1Dual Darboux Frame of a Timelike Ruled Surface and Darboux Approach to Mannheim Offsets of Timelike Ruled Surfaces Mehmet Önder 1 , H. Hüseyin Uğurlu 2 1 Celal Bayar University, Faculty of Science and Arts, Department of Mathematics, Muradiye, Manisa, Turkey. E-mail:
[email protected] 2 Gazi University, Faculty of Education, Department of Secondary Education Science and Mathematics Teaching, Mathematics Teaching Program, Ankara, Turkey. Abstract In this paper, we give the dual geodesic trihedron(dual Darboux frame) of a timelike ruled surface. Then, we study Mannheim offsets of timelike ruled surfaces in dual Lorentzian space. By the aid of the E. Study Mapping, we consider the timelike ruled surfaces as dual hyperbolic unit spherical curves and define the Mannheim offsets of the timelike ruled surfaces by means of dual Darboux frame. We obtain the relationships between the invariants of Mannheim timelike offset surfaces. Furthermore, we give the conditions for these surface offset to be developable. AMS: 53A25, 53C40, 53C50. Keywords: Timelike ruled surface; Mannheim offset; dual Darboux frame; dual hyperbolic angle. 1. Introduction In the surface theory it is well-known that a surface is said to be “ruled” if it is generated by a continuously moving of a straight line in the space. Ruled surfaces are one of the simplest objects in geometric modeling that explains why these surfaces are one of the most fascinating topics of the surface theory and also used in many areas of science such as Computer Aided Geometric Design (CAGD), mathematical physics, moving geometry, kinematics for modeling the problems and model-based manufacturing of mechanical products. For example, the building materials such as wood are straight and they can be considered as straight lines. The result is that if engineers are planning to construct something with curvature, they can use a ruled surface since all the lines are straight[9]. An offset surface is offset a specified distance from the original along the parent surface's normal. Offsetting of curves and surfaces is one of the most important geometric operations in CAD/CAM due to its immediate applications in geometric modeling, NC machining, and robot navigation[5]. Especially, the offsets of the ruled surfaces have an important role in (CAGD)[14,17]. In [16], Ravani and Ku defined and given a generalization of the theory of Bertrand curve for Bertrand trajectory ruled surfaces on the line geometry. By considering the E. Study mapping, Küçük and Gürsoy have studied the integral invariants of closed Bertrand trajectory ruled surfaces in dual space[7]. They have given some characterizations of Bertrand offsets of trajectory ruled surfaces in terms of integral invariants (such as the angle of pitch and the pitch) of closed trajectory ruled surfaces and obtained the relationship between the area of projections of spherical images of Bertrand offsets of trajectory ruled surfaces and their integral invariants. Recently, a new offset of ruled surfaces has been defined by Orbay and et. al[9]. They have called this new surface offset as Mannheim offset and shown that every developable ruled surface have a Mannheim offset if and only if an equation should be satisfied between the geodesic curvature and the arc-length of spherical indicatrix of the reference surface. The corresponding characterizations of Mannheim offsets of ruled surfaces in Minkowski 3-space have been given in [10,13]. Furthermore, in [11] we have studied Mannheim offsets of ruled 2 surfaces in dual space with Blaschke approach. We have given the characterizations of Mannheim offsets of ruled surfaces in terms of integral invariants of closed trajectory ruled surfaces and obtained the relationship between the area of projections of spherical images of Mannheim offsets of trajectory ruled surfaces and their integral invariants. Moreover, we have studied the Mannheim offsets of timelike ruled surfaces in dual Lorentzian space by considering the Blaschke frame[12]. In this paper, we define the dual geodesic trihedron(dual Darboux frame) of a timelike ruled surface and give the real and dual curvatures of this surface. Then, we examine the Mannheim offsets of the timelike ruled surfaces in view of their dual Darboux frame. Using the dual representations of the timelike ruled surfaces, we give some theorems and new results which characterize the developable Mannheim timelike surface offsets. 2. Preliminaries Let 3 1 IR be a 3-dimensional Minkowski space over the field of real numbers IR with the Lorentzian inner product , given by 1 1 2 2 3 3 , a a a b a b a b = − + + , where 1 2 3 ( , , ) a a a a = and 3 1 2 3 ( , , ) b b b b IR = ∈ . A vector 1 2 3 ( , , ) a a a a = of 3 1 IR is said to be timelike if , 0 a a < , spacelike if , 0 a a > or 0 a = , and lightlike (null ) if , 0 a a = and 0 a ≠ . Similarly, an arbitrary curve ( ) s α in 3 1 IR can locally be spacelike, timelike or null (lightlike), if all of its velocity vectors ( ) s α′ are spacelike, timelike or null (lightlike), respectively[8]. The norm of a vector a is defined by , a a a = . Now, let 1 2 3 ( , , ) a a a a = and 1 2 3 ( , , ) b b b b = be two vectors in 3 1 IR , then the Lorentzian cross product of a and b is given by 2 3 3 2 1 3 3 1 2 1 1 2 ( , , ). a b a b a b a b a b a b a b × = − − − By using this definition it can be easily shown that , det( , , ) a b c a b c × = − [19]. The sets of the unit timelike and spacelike vectors are called hyperbolic unit sphere and Lorentzian unit sphere, respectively, and denoted by { } 2 3 0 1 2 3 1 ( , , ) : , 1 H a a a a E a a = = ∈ = − and { } 2 3 1 1 2 3 1 ( , , ) : , 1 S a a a a E a a = = ∈ = respectively(See [17]). A surface in the Minkowski 3-space 3 1 IR is called a timelike surface if the induced metric on the surface is a Lorentz metric and is called a spacelike surface if the induced metric on the surface is a positive definite Riemannian metric, i.e., the normal vector on the spacelike (timelike) surface is a timelike (spacelike) vector[1]. 3. Dual Numbers and Dual Lorentzian Vectors Let { } ( , ) : , D IR IR a a a a a IR ∗ ∗ = × = = ∈ be the set of the pairs ( , ) a a ∗ . For ( , ) a a a ∗ = , ( , ) b b b D ∗ = ∈ the following operations are defined on D: Equality: , a b a b a b ∗ ∗ = ⇔ = = Addition: ( , ) a b a b a b ∗ ∗ + ⇔ + + 3 Multiplication: ( , ) ab ab ab a b ∗ ∗ ⇔ + The element (0,1) D ε = ∈ satisfies the relationships 0 ε ≠ , 2 0 ε = , 1 1 ε ε ε = = (1) Let consider the element a D ∈ of the form ( , 0) a a = . Then the mapping : , ( , 0) f D IR f a a → = is a isomorphism. So, we can write ( , 0) a a = . By the multiplication rule we have that ( , ) ( , 0) (0, ) ( , 0) (0,1)( , 0) a a a a a a a a a ε ∗ ∗ ∗ ∗ = = + = + = + Then a a a ε ∗ = + is called dual number and ε is called dual unit. Thus the set of dual numbers is given by { } 2 : , , 0 D a a a a a IR ε ε ∗ ∗ = = + ∈ = (2) 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[3]. He derived the general expression for dual analytic (differentiable) function as follows ( ) ( ) ( ) ( ) f x f x x f x x f x ε ε ∗ ∗ ′ = + = + , (3) where ( ) f x ′ is derivative of ( ) f x and , x x IR ∗ ∈ . This definition allows us to write the dual forms of some well-known functions as follows cosh( ) cosh( ) cosh( ) sinh( ), sinh( ) sinh( ) sinh( ) cosh( ), , ( 0). 2 x x x x x x x x x x x x x x x x x x x ε ε ε ε ε ε ∗ ∗ ∗ ∗ ∗ ∗ ¦ ¦ = + = + ¦ ¦ = + = + ´ ¦ ¦ = + = + > ¦ ¹ (4) Let 3 D D D D = × × be the set of all triples of dual numbers, i.e., { } 3 1 2 3 ( , , ) : , 1, 2, 3 i D a a a a a D i = = ∈ = , (5) Then the set 3 D is 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 ( , ) a a a a a ε ∗ ∗ = + = , where a and a ∗ are the vectors of 3 IR . Then for any vectors a a a ε ∗ = + and b b b ε ∗ = + of 3 D , the scalar product and the vector product are defined by ( ) , , , , a b a b a b a b ε ∗ ∗ = + + , (6) and ( ) * a b a b a b a b ε ∗ × = × + × + × , (7) respectively, where , a b and a b × are the inner product and the vector 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 ε ∗ = + ≠ . (8) 4 A dual vector a with norm 1 0 ε + is called dual unit vector. The set of dual unit vectors is given by { } 2 3 1 2 3 ( , , ) : , 1 0 S a a a a D a a ε = = ∈ = + , (9) and called dual unit sphere[2,4]. The Lorentzian inner product of two dual vectors a a a ε ∗ = + , * 3 b b b ε = + ∈ D is defined by ( ) * , , , , , a b a b a b a b ε ∗ = + + where , a b is the Lorentzian inner product of the vectors a and b in the Minkowski 3- space 3 1 . IR Then a dual vector a a a ε ∗ = + is said to be timelike if a is timelike, spacelike if a is spacelike or 0 a = and lightlike (null) if a is lightlike (null) and 0 a ≠ [17]. The set of all dual Lorentzian vectors is called dual Lorentzian space and it is denoted by 3 1 : D { } 3 3 1 1 : , a a a a a IR ε ∗ ∗ = = + ∈ D . The Lorentzian cross product of dual vectors a and 3 1 b∈ D is defined by * ( ) , a b a b a b a b ε ∗ × = × + × + × where a b × is the Lorentzian cross product in 3 1 IR . Let a a a ε ∗ = + 3 1 ∈ D . Then a ~ is said to be dual unit timelike(resp. spacelike) vector if the vectors a and a ∗ satisfy the following equations: , 1 ( . , 1), , 0. a a resp a a a a ∗ < >=− < >= < > = (10) The set of all unit dual timelike vectors is called the dual hyperbolic unit sphere, and is denoted by 2 0 H , { } 2 3 0 1 2 3 1 ( , , ) : , 1 0 H a a a a D a a ε = = ∈ = − + (11) Similarly, the set of all unit dual spacelike vectors is called the dual Lorentzian unit sphere, and is denoted by 2 1 S , { } 2 3 1 1 2 3 1 ( , , ) : , 1 0 S a a a a D a a ε = = ∈ = + . (12) (For details see [17]) . Definition 3.1. i) Dual Hyperbolic angle: Let x and y be dual timelike vectors in 3 1 ID . Then the dual angle between x and y is defined by , cosh x y x y θ < >= − . The dual number θ θ εθ ∗ = + is called the dual hyperbolic angle. ii) Dual Lorentzian timelike angle: Let x be a dual spacelike vector and y be a dual timelike vector in 3 1 ID . Then the angle between x and y is defined by , sinh x y x y θ < >= . The dual number θ θ εθ ∗ = + is called the dual Lorentzian timelike angle[17]. 4. Dual Representation and Dual Darboux Frame of a Timelike Ruled Surface In the Minkowski 3-space 3 1 IR , an oriented timelike line L is determined by a point p L ∈ and a unit timelike vector a . Then, one can define a p a ∗ = × which is called moment vector. The value of a ∗ does not depend on the point p , because any other point q in L can 5 be given by q p a λ = + and then a p a q a ∗ = × = × . Reciprocally, when such a pair ( , ) a a ∗ is given, one recovers the timelike line L as { } 3 ( ) : , , L a a a a a E IR λ λ ∗ ∗ = × + ∈ ∈ , written in parametric equations. The vectors a and a ∗ are not independent of one another and they satisfy the following relationships , 1, , 0 a a a a ∗ = − = (13) The components , i i a a ∗ (1 3) i ≤ ≤ of the vectors a and a ∗ are called the normalized Plucker coordinates of the timelike line L . From (10), (11) and (13) we see that the dual timelike unit vector a a a ε ∗ = + corresponds to timelike line L . This correspondence is known as E. Study Mapping: There exists a one-to-one correspondence between the timelike vectors of dual hyperbolic unit sphere 2 0 H and the directed timelike lines of the Minkowski space 3 1 IR [17]. By the aid of this correspondence, the properties of the spatial motion of a timelike line can be derived. Hence, the geometry of timelike ruled surface is represented by the geometry of dual hyperbolic curves lying on the dual hyperbolic unit sphere 2 0 H . In [20], Veldkamp given the dual representation and dual geodesic trihedron of a ruled surface. Now, we use the similar procedure for a timelike ruled surface and introduce the dual Darboux frame of a timelike ruled surface. Let now ( ) k be a dual hyperbolic curve represented by the dual timelike unit vector ( ) ( ) ( ) e u e u e u ε ∗ = + . The real unit vector e draws a curve on the real hyperbolic unit sphere 2 0 H and is called the (real) indicatrix of ( ) k . We suppose throughout that it is not a single point. We take the parameter u as the arc-length parameter s of the real indicatrix and denote the differentiation with respect to s by primes. Then we have , 1 e e ′ ′ = . The vector e t ′ = is the unit vector parallel to the tangent of the indicatrix. The equation ( ) ( ) ( ) e s p s e s ∗ = × has infinity of solutions for the function ( ) p s . If we take ( ) o p s as a solution, the set of all solutions is given by ( ) ( ) ( ) ( ) o p s p s s e s λ = + , where λ is a real scalar function of s . Therefore we have , , o p e p e λ ′ ′ ′ ′ = + . By taking , o o p e λ λ ′ ′ = = − we see that ( ) ( ) ( ) ( ) o o p s s e s c s λ + = is the unique solution for ( ) p s with , 0 c e ′ ′ = . Then, the given dual curve ( ) k corresponding to the timelike ruled surface ( ) ( ) e c s ve s ϕ = + (14) may be represented by ( ) e s e c e ε = + × (15) where , 1 e e = − , , 1 e e ′ ′ = , , 0 c e ′ ′ = . and c is the direction vector of the striction curve. Then we have 1 det( , , ) 1 e c e t ε ε ′ ′ = − = − ∆ (16) where det( , , ) c e t ′ ∆ = which characterizes the developable timelike surface, i.e, the timelike surface is developable if and only if 0 ∆ = . Then, the dual arc-length s of the dual curve ( ) k is given by 0 0 0 ( ) (1 ) s s s s e u du du s du ε ε ′ = = − ∆ = − ∆ ∫ ∫ ∫ (17) From (17) we have 1 s ε ′ = − ∆. Therefore, the dual unit tangent to the dual curve ( ) e s is given by 6 ( ) 1 de e e t t c t ds s ε ε ′ ′ = = = = + × ′ − ∆ (18) Introducing the dual unit vector g e t g c g ε = − × = + × we have the dual frame { } , , e t g which is known as dual geodesic trihedron or dual Darboux frame of e ϕ (or ( ) e ). Also, it is well known that the real orthonormal frame { } , , e t g is called the geodesic trihedron of the indicatrix ( ) e s with the derivations , , e t t e g g t γ γ ′ ′ ′ = = + = − (19) where γ is called the conical curvature[6,18]. Lot now consider the derivations of vectors of dual geodesic trihedron and find the dual Darboux formulae of a timelike ruled surface. From (18) we have , 1 0 t t ε = + . By using this equality and considering that g e t = − × , we have , 0, dt dg dt t e ds ds ds = = − × . (20) For the derivative of t let write dt e t g ds α β γ = + + (21) where , , α β γ are the dual functions of dual arc-length s . The first equation of (20) gives that β =0. Thus from the second equation of (20) we have dg t ds γ = − . (22) Finally, from (22) and the equality t g e = × , we obtain dt e g ds γ = + . (23) Then from (18), (22) and (23) we have the following theorem Theorem 4.1. The derivatives of the vectors of dual frame { } , , e t g of a timelike ruled surface are obtained as follows , , de dt dg t e g t ds ds ds γ γ = = + = − (24) and called dual Darboux formulae of timelike ruled surface e (or e ϕ ). Then the dual Darboux vector of the trihedron is d e g γ = − − . Let now give the invariants of the surface. Since 1 s ε ′ = − ∆, it follows from (22) that (1 ) g t γ ε ′ = − − ∆ (25) On the other hand using that g g c g ε = + × , from (18) we have ( ) ( ) ( ) ( ) g t c g c t t c t c g t c g γ ε γ γ εγ ε γ ε ′ ′ = − + × − × ′ = − − × + × ′ = − + × (26) Therefore, from (25) and (26) we obtain (1 ) ( ) t t c g γ ε γ ε ′ − − ∆ = − + × (27) which gives us 7 (1 ) γ ε γ εδ − ∆ = + (28) where , c e δ ′ = . Then from (28) we have ( ) γ γ ε δ γ = + + ∆ (29) Moreover, since c′ as well as e is perpendicular to t , for the real scalar µ we may write c e t µ ′× = . Then det( , , ) , , c e t c e t t t µ µ ′ ′ ∆ = = − × = − = − . Hence ( ) e c e e t g ′ × × = −∆ × = ∆ and c e g δ ′ = − + ∆ . The functions ( ), ( ) s s γ δ and ( ) s ∆ are the invariants of the timelike ruled surface e ϕ . They determine the timelike ruled surface uniquely up to its position in the space. For example, if 0 δ = ∆ = we have c is constant. It means that the timelike ruled surface e ϕ is a timelike cone. 4.1. Elements of Curvature of a Dual Hyperbolic Curve The dual radius of curvature of dual hyperbolic curve(timelike ruled surface) ( ) e s is can be calculated analogous to common Lorentzian differential geometry of curves as follows 3 2 2 2 1 1 de ds R de d e ds ds γ = = − × (30) The unit vector o d with the same sense as the Darboux vector d e g γ = − − is given by 2 2 1 1 1 o d e g γ γ γ = − − − − (31) It is clear that o d is timelike (or spacelike) if 1 γ > (or 1 γ < ). Then, the dual angle between o d and e satisfies the followings 2 2 2 2 1 cosh , sinh , 1. 1 1 1 sinh , cosh , 1. 1 1 if if γ ρ ρ γ γ γ γ ρ ρ γ γ γ ¦ = − = − > ¦ − − ¦ ´ ¦ = − = − < ¦ − − ¹ (32) where ρ is the dual spherical radius of curvature. Hence sinh , 1, cosh , 1. if R if ρ γ ρ γ ¦− > ¦ = ´ − < ¦ ¹ and coth , 1, tanh , 1. if if ρ γ γ ρ γ ¦ > ¦ = ´ < ¦ ¹ 5. Darboux Approach to Mannheim Offsets of Timelike Ruled Surfaces Let e ϕ be a timelike ruled surface generated by dual timelike unit vector e and 1 e ϕ be a ruled surface generated by dual unit vector 1 e and let { } ( ), ( ), ( ) e s t s g s and { } 1 1 1 1 1 1 ( ), ( ), ( ) e s t s g s denote the dual Darboux frames of e ϕ and 1 e ϕ , respectively. Then e ϕ and 1 e ϕ are called Mannheim surface offsets, if 1 1 ( ) ( ) g s t s = (33) 8 holds, where s and 1 s are the dual arc-lengths of e ϕ and 1 e ϕ , respectively. By this definition, it is seen that the Mannheim offset of e ϕ is also a timelike ruled surface, but the generator of this surface can be timelike or spacelike. In this study, we consider the Mannheim offset surface with timelike ruling. If one considers the second situation, ruling is spacelike, and uses the Definition 3.1 (ii) similar results can be found. Let now e ϕ and 1 e ϕ be timelike ruled surfaces with timelike rulings and form a Mannheim surface offset. Then the relation between the trihedrons of the ruled surfaces e ϕ and 1 e ϕ can be given as follows 1 1 1 cosh sinh 0 0 0 1 sinh cosh 0 e e t t g g θ θ θ θ | | | | | | | | | = | | | | | | \ ¹ \ ¹ \ ¹ (34) where θ θ εθ ∗ = + , ( , ) θ θ ∗ ∈» is the dual hyperbolic angle between the dual timelike generators e and 1 e of Mannheim timelike ruled surfaces e ϕ and 1 e ϕ . The angle θ is called the offset angle which is the angle between the rulings e and 1 e , and θ ∗ is called the offset distance which is measured from the striction point c of e ϕ to striction point 1 c of 1 e ϕ . And from (37) we write 1 c c t θ ∗ = + . Then, θ θ εθ ∗ = + is called dual hyperbolic offset angle of the Mannheim timelike ruled surfaces e ϕ and 1 e ϕ . If 0 θ = , then the Mannheim timelike surface offsets are said to be oriented offsets. Now, we give some theorems and results characterizing Mannheim timelike surface offsets. When we write e ϕ and 1 e ϕ , we mean timelike ruled surfaces with timelike ruling and for short we don’t write the Lorentzian characters of the surfaces hereinafter. Theorem 5.1. Let e ϕ and 1 e ϕ form a Mannheim surface offset. The offset angle θ and the offset distance θ ∗ are given by 0 , s s c du c θ θ ∗ ∗ = − + = ∆ + ∫ (35) respectively, where c and c ∗ are real constants. Proof. Suppose that e ϕ and 1 e ϕ form a Mannheim offset. Then from (34) we have 1 cosh sinh e e t θ θ = + (36) By differentiating (36) with respect to s we have 1 sinh 1 cosh 1 sinh de d d e t g ds ds ds θ θ θ θ γ θ | | | | = + + + + | | \ ¹ \ ¹ (37) Since 1 de ds and g are linearly dependent, from (37) we get 1 d ds θ = − . Then for the dual constant c c c ε ∗ = + we write d ds s c s s c c θ θ θ εθ ε ε ∗ ∗ ∗ = − = − + + = − − + + and from (17) we have 9 0 , s s c du c θ θ ∗ ∗ = − + = ∆ + ∫ , where c and c ∗ are real constants. From (35), the following corollary can be given. Corollary 5.1. Let e ϕ and 1 e ϕ form a Mannheim surface offset. Then e ϕ is developable if and only if constant c θ ∗ ∗ = = . Theorem 5.2. Let e ϕ and 1 e ϕ form a Mannheim surface offset. Then there is the following differential relationship between the dual arc-length parameters of e ϕ and 1 e ϕ 1 sinh ds ds γ θ = . (38) Proof. Suppose that e ϕ and 1 e ϕ form a Mannheim offset. Then, from Theorem 5.1 we have 1 1 1 1 sinh de ds t g ds ds γ θ = = (39) From (33) we have 1 t g = . Then (39) gives us 1 sinh 1 ds ds γ θ = (40) and from (40) we get (38). Corollary 5.2. Let e ϕ and 1 e ϕ form a Mannheim surface offset. Then there are the following relationships between the real arc-length parameters of e ϕ and 1 e ϕ 1 1 1 2 sinh , cosh ( ) sinh ds dsds ds ds ds ds γ θ θ γ θ δ γ θ ∗ ∗ ∗ − = = + + ∆ (41) Proof. Let e ϕ and 1 e ϕ form a Mannheim surface offset. Then by Theorem 5.2, (38) holds. By considering (4), the real and dual parts of (38) are 1 1 1 2 sinh , cosh ( ) sinh ds dsds ds ds ds ds γ θ θ γ θ δ γ θ ∗ ∗ ∗ − = = + + ∆ . (42) In Corollary 5.1, we give the relationship between the offset distance θ ∗ and developable timelike ruled surface e ϕ . Now we give the condition for 1 e ϕ to be developable according to θ ∗ . From (17) we have 1 1 1 , ds ds ds ds ∗ ∗ = −∆ = −∆ . (43) Then writing (43) in (42) and using (41) we get 1 coth δ θ θ γ ∗ | | ∆ = − + | \ ¹ and give the following corollaries: Corollary 5.3. Let e ϕ and 1 e ϕ form a Mannheim surface offset. Then 10 1 coth δ θ θ γ ∗ | | ∆ = − + | \ ¹ (44) holds. Corollary 5.4. Let e ϕ and 1 e ϕ form a Mannheim surface offset. Then 1 e ϕ is developable if and only if tanh δ θ θ γ ∗ = − holds. Theorem 5.3. Let e ϕ and 1 e ϕ form a Mannheim surface offset. There exists the following relationship between the invariants of the surfaces and offset angle θ , offset distance θ ∗ , ( ) 1 1 ( ) coth δ δ θ θ γ ∗ = − + ∆ . (45) Proof. Let the striction lines of e ϕ and 1 e ϕ be ( ) c s and 1 1 ( ) c s , respectively, and let e ϕ and 1 e ϕ form a Mannheim surface offset. Then, from the Mannheim condition we can write 1 c c t θ ∗ = + . (46) Differentiating (46) with respect to 1 s we have 1 1 1 dc dc d ds e t g ds ds ds ds θ θ θ γ ∗ ∗ ∗ | | = + + + | \ ¹ . (47) We know that 1 1 1 1 / , dc ds e δ = . Then from (34) and (47) we obtain 1 1 cosh / , sinh / , cosh sinh d ds dc ds e dc ds t ds ds θ δ θ θ θ θ θ ∗ ∗ | | = + − + | \ ¹ (48) Since / , dc ds e δ = and / , 0 dc ds t = , from (48) we write 1 1 cosh cosh sinh d ds ds ds θ δ δ θ θ θ θ ∗ ∗ | | = − + | \ ¹ (49) Furthermore, from (35) and (41) we have d ds θ ∗ = ∆ , 1 1 sinh ds ds γ θ = (50) respectively. Substituting (50) in (49) we obtain ( ) 1 1 ( ) coth δ δ θ θ γ ∗ = − + ∆ . Then we give the following corollary. Corollary 5.5. Let e ϕ and 1 e ϕ form a Mannheim surface offset. Then e ϕ is developable if and only if 1 coth δ θ δ θ γ ∗ − = holds. Theorem 5.4. Let e ϕ and 1 e ϕ form a Mannheim surface offset. Then the relationship between the invariants of the surfaces and offset angle θ , offset distance θ ∗ is given by 1 1 ( coth ) θ δ θ γ ∗ ∆ = − − ∆ . (51) 11 Proof. We know that 1 1 1 1 1 det( / , , ) dc ds e t ∆ = . Then from (34) and (47) we obtain 1 1 1 det , cosh sinh , det , , cosh det , , sinh det( , , ) sinh det( , , ) cosh ds dc e t g e t g ds ds ds dc dc e g t g e t g t e g ds ds ds θ θ γ θ θ θ θ θ θ θ ∗ ∗ ∗ ∆ = + ∆ + + = + + + ∆ | | + | \ ¹ | | | | ( | | ( ¸ \ ¹ \ ¹ ¸ (52) Since det( , , ) det( , , ) , 1, det , , , 0, det , , , , t e g e t g e t g dc dc dc dc e g t t g e ds ds ds ds δ ¦ = − = × = − ¦ ´ | | | | = = = = ¦ | | \ ¹ \ ¹ ¹ by using the first equality of (41) from (52) we have (51). By Corollary 5.1, from (51) we have the following corollary. Corollary 5.6. Let e ϕ and 1 e ϕ form a Mannheim surface offset and let e ϕ be a developable ruled surface. Then, the Mannheim offset 1 e ϕ is developable if and only if constant θ δ ∗ = = . Moreover, comparing (44) and (51) we have the following corollary. Corollary 5.7. Let e ϕ and 1 e ϕ form a Mannheim surface offset. Then the offset distance is given by coth 1 coth θ θ γ θ ∗ ∆ = + . By Corollary 5.1, we know that e ϕ is developable if and only if constant c θ ∗ ∗ = = . Let now assume that e ϕ is developable and θ ∗ is a non-zero constant. Then by Corollary 5.7 we obtain (1 coth ) coth θ γ θ θ ∗ + ∆ = (53) Thus we have the following corollary. Corollary 5.8. Let e ϕ and 1 e ϕ form a Mannheim surface offset with non-zero constant offset distance θ ∗ . Then e ϕ is developable if and only if tanh γ θ = − . Theorem 5.5. If e ϕ and 1 e ϕ form a Mannheim surface offset, then the relationship between the conical curvature 1 γ of 1 e ϕ and offset angle θ is given by 1 coth γ θ = − . (54) Proof. From (19) and (34) we have 1 1 1 1 1 , (sinh cosh ), cosh g t d e t g ds ds ds γ θ θ γ θ ′ = − = − + = − (55) 12 From the first equality of (41) and (55) we have 1 coth γ θ = − . Theorem 5.6. If the surfaces e ϕ and 1 e ϕ form a Mannheim surface offset, then the dual conical curvature 1 γ of 1 e ϕ is obtained as ( ) 2 1 1 coth 2( ) coth (1 coth ) γ θ ε δ θ θ θ γ ∗ = − + − + ∆ + . (56) Proof. From (29), (45), (51) and (55) by direct calculation we have (56). Corollary 5.9. Let e ϕ and 1 e ϕ form a Mannheim surface offset. Then, the surface e ϕ is developable if and only if 1 γ is given by 1 2( ) coth 1 θ δ γ θ ε γ ∗ | | − = − + | \ ¹ . Theorem 5.7. If the surfaces e ϕ and 1 e ϕ form a Mannheim surface offset, then the dual curvature of 1 e ϕ is given by ( ) 2 2 1 cosh sinh sinh ( ) coth (1 coth ) R θ θ θ ε δ θ θ θ γ ∗ = − − + ∆ + (57) Proof. From (56) we have ( ) 2 2 1 cosh 1 cos ech ( ) coth (1 coth ) θ γ θ ε δ θ θ θ γ ∗ − = + − + ∆ + . Then from (30) we have ( ) 2 2 1 2 1 1 cosh sinh sinh ( ) coth (1 coth ) 1 R θ θ θ ε δ θ θ θ γ γ ∗ = = − − + ∆ + − . Corollary 5.10. Let e ϕ and 1 e ϕ form a Mannheim surface offset. Then the surface e ϕ is developable if and only if 2 1 sinh 1 cosh R θ δ θ ε θ γ ∗ | | − = + | \ ¹ holds. Theorem 5.8. Let e ϕ and 1 e ϕ form a Mannheim surface offset and let 1 1 γ > . Then, the dual spherical radius of curvature of 1 e ϕ is given by ( ) 3 2 1 sinh cos cosh ( ) coth (1 coth ) θ ρ θ ε δ θ θ θ γ ∗ = − − + ∆ + (58) Proof. From (32) we have 1 1 sinh R ρ = − . Then by (4) and (57) we get ( ) 2 2 1 1 1 cosh sinh sinh sinh , cosh ( ) coth (1 coth ) θ θ ρ θ ρ ρ δ θ θ θ γ ∗ ∗ = − = − + ∆ + (59) By using the trigonometric relationship 2 2 1 1 cosh sinh 1 ρ ρ − = , from the first equality of (59) we have 1 cosh cosh ρ θ = . (60) Then by writing (60) in the second equality of (59) we obtain 13 ( ) 2 2 1 sinh ( ) coth (1 coth ) θ ρ δ θ θ θ γ ∗ ∗ = − + ∆ + (61) Thus by the aid of the extension 1 1 1 1 cosh cosh sinh ρ ρ ερ ρ ∗ = + , from (59)-(61) we have (58). From (60) we have the following corollary. Corollary 5.11. Let e ϕ and 1 e ϕ form a Mannheim surface offset. Then the offset angle θ is equal to the real spherical radius of curvature 1 ρ , i.e., 1 ρ θ = . And from (61) we have the following. Corollary 5.12. Let e ϕ and 1 e ϕ form a Mannheim surface offset. Then e ϕ is developable if and only if 1 sinh 2 2 δ θ ρ θ γ ∗ ∗ − = . If we assume that 1 1 γ < , then we have equalities for a timelike ruled surface whose Darboux vector is spacelike and the obtained equalities will be analogue to given ones in Theorem 5.8, Corollary 5.11 and Corollary 5.12. 6. Conclusions The dual geodesic trihedron(dual Darboux frame) of a timelike ruled surface is introduced. Then the characterizations of Mannheim offsets of timelike ruled surfaces are given in view of dual Darboux frame and new relations between the invariants of Mannheim timelike surface offsets are obtained. The results of the paper are new characterizations of Mannheim timelike surface offsets and also give the relationships for Mannheim timelike surface offsets to be developable according to offset angle and offset distance. References [1] Abdel-All N. H., Abdel-Baky R. A., Hamdoon F. M., Ruled surfaces with timelike rulings, App. Math. and Comp., 147 (2004) 241–253. [2] Blaschke, W., Differential Geometrie and Geometrischke Grundlagen ven Einsteins Relativitasttheorie Dover, New York, (1945). [3] Dimentberg, F. M., The Screw Calculus and its Applications in Mechanics, (Izdat. Nauka, Moscow, USSR, 1965) English translation: AD680993, Clearinghouse for Federal and Scientific Technical Information. [4] Hacısalihoğlu. H. H., Hareket Geometrisi ve Kuaterniyonlar Teorisi, Gazi Üniversitesi Fen-Edb. Fakültesi, (1983). [5] Hoschek, J., Lasser, D., Fundamentals of computer aided geometric design, Wellesley, MA:AK Peters, (1993). [6] Karger, A., Novak, J., Space Kinematics and Lie Groups, STNL Publishers of Technical Lit., Prague, Czechoslovakia (1978). [7] Kucuk, A., Gursoy O., On the invariants of Bertrand trajectory surface offsets, App. Math. and Comp., 151 (2004) 763-773. [8] O’Neill, B., Semi-Riemannian Geometry with Applications to Relativity, Academic Press, London, (1983). [9] Orbay, K., Kasap, E., Aydemir, I, Mannheim Offsets of Ruled Surfaces, Mathematical Problems in Engineering, Volume 2009, Article ID 160917. 14 [10] Önder, M., Uğurlu, H.H., On the Developable of Mannheim offsets of timelike ruled surfaces in Minkowski 3-space, arXiv:0906.2077v5 [math.DG]. [11] Önder, M., Uğurlu, H.H., Some Results and Characterizations for Mannheim Offsets of the Ruled Surfaces, arXiv:1005.2570v3 [math.DG]. [12] Önder, M., Uğurlu, H.H., Mannheim Offsets of the Timelike Ruled Surfaces with Spacelike Ruligns in Dual Lorentzian Space, arXiv:1007.2041v2 [math.DG]. arXiv:1005.2570v3 [math.DG]. [13] Önder, M., Uğurlu, H.H., Kazaz, M., Mannheim offsets of spacelike ruled surfaces in Minkowski 3-space, arXiv:0906.4660v3 [math.DG]. [14] Papaioannou, S.G., Kiritsis, D., An application of Bertrand curves and surfaces to CAD/CAM, Computer Aided Design, 17 (8) (1985) 348-352. [15] Pottmann, H., Lü, W., Ravani, B., Rational ruled surfaces and their offsets, Graphical Models and Image Processing, 58 (6) (1996) 544-552. [16] Ravani, B., Ku, T. S., Bertrand Offsets of ruled and developable surfaces, Comp. Aided Geom. Design, 23 (2) (1991) 145-152. [17] Uğurlu, H.H., Çalışkan, A,. The Study Mapping for Directed Spacelike and Timelike Lines in Minkowski 3-Space 3 1 IR , Mathematical and Computational Applications, 1 (2) (1996) 142-148. [18] Ugurlu, H. H., Onder, M., Instantaneous Rotation vectors of Skew Timelike Ruled Surfaces in Minkowski 3-space, VI. Geometry Symposium, 01-04 July 2008, Bursa, Turkey. [19] Ugurlu, H. H., The relations among instantaneous velocities of trihedrons depending on a spacelike ruled surface, Hadronic Journal, 22, (1999) 145-155. [20] Veldkamp, G.R., On the use of dual numbers, vectors and matrices in instantaneous spatial kinematics, Mechanism and Machine Theory, II (1976) 141-156. spacelike if a . Using the dual representations of the timelike ruled surfaces. a = − a1b1 + a2b 2 + a3b3 . b = (b. a = −1} and S12 = {a = ( a1 . a = 0 and a ≠ 0 . a 3 ) and b = ( b1 . Now. The norm of a vector a is defined by a = a. where a = ( a 1 . b2 . a ∗ ) : a . a 3 ) ∈ E13 : a .e. a ∗ ∈ IR} be the set of the pairs (a. a = 1} respectively(See [17]). a . the normal vector on the spacelike (timelike) surface is a timelike (spacelike) vector[1]. b∗ ) ∈ D the following operations are defined on D : Equality: a = b ⇔ a = b. and lightlike (null ) if a . 2. In this paper. timelike or null (lightlike). a 2 . Preliminaries Let IR13 be a 3-dimensional Minkowski space over the field of real numbers IR with the Lorentzian inner product . Similarly. b .. we have studied the Mannheim offsets of timelike ruled surfaces in dual Lorentzian space by considering the Blaschke frame[12]. let a = ( a 1 . By using this definition it can be easily shown that a × b . c ) [19]. i.surfaces in dual space with Blaschke approach. a 2 . Then. We have given the characterizations of Mannheim offsets of ruled surfaces in terms of integral invariants of closed trajectory ruled surfaces and obtained the relationship between the area of projections of spherical images of Mannheim offsets of trajectory ruled surfaces and their integral invariants. a ∗ + b∗ ) 2 . c = − det(a . a 2 . Dual Numbers and Dual Lorentzian Vectors Let D = IR × IR = {a = (a. The sets of the unit timelike and spacelike vectors are called hyperbolic unit sphere and Lorentzian unit sphere. a ∗ = b∗ Addition: a + b ⇔ ( a + b. a 3 ) of IR13 is said to be timelike if a . a∗ ) . we give some theorems and new results which characterize the developable Mannheim timelike surface offsets. given by a . a < 0 . A surface in the Minkowski 3-space IR13 is called a timelike surface if the induced metric on the surface is a Lorentz metric and is called a spacelike surface if the induced metric on the surface is a positive definite Riemannian metric. a > 0 or a = 0 . a1b3 − a 3b1 . an arbitrary curve α (s) in IR13 can locally be spacelike. then the Lorentzian cross product of a and b is given by a × b = ( a2b3 − a3b 2 . we examine the Mannheim offsets of the timelike ruled surfaces in view of their dual Darboux frame. a 3 ) and b = ( b1 . a2 b1 − a1b2 ). a∗ ) . a 2 . if all of its velocity vectors α ′( s ) are spacelike. respectively[8]. timelike or null (lightlike). For a = (a. and denoted by 2 H 0 = {a = ( a 1 . a 3 ) ∈ E13 : a . b2 . A vector a = ( a 1 . b3 ) be two vectors in IR13 . 3. a 2 . we define the dual geodesic trihedron(dual Darboux frame) of a timelike ruled surface and give the real and dual curvatures of this surface. respectively. Moreover. b3 )∈ IR3 . 0) + (0. a∗ ∈ IR. Dual function of dual number presents a mapping of a dual numbers space on itself. ( a ≠ 0) . where a and a ∗ are the vectors of IR3 . ( ) respectively. The elements of D3 are called dual vectors.0) + (0. The associative laws hold for multiplication. f (a.e. Then the mapping f : D → IR. D3 = {a = (a1 . x ∈ IR . (6) (7) and a × b = a × b + ε a × b * + a∗ × b . 0) = a + ε a∗ Then a = a + ε a ∗ is called dual number and ε is called dual unit. b ). ( x > 0).. 2. ε 2 = 0} The set D forms a commutative group under addition. a ∗ a = a +ε . where a. By the multiplication rule we have that a = ( a. i = 1. a dual vector a may be expressed in the form a = a + ε a ∗ = (a . a (8) 3 . This definition allows us to write the dual forms of some well-known functions as follows cosh( x ) = cosh( x + ε x∗ ) = cosh( x) + ε x∗ sinh( x).Multiplication: ab ⇔ ( ab. the scalar product and the vector product are defined by a. respectively. ε 2 = 0 .3} . b = a . 2 x 3 Let D = D × D × D be the set of all triples of dual numbers.1) ∈ D satisfies the relationships ε ≠ 0 . Properties of dual functions were thoroughly investigated by Dimentberg[3]. a ∗ ) . a ∗ ) = (a . The norm of a dual vector a is given by a. 0) . He derived the general expression for dual analytic (differentiable) function as follows f ( x ) = f ( x + ε x∗ ) = f ( x) + ε x∗ f ′( x) . i. a3 ) : ai ∈ D. ε 1 = 1ε = ε (1) Let consider the element a ∈ D of the form a = (a. b and a × b are the inner product and the vector product of the vectors a and a ∗ in IR3 . ab∗ + a ∗b ) The element ε = (0. (5) Then the set D 3 is called dual space. 0) . Then for any vectors a = a + ε a ∗ and b = b + ε b ∗ of D3 . 0) = a is a isomorphism. a∗ ) = (a . Thus the set of dual numbers is given by (2) D = {a = a + ε a ∗ : a. ∗ ∗ (4) sinh( x ) = sinh( x + ε x ) = sinh( x) + ε x cosh( x). a2 .1)(a ∗ . Dual numbers are distributive and form a ring over the real number field[4]. ∗ x = x + ε x∗ = x + ε x . (3) ∗ where f ′( x ) is derivative of f ( x ) and x . b + ε ( a. So. b ∗ + a∗ . Similar to the dual numbers. we can write a = (a. Then a dual vector a = a + ε a ∗ is said to be timelike if a is timelike. a 3 ) ∈ D13 : a . < a . a = 1 + ε 0} . b . S12 = a = ( a 1 .A dual vector a with norm 1 + ε 0 is called dual unit vector. a 2 . i) Dual Hyperbolic angle: Let x and y be dual timelike vectors in ID13 . a > = − 1 ( resp. 2 H 0 = a = ( a 1 .4]. an oriented timelike line L is determined by a point p ∈ L and a unit timelike vector a . ) is the Lorentzian inner product of the vectors a and b in the Minkowski 3- space IR13 . b = a . The Lorentzian inner product of two dual vectors a = a + ε a ∗ . a = −1 + ε 0 { } (11) Similarly. The dual number θ = θ + εθ ∗ is called the dual Lorentzian timelike angle[17]. Then the dual angle between x and y is defined by < x. spacelike) vector if the vectors a and a ∗ satisfy the following equations: (10) < a . The set of dual unit vectors is given by S 2 = {a = ( a1 . b + ε where a . 4. the set of all unit dual spacelike vectors is called the dual Lorentzian unit sphere. because any other point q in L can 4 . one can define a ∗ = p × a which is called moment vector. Dual Representation and Dual Darboux Frame of a Timelike Ruled Surface In the Minkowski 3-space IR13 . Then the angle between x and y is defined by < x. a = 1 + ε 0 . { } (12) (For details see [17]) . a ∗ > = 0. The dual number θ = θ + εθ ∗ is called the dual hyperbolic angle. where a × b is the Lorentzian cross product in IR13 .b * + a∗ . and is denoted by H 02 . y >= x y sinh θ . a3 ) ∈ D 3 : a. b ( a . a 3 ) ∈ D13 : a. The value of a ∗ does not depend on the point p . a > = 1). The set of all dual Lorentzian vectors is called dual Lorentzian space and it is denoted by 3 D1: 3 D 1 = { a = a + ε a ∗ : a . < a . (9) and called dual unit sphere[2. a 2 . 3 The Lorentzian cross product of dual vectors a and b ∈D 1 is defined by a × b = a × b + ε (a ∗ × b + a × b * ) . y >= − x y cosh θ . spacelike if a is spacelike or a = 0 and lightlike (null) if a is lightlike (null) and a ≠ 0 [17]. ii) Dual Lorentzian timelike angle: Let x be a dual spacelike vector and y be a dual timelike vector in ID13 . Definition 3. Then.1. b = b + ε b * ∈D by 3 is defined a. 3 ~ Let a = a + ε a ∗ ∈D 1 . a2 . and is denoted by S12 . The set of all unit dual timelike vectors is called the dual hyperbolic unit sphere. Then a is said to be dual unit timelike(resp. a ∗ ∈ IR13 } . In [20]. a ∗ ∈ E 3 . the set of all solutions is given by p ( s) = po ( s ) + λ ( s ) e ( s ) .be given by q = p + λ a and then a ∗ = p × a = q × a . and c is the direction vector of the striction curve. one recovers the timelike line L as L = {( a × a ∗ ) + λ a : a . the timelike surface is developable if and only if ∆ = 0 . e′ = po . The real unit vector e draws a curve on the real hyperbolic unit sphere H 02 and is called the (real) indicatrix of ( k ) . The equation e ∗ ( s) = p ( s ) × e ( s ) has infinity of solutions for the function p( s ) . the geometry of timelike ruled surface is represented by the geometry of dual hyperbolic curves lying on the dual hyperbolic unit sphere H 02 . we use the similar procedure for a timelike ruled surface and introduce the dual Darboux frame of a timelike ruled surface. This correspondence is known as E.e. The vectors a and a ∗ are not independent of one another and they satisfy the following relationships a . If we take po ( s) as a solution. Therefore. a ∗ ) is given. e = −1 . ai∗ (1 ≤ i ≤ 3) of the vectors a and a ∗ are called the normalized Plucker coordinates of the timelike line L . e′ we see that po ( s ) + λo ( s ) e ( s ) = c ( s ) is the unique solution for p ( s ) with c′. e′ = 0 . λ ∈ IR } . written in parametric equations. (11) and (13) we see that the dual timelike unit vector a = a + ε a ∗ corresponds to timelike line L . Reciprocally. Therefore we have ′ ′ p′. t ) which characterizes the developable timelike surface. e′ = 1 . Veldkamp given the dual representation and dual geodesic trihedron of a ruled surface. By the aid of this correspondence. e . the properties of the spatial motion of a timelike line can be derived. i. Hence. By taking λ = λo = − po . From (10). e′ = 1 . Then we have e′ = 1 − ε det(c′. the given dual curve ( k ) corresponding to the timelike ruled surface ϕ e = c ( s ) + ve (s ) may be represented by e( s) = e + ε c × e where e . a ∗ = 0 (13) The components ai . Study Mapping: There exists a one-to-one correspondence between the timelike vectors of dual 2 hyperbolic unit sphere H 0 and the directed timelike lines of the Minkowski space IR13 [17]. Now. t ) = 1 − ε∆ (14) (15) (16) where ∆ = det(c′. e′ = 0 . Then. Then. We suppose throughout that it is not a single point. We take the parameter u as the arc-length parameter s of the real indicatrix and denote the differentiation with respect to s by primes. The vector e′ = t is the unit vector parallel to the tangent of the indicatrix. where λ is a real scalar function of s . a = −1. when such a pair ( a . c′. e′ + λ . the dual unit tangent to the dual curve e (s ) is given by 5 . the dual arc-length s of the dual curve ( k ) is given by s = ∫ e′(u ) du = ∫ (1 − ε∆)du =s − ε ∫ ∆du 0 0 0 s s s (17) From (17) we have s′ = 1 − ε∆ . a . Let now (k ) be a dual hyperbolic curve represented by the dual timelike unit vector e (u ) = e (u ) + ε e ∗ (u ) . Then we have e′. e′. e . g ′ = −γ t (19) where γ is called the conical curvature[6.1. = e + γ g. ds Then from (18).de e′ e′ = = = t = t + ε (c × t ) ds s ′ 1 − ε∆ Introducing the dual unit vector g = −e × t = g + ε c × g we have the dual frame (18) {e. t = 1 + ε 0 . β . it follows from (22) that g ′ = −γ (1 − ε∆)t (25) On the other hand using that g = g + ε c × g .18]. ds Finally. t . g} of a timelike ruled surface are obtained as follows de dt dg (24) =t. The first equation of (20) gives that β =0. t . From (18) we have t . = −e × . ds ds ds For the derivative of t let write dt = αe + βt +γ g (21) ds where α . we have dt dg dt (20) t. γ are the dual functions of dual arc-length s . = −γ t ds ds ds and called dual Darboux formulae of timelike ruled surface e (or ϕ e ). Since s′ = 1 − ε∆ . it is well known that the real orthonormal frame {e . t ′ = e + γ g . from (18) we have g ′ = −γ t + ε (c′ × g − γ c × t ) = −γ t − εγ (c × t ) + ε (c′ × g ) (26) = −γ t + ε (c′ × g ) Therefore. g } is called the geodesic trihedron of the indicatrix e ( s ) with the derivations e′ = t . from (25) and (26) we obtain −γ (1 − ε∆)t = −γ t + ε (c′ × g ) (27) which gives us 6 . By using this equality and considering that g = −e × t . Let now give the invariants of the surface. from (22) and the equality t = g × e . The derivatives of the vectors of dual frame {e. Lot now consider the derivations of vectors of dual geodesic trihedron and find the dual Darboux formulae of a timelike ruled surface. g} which is known as dual geodesic trihedron or dual Darboux frame of ϕe (or (e ) ). we obtain dt = e+γ g . Thus from the second equation of (20) we have dg = −γ t . Then the dual Darboux vector of the trihedron is d = −γ e − g . t . Also. = 0. (22) and (23) we have the following theorem (22) (23) Theorem 4. t ( s ). It means that the timelike ruled surface ϕe is a timelike cone. 4. coth ρ . the dual angle between d o and e satisfies the followings γ 1 . 1− γ 2 1− γ 2 where ρ is the dual spherical radius of curvature. tanh ρ . δ (s ) and ∆ ( s ) are the invariants of the timelike ruled surface ϕ e . if γ > 1. t ) = − c′ × e . Elements of Curvature of a Dual Hyperbolic Curve The dual radius of curvature of dual hyperbolic curve(timelike ruled surface) e ( s ) is can be calculated analogous to common Lorentzian differential geometry of curves as follows 3 de 1 ds (30) R= = 2 de d e 1− γ 2 × ds ds 2 γ (1 − ε∆) = γ + εδ where δ = c′. if γ < 1. Then ϕe (33) and ϕ e1 are called Mannheim surface offsets. cosh ρ = − 1− γ 2 1− γ 2 γ 1 sinh ρ = − . The functions γ (s ).(29) Moreover. They determine the timelike ruled surface uniquely up to its position in the space. g ( s )} and {e1 ( s1 ). since c′ as well as e is perpendicular to t . Then from (28) we have γ = γ + ε (δ + γ∆) (28) The unit vector d o with the same sense as the Darboux vector d = −γ e − g is given by γ 1 (31) do = − e− g 2 2 1− γ 1− γ It is clear that d o is timelike (or spacelike) if γ > 1 (or γ < 1 ). sinh ρ = − . Then.1. for the real scalar µ we may write c′× e = µ t . if 7 . R = and γ = − cosh ρ . t1 ( s1 ). Then ∆ = det(c′. Hence e × (c′ × e ) = −∆e × t = ∆g and c′ = −δ e + ∆g . For example. cosh ρ = − . if γ < 1. e . g1 ( s1 )} g ( s ) = t1 ( s1 ) denote the dual Darboux frames of ϕ e and ϕ e1 . (32) 5. e . if δ = ∆ = 0 we have c is constant. Hence − sinh ρ . Darboux Approach to Mannheim Offsets of Timelike Ruled Surfaces Let ϕ e be a timelike ruled surface generated by dual timelike unit vector e and ϕ e1 be a ruled surface generated by dual unit vector e1 and let {e ( s ). t = − µ . if γ > 1. respectively. t = − µ t . if γ > 1. if γ < 1. Theorem 5. Proof. respectively. where c and c∗ are real constants. Then the relation between the trihedrons of the ruled surfaces ϕ e and ϕ e1 can be given as follows e1 cosh θ sinh θ 0 e 0 1 t (34) t1 = 0 g sinh θ cosh θ 0 g 1 ∗ ∗ where θ = θ + εθ . If θ = 0 . from (37) we get = −1. Then for the dual ds ds constant c = c + ε c∗ we write dθ = −ds Since θ = −s + c θ + εθ ∗ = − s − ε s ∗ + c + ε c∗ and from (17) we have 8 .1 (ii) similar results can be found. When we write ϕe and ϕ e1 . And from (37) we write c1 = c + θ ∗t . we mean timelike ruled surfaces with timelike ruling and for short we don’t write the Lorentzian characters of the surfaces hereinafter. θ = θ + εθ ∗ is called dual hyperbolic offset angle of the Mannheim timelike ruled surfaces ϕe and ϕ e1 . ruling is spacelike. Now. it is seen that the Mannheim offset of ϕ e is also a timelike ruled surface. where s and s1 are the dual arc-lengths of ϕ e and ϕ e1 . The offset angle θ and the offset distance θ ∗ are given by s θ = − s + c. θ ∗ = ∫ ∆du + c∗ 0 (35) respectively. Then from (34) we have e1 = cosh θ e + sinh θ t By differentiating (36) with respect to s we have dθ dθ de1 = sinh θ 1 + e + cosh θ 1 + t + γ sinh θ g ds ds ds (36) (37) dθ de1 and g are linearly dependent. Then. and θ ∗ is called the offset distance which is measured from the striction point c of ϕ e to striction point c1 of ϕ e1 . Let now ϕe and ϕ e1 be timelike ruled surfaces with timelike rulings and form a Mannheim surface offset.holds. If one considers the second situation. Let ϕe and ϕ e1 form a Mannheim surface offset. θ ∈ ») is the dual hyperbolic angle between the dual timelike generators e and e1 of Mannheim timelike ruled surfaces ϕ e and ϕ e1 . The angle θ is called the offset angle which is the angle between the rulings e and e1 . By this definition. we consider the Mannheim offset surface with timelike ruling. In this study. we give some theorems and results characterizing Mannheim timelike surface offsets.1. then the Mannheim timelike surface offsets are said to be oriented offsets. (θ . and uses the Definition 3. Suppose that ϕe and ϕ e1 form a Mannheim offset. but the generator of this surface can be timelike or spacelike. θ = − s + c. Then (39) gives us ds γ sinh θ =1 ds1 and from (40) we get (38). Then there are the following relationships between the real arc-length parameters of ϕe and ϕ e1 ∗ ds1 dsds1 − ds ∗ds1 (41) = γ sinh θ .2. Suppose that ϕe and ϕ e1 form a Mannheim offset. (38) holds.2. 0 s where c and c are real constants. = θ ∗γ cosh θ + (δ + γ∆) sinh θ ds ds 2 Proof. Then.2. Then (43) 9 . = θ ∗γ cosh θ + (δ + γ∆) sinh θ .1 we have de1 ds = t1 = γ sinh θ g ds1 ds1 From (33) we have t1 = g . Then there is the following differential relationship between the dual arc-length parameters of ϕe and ϕ e1 ds1 (38) = γ sinh θ . the following corollary can be given.1.1. Let ϕ e and ϕ e1 form a Mannheim surface offset. Now we give the condition for ϕ e1 to be developable according to θ∗ .3. Let ϕe and ϕ e1 form a Mannheim surface offset. By considering (4). Let ϕe and ϕ e1 form a Mannheim surface offset. Then by Theorem 5. (39) (40) Corollary 5. ds Proof. from Theorem 5. Then ϕe is developable if and ∗ only if θ ∗ = c∗ = constant . ds1 = −∆1ds1 . 2 ds ds (42) In Corollary 5. Then writing (43) in (42) and using (41) we get δ ∆1 = − θ ∗ coth θ + γ and give the following corollaries: Corollary 5. From (17) we have ∗ ds∗ = −∆ds. we give the relationship between the offset distance θ ∗ and developable timelike ruled surface ϕ e . Let ϕe and ϕ e1 form a Mannheim surface offset. the real and dual parts of (38) are ∗ ds1 dsds1 − ds ∗ds1 = γ sinh θ . θ ∗ = ∫ ∆du + c∗ . Let ϕ e and ϕe1 form a Mannheim surface offset. From (35). Corollary 5. Theorem 5. There exists the following relationship between the invariants of the surfaces and offset angle θ . Then ϕ e1 is developable if and only if θ ∗ = − δ tanh θ holds. from the Mannheim condition we can write 1 1 c1 = c + θ ∗t . Let ϕe and ϕ e1 form a Mannheim surface offset. Differentiating (46) with respect to s1 we have ds dc1 dc dθ ∗ . δ1 = cosh θ dc / ds. Corollary 5. γ Theorem 5. t − θ ∗ cosh θ + ds dθ ∗ sinh θ ds ds1 (48) δ1 = δ cosh θ − θ ∗ cosh θ + (49) (50) γ Then we give the following corollary. γ (51) 10 . Substituting (50) in (49) we obtain 1 δ1 = ( (δ − θ ∗ ) coth θ + ∆ ) . from (48) we write ds dθ ∗ sinh θ ds ds1 Furthermore. e + sinh θ dc / ds. Then from (34) and (47) we obtain (46) (47) Since δ = dc / ds. Then ϕe is developable if and only if δ1 = δ −θ ∗ coth θ holds. t = 0 . Then the relationship between the invariants of the surfaces and offset angle θ . = + θ ∗e + t + θ ∗γ g ds1 ds ds ds1 We know that δ1 = dc1 / ds1 . (45) γ Proof. and let ϕ e and ϕ e form a Mannheim surface offset. Corollary 5. offset distance θ ∗ is given by 1 ∆1 = (θ ∗ − δ − ∆ coth θ ) . Let ϕe and ϕ e1 form a Mannheim surface offset. Then. offset distance θ ∗ .3. = ds ds1 γ sinh θ respectively. Let ϕ e and ϕ e1 form a Mannheim surface offset. Let the striction lines of ϕ e and ϕ e be c( s ) and c1 ( s1 ) . respectively.4. δ ∆1 = − θ ∗ coth θ + γ (44) holds. Let ϕ e and ϕ e1 form a Mannheim surface offset. 1 δ1 = ( (δ − θ ∗ ) coth θ + ∆ ) . e and dc / ds. e1 . from (35) and (41) we have dθ ∗ ds 1 =∆.4. γ Theorem 5.5. t1 =− d (sinh θ e + cosh θ t ).7. g )∆ cosh θ = det ds1 ds ds ds (52) Since det(t . e1 . g = ds ds ds ds by using the first equality of (41) from (52) we have (51). t . Then.7 we obtain θ ∗ (1 + γ coth θ ) ∆= (53) coth θ Thus we have the following corollary.1.1. comparing (44) and (51) we have the following corollary. t = 0. we know that ϕe is developable if and only if θ ∗ = c∗ = constant . Then ϕ e is developable if and only if γ = − tanh θ . Theorem 5. cosh θ e + sinh θ t . Then the offset distance is given by θ ∗ = ∆ coth θ . g cosh θ + det dc . e . By Corollary 5. e . Let now assume that ϕe is developable and θ ∗ is a non-zero constant.5. Corollary 5. Corollary 5. g sinh θ + det(e . t . det . If ϕe and ϕ e1 form a Mannheim surface offset. g ) = e × t . e . g ds1 ds ds1 (54) (55) = −γ cosh θ 11 . e . then the relationship between the conical curvature γ 1 of ϕ e1 and offset angle θ is given by γ 1 = − coth θ . g ) = − det(e . From (19) and (34) we have γ 1 = − g1′. g )θ ∗ sinh θ + det(t . the Mannheim offset ϕ e1 is developable if and only if θ ∗ = δ = constant . t . Moreover.e = δ . Then from (34) and (47) we obtain ∆1 = ds ds1 det dc + θ ∗e + ∆t + θ ∗γ g . 1 + γ coth θ By Corollary 5. t . Let ϕ e and ϕ e1 form a Mannheim surface offset with non-zero constant offset distance θ ∗ . dc dc dc dc . from (51) we have the following corollary. g ds dc . Proof.8. g = −1. Let ϕe and ϕ e1 form a Mannheim surface offset and let ϕe be a developable ruled surface. Then by Corollary 5. We know that ∆1 = det(dc1 / ds1 . g = . t1 ) . det .Proof. Corollary 5.6. Let ϕe and ϕ e1 form a Mannheim surface offset. the dual spherical radius of curvature of ϕ e1 is given by cos ρ1 = cosh θ − ε sinh 3 θ γ ( (δ − θ ∗ ) coth θ + ∆(1 + coth 2 θ ) ) (58) Proof. γ Theorem 5. 2 γ 1− γ1 Corollary 5. Let ϕe and ϕ e1 form a Mannheim surface offset. Then by (4) and (57) we get ( (δ − θ ∗ ) coth θ + ∆(1 + coth 2 θ ) ) (59) γ By using the trigonometric relationship cosh 2 ρ1 − sinh 2 ρ1 = 1 . Then the surface ϕ e is θ ∗ −δ developable if and only if R1 = sinh θ 1 + ε cosh 2 θ holds. from the first equality of (59) sinh ρ1 = − sinh θ . Let ϕ e and ϕ e1 form a Mannheim surface offset and let γ 1 > 1 . Theorem 5. If the surfaces ϕe and ϕ e1 form a Mannheim surface offset. ρ1∗ cosh ρ1 = cosh θ sinh 2 θ we have cosh ρ1 = cosh θ .6. (51) and (55) by direct calculation we have (56). the surface ϕ e is 2(θ ∗ − δ ) developable if and only if γ 1 is given by γ 1 = − coth θ 1 + ε . Then from (30) we have 1 cosh θ sinh 2 θ = sinh θ − ε R1 = ( (δ − θ ∗ ) coth θ + ∆(1 + coth 2 θ ) ) .9. Then by writing (60) in the second equality of (59) we obtain (60) 12 . Then. then the dual curvature of ϕ e1 is given by R1 = sinh θ − ε cosh θ sinh 2 θ γ ( (δ − θ ∗ ) coth θ + ∆(1 + coth 2 θ ) ) (57) Proof. γ Theorem 5.8. Then. If the surfaces ϕe and ϕ e1 form a Mannheim surface offset. γ Proof. From (32) we have sinh ρ1 = − R1 . From (29).7. (45).From the first equality of (41) and (55) we have γ 1 = − coth θ . Let ϕe and ϕ e1 form a Mannheim surface offset.10. then the dual conical curvature γ 1 of ϕ e1 is obtained as γ 1 = − coth θ + ε ( 2(δ − θ ∗ ) coth θ + ∆(1 + coth 2 θ ) ) . From (56) we have 1 − γ 12 = cos echθ + ε cosh θ γ ( (δ − θ ∗ ) coth θ + ∆(1 + coth 2 θ ) ) . 1 (56) Corollary 5. and Comp. Corollary 5. 13 .. M. Wellesley. Semi-Riemannian Geometry with Applications to Relativity. F.. The results of the paper are new characterizations of Mannheim timelike surface offsets and also give the relationships for Mannheim timelike surface offsets to be developable according to offset angle and offset distance. 147 (2004) 241–253. (1945).. Space Kinematics and Lie Groups. (1983). New York. 151 (2004) 763-773. J. Volume 2009. A. Math.. 2γ If we assume that γ 1 < 1 . ρ1 = θ . H. Math. and Comp. (1993). [6] Karger.11. A. Mannheim Offsets of Ruled Surfaces. The Screw Calculus and its Applications in Mechanics. Conclusions The dual geodesic trihedron(dual Darboux frame) of a timelike ruled surface is introduced. Corollary 5.. Let ϕ e and ϕ e1 form a Mannheim surface offset.. [7] Kucuk.11 and Corollary 5. London. Fundamentals of computer aided geometric design. Lasser. Kasap. ρ1∗ = sinh 2 θ From (60) we have the following corollary.. Nauka. 1965) English translation: AD680993. Then the offset angle θ is equal to the real spherical radius of curvature ρ1 ...e.. Gazi Üniversitesi Fen-Edb. And from (61) we have the following. Abdel-Baky R.8. STNL Publishers of Technical Lit.. W.12. M. B. Gursoy O. 6. (1983).. K.(61) ( (δ − θ ∗ ) coth θ + ∆(1 + coth 2 θ ) ) γ Thus by the aid of the extension cosh ρ1 = cosh ρ1 + ερ1∗ sinh ρ1 . Article ID 160917. Differential Geometrie and Geometrischke Grundlagen ven Einsteins Relativitasttheorie Dover. [9] Orbay. Aydemir. (Izdat.. D. Then the characterizations of Mannheim offsets of timelike ruled surfaces are given in view of dual Darboux frame and new relations between the invariants of Mannheim timelike surface offsets are obtained. Mathematical Problems in Engineering. [5] Hoschek.. [2] Blaschke. On the invariants of Bertrand trajectory surface offsets. [4] Hacısalihoğlu. i. A. App. [3] Dimentberg.. [8] O’Neill.. J. then we have equalities for a timelike ruled surface whose Darboux vector is spacelike and the obtained equalities will be analogue to given ones in Theorem 5. E. References [1] Abdel-All N. MA:AK Peters. App. Prague.. Fakültesi. Ruled surfaces with timelike rulings.12. Hamdoon F.. Clearinghouse for Federal and Scientific Technical Information. Hareket Geometrisi ve Kuaterniyonlar Teorisi. Corollary 5. USSR.. H. Let ϕ e and ϕ e1 form a Mannheim surface offset. Academic Press. I. Novak. Czechoslovakia (1978). Then ϕ e is developable if and only if ρ1∗ = δ −θ ∗ sinh 2θ . H. Moscow. from (59)-(61) we have (58). .. vectors and matrices in instantaneous spatial kinematics. Mannheim Offsets of the Timelike Ruled Surfaces with Spacelike Ruligns in Dual Lorentzian Space. Onder.. On the use of dual numbers. Instantaneous Rotation vectors of Skew Timelike Ruled Surfaces in Minkowski 3-space. Çalışkan.. Uğurlu. 17 (8) (1985) 348-352. Rational ruled surfaces and their offsets. M.2077v5 [math. G.DG]. H. Computer Aided Design.H. (1999) 145-155. A. M. Mannheim offsets of spacelike ruled surfaces in Minkowski 3-space. [17] Uğurlu. 23 (2) (1991) 145-152..2570v3 [math.. [20] Veldkamp. arXiv:1005.DG]. An application of Bertrand curves and surfaces to CAD/CAM. H.2570v3 [math. Geometry Symposium. Uğurlu. S.H. arXiv:1007. Graphical Models and Image Processing.H.. The relations among instantaneous velocities of trihedrons depending on a spacelike ruled surface. 1 (2) (1996) 142-148. H. Bertrand Offsets of ruled and developable surfaces. H. M. [14] Papaioannou.G. H. Design. H. Aided Geom.DG].DG]. S. [15] Pottmann. Uğurlu.. arXiv:0906. [18] Ugurlu. Turkey. 14 ... [12] Önder. M..R. D. H. Lü. 58 (6) (1996) 544-552. Uğurlu. II (1976) 141-156... Ravani... [16] Ravani. Kazaz. Mathematical and Computational Applications. Mechanism and Machine Theory. On the Developable of Mannheim offsets of timelike ruled surfaces in Minkowski 3-space.. Bursa.. H. W. Kiritsis.DG]. T. Hadronic Journal. M. B.4660v3 [math.H. [19] Ugurlu. Comp.. H. Ku. H. [13] Önder. arXiv:0906. VI.H.. B. 22. [11] Önder. The Study Mapping for Directed Spacelike and Timelike Lines in Minkowski 3-Space IR13 . 01-04 July 2008..2041v2 [math. arXiv:1005..[10] Önder. M. Some Results and Characterizations for Mannheim Offsets of the Ruled Surfaces.. 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