On some characterizations of ruled surface of a closed timelike curve in dual Lorentzian space
aa r X i v : . [ m a t h . DG ] S e p ON SOME CHARACTERIZATIONS OFRULED SURFACE OF A CLOSED TIMELIKECURVE IN DUAL LORENTZIAN SPACE ¨Ozcan BEKTAS¸ ∗ S¨uleyman S¸ENYURT ∗ Abstract
In this paper, we investigate the relations between the pitch, the an-gle of pitch and drall of parallel ruled surface of a closed curve in dualLorentzian space.
Keywords:
Timelike dual curve; ruled surface; Lorentzian space; dual num-bers.
Dual numbers were introduced by W.K. Clifford (1849-79) as a tool for his geo-metrical investigations. After him E.Study used dual numbers and dual vectorsin his research on the geometry of lines and kinematics [4]. The pitches and theangles of the pitches of the closed ruled surfaces corresponds to the one param-eter dual unit spherical curves in space of lines IR were calculated respectivelyby Hacısalihoglu [7] and Gursoy [5].Definition of the parallel ruled surface werepresented by Blaschke (translated by Erim [3]). The integral invariants of theparallel ruled surfaces in the 3-dimensional Euclidean space E correspondingto the unit dual spherical parallel curves were calculated by Senyurt [11]. Theset ID = { ˆ λ = λ + ελ ∗ (cid:12)(cid:12) λ, λ ∗ ∈ IR, ε = 0 } is called dual numbers set [2].On this set product and addition operations are respectively( λ + ελ ∗ ) + ( β + εβ ∗ ) = ( λ + β ) + ε ( λ ∗ + β ∗ )and ( λ + ελ ∗ ) ( β + εβ ∗ ) = λβ + ε ( λβ ∗ + λ ∗ β ) .ID = n −→ A = −→ a + ε −→ a ∗ (cid:12)(cid:12) −→ a , −→ a ∗ ∈ IR o the elements of ID are called dualvectors . On this set addition and scalar product operations are respectively ∗ Ordu University, Faculty of Art and Science, Department of Mathematics, 52750,Per¸sembe, Ordu, Turkey, [email protected], [email protected]. : ID × ID → ID (cid:16) −→ A , −→ B (cid:17) → −→ A ⊕ −→ B = −→ a + −→ b + ε (cid:16) −→ a ∗ + −→ b ∗ (cid:17) ⊙ : ID × ID → ID (cid:16) λ, −→ A (cid:17) → λ ⊙ −→ A = ( λ + ελ ∗ ) ⊙ ( −→ a + ε −→ a ∗ ) = λ −→ a + ε ( λ −→ a ∗ + λ ∗ −→ a )The set (cid:0) ID , ⊕ (cid:1) is a module over the ring ( ID, + , · ). ( ID − M odul ).The Lorentzian inner product of dual vectors −→ A , −→ B ∈ ID is defined by D −→ A , −→ B E = D −→ a , −→ b E + ε (cid:16)D −→ a , −→ b ∗ E + D −→ a ∗ , −→ b E(cid:17) with the Lorentzian inner product −→ a = ( a , a , a ) and −→ b = ( b , b , b ) ∈ IR D −→ a , −→ b E = − a b + a b + a b . Therefore, ID with the Lorentzian inner product D −→ A , −→ B E is called 3-dimensionaldual Lorentzian space and denoted by of ID = n −→ A = −→ a + ε −→ a ∗ (cid:12)(cid:12) −→ a , −→ a ∗ ∈ IR o [14].A dual vector −→ A = −→ a + ε −→ a ∗ ∈ ID is called A dual space-like vectorif D −→ A , −→ A E > −→ A = 0, A dual time-like vector if D −→ A , −→ A E < D −→ A , −→ A E = 0 for −→ A = 0 . For −→ A = 0, the norm (cid:13)(cid:13)(cid:13) −→ A (cid:13)(cid:13)(cid:13) of −→ A = −→ a + ε −→ a ∗ ∈ ID is defined by (cid:13)(cid:13)(cid:13) −→ A (cid:13)(cid:13)(cid:13) = r(cid:12)(cid:12)(cid:12)D −→ A , −→ A E(cid:12)(cid:12)(cid:12) = k−→ a k + ε h−→ a , −→ a ∗ ik−→ a k , k−→ a k 6 = 0 . The dual Lorentzian cross-product of −→ A , −→ B ∈ ID is defined as −→ A ∧ −→ B = −→ a ∧ −→ b + ε (cid:16) −→ a ∧ −→ b ∗ + −→ a ∗ ∧ −→ b (cid:17) with the Lorentzian cross-product −→ a , −→ b ∈ IR −→ a ∧ −→ b = ( a b − a b , a b − a b , a b − a b ) [14] . Theorem(E. Study):
The oriented lines in are in one to one correspondencewith the points of the dual unit sphere where ID-Modul , see [9].Dual number Φ = ϕ + εϕ ∗ is called dual angle between −→ A ve −→ B unit dualvectors. In this placesinh ( ϕ + εϕ ∗ ) = sinh ϕ + εϕ ∗ cosh ϕ and cosh ( ϕ + εϕ ∗ ) = cosh ϕ + εϕ ∗ sinh ϕ. ON SOME CHARACTERIZATIONS OF RULEDSURFACE OF A CLOSED TIMELIKE CURVEIN DUAL LORENTZIAN SPACE (cid:0)
I D (cid:1) −→ U = −→ U ( t ) , (cid:13)(cid:13)(cid:13) −→ U ( t ) (cid:13)(cid:13)(cid:13) = 1 is a differentiable timelike curve in the one parameterdual unit spherical motion K/K ′ . The closed ruled surface ( −→ U ) corresponds tothis curve in IR .Let the dual orthonormal system of curve −→ U = −→ U ( t ) as −→ U = −→ U ( t ) , −→ U = −→ U ′ ( t ) (cid:13)(cid:13)(cid:13) −→ U ( t ) (cid:13)(cid:13)(cid:13) , −→ U = −→ U ∧ −→ U Let −→ U ( t )be a closed timelike curve with curvature κ = k + εk ∗ and torsion τ = k + εk ∗ . Let Frenet frames of −→ U ( t ) be n −→ U , −→ U , −→ U o . In this trihedron, −→ U is timelike vector , −→ U and −→ U are spacelike vectors. For this vectors, we canwrite −→ U ∧ −→ U = −−→ U , −→ U ∧ −→ U = −→ U , −→ U ∧ −→ U = −−→ U (2.1)where ∧ is the Lorentzian cross product, in space ID . In this situation, theFrenet formulas are given by −→ U ′ = κ −→ U , −→ U ′ = κ −→ U − τ −→ U , −→ U ′ = τ −→ U , [15] . (2.2)If the last equation is separated into the dual and real part, we can obtain −→ u ′ = k −→ u −→ u ′ = k −→ u − k −→ u −→ u ′ = k −→ u −→ u ∗ ′ = k ∗ −→ u + k −→ u ∗ −→ u ∗ ′ = k ∗ −→ u − k ∗ −→ u + k −→ u ∗ − k −→ u ∗ −→ u ∗ ′ = k ∗ −→ u + k −→ u ∗ (2.3)The Frenet instantaneous rotation vector for the timelike curve is given by −→ Ψ = τ −→ U − κ −→ U , [13] . (2.4)In this situation for the Steiner vector of the motion, we may write −→ D = I −→ Ψ (2.5)or −→ D = −→ U I τ dt − −→ U I κdt (2.6)3he equation (2.6) can be written type of the dual and real part as follow ( −→ d = −→ u H k dt − −→ u H k dt, −→ d ∗ = −→ u ∗ H k dt + −→ u H k ∗ dt − −→ u ∗ H k dt − −→ u H k ∗ dt (2.7)Now, let is calculate the integral invariants of the closed ruled surfaces respec-tively. The pitch of the closed surface is obtained as L u = D −→ d , −→ u ∗ E + D −→ d ∗ , −→ u E ,L u = − I k ∗ dt. (2.8)For the dual angle of the pitch of the closed surface , we may writeΛ U = − D −→ D , −→ U E , Because of the equation (2.6) we can obtainΛ U = I τ dt. (2.9)If the equation (2.9) is separated into the dual and real part, we can obtain λ u = I k dt , L u = − I k ∗ dt (2.10)For the drall of the closed surface , we may write P U = h d −→ u , d −→ u ∗ ih d −→ u , d −→ u i Setting by the values of the statements d −→ u and d −→ u ∗ as the equations (2.3) intothe last equations, we get P U = k ∗ k (2.11)The pitch of the closed surface is obtained as L u = D −→ d , −→ u ∗ E + D −→ d ∗ , −→ u E ,L u = 0 . (2.12)For the dual angle of the pitch of the closed surface , we may writeΛ U = − D −→ D , −→ U E , Λ U = 0 . (2.13)4or the drall of the closed surface , we may write P U = h d −→ u , d −→ u ∗ ih d −→ u , d −→ u i Setting by the values of the statements d −→ u and d −→ u ∗ as the equations (2.3) intothe last equations, we get P U = k k ∗ − k k ∗ k − k (2.14)The pitch of the closed surface is obtained as L u = D −→ d , −→ u ∗ E + D −→ d ∗ , −→ u E ,L u = − I k ∗ dt (2.15)For the dual angle of the pitch of the closed surface , we may writeΛ U = − D −→ D , −→ U E Because of the equation (2.6) we can obtainΛ U = I κdt (2.16)If the equation (2.16) is separated into the dual and real part, we can obtain λ u = I k dt , L u = − I k ∗ dt (2.17)For the drall of the closed surface , we may write P U = h d −→ u , d −→ u ∗ ih d −→ u , d −→ u i Setting by the values of the statements d −→ u and d −→ u ∗ as the equations (2.3) intothe last equations, we get P U = k ∗ k . (2.18)Let Ω ( t ) = ω ( t ) + εω ∗ ( t )be Lorentzian timelike angle of between the instanta-neous dual Pfaffion vector −→ Ψ and the vector −→ U . a) If the instantaneous dual Pfaffion vector −→ Ψ is spacelike ( | κ | > | τ | ) κ = (cid:13)(cid:13)(cid:13) −→ Ψ (cid:13)(cid:13)(cid:13) cosh Ω , τ = (cid:13)(cid:13)(cid:13) −→ Ψ (cid:13)(cid:13)(cid:13) sinh ΩOn the way −→ C = −→ c + ε −→ c ∗ , unit vector about the vector −→ Ψ direction is5 → C = sinh Ω −→ U − cosh Ω −→ U (2.19)If the equation (2.19) is separated into the dual and real part, we can obtain (cid:26) −→ c = sinh ω −→ u − cosh ω −→ u −→ c ∗ = sinh ω −→ u ∗ − cosh ω −→ u ∗ + ω ∗ cosh ω −→ u − ω ∗ sinh ω −→ u (2.20)The pitch of the closed surface is obtained as L C = D −→ d , −→ c ∗ E + D −→ d ∗ , −→ c E L C = cosh ω I k ∗ dt − sinh ω I k ∗ dt −− ω ∗ (cid:18) cosh ω I k dt − sinh ω I k dt (cid:19) (2.21)If we use the equations (2.10) and (2.17) into the equation (2.21) we get L C = sinh ωL u − cosh ωL u − ω ∗ (cosh ωλ u − sinh ωλ u ) (2.22)For the dual angle of the pitch of the closed ruled surface , we may writeΛ C = − D −→ D , −→ C E Because of the equations (2.6) and (2.19) we can obtainΛ C = sinh Ω I τ dt − cosh Ω I κdt (2.23)If we use the equations (2.9) and (2.16) into the last equations, we getΛ C = sinh Ω Λ U − cosh Ω Λ U (2.24)For the drall of the closed surface , we may write P C = h d −→ c , d −→ c ∗ ih d −→ c , d −→ c i P C = − ω ′ ω ∗ ′ + ( k sinh ω − k cosh ω ) [( k ∗ − k ω ∗ ) sinh ω + ( k ω ∗ − k ∗ ) cosh ω ]( k sinh ω − k cosh ω ) − ω ′ (2.25) b) If the instantaneous dual Pfaffion vector −→ Ψ is timelike ( | κ | < | τ | ) κ = (cid:13)(cid:13)(cid:13) −→ Ψ (cid:13)(cid:13)(cid:13) sinh Ω , τ = (cid:13)(cid:13)(cid:13) −→ Ψ (cid:13)(cid:13)(cid:13) cosh Ω6n the way −→ C = −→ c + ε −→ c ∗ , unit vector about the vector −→ Ψ direction is −→ C = cosh Ω −→ U − sinh Ω −→ U (2.26)If the equation (2.26) is separated into the dual and real part, we can obtain (cid:26) −→ c = cosh ω −→ u − sinh ω −→ u −→ c ∗ = cosh ω −→ u ∗ − sinh ω −→ u ∗ + ω ∗ sinh ω −→ u − ω ∗ cosh ω −→ u (2.27)The pitch of the closed surface is obtained as L C = D −→ d , −→ c ∗ E + D −→ d ∗ , −→ c E L C = sinh ω I k ∗ dt − cosh ω I k ∗ dt −− ω ∗ (cid:18) sinh ω I k dt − cosh ω I k dt (cid:19) (2.28)If we use the equations (2.10) and (2.17) into the equation (2.21) we get L C = cosh ωL u − sinh ωL u − ω ∗ (sinh ωλ u − cosh ωλ u ) (2.29)For the dual angle of the pitch of the closed ruled surface , we may writeΛ C = − D −→ D, −→ C E Because of the equations (2.6) and (2.19) we can obtainΛ C = cosh Ω I τ dt − sinh Ω I κdt (2.30)If we use the equations (2.9) and (2.16) into the last equations, we getΛ C = cosh Ω Λ U − sinh Ω Λ U (2.31)For the drall of the closed surface , we may write P C = h d −→ c , d −→ c ∗ ih d −→ c , d −→ c i P C = ω ′ ω ∗ ′ + ( k cosh ω − k sinh ω ) [( k ∗ − k ω ∗ ) cosh ω + ( k ω ∗ − k ∗ ) sinh ω ]( k sinh ω − k cosh ω ) − ω ′ (2.32) Definition:
The closed ruled surface ( −→ U ) corresponds to the dual timelike curve −→ U ( t ) which makes the fixed dual angle Φ = ϕ + εϕ ∗ with −→ U ( t ) and defines by7 → V = cosh Φ −→ U + sinh Φ −→ U (2.33)This surface ( −→ V ) corresponds to dual timelike vector −→ V is called the parallelruled surface of surface ( −→ U ) in dual lorentzian space ID . Let be −→ V = −→ V . Differentiating of the vector −→ V with respect the parameterand using the equation (2.3) we get −→ V ′ = ( κ cosh Φ + τ sinh Φ) −→ U (2.34)If the norm of the vector denotes by P , we get −→ P = κ cosh Φ + τ sinh Φ (2.35)Then if is known thatSubstituting by the values of the equations (2.34) and (2.35) into , we get −→ V = −→ U (2.36)Then, fort he vector , we get −→ V = − sinh Φ −→ U − cosh Φ −→ U (2.37)If the equation (2.33) , (2.36) and (2.37) are written matrix form, we have −→ V −→ V −→ V = cosh Φ 0 sinh Φ0 1 0 − sinh Φ 0 − cosh Φ · −→ U −→ U −→ U (2.38)or −→ U −→ U −→ U = cosh Φ 0 sinh Φ0 1 0 − sinh Φ 0 − cosh Φ · −→ V −→ V −→ V (2.39)If the equation (2.39) is separated into real and dual parts, we get −→ u = cosh ϕ −→ v + sinh ϕ −→ v −→ u = −→ v −→ u = − sinh ϕ −→ v − cosh ϕ −→ v −→ u ∗ = cosh ϕ −→ v ∗ + sinh ϕ −→ v ∗ + ϕ ∗ (sinh ϕ −→ v + cosh ϕ −→ v ) −→ u ∗ = −→ v ∗ −→ u ∗ = − sinh ϕ −→ v ∗ − cosh ϕ −→ v ∗ − ϕ ∗ (cosh ϕ −→ v + sinh ϕ −→ v ) (2.40)Let be curvature P = p + εp ∗ and torsion Q = q + εq ∗ of curve −→ V ( t ) . Between thevectors −→ V , −→ V , −→ V and the derivate vectors −→ V ′ , −→ V ′ , −→ V ′ there is following relation8 −→ V ′ = P −→ V , −→ V ′ = P −→ V − Q −→ V , −→ V ′ = Q −→ V P = q < −→ V ′ , −→ V ′ >, Q = det ( −→ V , −→ V ′ , −→ V ′′ ) < −→ V ′ , −→ V ′ > . [15] . (2.41)If the equation (2.41) is separated into the real and dual parts, we can write −→ v ′ = p −→ v , −→ v ′ = p −→ v − q −→ v , −→ v ′ = q −→ v −→ v ′ ∗ = p −→ v ∗ + p ∗ −→ v , −→ v ′ ∗ = p −→ v ∗ + p ∗ −→ v − q −→ v ∗ − q ∗ −→ v , −→ v ′ ∗ = q −→ v ∗ + q ∗ −→ v (2.42)Now, we can calculate the value of Q relative to κ and τ . Derivative the equation(2.34) with respect to the parameter an of making the recesiory operations wemay write −→ V ′′ =( κ cosh Φ + κτ sinh Φ) −→ U ++ ( κ cosh Φ + τ sinh Φ) ′ −→ U + ( − κτ cosh Φ − τ sinh Φ) −→ U (2.43)Substituting by the equations (2.33) , (2.34) and (2.43) into the equation (2.41),we get Q = − κ sinh Φ − τ cosh Φ (2.44)The equations (2.35) and (2.44) are separated into the dual and real parts, wehave p = k cosh ϕ + k sinh ϕp ∗ = k ∗ cosh ϕ + k ∗ sinh ϕ + ϕ ∗ ( k sinh ϕ + k cosh ϕ ) q = − k sinh ϕ − k cosh ϕq ∗ = − k ∗ sinh ϕ − k ∗ cosh ϕ − ϕ ∗ ( k cosh ϕ + k sinh ϕ ) (2.45)In the dual unit spharical motion K/K ′ , the dual orthonormal system n −→ V , −→ V , −→ V o each time t is make a dual rotation motion around the instantaneous dual Pfaf-fion vector. This vector is determined the following equation −→ Ψ = Q −→ V − P −→ V , [13] (2.46)For the Steiner vector of the motion, we can write −→ D = I −→ Ψ (2.47)or −→ D = −→ V I Qdt − −→ V I P dt (2.48)Setting by the values of the vectors −→ U and −→ U as the equations (2.40) into theequations (2.4), we get −→ Ψ = − Q −→ V + P −→ V
9n this case, we consider the last equation and equation (2.46), we can write −→ Ψ = −−→
Ψ (2.49)Because of the equations −→ D = H −→ Ψ and −→ D = H −→ Ψ we can obtain −→ D = −−→ D .Then, for the dual Steiner vector of the motion, we may write −→ D = −−→ V I Qdt + −→ V I P dt (2.50)It is separated into the dual and real part as ( −→ d = −−→ v H qdt + −→ v H pdt, −→ d ∗ = −−→ v H q ∗ dt − −→ v ∗ H qdt + −→ v H p ∗ dt + −→ v ∗ H pdt (2.51)Now, let is calculate the integral invariants of the closed ruled surfaces respec-tively. The pitch of the closed surface is obtained as L V = D −→ d , −→ v ∗ E + D −→ d ∗ , −→ v E ,L V = I q ∗ dt. (2.52)Substituting by the value into the equation (2.52) L V = − sinh ϕ I k ∗ dt − cosh ϕ I k ∗ dt −− ϕ ∗ (cid:18) cosh ϕ I k dt + sinh ϕ I k dt (cid:19) . (2.53)or L V = cosh ϕL u + sinh ϕL u − ϕ ∗ (cid:0) sinh ϕλ u + cosh ϕλ u (cid:1) . (2.54)For the dual angle of the pitch of the closed ruled surface , we may writeΛ V = − D −→ D , −→ V E Because of the equation (2.50) we can obtainΛ V = − I Qdt. (2.55)Substituting by the equation (2.44) into the last equation, we getΛ V = sinh Φ I κdt + cosh Φ I τ dt or ∧ V = cosh Φ ∧ U + sinh Φ ∧ U (2.56)10f the equation (2.56) is separated into the dual and real part, we can obtain (cid:26) λ v = cosh ϕλ u + sinh ϕλ u L v = cosh ϕL u + sinh ϕL u − ϕ ∗ (sinh ϕλ u + cosh ϕλ u ) (2.57)For the drall of the closed surface , we may write P V = h d −→ v , d −→ v ∗ ih d −→ v , d −→ v i Setting by the values of the statements d −→ v and d −→ v ∗ as the equations (2.42)into the last equations, we get P V = p ∗ p (2.58)Setting by the values of p and p ∗ as the equations (2.45) into the last equations,we get P V = k ∗ cosh ϕ + k ∗ sinh ϕk cosh ϕ + k sinh ϕ + ϕ ∗ k sinh ϕ + k cosh ϕk cosh ϕ + k sinh ϕ (2.59) Theorem 2.1:
Let ( V ) be the parallel surface of the surface ( U ). The pitch,drall and the dual of the pitch of the ruled surface ( V ) are1 − ) L V = I q ∗ dt − )Λ V = − I Qdt − ) P V = p ∗ p . Corollary 2.1:
Let ( V ) be the parallel surface of the surface ( U ). The pitchand the dual of the pitch of the ruled surface ( V ) related to the invariants ofthe surface ( U ) are written as follow1 − ) L V = cosh ϕL u + sinh ϕL u − ϕ ∗ (cid:0) sinh ϕλ u + cosh ϕλ u (cid:1) − ) ∧ V = cosh Φ ∧ U + sinh Φ ∧ U The pitch of the closed surface ( V ) is obtained as L V = D −→ d , −→ v ∗ E + D −→ d ∗ , −→ v E L V = 0 (2.60)For the dual angle of the pitch of the closed ruled surface ( V ), we may writeΛ V = − D −→ D , −→ V E Because of the equation (2.32) we can obtainΛ V = 0 (2.61)For the drall of the closed surface ( V ), we may write P V = h d −→ v , d −→ v ∗ ih d −→ v , d −→ v i d −→ v and d −→ v ∗ as the equations (2.24)into the last equations, we get P V = qq ∗ − pp ∗ q − p (2.62)Setting by the values of p, p ∗ , q and q ∗ as the equations (2.45) into the last equa-tions, we get P V = k k ∗ − k k ∗ k − k (2.63) Theorem 2.2:
Let ( V ) be the parallel surface of the surface ( U ). Thepitch, drall and the dual of the pitch of the ruled surface ( V ) are1 − ) L V = 0 2 − )Λ V = 0 3 − ) P V = qq ∗ − pp ∗ q − p The pitch of the closed surface ( V ) is obtained as L V = D −→ d , −→ v ∗ E + D −→ d ∗ , −→ v E ,L V = I p ∗ dt (2.64)Substituting by the value into the equation (2.64) L V = cosh ϕ I k ∗ dt +sinh ϕ I k ∗ dt + ϕ ∗ (sinh ϕ I k dt +cosh ϕ I k dt ) (2.65)or L V = − sinh ϕL u − cosh ϕL u + ϕ ∗ (cosh ϕλ u + sinh ϕλ u ) (2.66)For the dual angle of the pitch of the closed ruled surface , we may writeΛ V = − D −→ D , −→ V E Because of the equation (2.50) we can obtainΛ V = − I P dt. (2.67)Substituting by the equation (2.35) into the last equation, we getΛ V = − cosh Φ I κdt − sinh Φ I τ dt or ∧ V = − sinh Φ ∧ U − cosh Φ ∧ U (2.68)12f the equation (2.68) is separated into the dual and real part, we can obtain (cid:26) λ v = − sinh ϕλ u − cosh ϕλ u L v = − sinh ϕL u − cosh ϕL u + ϕ ∗ (cosh ϕλ u + sinh ϕλ u ) (2.69)For the drall of the closed surface , we may write P V = h d −→ v , d −→ v ∗ ih d −→ v , d −→ v i Setting by the values of the statements d −→ v and d −→ v ∗ as the equations (2.42)into the last equations, we get P V = q ∗ q (2.70)Setting by the values of q and q ∗ as the equations (2.45) into the last equations,we get P V = − k ∗ sinh ϕ − k ∗ cosh ϕ − k sinh ϕ − k cosh ϕ − ϕ ∗ (cid:18) k cosh ϕ + k sinh ϕ − k sinh ϕ − k cosh ϕ (cid:19) (2.71) Theorem 2.3:
Let ( V ) be the parallel surface of the surface ( U ). The pitch, drall and the dual of the pitch of the ruled surface ( V ) are1 − ) L V = I p ∗ dt − )Λ V = − I P dt − ) P V = q ∗ q Corollary 2.2:
Let ( V ) be the parallel surface of the surface ( U ). The pitchand the dual of the pitch of the ruled surface ( V ) related to the invariants ofthe surface ( U ) are written as follow1 − ) L V = − sinh ϕL u − cosh ϕL u + ϕ ∗ (cosh ϕλ u + sinh ϕλ u )2 − ) ∧ V = − sinh Φ ∧ U − cosh Φ ∧ U Let Θ( t ) = θ ( t ) + εθ ∗ ( t ) be Lorentzian timelike angle of between the instanta-neous dual Pfaffion vector −→ Ψ and the vector −→ V . a) If the instantaneous dual Pfaffion vector −→ Ψ is spacelike ( | P | > | Q | ) P = (cid:13)(cid:13)(cid:13)(cid:13) −→ Ψ (cid:13)(cid:13)(cid:13)(cid:13) cosh Θ , Q = (cid:13)(cid:13)(cid:13)(cid:13) −→ Ψ (cid:13)(cid:13)(cid:13)(cid:13) sinh ΘOn the way −→ C = −→ c + ε −→ c ∗ , unit vector about the vector −→ Ψ direction is −→ C = sinh Θ −→ V − cosh Θ −→ V (2.72)Setting by the values of the vectors −→ V and −→ V as the equations (2.38) into theequation (2.72), we get −→ C = (sinh Θ cosh Φ + cosh Θ sinh Φ) −→ U + (cosh Θ cosh Φ + sinh Θ sinh Φ) −→ U → C = sinh (Θ + Φ) −→ U + cosh (Θ + Φ) −→ U (2.73)If the equation (2.72) is separated into the dual and real part, we can obtain ( −→ c = sinh θ −→ v − cosh θ −→ v −→ c ∗ = sinh θ −→ v ∗ − cosh θ −→ v ∗ + θ ∗ cosh θ −→ v − θ ∗ sinh θ −→ v (2.74)The pitch of the closed surface is obtained as L C = D −→ d , −→ c ∗ E + D −→ d ∗ , −→ c E Setting by the values of the statements −→ d and −→ d ∗ as the equations (2.51) intothe last equations and if we do the necessary operations , we get L C = − cosh θ I p ∗ dt +sinh θ I q ∗ dt + θ ∗ (cid:18) cosh θ I qdt − sinh θ I pdt (cid:19) (2.75)If we use the equations (2.52) , (2.64) into the equation (2.75) we get L C = sinh θL V − cosh θL V + θ ∗ ( − cosh θλ V + sinh θλ V ) (2.76)If we use the equations (2.57) and (2.69) into the equation (2.76) and necessaryoperations have been done, we get L C = sinh ( θ + ϕ ) L U + cosh ( θ + ϕ ) L U −− ( ϕ ∗ + θ ∗ ) (cosh ( θ + ϕ ) λ U + sinh ( θ + ϕ ) λ U ) (2.77)For the dual angle of the pitch of the closed ruled surface , we may writeΛ C = − (cid:28) −→ D , −→ C (cid:29) Because of the equations (2.21) and (2.71) we can obtainΛ C = − sinh Θ I Qdt + cosh Θ I P dt (2.78)If we use the equations (2.55) and (2.67) into the last equations, we getΛ C = sinh Θ Λ V − cosh Θ Λ V (2.79)If we use the equations (2.56) and (2.68) into the equations (2.79), we getΛ C = sinh (Θ + Φ) Λ U + cosh (Θ + Φ) Λ U (2.80)14or the drall of the closed surface , we may write P C = D d −→ c , d −→ c ∗ ED d −→ c , d −→ c E P C = − θ ′ θ ∗ ′ + ( p sinh θ − q cosh θ ) [( p ∗ − qθ ∗ ) sinh θ + ( pθ ∗ − q ∗ ) cosh θ ]( p sinh θ − q cosh θ ) − θ ′ (2.81) b) If the instantaneous dual Pfaffion vector −→ Ψ is timelike ( | P | < | Q | ) P = (cid:13)(cid:13)(cid:13)(cid:13) −→ Ψ (cid:13)(cid:13)(cid:13)(cid:13) sinh Θ , Q = (cid:13)(cid:13)(cid:13)(cid:13) −→ Ψ (cid:13)(cid:13)(cid:13)(cid:13) cosh ΘOn the way −→ C = −→ c + ε −→ c ∗ , unit vector about the vector −→ Ψ direction is −→ C = cosh Θ −→ V − sinh Θ −→ V (2.82)Setting by the values of the vectors −→ V and −→ V as the equations (2.38) into theequation (2.82), we get −→ C = (cosh Θ cosh Φ + sinh Θ sinh Φ) −→ U + (sinh Θ cosh Φ + cosh Θ sinh Φ) −→ U −→ C = cosh (Θ + Φ) −→ U + sinh (Θ + Φ) −→ U (2.83)If the equation (2.82) is separated into the dual and real part, we can obtain ( −→ c = cosh θ −→ v − sinh θ −→ v −→ c ∗ = cosh θ −→ v ∗ − sinh θ −→ v ∗ + θ ∗ sinh θ −→ v − θ ∗ cosh θ −→ v (2.84)The pitch of the closed surface is obtained as L C = D −→ d , −→ c ∗ E + D −→ d ∗ , −→ c E Setting by the values of the statements −→ d and −→ d ∗ as the equations (2.51) intothe last equations and if we do the necessary operations , we get L C = − sinh θ I p ∗ dt +cosh θ I q ∗ dt + θ ∗ (cid:18) sinh θ I qdt − cosh θ I pdt (cid:19) (2.85)or L C = cosh θL V − sinh θL V + θ ∗ ( − sinh θλ V + cosh θλ V ) (2.86)15f we use the equations (2.57) and (2.69) into the equation (2.86) and necessaryoperations have been done, we get L C = cosh( θ + ϕ ) L U + sinh( θ + + ϕ ) L U −− ( ϕ ′ ∗ + θ ∗ )(sinh( θ + ϕ ) λ U + cosh( θ + ϕ ) λ U ) (2.87)For the dual angle of the pitch of the closed ruled surface , we may writeΛ C = − (cid:28) −→ D , −→ C (cid:29) Because of the equations (2.50) and (2.82) we can obtainΛ C = − cosh Θ I Qdt + sinh Θ I P dt (2.88)If we use the equations (2.55) and (2.67) into the last equations, we getΛ C = cosh Θ Λ V − sinh Θ Λ V (2.89)If we use the equations (2.57) and (2.68) into the equations (2.89), we getΛ C = cosh (Θ + Φ) Λ U + sinh (Θ + Φ) Λ U (2.90)For the drall of the closed surface , we may write P C = D d −→ c , d −→ c ∗ ED d −→ c , d −→ c E P C = θ ′ θ ∗ ′ + ( p cosh θ − q sinh θ ) [( pθ ∗ − q ∗ ) sinh θ + ( p ∗ − qθ ∗ ) cosh θ ]( p cosh θ − q sinh θ ) + θ ′ (2.91) References [1] Birman, G.S. and Nomizu, K.,
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