DUAL SMARANDACHE CURVES AND SMARANDACHE RULED SURFACES

Transcription

1 Mathematical Sciences And Applications E-Notes Volume No pp ) c MSAEN DUAL SMARANDACHE CURVES AND SMARANDACHE RULED SURFACES TANJU KAHRAMAN AND HASAN HÜSEYİN UĞURLU Communicated by Johann DAVIDOV) Abstract In this paper, by considering dual Darboux frame, we define dual Smarandache curves lying fully on unit dual sphere S and corresponding ruled surfaces We obtain the relationships between the elements of curvature of dual spherical curve ruled surface) αs) and its dual Smarandache curve Smarandache ruled surface) α s) and we give an example for dual Smarandache curves of a dual spherical curve Introduction In the Euclidean space E, an oriented line L can be determined by a point p L and a normalized direction vector a of L, ie a = The components of L are obtained by the moment vector a = p a with respect to the origin at cartesian coordinate system in E The two vectors a and a are not independent of one another; they satisfy the relationships a, a =, a, a = 0 The pair a, a ) of the vectors a and a, which satisfies those relationships, is called unit dual vector[] The most important properties of real vector analysis are valid for the dual vectors Since each dual unit vector corresponds to a oriented line of E, there is a one-to-one correspondence between the points of unit dual sphere S and the oriented lines of E This correspondence is known as E Study Mapping[] As a sequence of that, a differentiable curve lying fully on unit dual sphere in dual space D represents a ruled surface which is a surface generated by moving of a line L along a curve αs) in E and has the parametrization rs, u) = αs) + u ls), where αs) is called generating curve and ls), 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 Date: Received: September 9, 0; Accepted: November, 0 00 Mathematics Subject Classification 5A5, 4J6 Key words and phrases E Study Mapping; dual Smarandache curves; dual unit sphere; dual curvature 8

2 84 TANJU KAHRAMAN AND HASAN HÜSEYİN UĞURLU Turgut and Yılmaz in Minkowski space-time[6] Then Ali have studied Smarandache curves in the Euclidean -space E [] Moreover, Önder has studied the Bertrand offsets of ruled surface according to the dual geodesic trihedrondarboux 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 unit dual sphere S Firstly, we define the four types of dual Smarandache curves Smarandache ruled surfaces) of a dual spherical curveruled surface) Then, we obtain the relationships between the dual curvatures of dual spherical curve αs) and its dual Smarandache curves Furthermore, we show that dual Smarandache g-curve of a dual curve is always its Bertrand offset Finally, we give an example for Smarandache curves of an arbitrary curve on unit dual sphere S Dual Numbers and Dual Vectors Let D = IR IR = {ā = a, a ) : a, a IR} be the set of the pairs a, a ) For ā = a, a ), b = b, b ) D the following operations are defined on D: Equality: ā = b a = b, a = b Addition: ā + b = a + b, a + b ) Multiplication: ā b = ab, ab + a b) The element ε = 0, ) D satisfies the relationships ) ε 0, ε = 0, ε = ε = ε Let consider the element ā D of the form ā = a, 0) Then the mapping f : D IR, fa, 0) = a is a isomorphism So, we can write a = a, 0) By the multiplication rule we have that ) ā = a, a ) = a, 0) + 0, a ) = a, 0) + 0, )a, 0) = a + εa Then ā = a + εa is called dual number and ε is called dual unit Thus the set of all dual numbers is given by ) D = { ā = a + εa : a, a IR, ε = 0 } 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[,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[] He derived the general expression for dual analytic differentiable) function as follows: 4) f x) = fx + εx ) = fx) + εx f x), where f x) is derivative of fx) and x, x IR Let D = D D D be the set of all triples of dual numbers, ie, 5) D = {ã = ā, ā, ā ) : ā i D, i =,, }, Then the set D is module together with addition and multiplication operations on the ring D and called dual space The elements of D are called dual vectors Similar to the dual numbers, a dual vector ã may be expressed in the form ã =

3 DUAL SMARANDACHE CURVES AND SMARANDACHE RULED SURFACES 85 a + ε a = a, a ), where a and a are the vectors of IR Then for any vectors ã = a + ε a and b = b + ε b of D, the scalar product and the cross product are defined by 6) ã, b = a, b + ε a, b + a, ) b, and 7) ã b = a b + ε a b + a ) b, respectively, where a, b and a b are the inner product and the cross product of the vectors a and a in IR, respectively The norm of a dual vector ã is given by 8) ã = a + ε a, a, a 0) a A dual vector ã with norm + ε0 is called unit dual vector The set of all unit dual vectors is given by 9) S = { ã = a, a, a ) D : ã, ã = + ε0 }, and called dual unit sphere[,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: Theorem E Study Mapping) There exists one-to-one correspondence between the vectors of unit dual sphere S and the directed lines of space R [,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 given by the geometry of dual curves lying fully on the unit dual sphere in D The angle θ = θ + εθ between two unit dual vectors ã, b is called dual angle and defined by ã, b = cos θ = cos θ εθ sin θ By considering the E Study Mapping, the geometric interpretation of dual angle is that θ is the real angle between lines L, L corresponding to the dual unit vectors ã, b respectively, and θ is the shortest distance between those lines[,4] Dual Representation of Ruled Surfaces In this section, we introduce dual representation of a ruled surface which is given by Veldkamp in [7] as follows: Let k be a dual curve represented by x = u) or x + ε x = eu) + ε e u) The real curve x = eu) on the unit real sphere is called the real) indicatrix of k; we suppose throughout that it does not exist of a single point We take as the parameter u the arc-length s on the real indicatrix and we denote differentiation with respect to s by primes Then x = s) and e, e = The vector e = t is the unit vector parallel to the tangent at the indicatrix It is well known that given dual curve k may be represented by ) x = s) = e + ε c e,

4 86 TANJU KAHRAMAN AND HASAN HÜSEYİN UĞURLU where e, e =, e, e =, c, e = 0 We observe that c is unambiguously determined by k It follows from ) that ) ) = t + ε c t + c e Hence by means of x = x + ε x, x x x 0): ) ) = + ε det c, e, t = + ε ) where = det c, e, t Since c as well as e is perpendicular to t we may write c e = µ t; then we obtain = c e, t = µ Therefore c e = t and we obtain in view of ): 4) = t + ε c t + t ) Let t be dual unit vector with the same sense as ; then we find as a consequence of ): = + ε ) t This leads in view of 4) to: 5) t = t + ε c t Guided by elementary differential geometry of real curves we introduce the dual arc-length s of the dual curve k by s s s s = σ) dσ = + ε )dσ = s + ε dσ 0 Then s d = + ε We define furthermore: therefore d 6) d s = t 0 d s = 0 s ; hence d d s = + ε and Introducing the dual unit vector t = g = g + ε g we observe e t = g; hence by means of ) and 5): 7) g = g + ε c g Then the dual frame {, t, g } is called dual geodesic trihedron or dual Darboux frame) of the ruled surface corresponding to dual curve Thus, the derivative formulae of this frame are given as follows, d 8) d s = d t d g t, = g, d s d s = t where is called dual spherical curvature and given by 9) = γ + ε δ γ ) ; and δ = c, e, γ = g, t From 8) introducing the dual Darboux vector d = + g we have 0) See [7]) d d s = d, d t d s = d t, d g d s = d g

5 DUAL SMARANDACHE CURVES AND SMARANDACHE RULED SURFACES 87 Analogous to common differential geometry the dual radius of curvature R of the dual curve x = s) is given by d d s R = d d s d d s Then from 6) and 8), ) R = + ) / The unit vector d 0 with the same sense as the Darboux vector d = + g is given by ) d0 = + g + + The dual angle ρ between d 0 and satisfies therefore: cos ρ =, sin ρ = + + Hence: ) R = sin ρ, = cot ρ The point M on the dual unit sphere indicated by d 0 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] 4 Dual Smarandache curves and Smarandache Ruled Surfaces From E Study Mapping, it is well-known that dual curves lying on unit dual sphere correspond to ruled surfaces of the line space IR Thus, by defining the dual smarandache curves lying fully on dual unit sphere, we also obtain the smarandache ruled surfaces corresponding at line spaceir Then, the differential geometry of smarandache ruled surfaces can be investigated by considering the corresponding dual smarandache curves on unit dual sphere In this section, we first define the four different types of the dual smarandache curves on unit dual sphere Then by the aid of dual geodesic trihedrondual Darboux frame), we give the characterizations of these dual curvesruled surfaces) 4 Dual Smarandache t -curve of a unit dual spherical curve In this section, we define the first type of dual Smarandache curves as dual Smarandache t-curve Then, we give the relationships between the dual curve and its dual Smarandache t-curve Using the found results and relationships we study the developability of the corresponding ruled surface and its Smaranadache ruled surface Definition 4 Let α = α s) be a unit speed regular dual curve lying fully on unit dual sphere S and {, t, g } be its moving dual Darboux frame The dual curve α defined by 4) α = + t ) is called the dual Smarandache t-curve of α and fully lies on S Then the ruled surface corresponding to α is called the Smarandache e t-ruled surface of the surface corresponding to dual curve α

6 88 TANJU KAHRAMAN AND HASAN HÜSEYİN UĞURLU Now we can give the relationships between α and its dual Smarandache t-curve α as follows Theorem 4 Let α = α s) be a unit speed regular dual curve lying on dual unit sphere S Then the relationships between the dual Darboux frames of α and its dual Smarandache t-curve α are given by 0 4) t = g t g where is as given in 9) Proof Let us investigate the dual Darboux frame fields of dual Smarandache tcurve according to α = α s) Since α =, we have 4) = + t ) Differentiating 4) with respect to s, we get and hence d d s = d d s d s d s = t d s d s = + t + g ) 44) t = + t + g ) + where d s + d s = Thus, since g = t, we have 45) g = t + + g From 4)-45) we have 4) If we represent the dual Darboux frames of α and α by the dual matrixes Ẽ and Ẽ, respectively, then 4) can be written as follows: where 46) à = + Ẽ = ÃẼ It is easily seen that detã) = and ÃÃT = ÃT à = I, where I is the unitary matrix It means that à is a dual orthogonal matrix Then we can give the following corollary Corollary 4 The relationship between Darboux frames of the dual curvesruled surfaces) α and α is given by dual orthogonal matrix 46)

7 DUAL SMARANDACHE CURVES AND SMARANDACHE RULED SURFACES 89 Theorem 4 The relationship between the dual Darboux formulae of dual Smarandache t -curve α and dual Darboux frame of α is as follows: 47) d d s d t d s d g d s = + + ) ) + + )+ ) + ) + ) 4 + ) + ) + +)+ ) Proof Differentiating 4), 44) and 45) with respect to s, we have the desired equation 47) Theorem 4 Let α = α s) be a unit speed regular curve on unit dual sphere Then the relationship between the dual curvatures of α and its dual Smarandache t-curve α is given by 48) = ) Proof Since d g d s α s ) as follows: = t, from 44) and 46), we get dual curvature of the curve = ) Corollary 4 The Darboux vector of dual Smarandache t-curve is given by [ 49) d = + + 4) + t + + 4) g ] + ) t g Proof It is known that the dual Darboux instantaneous vector of dual Smarandache t-curve is d = + g Then, from 4) and 48) we have 49) Theorem 44 Let α s) = α be a unit speed regular dual curve on unit dual sphere and α be its dual Smarandache t-curve If the ruled surface corresponding to dual curve α is developable then the ruled surface corresponding to dual curve α is also developable if and only if Proof From 9) we have δ = δγ + δ + δ + γ ) δγγ + γ + γ) + γ ) 5 = γ + ε δ γ ) = γ + ε δ γ ) Then substituting these equalities into to equation 48), we have δ γ = γ + )δ γ ) + δ γ γ + γ ) γδ γ )γ + γ + γ) + γ ) 5 Since the ruled surface corresponding to dual curve α is developable, = 0 Hence, = δ γ γ + )δ + δ γ + γ ) + γδγ + γ + γ) γ + γ ) 5

8 90 TANJU KAHRAMAN AND HASAN HÜSEYİN UĞURLU if Thus, the ruled surface corresponding to dual curve α is developable if and only δ = δγ + δ + δ + γ ) δγγ + γ + γ) + γ ) 5 Theorem 45 The relationship between the radius of dual curvature of dual Smarandache t-curve α and the dual curvature of α is given by ) + 40) R = Proof From ), R = Then from 48), the radius of dual curvature is + ) ) + R = = + + +) ) Theorem 46 The relationship between the radius of dual spherical curvature of dual Smarandache t-curve α and the elements of dual curvature of α is, ) + 4) ρ = arcsin Proof Let ρ be the radius of dual spherical curvature and R be the radius of dual curvature of α From equation ) we have sin ρ = R Thus, we get radius of dual spherical curvature ) + ρ = arcsin In the following sections we define dual Smarandache g, t g and t g curves The proofs of the theorems and corollaries of these sections can be given by using the similar way used in previous section 4 Dual Smarandache g-curve of a unit dual spherical curve In this section, we define the second type of dual Smarandache curves as dual Smarandache g-curve Then, we give the relationships between the dual curve and its dual Smarandache g-curve Using obtained results and relationships we study the developable of the corresponding ruled surface and its Smarandache ruled surface Definition 4 Let α s) = α be a unit speed regular dual curve lying fully on unit dual sphere and {, t, g } be its moving Darboux frame The dual curve α defined by 4) α = + g)

9 DUAL SMARANDACHE CURVES AND SMARANDACHE RULED SURFACES 9 is called the dual Smarandache g-curve of α and fully lies on S Then the ruled surface corresponding to α is called the Smarandache e g-ruled surface of the surface corresponding to dual curve α Now we can give the relationships between α and its dual smarandache g-curve α as follows Theorem 47 Let α s) = α be a unit speed regular dual curve lying on unit dual sphere S Then the relationships between the dual Darboux frames of α and its dual Smarandache g-curve α are given by 0 4) t = 0 0 t g 0 g From 4) we have t = t, ie, 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 t = cos θ 0 sin θ 0 0 t g sin θ 0 cos θ g where θ = θ + εθ, 0 θ π, θ R) is the dual angle between the generators and of Bertrand ruled surface ϕ e and ϕ e The angle θ is called the offset angle and θ is called the offset distance[5] Then from 4) we have that offset angle is θ = π/4 and offset distance is θ = 0 Then we have the following corollary Corollary 4 The dual Smarandache g-curve of a dual curve α is always its Bertrand offset with dual offset angle θ = π/4 + ε0 Theorem 48 Let α s) = α be a unit speed regular dual curve on unit dual sphere S Then according to dual Darboux frame of α, the dual Darboux formulae of dual Smarandache g-curve α are as follows: 44) d ds d t ds d g ds = Theorem 49 Let α s) = α be a unit speed regular curve on unit dual sphere Then the relationship between the dual curvatures of α and its dual Smarandache g-curve α is given by = + Corollary 44 The dual curvature of α is zero if and only if the dual curvature of dual Smarandache t-curve α is Corollary 45 is given by The Darboux instantaneous vector of dual Smarandache g-curve d = + g t g

10 9 TANJU KAHRAMAN AND HASAN HÜSEYİN UĞURLU Theorem 40 Let α s) = α be a unit speed regular curve on unit dual sphere and α be the dual Smarandache g-curve of α If the ruled surface corresponding to the dual curve α is developable then the ruled surface corresponding to dual curve α is developable if and only if γ ) δ δ = 0 Theorem 4 Let α s) = α be a unit speed regular curve on unit dual sphere Then the relationship between the radius of dual curvature of dual Smarandache g-curve α and the dual curvature of α s) = α is given by R = + Theorem 4 Let α s) = α be a unit speed regular curve on unit dual sphere Then the relationship between the radius of dual spherical curvature of dual Smarandache g-curve α and the elements of dual curvature of α is given by ) γ γ + γγ ρ = arcsin ε )) + γ + γ ) γ cos arcsin +γ 4 Dual Smarandache t g -curve of a unit dual spherical curve In this section, we define the second type of dual Smarandache curves as dual Smarandache t g-curve Then, we give the relationships between the dual curve and its dual smarandache t g-curve Using the found results and relationships we study the developable of the corresponding ruled surface and its Smaranadache ruled surface Definition 4 Definition 5 Let α s) = α be a unit speed regular dual curve lying fully on unit dual sphere and {, t, g } be its moving Darboux frame The dual curve α defined by α = t + g ) is called the dual Smarandache t g-curve of α and fully lies on S Then the ruled surface corresponding to α is called the Smarandache t g-ruled surface of the surface corresponding to dual curve α Now we can give the relationships between α and its dual Smarandache t g-curve α as follows: Theorem 4 Let α s) = α be a unit speed regular dual curve lying on unit dual sphere S Then the relationships between the dual Darboux frames of α and its dual Smarandache t g-curve α are given by 0 45) t = t g g If we represent the dual darboux frames of α and α by the dual matrixes Ẽ and Ẽ, respectively, then 45) can be written as follows Ẽ = ÃẼ

11 DUAL SMARANDACHE CURVES AND SMARANDACHE RULED SURFACES 9 where 46) à = It is easily seen that detã) = and ÃÃT = ÃT à = I, where I is the unitary matrix It means that à is a dual orthogonal matrix Then we can give the following corollary: Corollary 46 The relationship between the Darboux frames of the dual curvesruled surfaces) α and α is given by a dual orthogonal matrix defined by 46) Theorem 44 The relationship between the dual Darboux formulae of dual Smarandache t g -curve α and dual Darboux frame of α is given by 47) d ds d t ds d g ds = + + ) + +)+ ) + ) + ) )+ ) + ) )+4 ) ) 8 +4 ) + ) + ) + ) Theorem 45 Let α s) = α be a unit speed regular curve on unit dual sphere Then the relationship between the dual curvatures of α and its dual Smarandache t g-curve α is given by t g = ) Corollary 47 If the dual curvature of α is zero, the dual curvature of dual Smarandache t-curve α is Corollary 48 The Darboux instantaneous vector of dual Smarandache t g-curve is given by d = + 4 t g ) +4 ) Theorem 46 Let α s) = α be a unit speed regular curve on unit dual sphere and α be the dual Smarandache t g-curve of α If the ruled surface corresponding to the dual curve α is developable then the ruled surface corresponding to dual curve α is developable if and only if δ = 4 δγ + 4 γδ + 8 δγ + γ ) + δγ4 γγ + 4 γ + ) + γ ) 5 Theorem 47 Let α s) = α be a unit speed regular curve on unit dual sphere Then the relationship between the radius of dual curvature of dual Smarandache t g-curve α and the dual curvature of α s) = α is given by R = + 4 ) + 4 ) )

12 94 TANJU KAHRAMAN AND HASAN HÜSEYİN UĞURLU Theorem 48 Let α s) = α be a unit speed regular curve on unit dual sphere Then the relationship between the radius of dual spherical curvature of dual Smarandache t g-curve α and the elements of dual curvature of α is, ) + 4 ρ = arcsin + 4 ) ) 44 Dual Smarandache t g -curve of a unit dual spherical curve In this section, we define the second type of dual Smarandache curves as dual Smarandache t g-curve Then, we give the relationships between the dual curve and its dual smarandache t g-curve Using the found results and relationships we study the developable of the corresponding ruled surface and its Smaranadache ruled surface Definition 44 Let α s) = α be a unit speed regular dual curve lying fully on unit dual sphere and {, t, g } be its moving Darboux frame The dual curve α 4 defined by α 4 = + t + g ) is called the dual Smarandache t g-curve of α and fully lies on S Then the ruled surface corresponding to α 4 is called the Smarandache e t g-ruled surface of the surface corresponding to dual curve α Now we can give the relationships between α and its dual smarandache t g-curve α 4 as follows: Theorem 49 Let α s) = α be a unit speed regular dual curve lying on unit dual sphere S Then the relationships between the dual Darboux frames of α and its dual Smarandache t g-curve α 4 are given by 4 48) t 4 = γ γ t g 4 g 6 + +) If we represent the dual darboux frames of α and α 4 by the dual matrixes Ẽ and Ẽ, respectively, then 48) can be written as follows where 49) à = Ẽ = ÃẼ γ + +) 6 + γ It is easily seen that detã) = and ÃÃT = ÃT à = I, where I is the unitary matrix It means that à is a dual orthogonal matrix Then we can give the following corollary Corollary 49 The relationship between the Darboux frames of the dual curvesruled surfaces) α and α 4 is given by a dual orthogonal matrix defined by 49)

13 DUAL SMARANDACHE CURVES AND SMARANDACHE RULED SURFACES 95 Theorem 40 The relationship between the dual Darboux formulae of dual Smarandache t g-curve α 4 and dual Darboux frame of α is given by 40) d 4 ds 4 d t 4 ds 4 d g 4 ds 4 = )+ ) ) ) ) ) 4 + t g Theorem 4 Let α s) = α be a unit speed regular curve on unit dual sphere Then the relationship between the dual curvatures of α and its dual Smarandache t g-curve α 4 is given by 4 = ) Corollary 40 If the dual curvature of α is zero, then the dual curvature 4 of dual Smarandache t g-curve α 4 is Corollary 4 The Darboux vector of dual Smarandache t g-curve is given by d 4 = ) ) t ) g Theorem 4 Let α s) = α be a unit speed regular curve on unit dual sphere and α 4 be the dual Smarandache t g-curve of α If the ruled surface corresponding to the dual curve α is developable then the ruled surface corresponding to dual curve α 4 is developable if and only if δ 4 = 6γ δ + δ γ γ + ) δ γ ) γ + γ + ) 4 γ γ + ) 5 Theorem 4 Let α s) = α be a unit speed regular curve on unit dual sphere Then the relationship between the radius of dual curvature of dual Smarandache t g-curve α 4 and the dual curvature of α s) = α is given by R 4 = + ) 8 + ) ) Theorem 44 Let α s) = α be a unit speed regular curve on dual unit sphere Then the relationship between the radius of dual spherical curvature of dual Smarandache t g-curve α 4 and the elements of dual curvature of α is ρ 4 = arcsin + ) 8 + ) ) Example 4 Let consider the dual spherical curve α s) given by the parametrization αs) = cos s, sin s, 0) + ε s sin s, s cos s, 0) The curve αs) represents the ruled surface rs, v) = v cos s, v sin s, s)

14 96 TANJU KAHRAMAN AND HASAN HÜSEYİN UĞURLU which is a helicoids surface rendered in Fig [7] Then the dual Darboux frame of α is obtained as follows, s) = cos s, sin s, 0) + ε s sin s, s cos s, 0) ts) = sin s, cos s, 0) + ε s cos s, s sin s, 0) gs) = 0, 0, ) The Smarandache t, g, t g, and t g curves of the dual curve α are given by α s) = [cos s sin s, cos s + sin s, 0) + ε s cos s s sin s, s cos s s sin s, 0)] α s) = [cos s, sin s, ) + ε s sin s, s cos s, 0)] α s) = [ sin s, cos s, ) + ε s cos s, s sin s, 0)] α 4 s) = [cos s sin s, cos s + sin s, ) + ε s cos s s sin s, s cos s s sin s, 0)] respectively From E Study mapping, these dual spherical curves correspond to the following ruled surfaces ) r s, v) = 0, 0, s) + v cos s sin s, cos s + sin s, 0 ) r s, v) = 0, 0, s) + v cos s, sin s, ) r s, v) = 0, 0, s) + v sin s, cos s, ) r 4 s, v) = 0, 0, s) + v cos s sin s, cos s + sin s, respectively These surfaces are rendered in Fig, Fig, Fig 4 and Fig 5, respectively Figure Helicoid surface corresponding to dual curve α

15 DUAL SMARANDACHE CURVES AND SMARANDACHE RULED SURFACES 97 Figure Smarandache t ruled surface Figure Smarandache g ruled surface Figure 4 Smarandache t g ruled surface

16 98 TANJU KAHRAMAN AND HASAN HÜSEYİN UĞURLU Figure 5 Smarandache t g ruled surface References [] Ali, A T, Special Smarandache Curves in the Euclidean Space, International Journal of Mathematical Combinatorics,00, Vol,pp0-6 [] Blaschke, W, Differential Geometrie and Geometrischke Grundlagen ven Einsteins Relativitasttheorie Dover, New York, 945 [] Dimentberg, F M, The Screw Calculus and its Applications in Mechanics, Foreign Technology Division, Wright-Patterson Air Force Base, Ohio Document NoFTD-HT-6-67, 965 [4] Hacısaliholu H H, Hareket Gometrisi ve Kuaterniyonlar Teorisi, Gazi Üniversitesi Fen-Edb Fakültesi, 98 [5] Önder, M, Darboux Approach to Bertrand Surface Offsets, Int Journal of Pure and App Math IJPAM), 0, Vol 74 No, -4 [6] Turgut, M, Yilmaz, S, Smarandache Curves in Minkowski Space-time, Int J Math Comb,, 008, 5{55 [7] Uğurlu, H H, Çalışkan, A, Kılıç, O, Öklid ve Lorentz Uzaylarında Doğrular Geometrisi Ders Notları, Celal Bayar Üniversitesi, Manisa in press) [8] Veldkamp, GR, On the use of dual numbers, vectors and matrices in instantaneous spatial kinematics, Mechanism and Machine Theory, II 976) 4-56 Celal Bayar University, Department of Mathematics,Faculty of Arts and Sciences, Manisa/Turkey address: tanjukahraman@bayaredutr Gazi University, Gazi Faculty of Education, Department of Secondary Education Science and Mathematics Teaching, Mathematics Teaching Program, Ankara, Turkey address: hugurlu@gaziedutr