inextensible flows of timelike curves with sabban frame in s2

’iauliai Math. Semin.,7 (15), 2012, 512INEXTENSIBLE FLOWS OF TIMELIKECURVES WITH SABBAN FRAME IN S 2 1Vedat AS L, Talat KÖRPINAR, Selçuk BA“Department of Mathematics, Frat University, 23119 Elaz§, Turkey;e-mails: vasil@rat.edu.tr, talatkorpinar@gmail.com, selcukbas79@gmail.comAbstract. In this paper, we study timelike curves according to Sabbanframe in S 2 1. We research inextensible ows of timelike curves according toSabban frame in S 2 1.Key words and phrases: inextensible ows, Sabban frame.2010 Mathematics Subject Classication: 53A35.1. IntroductionPhysically, inextensible curve and surface ows give rise to motions in whichno strain energy is induced. The swinging motion of a cord of xed length,for example, or of a piece of paper carried by the wind, can be describedby inextensible curve and surface ows. Such motions arise quite naturallyin a wide range of physical applications. For example, both Chirikjian andBurdick [1] and Mochiyama et al. [6] study the shape control of hyperredundant,or snake-like, robots. Inextensible curve and surface ows alsoarise in the context of many problems in computer vision [5] and computeranimation [2], [4].This paper is organized as follows. Firstly, we study timelike curvesaccording to the Sabban frame in S 2 1 . Finally, we research inextensible owsof timelike curves according to the Sabban frame in S 2 1 .2. PreliminariesThe Minkowski 3-space E 3 1 provided with the standard at metric given by

6 V. Asil, T. Körpinar, S. Ba³⟨, ⟩ = −dx 2 1 + dx 2 2 + dx 2 3,where (x 1 , x 2 , x 3 ) is a rectangular coordinate system of E 3 1 . Recall that, thenorm of an arbitrary vector a ∈ E 3 1 is given by ∥a∥ = √ ⟨a, a⟩. γ is called aunit speed curve if velocity vector v of γ satises ∥a∥ = 1.Denote by {T, N, B} the moving Frenet-Serret frame along the timelikecurve γ in the space E 3 1 . For an arbitrary timelike curve γ with rst and secondcurvature, κ and τ in the space E 3 1 , the following Frenet-Serret formulaeare givenwhereT ′ = κN,N ′ = κT + τB,B ′ = −τN,⟨T, T⟩ = −1, ⟨N, N⟩ = ⟨B, B⟩ = 1,⟨T, N⟩ = ⟨T, B⟩ = ⟨N, B⟩ = 0.Here, the curvature functions are dened by κ = κ(s) = ∥T ′ (s)∥ and τ(s) =− ⟨N, B ′ ⟩.Torsion of the timelike curve γ is given by the aid of the mixed productτ = [γ′ , γ ′′ , γ ′′′ ]κ 2 .Now we give a new frame dierent from the Frenet frame. Let α : I → S 2 1be an unit speed spherical timelike curve. We denote σ as the arc-lengthparameter of α. Let us denote t (σ) = α ′ (σ), and we call t (σ) a unit tangentvector of α. We now set a vector s (σ) = α (σ) × t (σ) along α. This frameis called the Sabban frame of α on S 2 1 . Then we have the following sphericalFrenet-Serret formulae of α [3]:α ′ = t,t ′ = α + κ g s,s ′ = κ g t,where κ g is the geodesic curvature of the timelike curve α on the S 2 1 , andg (t, t) = −1, g (α, α) = 1, g (s, s) = 1,g (t, α) = g (t, s) = g (α, s) = 0.

Inextensible ows of timelike curves... 73. Inextensible ows of timelike curves according tothe Sabban frame in S 2 1Let α (u, t) is a one-parameter family of smooth timelike curves in S 2 1 .The arc length of α is given byσ(u) = u ∂α0 ∣ ∂u ∣ du,whereThe operator ∂∂σ∣⟨ ∂α∣∣∣ ∂α∣ ∂u ∣ = ∂u , ∂α ⟩∣ ∣∣∣1/2.∂uis given in terms of u by∂∂σ = 1 ∂ν ∂u ,where v =∂α∣ ∂u ∣ , and the arc length parameter is dσ = vdu.Any ow of α can be represented asLet the arc length variation be∂α∂t = fS 1 α + f S 2 t + f S 3 s. (3.1)σ(u, t) = u 0vdu.In the S 2 1 , the requirement that the curve not be subject to any elongationor compression can be expressed by the conditionfor all u ∈ [0, l] .∂∂t σ(u, t) = u ∂v0 du = 0 (3.2)∂tDefinition 3.1. The ow ∂α∂t in S2 1 is said to be inextensible if∂∂t∂α∣ ∂u ∣ = 0.

8 V. Asil, T. Körpinar, S. Ba³Lemma 3.2. Let ∂α∂t = fS 1 α + fS 2 t + fS 3 s be a smooth ow of the timelike curveα. The ow is inextensible if and only if∂v∂t + ∂fS 2∂u = −fS 1 v − f S 3 vκ g . (3.3)Proof. Suppose that ∂α be a smooth ow of the timelike curve α. Using∂tdenition of α, we have⟨ ∂αv 2 =∂u , ∂α ⟩. (3.4)∂uBy dierentiating of the formula (??), we get2v ∂v∂t = ∂ ⟨ ∂α∂t ∂u , ∂α ⟩.∂uOn the other hand, changing ∂∂u and ∂ , we have∂tv ∂v ⟨ ⟩∂α∂t = ∂u , ∂∂u (∂α ∂t ) .From (3.1), we obtainv ∂v ⟨ ∂α∂t = ∂u , ∂ (fS∂u 1 α + f S 2 t + f S 3 s )⟩ .By the Sabban formula, we have⟨ ( ) () ( ) ⟩∂v ∂fS∂t = t, 1∂u + fS 2 v α + f S 1 v + ∂fS 2∂fS∂u + fS 3 vκ g t + 3∂u + fS 2 vκ g s .Making necessary calculations, from above equation we obtain (3.3) whichproves the lemma.Theorem 3.3. Let ∂α∂t = fS 1 α + fS 2 t + fS 3 s be a smooth ow of the timelikecurve α. The ow is inextensible if and only if∂f S 2∂u = −fS 3 vκ g − f S 1 v.

Inextensible ows of timelike curves... 9Proof. Assume that ∂α∂t∂∂t σ(u, t) = u ∂v0∂t du = u 0is inextensible. From (3.2), we have(−f S 1 v − ∂fS 2∂u − fS 3 vκ g)du = 0. (3.5)Substituting (3.3) in (3.5) completes the proof of the theorem.Now we restrict ourselves to arc length parametrized curves. That is,v = 1 and the local coordinate u corresponds to the curve arc length σ. Werequire the following lemma.Lemma 3.4. The following relations hold( )∂t ∂fS= 1∂t ∂σ + fS 2 α +(∂α ∂fS= 1∂t ∂σ + fS 2where ψ =∂s∂t⟨ ⟩ ∂α∂t , s .=)t + ψs,( ∂fS3∂σ + fS 2 κ g)t − ψα,Proof. Using denition of α, we have∂t∂t = ( ∂fS1∂σ + fS 2( ∂fS3∂σ + fS 2 κ g)s,∂t∂t = ∂ ∂α∂t ∂σ = ∂∂σ (fS 1 α + f S 2 t + f S 3 s).Using the Sabban equations, we nd that)α +Substituting (3.3) in (3.6), we get∂t∂t = ( ∂fS1∂σ + fS 2(f S 1 + ∂fS 2∂σ + fS 3 κ g)t +)α +( ∂fS3∂σ + fS 2 κ g)s.Now dierentiate the Sabban frame by t:∂f S ⟨1∂σ + fS 2 + t, ∂α ⟩= 0,∂t∂f S ⟨3∂σ + fS 2 κ g + t, ∂s ⟩= 0,∂t⟨ψ + α, ∂s ⟩= 0.∂t( ∂fS3∂σ + fS 2 κ g)s. (3.6)

10 V. Asil, T. Körpinar, S. Ba³Then, a straightforward computation using above system gives( )∂α ∂fS= 1∂t ∂σ + fS 2 t + ψs,( )∂s ∂fS= 3∂t ∂σ + fS 2 κ g t − ψα,⟨ ⟩ ∂αwhere ψ =∂t , s . Thus, we obtain the assertion of the theorem.The following theorem states the conditions on the curvature and torsionfor the ow to be inextensible.Theorem 3.5. Let ∂α be inextensible. Then the system of partial dierential∂tequationsholds.∂κ g∂σ + ψ = ∂2 f S 3∂σ 2 + ∂ ∂s (fS 2 κ g ),( ∂ 2 f S )1κ g ψ = −∂σ 2 + ∂fS 2∂σProof. Assume that ∂α is inextensible. Then∂t∂ ∂t= ∂ [( ) ( ) ]∂fS1∂fS∂σ ∂t ∂σ ∂σ + fS 2 α + 3∂σ + fS 2 κ g s( ∂ 2 f S ) ( ) ( )1=∂σ 2 + ∂fS 2 ∂fSα + [ 1∂fS∂σ ∂σ + fS 2 +κ 3 g∂σ + fS 2 κ g ]t( ∂ 2 f S 3+∂σ 2 + ∂ )∂σ (fS 2 κ g ) s.From the Sabban frame, we have∂ ∂t∂t ∂σ = ∂ ∂t (α+κ gs)=[ ∂κg∂σ + ψ ]s +[ ( ) ( ) ]∂fS1∂fS∂σ + fS 2 + κ 3 g∂σ + fS 2 κ g t − κ g ψα.Therefore,∂κ g∂σ + ψ = ∂2 f S 3∂σ 2 + ∂∂σ (fS 2 κ g )

Inextensible ows of timelike curves... 11andThus, we obtain the theorem.( ∂ 2 f S )1κ g ψ = −∂σ 2 + ∂fS 2.∂σCorollary 3.6.( ) ∂fSκ 1 g∂σ + fS 2 = − ∂fS 3∂σ − fS 2 κ g − ∂ψ∂σ .Proof. Similarly, we have∂ ∂s∂σ ∂t= ∂ [( ) ]∂fS3∂σ ∂σ + fS 2 κ g t − ψα[ ( ∂ 2 f S 3=∂σ + ∂) ( )( )fS ∂fS∂σ 2 κ g − ψ t+κ 3 g∂σ + fS 2 κ g s[ ( )∂fS+ 3∂σ + fS 2 κ g − ∂ψ ] ]α .∂σOn the other hand, a straightforward computation gives∂ ∂s∂t ∂σ = ∂ ∂t (κ gt)= ∂κ g∂t t + κ g[( ) ( ) ]∂fS1∂fS∂σ + fS 2 α + 3∂σ + fS 2 κ g s .Combining these equalities, we obtain the corollary.In the light of Theorem 3.5, we express the following corollaries withoutproofs.Corollary 3.7.Corollary 3.8.∂κ g∂t = ∂2 f S 3∂σ + ∂ ( )fS∂σ 2 κ g − ψ.∂f S 3∂σ + fS 2 κ g − ∂ψ∂σ = ∂fS 1∂σ + fS 2 .References[1] G. Chirikjian, J. Burdick, A modal approach to hyper-redundant manipulatorkinematics, IEEE Trans. Robot. Autom., 10, 343354 (1994).

12 V. Asil, T. Körpinar, S. Ba³[2] M. Desbrun, M.-P. Cani-Gascuel, Active implicit surface for animation, in:Proc. Graphics Interface-Canadian Inf. Process. Soc., 143150, 1998.[3] S. Izumiya, D. H. Pei, T. Sano, E. Torii, Evolutes of hyperbolic plane curves,Acta Math. Sinica (English Series), 20 (3), 543550 (2004).[4] D. Y. Kwon , F. C. Park, D. P. Chi, Inextensible ows of curves and developablesurfaces, Appl. Math. Lett., 18, 11561162 (2005).[5] H. Q. Lu, J. S. Todhunter, T. W. Sze, Congruence conditions for nonplanardevelopable surfaces and their application to surface recognition, CVGIP,Image Underst., 56, 265285 (1993).[6] H. Mochiyama, E. Shimemura, H. Kobayashi, Shape control of manipulatorswith hyper degrees of freedom, Int. J. Robot. Res., 18, 584600 (1999).Received26 January 2012