship between the dual Darboux formulae of dual Smarandache tg -curve 3 α and dual Darboux frame of α is given by ( Theorem 5.15.Let ( ) s α α = be a unit speed regular curve on dual unit sphere.Then the relationship between the dual curvatures of α and its dual Smarandache tg -curve 3 α is given by ( ) .

Dual Smarandache etg -curve of a unit dual spherical curve(ruled surface)
In this section, we define the second type of dual Smarandache curves as dual Smarandache etg -curve.Then, we give the relationships between the dual curve and its dual smarandache etg -curve.Using the found results and relationships we study the developable of the corresponding ruled surface and its Smaranadache ruled surface.
If we represent the dual darboux frames of α and 4 α by the dual matrixes E and 1 E , respectively, then (40) can be written as follows It is easily seen that det( ) 1 A = and α is given by a dual orthogonal matrix defined by (41).
Theorem 5.20.The relationship between the dual Darboux formulae of dual Smarandache etg -curve 4 α and dual Darboux frame of α is given by Theorem 5.21.Let ( ) s α α = be a unit speed regular curve on dual unit sphere.Then the relationship between the dual curvatures of α and its dual Smarandache etg -curve 4 α is given by ( )

*
) Guided by elementary differential geometry of real curves we introduce the dual arc-length s of the dual curve k by differential geometry the dual radius of curvature R of the dual curve speed regular dual curve lying fully on dual unit sphere 2S and { }, , e t g be its moving dual Darboux frame.The dual curve 1 is called the dual Smarandache et -curve of α and fully lies on 2 S .Then the ruled surface corresponding to 1 α is called the Smarandache et -ruled surface of the surface corresponding to dual curve α .Now we can give the relationships between α and its dual Smarandache et -curve 1 speed regular dual curve lying on dual unit sphere 2 S .Then the relationships between the dual Darboux frames of α and its dual Smarandache et - curve 1 29)Proof.Differentiating (25), (26) and (27) with respect to s , we have the desired equation (29).Theorem 5.3.Let ( ) s α α = be a unit speed regular curve on dual unit sphere.Then the relationship between the dual curvatures of α and its dual Smarandache et -curve 1 α is

Corollary 5 . 2 .
26) and (28), we get dual curvature of the curve 1 If the dual curvature γ of α is zero, then the dual curvature 1 γ of dual Smarandache et -curve 1

Definition 5 . 2 .
Let ( ) s α α = be a unit speed regular dual curve lying fully on dual unit sphere and { } the dual angle between the generators e and 2 e of Bertrand ruled surface e ϕ and 2 e ϕ .The angle θ is called the offset angle and θ * is called the offset distance[5].Then from (35) we have that offset angle is / 4 θ π = and offset distance is 0 θ * = .Then we have the following corollary.

Corollary 5 . 4 .= 2 α
The dual Smarandache eg -curve of a dual curve α is always its Bertrand offset with dual offset angle / be a unit speed regular dual curve on dual unit sphere 2 S .Then according to dual Darboux frame of α , the dual Darboux formulae of dual Smarandache eg -curve

=Corollary 5 . 5 . 1 α is 1 . 5 . 6 .==.
be a unit speed regular curve on dual unit sphere.Then the relationship between the dual curvatures of α and its dual Smarandache eg -curve 2 The dual curvature γ of α is zero if and only if the dual curvature 2 γ of dualSmarandache et -curveCorollaryThe Darboux instantaneous vector of dual Smarandache eg -curve is given by be a unit speed regular curve on dual unit sphere and 2 α be the dual Smarandache eg -curve of α .If the ruled surface corresponding to the dual curve α is developable then the ruled surface corresponding to dual curve 2 α is developable if and only if be a unit speed regular curve on dual unit sphere.Then the relationship between the radius of dual curvature of dual Smarandache eg -curve 2 Theorem 5.12.Let ( ) s α α

Definition 5 . 3 .
Let ( ) s α α = be a unit speed regular dual curve lying fully on dual unit sphere and { }

I is the 3 3 ×Corollary 5 . 7 .
unitary matrix.It means that A is a dual orthogonal matrix.Then we can give the following corollary.The relationship between the Darboux frames of the dual curves(ruled surfaces) α and 3

Definition 5 . 4 . 4 α=
Let ( ) s α α = be a unit speed regular dual curve lying fully on dual unit sphere and { }, , e t g be its moving Darboux frame.The dual curve 4 dual Smarandache etg -curve of α and fully lies on 2 S .Then the ruled surface corresponding to is called the Smarandache etg -ruled surface of the surface corresponding to dual curve α .Now we can give the relationships between α and its dual smarandache etg -curve 4 α be a unit speed regular dual curve lying on dual unit sphere 2 S .Then the relationships between the dual Darboux frames of α and its dual Smarandache etg -curve 4 α are given by ( )

I is the 3 3 ×
unitary matrix.It means that A is a dual orthogonal matrix.Then we can give the following corollary.Corollary 5.10.The relationship between the Darboux frames of the dual curves(ruled surfaces) α and 4

Figure 4 . 5 .
Figure 4. Smarandache tg ruled surface Figure 5. Smarandache etg ruled surface us investigate the dual Darboux frame fields of dual Smarandache et -curve If the dual curvature γ of α is zero, then the dual curvature 4 γ of dual Smarandache etg -curve 4 be a unit speed regular curve on dual unit sphere and 4 α be the dual Smarandache etg -curve of α .If the ruled surface corresponding to the dual curve α is developable then the ruled surface corresponding to dual curve 4