On the Invariants of Mannheim Offsets of Timelike Ruled Surfaces with Timelike Rulings

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1 Gen Math Notes, Vol, No, June 04, pp 0- ISSN 9-784; Copyright ICSRS Publication, 04 wwwi-csrsorg Available free online at On the Invariants of Mannheim Offsets of Timelike Ruled Surfaces with Timelike Rulings Mehmet Önder H Hüseyin Uğurlu Celal Bayar University, Faculty of Arts Sciences Department of Mathematics, 4540, Muradiye Campus Muradiye, Manisa, Turkey mehmetonder@cbuedutr Gazi University, Faculty of Education Department of Secondary School Science Mathematics Education Teknikokullar 06500, Ankara, Turkey hugurlu@gaziedutr (Received: 4--4 / Accepted: 9--4) Abstract In this study, we give dual characterizations for Mannheim offsets of timelike ruled surfaces with timelike rulings in terms of their integral invariants Also, we give new characterization for the Mannheim offsets of developable timelike ruled surfaces We show that if the offset surfaces are developable then the striction lines of surfaces are Mannheim curves Moreover, we obtain relationships between areas of projections of spherical images for Mannheim offsets of timelike ruled surfaces their integral invariants Keywords: Integral invariants, Mannheim offset, Timelike Ruled surface

2 04 Mehmet Önder et al Introduction In the space, a continuously moving of a straight line generates a surface which is called ruled surface These surfaces are one of the most important topics of differential geometry Because of their simple generation, these surfaces arise in a variety of applications including Computer Aided Geometric Design (CAGD), mathematical physics, kinematics for modeling the problems model-based manufacturing of mechanical products Furthermore, in general, offset surface, the surface which is offset a specified distance from the original along the parent surface's normal, is used in CAGD widely Some studies dealing with offsets of surfaces have been given in references [4, 8, 9, 0] Ravani Ku have defined given a generalization of theory of Bertr curves to ruled surfaces called Bertr trajectory ruled surfaces on the line geometry [0] By using the invariants of ruled surfaces given in [5, 6], Küçük Gürsoy have given the characterizations of Bertr trajectory ruled surfaces in dual space have obtained the relations between the invariants [0] Furthermore, recently, a new definition of curve pairs has been given by Liu Wang: Let C C be two space curves C is said to be a Mannheim partner curve of C if there exists a one to one correspondence between their points such that the binormal vector of C is the principal normal vector of C [] Orbay, Kasap Aydemir have given a generalization of Mannheim curves to ruled surfaces called Mannheim offsets [] Mannheim offsets of timelike ruled surfaces in the Minkowski -space IR have been studied in [4] In this paper, we examine the Mannheim offsets of trajectory timelike ruled surfaces with timelike rulings in view of their integral invariants We give a result obtained in [4] in short form Furthermore, using the dual representations of timelike ruled surfaces, we obtain some new results which are not obtained in [4] Moreover, we obtain that the striction lines of Mannheim offsets of developable timelike trajectory ruled surfaces are Mannheim partner curves in the Minkowski -space IR Furthermore, we give relations between the integral invariants (such as the angle of pitch the pitch) of closed timelike trajectory ruled surfaces Finally, we obtain the relationships between the areas of projections of spherical images of Mannheim offsets of timelike trajectory ruled surfaces their integral invariants Differential Geometry of the Ruled Surfaces in the Minkowski -Space Let IR be a -dimensional Minkowski space over the field of real numbers IR with the Lorentzian inner product, given by

3 On the Invariants of Mannheim Offsets of 05 a, b = a b + a b + a b where a = ( a, a, a) b = ( b, b, b ) IR A vector a = ( a, a, a) of IR is said to be timelike if a, a < 0, spacelike if a, a > 0 or a = 0, lightlike (null) if a, a = 0 a 0 Similarly, an arbitrary curve α ( s) in IR can locally be spacelike, timelike or null (lightlike), if all of its velocity vectors α ( s) are spacelike, timelike or null (lightlike), respectively [] The norm of a vector a is defined by a = a, a A vector a IR is called a unit vector if a = a, a = the sets of the unit timelike spacelike vectors are called hyperbolic unit sphere Lorentzian unit sphere, respectively, denoted by H = a = ( a, a, a ) IR : a, a = respectively [] { } 0 S = a = a a a IR a a = { (,, ) :, } A surface in the Minkowski -space IR is called a timelike surface if the induced metric on the surface is a Lorentz metric is called a spacelike surface if the induced metric on the surface is a positive definite Riemannian metric, ie, the normal vector on the spacelike (timelike) surface is a timelike (spacelike) vector [] Let I be an open interval in the real line IR, k = k ( s) be a curve in on I = ( s) be a unit direction vector of an oriented line in have the following parametrization for a ruled surface N ( s, v) = k ( s) + v ( s) IR defined IR Then we where ( s) is called ruling k = k ( s) is called base curve or generating curve of the surface In particular, if the direction of is constant, the ruled surface is said to be cylindrical, non-cylindrical otherwise The striction point on a ruled surface N is the foot of the common normal between two consecutive rulings The set of the striction points constitute a curve c = c( s) lying on the ruled surface is called striction curve The parametrization of the striction curve c = c( s) on a ruled surface is given by ()

4 06 Mehmet Önder et al d, dk c( s) = k ( s) () d, d So that, the base curve of the ruled surface is its striction curve if only if d, dk = 0 The distribution parameter (or drall) of the ruled surface in () is given as δ = dk,, d () d, d a ruled surface is developable if only if at all its points the distribution parameter is δ = 0 [, 9] For the unit normal vector m s v of the ruled surface N we have m = So, ( d / ds) at the points of a nontorsal ruling s = s we have a = lim m( s, v) = v d / ds The plane of ruled surface N which passes through its ruling s is perpendicular to the vector a is called the asymptotic plane α The tangent plane γ passing through the ruling s which is perpendicular to the asymptotic plane α is called the central plane Its point of contact C is central point of the ruling The straight lines which pass through point C are perpendicular to the planes α γ are called the central tangent central normal, respectively Since the vectors, d / ds are perpendicular to the vector a, representation of the unit vector h d / ds of central normal is given by h = The orthonormal d / ds system { C;, h, a} is called Frenet frame of the ruled surfaces N such that h a = h are the central normal the central tangent of N, respectively, C is the striction point Let now consider the ruled surface N According to the Lorentzian characters of ruling central normal, we can give the following classifications for the ruled surface N : s v

5 On the Invariants of Mannheim Offsets of 07 i) If the central normal vector h is spacelike the ruling is timelike, ii) iii) then the ruled surface N is said to be of type If both the central normal vector h the ruling are spacelike, then the ruled surface N is said to be of type If the central normal vector h is timelike the ruling is spacelike, then the ruled surface N is said to be of type The ruled surfaces of types of type is spacelike [8] [6, ] are clearly timelike the ruled surface By using these classifications, the parametrization of N can be given as follows, where ( s, v) = k ( s) + v ( s) h, h = ε ( = ± ),, = ε ( = ± ) (4) (5) The set of all bound vectors ( s) at the origin 0 constitutes the directing cone of the ruled surface N If ε = (resp ε = ), the end points of the vectors ( s) drive a spherical spacelike (resp spacelike or timelike) curve k on hyperbolic unit sphere H 0 (resp on Lorentzian unit sphere S ), called the hyperbolic (resp Lorentzian) spherical image of the ruled surface N [6,] Let {, h, a} closed space curve k ( s) be a moving orthonormal trihedron making a spatial motion along a in IR where s IR h is assumed spacelike In this motion, the oriented line generates a closed timelike ruled surface called closed timelike trajectory ruled surface (CTTRS) A parametric euation of a closed trajectory timelike ruled surface generated by -axis is ( s, v ) = k ( s ) + v ( s ), ( s + π, v ) = ( s, v ), s, v IR (6) Consider the moving orthonormal system {, h, a} ruled surface of the type or which represents a timelike generated by the vector Then, the axes of trihedron intersect at striction point of -generator of -CTTRS The structural euations of this motion are d = k h, dh = ε k + k a, da = ε k h (7)

6 08 Mehmet Önder et al And db = µ ( σ ) + ϑ( σ ) a ds where b = b( s) is the striction line of -CTTRS µ ( σ ) = coshσ, ϑ( σ ) = sinhσ, if both b( s) are timelike; µ ( σ ) = sinhσ, ϑ( σ ) = coshσ, if both b( s) are spacelike The differential forms k, k σ are the natural curvature, the natural torsion the striction of - CTTRS, respectively Here, s is the arclength of the striction line [6] The pole vector the Steiner vector of the motion are given by p (8) ψ, d ψ ψ (9) = = respectively, where ψ = ε k ka is the instantaneous Pfaffian vector of the motion The real invariants of -CTTRS are defined by which is the pitch, l = dµ = ε dk, (0) λ = θ = ε = ε = π d dh, a, d a () which is the angle of pitch of -CTTRS, respectively, where a is the measure of the spherical surface area bounded by the spherical image of -CTTRS The pitch the angle of pitch are well-known real integral invariants of closed timelike trajectory ruled surfaces [, 6] The area vector of a x -closed space curve in IR is given by v x = x dx () the area of projection of a x -closed space curve in direction of the generator of a y -CTRS is f = v, y () x, y x (See [5])

7 On the Invariants of Mannheim Offsets of 09 Dual Lorentzian Vectors E Study Mapping In this section, we give a brief summary of the theory of dual numbers dual Lorentzian vectors A dual number has the form λ = λ + ελ, where λ sts for dual unit which is subject to the rules 0ε = ε 0= 0, ε = ε = ε We denote set of all dual numbers by D : { λ λ ελ : λ, λ IR, ε 0} D = = + = Euality, addition multiplication are defined in D by λ are real numbers ε ε = 0, ε 0, λ + + ελ = β εβ if only if β λ = λ = β, ( λ + ε λ ) + ( β + εβ ) = ( λ + β ) + ε ( λ + β ) ( λ + ελ )( β + ε β ) = λβ + ε ( λ β + λ β ), respectively Then it is easy to show that ( D, +, ) is a commutative ring with unity The numbers ελ ( λ IR) are divisors of zero [7] Now let f be a differentiable function with dual variable x = x + x Then the Maclaurin series generated by f is ε f ( x) = f ( x + ε x ) = f ( x ) + ε x f ( x ), where f ( x ) is the derivative of f with respect to x [7] Let D be the set of all triples of dual numbers, ie D = { a ɶ = a a a a D i } (,, ), elements of D are called dual vectors A dual vector aɶ may be expressed in the form aɶ = a +ε a, where a a are the vectors of IR Now let aɶ = a +ε a, bɶ = b + εb D λ = λ + ε λ D Then we define aɶ + bɶ = a + b + a + b aɶ = a + a + a ( ) ε ( ), λ λ ε ( λ λ ) Then, D becomes a unitary module with these operations It is called D -module or dual space [7] i The

8 0 Mehmet Önder et al The Lorentzian inner product of two dual vectors aɶ = a +ε a, bɶ = b + εb D is defined by a, b = a, b + ε a, b ɶ ɶ + a, b, ( ) where a, b is the Lorentzian inner product of the vectors a b in the Minkowski -space IR Then a dual vector a = a +ε ɶ a is said to be timelike if a is timelike, spacelike if a is spacelike or a = 0 lightlike (null) if a is lightlike (null) a 0 [, 4] The set of all dual Lorentzian vectors is called dual Lorentzian space it is D = aɶ = a + a : a, a IR denoted by { ε } The Lorentzian cross product of dual vectors where a b aɶ, bɶ D is defined by aɶ bɶ = a b + ε a b + a b is the Lorentzian cross product in ( ), IR Let aɶ = a + ε a D Then aɶ is said to be dual unit timelike (resp spacelike) vector if the vectors a a satisfy the following euations: < a, a > = ( resp < a, a > = ), < a, a > = 0 The set of all dual unit timelike vectors is called the dual hyperbolic unit sphere, is denoted by Hɶ Similarly, the set of all dual unit spacelike vectors is called 0 the dual Lorentzian unit sphere, is denoted by S ɶ [, 4] Theorem (E Study Mapping) ([]): The dual timelike (respectively spacelike) unit vectors of the dual hyperbolic (respectively Lorentzian) unit sphere Hɶ 0 (respectively S ɶ ) are in on-to-one correspondence with the directed timelike (respectively spacelike) lines of the Minkowski -space Definition ([, 4]): IR i) Dual Hyperbolic Angle: Let xɶ yɶ be dual timelike vectors in D Then the dual angle between xɶ yɶ is defined by < x ɶ, y ɶ >= x ɶ y ɶ coshθ The dual number θ = θ + εθ is called the dual hyperbolic angle

9 On the Invariants of Mannheim Offsets of ii) Dual Lorentzian Timelike Angle: Let xɶ be a dual spacelike vector yɶ be a dual timelike vector in D Then the angle between xɶ yɶ is defined by < x ɶ, y ɶ > = x ɶ y ɶ sinhθ The dual number θ = θ + εθ is called the dual Lorentzian timelike angle Let now K ɶ be a moving dual unit hyperbolic or Lorentzian sphere generated by a dual orthonormal system dɶ ɶ, hɶ = ( spacelike), aɶ = ɶ hɶ, ɶ = + ε, hɶ = h + ε h, aɶ = a + ε a dɶ (4) K ɶ be a fixed dual unit sphere with the same center Then, the derivative euations of the dual spherical closed motion of K ɶ with respect to K ɶ are dɶ = k hɶ, dhɶ = ε k ɶ + k aɶ, daɶ = ε k hɶ (5) where k( s) = k( s) + εk ( s), k( s) = k( s) + ε k ( s), ( s IR) are dual curvature dual torsion, respectively From the E Study mapping, during the spherical motion of K ɶ with respect to K ɶ, the dual unit timelike (resp spacelike) vector ɶ draws a dual curve on dual unit hyperbolic (resp Lorentzian) sphere K this curve represents a timelike ruled surface with timelike (resp spacelike) ruling in line space IR [5] Dual vector ψɶ = ψ + εψ = ε kɶ kaɶ is called the instantaneous Pfaffian vector of the motion the vector P ɶ given by ψɶ = ψɶ Pɶ is called the dual pole vector of the motion Then the vector = is the dual Steiner vector of the closed motion dɶ ψɶ (6) By considering the E Study mapping, the dual euations (5) correspond to real euations (7) (8) of a closed spatial motion in IR So, the differentiable dual closed curve ɶ = ɶ ( s) corresponds to a closed trajectory timelike ruled surface in the line space IR denoted by -CTTRS A dual integral invariant of a -CTTRS can be given in terms of real integral invariants as follows is called the dual angle of pitch of a -CTTRS = ε dhɶ, a =, dɶ ɶ ɶ = π a = λ + ε εl (7)

10 Mehmet Önder et al where d ɶ = d + εd a = a + εa are the dual Steiner vector of the motion the measure of dual spherical surface area of -CTTRS, respectively Here, ε =, ε is dual unit [7] Analogue to the real area vector given in (), the dual area vector of a ɶ -closed dual curve is given by wɶ ɶ = ɶ dɶ (8) from ref [0], the dual area of projection of a ɶ -closed dual curve in direction of the generator of a ɶ -CTRS is f = wɶ, ɶ (9) ɶ, ɶ ɶ 4 Mannheim Offsets of Timelike Trajectory Ruled Surfaces with Timelike Rulings Let be a timelike trajectory ruled surface of the type dual unit vector ɶ let dual orthonormal frame of or generated by be { ( s), hɶ ( s), a( s) } ɶ ɶ The trajectory ruled surface, generated by dual unit vector ɶ with dual ɶ ( s ), hɶ ( s ), aɶ ( s ) is said to be Mannheim offset of the orthonormal frame { } timelike trajectory ruled surface, if aɶ ( s) = h ɶ ( s ) (0) holds, where s s are arc-lengths of striction lines of, respectively By this definition considering the classifications of the timelike ruled surfaces we have the following cases: Case : If the trajectory ruled surfaces is of the type, then by considering (0), the Mannheim offset of is a timelike trajectory ruled surface of the type or If is of the type is of the type, then we have ɶ sinhθ coshθ 0 ɶ hɶ = 0 0 hɶ a coshθ sinhθ 0 a ɶ ɶ ()

11 On the Invariants of Mannheim Offsets of Case : Similarly, if both are of the type, we have ɶ coshθ sinhθ 0 ɶ hɶ = 0 0 hɶ a sinhθ coshθ 0 a ɶ ɶ () In () (), θ = θ + εθ, ( θ, θ IR) is dual angle between the generators ɶ ɶ of Mannheim trajectory ruled surfaces The angle θ is called offset angle real number θ is called offset distance Then, θ = θ + εθ is called dual offset angle of the Mannheim trajectory ruled surfaces Thus, we can give the followings 5 -Type Mannheim Offsets of Timelike Ruled Surfaces with Timelike Rulings Let the timelike trajectory ruled surface of the type be a Mannheim offset of timelike trajectory ruled surface with timelike rulings Then we can give the followings Theorem 5: Let trajectory ruled surface of the type be a Mannheim offset of timelike trajectory ruled surface with timelike rulings Then offset angle offset distance are given by kds, θ k ds () θ = = respectively, where k = k + εk is the dual curvature of Proof: Let form a Mannheim offset where, respectively From () we have are of the type ɶ = sinhθ ɶ + coshθ hɶ (4) Differentiating (4) by using (5) (0), it follows dɶ ds dθ = + k aɶ + k coshθ hɶ (5) ds

12 4 Mehmet Önder et al dɶ Since ds is orthogonal to aɶ, from (5) we get θ = kds (6) Separating the last euation into real dual parts we have kds, θ k ds (7) θ = = Theorem 5: The closed timelike trajectory ruled surface with timelike rulings the closed timelike trajectory ruled surface of the type form a Mannheim offset with a constant dual offset angle if only if the following relationship holds = sinhθ + coshθ (8) h Proof: Let the closed timelike trajectory ruled surfaces (of the type ) form a Mannheim offset with a constant dual offset angle θ Then, from (7) (), the dual angle of pitch of -CTTRS is given by = dhɶ, aɶ = daɶ, (cosh θ ) ɶ + (sinh θ ) hɶ = daɶ, ɶ cosh θ + daɶ, hɶ sinhθ = daɶ, ɶ cosh θ dhɶ, aɶ sinhθ = sinhθ + h coshθ Conversely, if (8) holds, it is easily seen that Mannheim offsets with a constant dual offset angle -CTTRS form a Euality (8) is a dual characterization of Mannheim offsets of CTTRS with a constant dual offset angle in terms of their dual integral invariants Separating (8) into real dual parts, we obtain λ = λ sinh cosh θ + λh θ l = ( l + θ λ )sinh θ + ( l (9) + θ λ )coshθ h h Then, we may give following result:

13 On the Invariants of Mannheim Offsets of 5 Result 5: If θ = 0, ie, the generators of the Mannheim offset surfaces intersect, then we have λ = λ sinh cosh, θ + λh θ l = l sinhθ + l (0) cosh θ h In this case, -CTRS are intersect along their striction lines It means that their striction lines are the same Let now consider that what the condition for the developable Mannheim offset of a CTTRS is Let -CTTRS be the Mannheim offset surfaces let α( s) β ( s) be striction lines of of the type is of the type -CTRS, respectively, where is Then, we can write β ( s) = α( s) + θ a( s) () where s is the arc-length of α( s) Assume that -CTTRS is developable Then from () (8) we have coshσ + sinh σ a, kh sinhσ δ = = = 0 () k h, k h k which gives that σ = 0 Thus, from (8) we get dα = ds Hence, along the striction line α( s), orthogonal frame {, h, a} coincides with the T, N, B differential forms k k turn into curvature κ α Frenet frame { } torsion τ α of the striction line α( s), respectively Then, by the aid of (8), () () we have dβ = θ τα h (4) ds () On the other h from (0) () we obtain d dθ dθ = + κα coshθ + + κα sinhθh + τα coshθ a ds ds ds (5)

14 6 Mehmet Önder et al By using (4) the fact that k From (4) (6) we obtain = κα, from (5) we have d =τα cosh θa ds (6) δ dβ, d coshθ + θ τα sinhθ = = (7) d, d τ coshθ α Thus, from () (7), it can be stated that if the Mannheim offsets are developable then the following relationship holds coshθ + θ τ sinhθ = 0 (8) In [4], euality (8) is also obtained with a different way If (8) holds, along the striction line β ( s), orthogonal frame {, h, a} coincides with the Frenet frame,, T N B of β ( s) Thus, the following theorem may be given: { } Theorem 5: Let the trajectory timelike ruled surfaces be of the type, respectively If are developable Mannheim offsets, then their striction lines are Mannheim partner curves in the Minkowski -space In this case from (8) we have the following corollary: Corollary 5: If form a developable Mannheim offset, then the relationship between the torsion of α ( s) offset distance offset angle is given by τ θ = cothθ α If -CTTRS is developable then from the euations (0), () (4) the pitch l of -CTTRS is l =, dβ = θ τ h, (sinh θ ) + (cosh θ ) h ds α = (sinhθ + θ τ cosh θ ) ds Then we can give the following corollary α α

15 On the Invariants of Mannheim Offsets of 7 Corollary 5: If -CTTRS is developable then the relation between the pitch l of -CTTRS the torsion of sitriction line α ( s) of -CTTRS is given by l = (sinhθ + θ τ cosh θ ) ds (9) Let now consider the area of projections of Mannheim offsets From (8), the dual area vectors of the spherical images of Mannheim offsets are α wɶ = dɶ + ɶ, wɶ = dɶ ɶ, (40) respectively Then, from (9), dual area of projection of spherical image of - CTTRS in the direction is f = wɶ, ɶ = dɶ + ɶ, ɶ ɶ, ɶ = dɶ, ɶ (sinh θ ) ɶ + (cosh θ ) hɶ, ɶ = dɶ, ɶ sinhθ, fɶ ɶ = sinhθ (4) Separating (4) into real dual parts we have the following theorem Theorem 54: Let the trajectory timelike ruled surfaces be of the type, respectively let they form a Mannheim offset The relationships between the area of projections of spherical images of the timelike Mannheim offsets their integral invariants are given as follows f = λ λ sinh θ, f = ( l + l sinhθ + λ θ cosh θ ) (4),, Similarly, the dual area of projection of spherical image of -CTTRS in the direction h ɶ is f = wɶ, hɶ = dɶ + ɶ, hɶ ɶ, hɶ ɶ, hɶ = dɶ, hɶ (sinh θ ) ɶ + (cosh θ ) hɶ, hɶ = dɶ, hɶ coshθ h f = wɶ, hɶ = cosh θ (4)

16 8 Mehmet Önder et al Separating (4) into real dual parts we have the following: Theorem 55: Let the trajectory timelike ruled surfaces be of the type, respectively let they form a Mannheim offset Then the relationships between the area of projections of spherical images of the surfaces their integral invariants are given as follows: h f = λ λ cosh θ, f = l l coshθ λ θ sinh θ, (44), h h, h h Similarly, dual area of projection of spherical image of -CTRS in the direction aɶ is f = wɶ, aɶ = dɶ + ɶ, aɶ Since aɶ = h ɶ, we have ɶ, aɶ = dɶ, aɶ ɶ, hɶ = dɶ, a ɶ = a f, a = a = h 0 ɶ ɶ (45) Separating (45) into real dual parts we have the followings: Theorem 56: Let the trajectory timelike ruled surfaces be of the type, respectively let they form a Mannheim offset Then the relationships between the area of projections of spherical images of the surfaces their integral invariants are given as follows: a f = λ = λ, f = l = l, a h a, a h a 6 -Type Mannheim Offsets of Timelike Ruled Surfaces with Timelike Rulings Let the timelike trajectory ruled surface of the type be a Mannheim offset of the timelike trajectory ruled surface with timelike rulings Then we can give the followings The proofs of this section can be given by the similar ways in Section 5

17 On the Invariants of Mannheim Offsets of 9 Theorem 6: Let the timelike trajectory ruled surfaces with timelike rulings form a Mannheim offset Then the offset angle offset distance are given by respectively kds, θ k ds, θ = = Theorem 6: The closed timelike trajectory ruled surfaces with timelike rulings form a Mannheim offsets with a constant dual offset angle if only if following relationship holds, = sinhθ + coshθ h Separating this euation into real dual parts, we obtain, λ = λ sinhθ + λ cosh θ, l = ( l + θ λ )sinh θ + ( l + θ λ )coshθ h h h Then we have the following results Result 6: If θ = 0, then the relationships between real integral invariants of h -CTTRS are given by λ = λ, h l = l h + θ λ of spherical surface areas bounded by spherical images of Mannheim offsets are the same, ie, a = a h Furthermore, the measure h a = a + θ ( π + a ) h -CTTRS Result 6: If θ = 0, ie, the generators of the Mannheim offset surfaces intersect, then we have λ = λ coshθ + λ sinh θ, l = l coshθ + l sinhθ h h In this case, -CTRS are intersect along their striction lines It means that their striction lines are the same Theorem 6: Let the trajectory timelike ruled surface of the type be the Mannheim offset of developable timelike ruled surface with timelike ruling Then is developable if only if coshθ + θ τ sinhθ = 0 holds Theorem 64: Let the developable trajectory timelike ruled surfaces be of the type If form a Mannheim offset then their striction lines are Mannheim partner curves in Minkowski -space IR α

18 0 Mehmet Önder et al Corollary 6: If -CTTRS is developable then the relation between the pitch l of -CTTRS the torsion of sitriction line α ( s) of -CTTRS is given by l = (coshθ + θ τ sinh θ ) ds Theorem 65: Let the trajectory timelike ruled surfaces of the type form a Mannheim offset The relationships between area of projections of spherical images of the surfaces their integral invariants are given as follows, f ɶ, ɶ = wɶ, ɶ = cosh + θ, or f = λ λ cosh θ, f = ( l l coshθ + λ θ sinh θ ),, Theorem 66: Let the trajectory timelike ruled surfaces of the type form a Mannheim offset Then the relationships between area of projections of spherical images of ruled surfaces h their integral invariants are given as follows f = wɶ, hɶ = + sinh θ, or ɶ, hɶ α h f = λ λ sinh θ, f = l l sinhθ + λ θ coshθ, h h, h h Theorem 67: Let the trajectory timelike ruled surfaces of the type form a Mannheim offset Then the relationships between area of projections of spherical images of ruled surfaces a their integral invariants are given as follows fɶ ɶ = wɶ, aɶ = = or, a a h f = λ = λ, f = l = l, a h a, a h a Acknowledgements The authors would like to thank the reviewers editor for their valuable comments suggestions to improve the uality of the paper References [] NH Abdel-All, RA Abdel-Baky FM Hamdoon, Ruled surfaces with timelike rulings, App Math Comp, 47(004), 4-5

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