In this paper, we study inextensible flows of curves in 3-dimensional pseudoGalilean space. We give necessary and sufficient conditions for inextensible flows of curves according to equiform geometry in pseudo-Galilean space.Publisher's Versio

Gün Bozok, Hülya

Öztekin, Handan

TWMS Journal of Applied and Engineering Mathematics

English

Isik University Academic Open Access

TWMS J. App. Eng. Math. V.6, N.2, 2016, pp. 175-184INEXTENSIBLE FLOWS OF CURVES IN THE EQUIFORMGEOMETRY OF THE PSEUDO-GALILEAN SPACE G13HANDAN ÖZTEKIN 1, HÜLYA GÜN BOZOK 2, §Abstract. In this paper, we study inextensible flows of curves in 3-dimensional pseudo-Galilean space. We give necessary and sufficient conditions for inextensible flows of curvesaccording to equiform geometry in pseudo-Galilean space.Keywords: inextensible flows, equiform differential geometry, Galilean space, pseudo-Galilean space.AMS Subject Classification: 53A351. IntroductionThe flow of a curve is said to be inextensible if its arclength is preserved. Physically,the inextensible curve flows give rise to motions in which no strain energy is induced.The flows of inextensible curve and surface are used to solve many problems in computervision [8], [12], computer animation [1] and even structural mechanics [17]. Especially themethods used in this paper developed by Gage and Hamilton [6] and Grayson [7]. Thedifferentiation between heat flows and inextensible flows of planar curves were elaboratedin detail, and some examples of the latter were given by [10]. Also, a general formulationfor inextensible flows of curves and developable surfaces in R3 are exposed by [9]. Latifiet al.[11] studied inextensible flows of curves in Minkowski 3-space. Ogrenmis et al.[13]studied inextensible curves in the Galilean space G3 , moreover inelastic flows of curvesaccording to equiform in Galilean space given in [18].The curves and the surfaces in G13 are described in [14, 2, 3, 4]. Theory of curves and thecurves of constant curvature in the equiform differential geometry of the isotropic space I13and I23 and the Galilean space G3 are described in [15] and [16], respectively. Also, Divjaket al.[5] studied the equiform differential geometry of curves in the pseudo-Galilean space.In this paper, we investigate inextensible flows of curves in the equiform geometry ofthe pseudo-Galilean space G13. Then we obtain partial differential equations in terms ofinextensible flows of curves according to equiform in G13.1 Firat University, Faculty of Science, Department of Mathematics, Elazığ, Turkey.e-mail: handanoztekin@gmail.com;2 Osmaniye Korkut Ata University, Faculty of Arts and Sciences, Department of Mathematics,Osmaniye, Turkey.e-mail: hulya-gun@hotmail.com;§ Manuscript received: June 10, 2014; accepted: December 15, 2014.TWMS Journal of Applied and Engineering Mathematics, Vol.6, No.2; c© Işık University, Departmentof Mathematics, 2016; all rights reserved.175176 TWMS J. APP. ENG. MATH. V.6, N.2, 20162. PreliminariesThe pseudo-Galilean geometry is one of the real Cayley-Klein geometries of projectivesignature (0,0,+,-), explained in [3]. As in [3], pseudo-Galilean inner product can bewritten as〈v1, v2〉 ={x1x2 , if x1 6= 0 ∨ x2 6= 0y1y2 − z1z2 , if x1 = 0 ∧ x2 = 0where v1 = (x1, y1, z1) and v2 = (x2, y2, z2). The pseudo-Galilean norm of the vectorv = (x, y, z) defined by‖v‖ ={x , if x 6= 0√|y2 − z2| , if x = 0In pseudo-Galilean space a curve is given by α : I → G13α (t) = (x (t) , y (t) , z (t)) (1)where I ⊆ R and x (t) , y (t) , z (t) ∈ C3 . A curve α given by (1) is admissible if x′ (t) 6= 0[3].The curves in pseudo-Galilean space are characterized as follows [2]An admissible curve in G13 can be parametrized by arclength t = s, given in coordinateformα (s) = (s, y (s) , z (s)) . (2)For an admissible curve α : I ⊆ R → G13, the curvature κ (s) and the torsion τ (s) aredefined byκ (x) =√|y′′2 − z′′2|, (3)τ (s) =1κ2 (s)det(α′ (s) , α′′ (s) , α′′′ (s)). (4)The associated trihedron is given byt(s) = α′ (s) =(1, y′ (s) , z′ (s))n(s) =1κ (s)α′′ (s) =1κ (s)(0, y′′ (s) , z′′ (s))(5)b(s) =1κ (s)(0, z′′ (s) , y′′ (s))[4].The vectors t (s),n (s) and b (s) are called the vectors of tangent, principal normal andbinormal line of α, respectively. The curve α given by (2) is timelike if n (s) is spacelikevector. For derivatives of tangent vector t (s), principal normal vector n (s) and binormalvector b (s), respectively, the following Frenet formulas holdt′ (s) = κ (s)n (s) ,n′ (s) = τ (s) b (s) , (6)b′ (s) = τ (s)n (s) .If the admissible curve β is given by β (x) = (x, y (x) , 0) and for this admissible curvethe curvature κ (s) and the torsion τ (s) are defined byκ (x) = y′′ (x)τ (x) =a′2 (x)a3 (x)H. ÖZTEKIN, H. GÜN BOZOK, : INEXTENSIBLE FLOWS OF CURVES IN THE G13 177where a (x) = (0, a2 (x) , a3 (x)). The associated trihedron is given byt (x) =(1, y′ (x) , 0),n (x) = (0, a2 (x) , a3 (x)) ,b (x) = (0, a3 (x) , a2 (x)) .For derivatives of tangent vector t (s), principal normal vector n (s) and binormal vectorb (s), respectively, the following Frenet formulas holdt′ (x) = κ (x) (coshφ (x)n (x)− sinhφ (x) b (x))n′ (x) = τ (x) b (x) (7)b′ (x) = τ (x)n (x)where φ is the angle between a (x) and the plane z = 0 [4].3. Frenet Formulas in Equiform Geometry in G13Let α : I → G13 be an admissible curve. We define the equiform parameter of α byσ :=∫dsρ=∫κds ,for ρ =1κis the radius of curvature of the curve α. Thendσds=1ρ, i.e.,dsdσ= ρ . (8)Let h be a homothety with the center in the origin and the coefficient λ. If we putα̃ = h (α), we obtains̃ = λs and ρ̃ = λρwhere s̃ is the arclength parameter of α̃ and ρ̃ the radius of curvature of this curve. Then,σ is an equiform invariant parameter of α [5].From now on, we define the Frenet formula of the curve α with respect to the equiforminvariant parameter σ in G13. The vectorT =dαdsis called a tangent vector of the curve α in the equiform geometry. Using (5) and (8) wehaveT =dαds· dsdσ= ρ · dαds= ρ · t . (9)Also, we have the principal normal vector and binormal vector byN = ρ · n , B = ρ · b . (10)One can say that the trihedron {T,N,B} is an equiform invariant trihedron of the curveα. On the other hand, the derivations of these vectors with respect to σ are given byT ′ =dTdσ= ρ̇T +N,N ′ =dNdσ= ρ̇N + ρτB,B′ =dBdσ= ρ̇B + ρτN.178 TWMS J. APP. ENG. MATH. V.6, N.2, 2016Definition 3.1. The function K : I → R defined byK = ρ̇is called the equiform curvature of the curve α.Definition 3.2. The function T : I → R defined byT = ρτ =τκis called the equiform torsion of the curve α.Then we can write the Frenet formulas in the equiform geometry of the pseudo-Galileanspace as follows [5]dTdσ= K.T +N,dNdσ= K.N + T.B (11)dBdσ= T.N + K.B4. Inextensible Flows of Curves According to Equiform inpseudo-Galilean Space G13Throughout this paper, we assume that F : [0, l]× [0, w]→ G13 is a one parameter familyof smooth curves in pseudo-Galilean space G13, where l is the arclength of initial curve.Let u be the curve parametrization variable, 0 ≤ u ≤ l. We put v =∥∥∥∥∂F∂u∥∥∥∥, from whichthe arclength of F is defined by s (u) =∫ u0vdu .Also, the operator∂∂sis given in terms of u by∂∂s=1v∂∂u, and the arclength parameteris given by ds = vdu.On the equiform invariant orthonormal frame {T,N,B} of a curve α in G13 any flow of Fcan be written by∂F∂t= fT + gN + hB (12)for f , g, h are the tangential, principal normal, binormal speeds of the curve in G13,respectively. We set s (u, t) =∫ u0vdu, it is called the arclength variation of F . Then,the requirement that the curve not be subject to any elongation or compression can beexpressed by the condition∂∂ss (u, t) =∫ u0∂v∂tdu = 0 , (13)for all u ∈ [0, l].Definition 4.1. A curve evolution F (u, t)and its flow∂F∂tin pseudo-Galilean space G13are said to be inextensible if∂∂t∥∥∥∥∂F∂u∥∥∥∥ = 0.H. ÖZTEKIN, H. GÜN BOZOK, : INEXTENSIBLE FLOWS OF CURVES IN THE G13 179Lemma 4.1. Let∂F∂t= fT +gN+hB be a smooth flow of the curve α in pseudo-Galileanspace G13. The flow is inextensible if and only if∂v∂t=∂f∂u+ fvK (14)Proof. Let∂F∂tbe a smooth flow of the curve F in pseudo-Galilean space G13. From thedefinition of v, we obtainv∂v∂t=〈∂F∂u,∂∂u(fT + gN + hB)〉. (15)From (11) we obtain∂v∂t=〈T,(∂f∂u+ fvK)T +(∂g∂u+ fv + hvT + gvK)N +(∂h∂u+ gvT + hvK)B〉.If we make necessary calculations, we get (14). Theorem 4.1. Suppose that∂F∂t= fT + gN + hB be a smooth flow of the curve F inpseudo-Galilean space G13. The flow is inextensible if and only if∂f∂u= −fK.Proof. Using (13), we have∂∂ts (u, t) =∫ u0∂v∂tdu =∫ u0(∂f∂u+ fvK)= 0 . (16)Substituting (14) in (16) complete the proof of the theorem. Suppose that, v = 1 and the local coordinate u corresponds to the curve arc length s.Lemma 4.2. Let∂F∂t= fT+gN+hB be a smooth flow of the curve α in pseudo-Galileanspace G13. Using (11) we havei) if 〈N,N〉 = −1 and 〈B,B〉 = 1 we have∂T∂t=(∂g∂s+ f + gK + hT)N +(∂h∂s+ gT + hK)B (17)∂N∂t=(∂g∂s+ f + gK + hT)T + ϕB (18)∂B∂t= −(∂h∂s+ gT + hK)T + ϕN (19)ii) if 〈N,N〉 = 1 and 〈B,B〉 = −1 we have∂T∂t=(∂g∂s+ f + gK + hT)N +(∂h∂s+ gT + hK)B (20)∂N∂t= −(∂g∂s+ f + gK + hT)T − ϕB (21)∂B∂t=(∂h∂s+ gT + hK)T − ϕN (22)where ϕ =〈∂N∂t,B〉provided that(∂g∂s+ f + gK + hT)= 0.180 TWMS J. APP. ENG. MATH. V.6, N.2, 2016Proof. Using definition of F , we have∂T∂t=∂∂t∂F∂s=∂∂s(fT + gN + hB) .Using (11), we have∂T∂t=(∂f∂s+ fK)T +(∂g∂s+ f + gK + hT)N +(∂h∂s+ gT + hK)B (23)On the other hand using theorem 4.1. in (23), we get∂T∂t=(∂g∂s+ f + gK + hT)N +(∂h∂s+ gT + hK)B .i) Now we differentiate the Frenet frame by t:0 =∂∂t〈T,N〉 = −(∂g∂s+ f + gK + hT)+〈T,∂N∂t〉0 =∂∂t〈T,B〉 =(∂h∂s+ gT + hK)+〈T,∂B∂t〉0 =∂∂t〈N,B〉 = ϕ+〈N,∂B∂t〉Then, a straight forward computation using above system gives∂N∂t=(∂g∂s+ f + gK + hT)T + ϕB∂B∂t= −(∂h∂s+ gT + hK)T + ϕNwhere ϕ =〈∂N∂t,B〉provided that(∂g∂s+ f + gK + hT)= 0. So, we obtain the lemma4.2. (i). ii) Now we differentiate the Frenet frame by t:0 =∂∂t〈T,N〉 =(∂g∂s+ f + gK + hT)+〈T,∂N∂t〉0 =∂∂t〈T,B〉 = −(∂h∂s+ gT + hK)+〈T,∂B∂t〉0 =∂∂t〈N,B〉 = ϕ+〈N,∂B∂t〉Then, a straight forward computation using above system gives∂N∂t= −(∂g∂s+ f + gK + hT)T − ϕB∂B∂t=(∂h∂s+ gT + hK)T − ϕNwhere ϕ =〈∂N∂t,B〉provided that(∂g∂s+ f + gK + hT)= 0. Thus, we obtain thelemma 4.2. (ii).H. ÖZTEKIN, H. GÜN BOZOK, : INEXTENSIBLE FLOWS OF CURVES IN THE G13 181Theorem 4.2. Let∂F∂tbe inextensible. Then, using (11), the following system of partialdifferential equations holds:i) if 〈N,N〉 = −1 and 〈B,B〉 = 1 we have∂K∂t= 0∂T∂t= −(∂h∂s+ gT + hK)+∂ϕ∂sϕ =(∂2h∂s2+∂∂s(gT) +∂∂s(hK))ii) if 〈N,N〉 = 1 and 〈B,B〉 = −1 we have∂K∂t= 0∂T∂t=(∂h∂s+ gT + hK)− ∂ϕ∂sϕ = −(∂2h∂s2+∂∂s(gT) +∂∂s(hK))where ϕ =〈∂N∂t,B〉provided that(∂g∂s+ f + gK + hT)= 0.Proof. i) Using (17), we have∂∂s∂T∂t=∂∂s[(∂g∂s+ f + gK + hT)N +(∂h∂s+ gT + hK)B]=[(∂2g∂s2+∂∂s(f) +∂∂s(gK) +∂∂s(hT))]N+(∂g∂s+ f + gK + hT)(KN + TB)+(∂2h∂s2+∂∂s(gT) +∂∂s(hK))B +(∂h∂s+ gT + hK)(TN + KB)On the other hand, from (11), we have∂∂s∂T∂t=∂∂t(KT +N)=∂K∂tT + K(∂g∂s+ f + gK + hT)N + K(∂h∂s+ gT + hK)B=(∂g∂s+ f + gK + hT)T + ϕBHence we see that∂K∂t= 0 ,andϕ =(∂2h∂s2+∂∂s(gT) +∂∂s(hK)).where ϕ =〈∂N∂t,B〉provided that(∂g∂s+ f + gK + hT)= 0.182 TWMS J. APP. ENG. MATH. V.6, N.2, 2016Also, we have from lemma 4.2.(i)∂∂s∂B∂t=∂∂s[−(∂h∂s+ gT + hK)T + ϕN]= −(∂2h∂s2+∂∂s(gT) +∂∂s(hK))T −(∂h∂s+ gT + hK)(KT +N)+∂ϕ∂sN + ϕ (KN + TB)On the other hand, from (11), we have∂∂t∂B∂s=∂∂t(TN + KB)=(∂T∂t+ Kϕ)N +[T(∂g∂s+ f + gK + hT)−K(∂h∂s+ gT + hK)]=(∂K∂t+ Tϕ)BHence we see that∂T∂t= −(∂h∂s+ gT + hK)+∂ϕ∂s.ii) Using (20), we have∂∂s∂T∂t=∂∂s[(∂g∂s+ f + gK + hT)N +(∂h∂s+ gT + hK)B]=[(∂2g∂s2+∂∂s(f) +∂∂s(gK) +∂∂s(hT))]N +(∂g∂s+ f + gK + hT)(KN + TB)+(∂2h∂s2+∂∂s(gT) +∂∂s(hK))B +(∂h∂s+ gT + hK)(TN + KB)On the other hand, from (11), we have∂∂s∂T∂t=∂∂t(KT +N)=∂K∂tT + K(∂g∂s+ f + gK + hT)N + K(∂h∂s+ gT + hK)B−(∂g∂s+ f + gK + hT)T − ϕBHence we see that∂K∂t= 0 ,andϕ = −(∂2h∂s2+∂∂s(gT) +∂∂s(hK)).where ϕ =〈∂N∂t,B〉provided that(∂g∂s+ f + gK + hT)= 0H. ÖZTEKIN, H. GÜN BOZOK, : INEXTENSIBLE FLOWS OF CURVES IN THE G13 183Also, we have from lemma 4.2.(ii)∂∂s∂B∂t=∂∂s[(∂h∂s+ gT + hK)T − ϕN]=(∂2h∂s2+∂∂s(gT) +∂∂s(hK))T +(∂h∂s+ gT + hK)(KT +N)− ∂ϕ∂sN − ϕ (KN + TB)On the other hand, from (11), we have∂∂t∂B∂s=∂∂t(TN + KB)=(∂T∂t−Kϕ)N +[−T(∂g∂s+ f + gK + hT)+ K(∂h∂s+ gT + hK)]=(∂K∂t− Tϕ)BHence we see that∂T∂t=(∂h∂s+ gT + hK)− ∂ϕ∂s.Thus, we prove the theorem. References[1] Desbrun,M. and Cani-Gascuel,M.P., (1998), Active implicit surface for animation, Proc. GraphicsInterface-Canadian Inf. Process. Soc., pp.143-150.[2] Divjak,B., (2003), Special curves on ruled surfaces in Galilean and pseudo-Galilean space, Acta Math.Hungar. 98(3), pp. 203-215.[3] Divjak,B., (1998), Curves in pseudo-Galilean geometry, Annales Univ. Sci. Budapest, 41, pp. 117-128.[4] Divjak,B. and Milin-Sipus,Z., (2003), Special curves on the ruled surfaces in Galilean and pseudo-Galilean space, Acta Math. Hungar., 98(3), pp. 203-215.[5] Divjak,B. and Erjavec,Z., (2008), The equiform differential geometry of curves in the pseudo-Galileanspace, Mathematical Communitations, 13, pp. 321-331.[6] Gage,M. and Hamilton,R.S., (1986), The heat equation shrinking convex plane curves, J. DifferentialGeom., 23, pp. 69-96.[7] Grayson,M., (1987), The heat equation shrinks embedded plane curves to round points, J. DifferentialGeom., 26, pp. 285-314.[8] Kass,M., Witkin,A. and Terzopoulos,D., (1987), Snakes: active contour models, Proc. 1st Int. Con-ference on Computer Vision, pp. 259-268.[9] Kwon,D.Y., Park,F.C. and Chi,D.P., (2005), Inextensible flows of curves and developable surfaces,Applied Mathematics Letters, 18, pp. 1156-1162.[10] Kwon,D.Y. and Park,F.C., (1999), Evolution of inelastic plane curves, Appl. Math. Lett., 12 pp.115-119.[11] Latifi,D. and Razavi,A., (2008), Inextensible flows of curves in Minkowskian Space, Adv. StudiesTheor. Phys. 2(16), pp. 761-768.[12] Lu,H.Q., Todhunter,J.S. and Sze,T.W., (1993), Congruence conditions for nonplanar developablesurfaces and their application to surface recognition, CVGIP, Image Underst., 56, pp. 265-285 .[13] Ogrenmis,A.O. and Yeneroglu,M., (2010), Inextensible curves in the Galilean space, InternationalJournal of the Physical Sciences, 5(9),pp. 1424-1427.[14] Ogrenmis,A.O., Ergut,M. and Bektas,M., (2007), On the Helices in the Galilean Space G3, Iran. J.Sci, Tech. Trans. A Sci., 31(2), pp. 177-181.184 TWMS J. APP. ENG. MATH. V.6, N.2, 2016[15] Pavkovic,B.J., (1986), Equiform geometry of curves in the isotropic space I13 and I23 , Rad JAZU., pp.39-44.[16] Pavkovic,B.J. and Kamenarovic,I., (1987) ,The equiform differential geometry of curves in the Galileanspace G3, Glasnik Mat. 22(42), pp. 449-457.[17] Unger,D.J., (1991), Developable surfaces in elastoplastic fracture mechanics, Int. J. Fract., 50, pp.33-38.[18] Yoon,D.W., (2011), Inelastic flows of curves according to equiform in Galilean space, Journal of theChungcheong Mathematical Society, 24(4).Handan Öztekin is currently working as an associate professor in the Departmentof Mathematics of Faculty of Science in Firat University, Turkey. She received herB.S. and M.S. degrees from Firat University. Her area of interest includes Lorentzianspace and Galilean space.Hülya Gün Bozok is working as a research assistant in the Department of Math-ematics of Faculty of Arts and Science in Osmaniye Korkut Ata University, Turkey.She received her Ph.D. degree from Firat University. Her area of interest includesGalilean space.

Inextensible flows of curves in the equiform geometry of the pseudo-Galilean space G13

In this paper, we study inextensible ows of curves in 3-dimensional pseudo- Galilean space. We give necessary and sucient conditions for inextensible ows of curves according to equiform geometry in pseudo-Galilean space

Inextensible flows of curves in the equiform geometry of the pseudo-Galilean space G13

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