International J.Math. Combin. Vol.4(2016), 29-43 Smarandache Curves of a Spacelike Curve According to the Bishop Frame of Type-2 Yasin U¨nlu¨tu¨rk DepartmentofMathematics,KırklareliUniversity,39100Kırklareli,Turkey Su¨ha Yılmaz DokuzEylu¨lUniversity,BucaFacultyofEducation DepartmentofElementaryMathematicsEducation,35150,Buca-Izmir,Turkey E-mail: [email protected],[email protected] Abstract: In this study, we introduce new Smarandache curves of a spacelike curve ac- cordingtotheBishop frameoftype-2inE3. Also, Smarandachebreadthcurvesaredefined 1 according to this frame in Minkowski 3-space. A third order vectorial differential equation of position vector of Smarandache breadth curveshas been obtained in Minkowski 3-space. Key Words: Smarandache curves, the Bishop frame of type-2, Smarandache breadth curves, Minkowski 3-space. AMS(2010): 53A05, 53B25, 53B30 §1. Introduction Bishop frame extended to study canal and tubular surfaces [1]. Rotating camera orientations relative to a stable forward-facing frame can be added by various techniques such as that of Hanson and Ma [2]. This special frame also extended to height functions on a space curve [3]. The constructionofthe Bishopframe is due to L. R.Bishopandthe advantagesofBishop frame,andcomparisonsofBishopframewiththeFrenetframeinEuclidean3-spaceweregiven by Bishop [4] and Hanson [5]. That is why he defined this frame that curvature may vanish at some points on the curve. That is, second derivative of the curve may be zero. In this situation, an alternative frame is needed for non continously differentiable curves on which Bishop (parallel transport frame) frame is well defined and constructed in Euclidean and its ambient spaces [6,7,8]. A regular curve in Euclidean 3-space, whose position vector is composed of Frenet frame vectors on another regular curve, is called a Smarandache curve. M. Turgut and S. Yılmaz have defined a special case of such curves and call it Smarandache TB curves in the space E4 2 1 ([9]) and Turgut also studied Smarandache breadth of pseudo null curves in E4 ([10]). A.T.Ali 1 hasintroducedsomespecialSmarandachecurvesintheEuclideanspace[11]. Moreover,special 1ReceivedApril3,2016,Accepted November6,2016. 30 YasinU¨nlu¨tu¨rkandSu¨haYılmaz Smarandache curves have been investigated by using Bishop frame in Euclidean space [12]. Special Smarandache curves according to Sabban frame have been studied by [13]. Besides, some special Smarandache curves have been obtained in E3 by [14]. 1 Curvesof constantbreadthwereintroduced by L.Euler [15]. Some geometric propertiesof planecurvesofconstantbreadthweregivenin[16]. And, inanotherwork[17],theseproperties werestudiedintheEuclidean3-spaceE3. Moreover,MFujivara[18]hadobtainedaproblemto determine whether there exist space curve of constant breadth or not, and he defined breadth forspacecurvesonasurfaceofconstantbreadth. In[19],thesekindcurveswerestudiedinfour dimensional Euclidean space E4. In [20], Yılmaz introduced a new version of Bishop frame in E3 and called it Bishop frame of type-2 of regular curves by using common vector field as the 1 binormal vector of Serret-Frenet frame. Also, some characterizations of spacelike curves were given according to the same frame by Yılmaz and U¨nlu¨tu¨rk [21]. A regular curve more than 2 breadths in Minkowski 3-space is called a Smarandache breadth curve. In the light of this definition, westudy specialcasesofSmarandachecurvesaccordingto the new frameinE3.We 1 investigate position vector of simple closed spacelike curves and give some characterizations in caseofconstantbreadthaccordingtotype-2BishopframeinE3. Thus,weextendthisclassical 1 topic in E3 into spacelike curves of constant breadth in E3, see [22] for details. 1 In this study, we introduce new Smarandache curves of a spacelike curve according to the Bishop frame of type-2 in E3. Also, Smarandache breadth curves are defined accordingto this 1 frame in Minkowski 3-space. A third order vectorial differential equation of position vector of Smarandache breadth curves has been obtained in Minkowski 3-space. §2. Preliminaries To meet the requirements in the next sections, here, the basic elements of the theory of curves intheMinkowski3–spaceE3 arebrieflypresented. Thereexistsavastliteratureonthesubject 1 including several monographs,for example [23,24]. The three dimensional Minkowski space E3 is a real vector space R3 endowed with the 1 standard flat Lorentzian metric given by , = dx2+dx2+dx2, h iL − 1 2 3 where (x ,x ,x ) is a rectangular coordinate system of E3. This metric is an indefinite one. 1 2 3 1 Let u = (u ,u ,u ) and v = (v ,v ,v ) be arbitrary vectors in E3, the Lorentzian cross 1 2 3 1 2 3 1 product of u and v is defined as i j k − u v = det u u u . × − 1 2 3  v v v   1 2 3    Recall that a vector v E3 has one of three Lorentziancharacters: it is a spacelike vector ∈ 1 if v,v > 0 or v = 0; timelike v,v < 0 and null (lightlike) v,v = 0 for v = 0. Similarly, an h i h i h i 6 SmarandacheCurvesofaSpacelikeCurveAccordingtotheBishopFrameofType-2 31 arbitrarycurveδ =δ(s) in E3 canlocally be spacelike,timelike or null(lightlike) if its velocity 1 vector α are ,respectively, spacelike, timelike or null (lightlike), for every s I R. The ′ ∈ ⊂ pseudo-norm of an arbitrary vector a E3 is given by a = a,a . The curve α =α(s) is ∈ 1 k k |h i| calledaunitspeedcurveifitsvelocityvectorα isunitonei.e., α =1.Forvectorsv,w E3, ′ pk ′k ∈ 1 they are said to be orthogonal each other if and only if v,w = 0. Denote by T,N,B the h i { } moving Serret-Frenet frame along the curve α=α(s) in the space E3. 1 For an arbitrary spacelike curve α = α(s) in E3, the Serret-Frenet formulae are given as 1 follows T 0 κ 0 T ′  N = γκ 0 τ . N , (2.1) ′  B   0 τ 0   B   ′            where γ = 1, and the functions κ and τ are, respectively, the first and second (torsion) ∓ T (s) det(α,α ,α ) ′ ′ ′′ ′′′ curvature. T(s)=α(s),N(s)= ,B(s)=T(s) N(s) and τ(s)= . ′ κ(s) × κ2(s) If γ = 1, then α(s) is a spacelike curve with spacelike principal normal N and timelike − binormal B, its Serret-Frenet invariants are given as κ(s)= T (s),T (s) and τ(s)= N (s),B(s) . h ′ ′ i −h ′ i p If γ = 1, then α(s) is a spacelike curve with timelike principal normal N and spacelike binormal B, also we obtain its Serret-Frenet invariants as κ(s)= T (s),T (s) and τ(s)= N (s),B(s) . −h ′ ′ i h ′ i p The LorentziansphereS2 ofradiusr >0 andwith the centerinthe originofthe spaceE3 1 1 is defined by S2(r)= p=(p ,p ,p ) E3 : p,p =r2 . 1 { 1 2 3 ∈ 1 h i } Theorem 2.1 Let α= α(s) be a spacelike unit speed curve with a spacelike principal normal. If Ω ,Ω ,B is an adapted frame, then we have 1 2 { } Ω 0 0 ξ Ω ′1 1 1  Ω = 0 0 ξ . Ω  (2.2) ′2 − 2 2  B   ξ ξ 0   B   ′   − 1 − 2          Theorem 2.2 Let T,N,B and Ω ,Ω ,B be Frenet and Bishop frames, respectively. There 1 2 { } { } exists a relation between them as T sinhθ(s) coshθ(s) 0 Ω 1  N = coshθ(s) sinhθ(s) 0 . Ω , (2.3) 2  B   0 0 1   B              32 YasinU¨nlu¨tu¨rkandSu¨haYılmaz where θ is the angle between the vectors N and Ω . 1 ξ =τ(s)coshθ(s),ξ =τ(s)sinhθ(s). 1 2 s Theframe Ω ,Ω ,B isproperlyoriented,andτ andθ(s)= κ(s)dsarepolarcoordinates 1 2 { } 0 R for the curve α = α(s). We shall call the set Ω ,Ω ,B,ξ ,ξ as type-2 Bishop invariants 1 2 1 2 { } of the curve α=α(s) in E3. 1 §3. Smarandache Curves of a Spacelike Curve Inthissection,wewillcharacterizealltypesofSmarandachecurvesofspacelikecurveα=α(s) according to type-2 Bishop frame in Minkowski 3-space E3. 1 3.1 Ω Ω Smarandache Curves 1 2 − Definition 3.1 Let α = α(s) be a unit speed regular curve in E3 and Ωα,Ωα,B be its 1 { 1 2 α} moving Bishop frame. ΩαΩα Smarandache curves are defined by 1 2− β(s )= 1 (Ωα+Ωα). (3.1) ∗ √2 1 2 Now we can investigate Bishop invariants of ΩαΩα Smarandachecurves of the curveα= 1 2− α(s). Differentiating (3.1) with respect to s gives dβ ds 1 β˙ = . ∗ = (ξα ξα)B , (3.2) ds ds √2 1 − 2 α and ds 1 T . ∗ = (ξα ξα)B , β ds √2 1 − 2 α where ds 1 ∗ = ξα ξα . (3.3) ds √2| 1 − 2| The tangent vector of the curve β can be written as follows T =β . (3.4) β α Differentiating (3.4) with respect to s, we obtain dT ds β. ∗ = (ξαΩα+ξαΩα). (3.5) ds ds − 1 1 2 2 ∗ Substituting (3.3) into (3.5) gives √2 T = (ξαΩα+ξαΩα). β′ − ξα ξα 1 1 2 2 | 1 − 2| SmarandacheCurvesofaSpacelikeCurveAccordingtotheBishopFrameofType-2 33 Then the first curvature and the principal normal vector field of β are, respectively, com- puted as √2 T =κ = (ξα)2+(ξα)2, β′ β ξα ξα − 1 2 (cid:13) (cid:13) | 1 − 2| (cid:13) (cid:13) p and (cid:13) (cid:13) 1 N = − (ξαΩα+ξαΩα). β (ξα)2+(ξα)2 1 1 2 2 − 1 2 On the other hand, we expresps Ωα Ωα β 1 − 1 2 α Bβ = (ξα−)2+(ξα)2 (cid:12)(cid:12) 0 0 1 (cid:12)(cid:12), − 1 2 (cid:12)(cid:12) ξα ξα 0 (cid:12)(cid:12) p (cid:12) 1 2 (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) So the binormal vector of β is computed as fo(cid:12)llows (cid:12) 1 B = − (ξαΩα+ξαΩα). β (ξα)2+(ξα)2 2 1 1 2 − 1 2 p Differentiating (3.2) with respect to s in order to calculate the torsion of the curve β, we obtain 1 β¨= [ (ξα+ξα)ξαΩα (ξα+ξα)ξαΩα+(ξ˙α+ξ˙α)B ], √2 − 1 2 1 1 − 1 2 2 2 1 2 α and similarly ... β = 1 [( 3ξαξ˙α 2ξαξ˙α ξ˙αξα ξαξ˙α)Ωα √2 − 1 1 − 1 2 − 1 2 − 1 1 1 +( 2ξ˙αξα 2(ξ˙α)2 ξαξ˙α ξαξ˙α)Ωα − 1 2 − 2 − 1 2 − 2 2 2 +(ξ¨α+ξ¨α (ξα)3 (ξα)2ξα ξα(ξα)2+(ξα)2)B ]. 1 2 − 1 − 1 2 − 1 2 2 α The torsion of the curve β is found 1 (ξα ξα)2 τ = 1 − 2 [(ξα+ξα)K (s) (ξα+ξα)ξαK (s)], β 4√2(ξα)2+(ξα)2 1 2 2 − 1 2 2 1 1 2 where K (s)= 3ξαξ˙α 2ξαξ˙α ξ˙αξα ξαξ˙α, 1 − 1 1 − 1 2 − 1 2 − 1 1 K (s)= 2ξ˙αξα 2(ξ˙α)2 ξαξ˙α ξαξ˙α, 2 − 1 2 − 2 − 1 2 − 2 2 K (s)=ξ¨α+ξ¨α (ξα)3 (ξα)2ξα ξα(ξα)2+(ξα)2. 3 1 2 − 1 − 1 2 − 1 2 2 3.2 Ω B Smarandache Curves 1 − Definition 3.2 Let α = α(s) be a unit speed regular curve in E3 and Ωα,Ωα,B be its 1 { 1 2 α} moving Bishop frame. ΩαB Smarandache curves are defined by 1 − 1 β(s∗)= √2(Ωα1 +Bα). (3.6) 34 YasinU¨nlu¨tu¨rkandSu¨haYılmaz Now we can investigate Bishop invariants of ΠαB Smarandache curves of the curve 1 α− α=α(s). Differentiating (3.6) with respect to s, we have dβ ds 1 β˙ = . ∗ = (ξαB ξαΩα ξαΩα), (3.7) ds ds √2 1 α− 1 1 − 2 2 and ds 1 T . ∗ = (ξαB ξαΩα ξαΩα), β ds √2 1 α− 1 1 − 2 2 where ds ξα ∗ = 2 . (3.8) ds 2 The tangent vector of the curve β can be written as follows √2 T = ( ξαΩα ξαΩα+ξαB ). (3.9) β ξα − 1 1 − 2 2 1 α 2 Differentiating (3.9) with respect to s gives dT ds ξα β. ∗ = 2 (L (s)Ωα+L (s)Ωα+L (s)B ), (3.10) ds ds √2 1 1 2 2 3 α ∗ where ξαξ˙α L (s)= ξα (ξα)2+ 1 2 ,L (s)=ξαξα, 1 − 1 − 1 ξα 2 1 2 2 ξαξ˙α L (s)= (ξα)2+(ξα)2+ξ˙α 1 2 . 3 − 1 2 1 − ξα 2 Substituting (3.8) into (3.10) gives 2√2 T = (L (s)Ωα+L (s)Ωα+L (s)B ), β′ (ξα)2 1 1 2 2 3 α 2 then the first curvature and the principal normal vector field of β are, respectively, 2√2 T =κ = L2(s)+L2(s) L2(s), β′ β (ξα)2 1 2 − 3 2 (cid:13) (cid:13) p (cid:13) (cid:13) and (cid:13) (cid:13) 1 N = − (L (s)Ωα+L (s)Ωα+L (s)B ). β L2(s)+L2(s) L2(s) 1 1 2 2 3 α 1 2 − 3 On the other hanpd, we have √2 B = [(ξαL (s)+ξαL (s))Ωα β ξα L2(s)+L2(s) L2(s) 1 2 2 3 1 2 1 2 − 3 +(ξαL (s)+ξαL (s))Ωα+(ξαL (s) ξαL (s))B ]. p1 1 1 3 2 2 1 − 1 2 α Differentiating (3.7) with respect to s in order to calculate the torsion of the curve β, we SmarandacheCurvesofaSpacelikeCurveAccordingtotheBishopFrameofType-2 35 find 1 β¨= [ ((ξα)2+ξ˙α)Ωα+( ξαξ˙α ξ˙α)Ωα (ξ¨α (ξα)2+(ξα)2)B ], √2 − 1 1 1 − 1 2 − 2 2 − 1 − 1 2 α and similarly ... 1 β = [( 2ξαξ˙α ξ¨α)Ωα+( ξ˙αξ˙α ξ¨α ξαξ¨α)Ωα √2 − 1 1 − 1 1 − 1 2 − 2 − 1 2 2 +( (ξα)3 ξαξ˙α+ξαξ˙αξα ξαξ˙α)B ]. − 1 − 1 1 1 2 2 − 2 2 α The torsion of the curve β is (εα)4 τ = 2 [ ξαM (s) ξαM (s) ξαM (s)], β 16√2 − 1 1 − 2 2 − 1 3 where M (s)= 3ξαξ˙α 2ξαξ˙α ξ˙αξα ξαξ˙α, 1 − 1 1 − 1 2 − 1 2 − 1 1 M (s)= 2ξ˙αξα 2(ξ˙α)2 ξαξ˙α ξαξ˙α, 2 − 1 2 − 2 − 1 2 − 2 2 M (s)=ξ¨α+ξ¨α (ξα)3 (ξα)2ξα ξα(ξα)2+(ξα)2. 3 1 2 − 1 − 1 2 − 1 2 2 3.3 Ω B Smarandache Curves 2 − Definition 3.3 Let α = α(s) be a unit speed regular curve in E3 and Ωα,Ωα,B be its 1 { 1 2 α} moving Bishop frame. ΩαB Smarandache curves are defined by 2 − 1 β(s )= (Ωα+B ). (3.11) ∗ √2 2 α Now we can investigate Bishop invariants of ΩαB Smarandache curves of the curve 1 α− α=α(s). Differentiating (3.11) with respect to s, we have dβ ds 1 β˙ = . ∗ = ( ξαB ξαΩα ξαΩα), (3.12) ds ds √2 − 2 α− 1 1 − 2 2 and ds 1 T . ∗ = ( ξαΩα ξαΩα ξαB ), β ds √2 − 1 1 − 2 2 − 2 α where ds 2(ξα)2 (ξα)2 ∗ = 2 − 1 . (3.13) ds 2 r The tangent vector of the curve β can be written as follows ξαΩα ξαΩα ξαB Tβ = − 1 1 − 2 2 − 2 α. (3.14) 2(ξα)2 (ξα)2 2 − 1 p Differentiating (3.14) with respect to s gives dT ds β. ∗ =(N (s)Ωα+N (s)Ωα+N (s)B ), (3.15) ds ds 1 1 2 2 3 α ∗ 36 YasinU¨nlu¨tu¨rkandSu¨haYılmaz where N1(s)= 12(4ξ2αξ˙2α−2ξ1αξ˙1α)(2(ξ2α)2−(ξ1α)2)−23ξ1α −(2(ξ2α)2−(ξ1α)2)−21ξ˙1α+(2(ξ2α)2−(ξ1α)2)−21ξ1αξ2α, N2(s)= 12(4ξ2αξ˙2α−2ξ1αξ˙1α)(2(ξ2α)2−(ξ1α)2)−23ξ1α −(2(ξ2α)2−(ξ1α)2)−21ξ˙1α+(2(ξ2α)2−(ξ1α)2)(ξ2α)2, N3(s)= 12(4ξ2αξ˙2α−2ξ1αξ˙1α)(2(ξ2α)2−(ξ1α)2)−23 −(2(ξ2α)2−(ξ1α)2)−21((ξ2α)2−(ξ1α)2−ξ˙2α). Substituting (3.13) into (3.15) gives 2 T = (N (s)Ωα+N (s)Ωα+N (s)B ), β′ 2(ξα)2 (ξα)2 1 1 2 2 3 α r 2 − 1 then the first curvature and the principal normal vector field of β are, respectively, found as follows 2 κ = T = N2(s)+N2(s)+N2(s), β β′ 2(ξα)2 (ξα)2 1 2 3 (cid:13) (cid:13) r 2 − 1 p (cid:13) (cid:13) and (cid:13) (cid:13) 1 Nβ = N2(s)+N−2(s)+N2(s)(N1(s)Ωα1 +N2(s)Ωα2 +L3(s)Bα). (3.16) 1 2 3 On the other hapnd, we have 1 B = [( ξαN (s)+ξαN (s))Ωα β 2(ξ2α)2−(ξ1α)2 N12(s)+N22(s)+N32(s) − 2 3 2 2 1 (3.17) +( ξαN (s)+ξαN (s))Ωα+(ξαN ξαN (s))B ]. p − 2 3 p 2 1 2 1 2− 2 1 α Differentiating (3.12) with respect to s in order to calculate the torsionof the curve β, we obtain 1 β¨= [(ξαξα+ξ˙α)Ωα+((ξα)2 ξ˙α)Ωα+( ξ˙α+ξα (ξα)2)B ], √2 2 1 1 1 2 − 2 2 − 2 2 − 1 α and similarly ... 1 β = [(2ξ˙αξα+ξαξ˙α ξ¨α ξαξα+(ξα)3)Ωα √2 2 1 2 1 − 1 − 2 1 1 1 +(3ξαξ˙α ξ¨α (ξα)2 (ξα)2ξα)Ωα 2 2 − 2 − 2 − 1 2 2 +((ξα)2ξα (ξα)3+ξαξ˙α ξ¨α+ξ˙α 3ξαξ˙α)B ]. 1 2 − 2 2 2 − 2 2 − 1 1 α The torsion of the curve β is 2(ξα)2 (ξα)2 τ = 2 − 1 [P (s)((ξα)2 ξ˙α) P (s)( ξ˙α+ξα (ξα)2)]ξα β 4√2 { 3 2 − 2 − 2 − 2 2 − 1 1 +[P (s)(ξαξα ξ˙α) P (s)( ξ˙α+ξα (ξα)2)]ξα 3 2 1 − 1 − 1 − 2 2 − 1 2 +[P (s)(ξαξα ξ˙α) P (s)((ξα)2 ξ˙α)]ξα , 2 2 1 − 1 − 1 2 − 2 3} SmarandacheCurvesofaSpacelikeCurveAccordingtotheBishopFrameofType-2 37 where P (s)=2ξ˙αξα+ξαξ˙α ξ¨α ξαξ˙α+ξαξα+(ξα)3, 1 2 1 2 1 − 1 − 1 1 1 2 1 P (s)=3ξ˙αξα ξ¨α (ξα)2 (ξα)2ξα, 2 2 2 − 2 − 2 − 1 2 P (s)=(ξα)2ξα (ξα)3+ξαξ˙α ξ¨α+ξ˙α 3ξ˙αξα. 3 1 2 − 2 2 2 − 2 2 − 1 1 3.4 Ω Ω B Smarandache Curves 1 2 − Definition 3.4 Let α = α(s) be a unit speed regular curve in E3 and Ωα,Ωα,B be its 1 { 1 2 α} moving Bishop frame. ΩαΩαB Smarandache curves are defined by 1 2 − 1 β(s∗)= √3(Ωα1 +Ωα2 +Bα). (3.18) Now we can investigate Bishop invariants of ΩαΩαB Smarandache curves of the curve 1 2 − α=α(s). Differentiating (3.18) with respect to s, we have dβ ds 1 β˙ = . ∗ = ( ξαΩα ξαΩα+(ξα εα)B ), (3.19) ds ds √3 − 1 1 − 2 2 1 − 2 α and ds 1 T . ∗ = ( ξαΩα ξαΩα+(ξα ξα)B ), β ds √3 − 1 1 − 2 2 1 − 2 α where ds (ξα ξα)2+(ξα)2 (ξα)2 ∗ = 1 − 2 2 − 1 . (3.20) ds 3 r The tangent vector of the curve β is found as follows 1 Tβ = (ξα ξα)2+(ξα)2 (ξα)2(−ξ1αΩα1 −ξ2αΩα2 +(ξ1α−ξ2α)Bα). (3.21) 1 − 2 2 − 1 p Differentiating (3.21) with respect to s, we find dT ds β. ∗ =[ Q(s)ξ˙α Q(s)(ξα)2+Q(s)ξαξα Q(s)ξα]Ωα ds ds − 1 − 1 1 2 − ′ 1 1 ∗ +[ Q(s)ξ˙α Q(s)ξαξα+Q(s)(ξα)2 Q(s)ξα]Ωα (3.22) − 2 − 1 2 2 − ′ 2 2 +[Q(s)(ξα ξα)-Q(s)(ξα)2+Q(s)(ξα ξα)]B , 1 − 2 ′ 1 ′ 1 − 2 α where 1 Q(s)= . (ξα ξα)2+(ξα)2 (ξα)2 1 − 2 2 − 1 Substituting (3.20) into (3.22) bpy using (3.23) gives √3 T = (M (s)Ωα+M (s)Ωα+M (s)B ), β′ K(s) 1 1 2 2 3 α 38 YasinU¨nlu¨tu¨rkandSu¨haYılmaz where R (s)= Q(s)ξ˙α Q(s)(ξα)2+Q(s)ξαξα Q(s)ξα, 1 − 1 − 1 1 2 − ′ 1 R (s)= Q(s)ξ˙α Q(s)ξαξα+Q(s)(ξα)2 Q(s)ξα, (3.23) 2 − 2 − 1 2 2 − ′ 2 R (s)=Q(s)(ξα ξα) Q(s)(ξα)2+Q(s)(ξα ξα). 3 1 − 2 ′− 1 ′ 1 − 2 Thenthefirstcurvatureandtheprincipalnormalvectorfieldofβare,respectively,obtained as follows √3 κ = T = R2(s)+R2(s)+R2(s), β β′ K(s) − 1 2 3 (cid:13) (cid:13) p and (cid:13) (cid:13) (cid:13) (cid:13) 1 B = − [(M (ξα ξα)+M ξα)Ωα β K(s) R2(s)+R2(s)+R2(s) 2 1 − 2 3 2 1 (3.24) − 1 2 3 +(M (ξα ξα)+M ξα)Ωα+(ξαM (s) ξαM (s))B ]. 1 p1 − 2 3 2 2 2 1 − 1 2 α Differentiating (3.19) with respect to s in order to calculate the torsionof the curve β, we obtain 1 β¨= [( ξ˙α (ξα)2+ξαξα)Ωα √3 − 1 − 1 1 2 1 +( ξ˙α+(ξα)2 ξαξα+(ξα)2)Ωα+(ξ˙α ξ˙α (ξα)2)B ], − 2 2 − 1 2 2 2 1 − 2 − 1 α and similarly ... 1 β = [( ξ¨α 2ξαξ˙α+ξ˙αξα+ξαξ˙α)Ωα √3 − 1 − 1 1 1 2 1 2 1 +( ξ¨α+4ξαξ˙α ξ˙αξα 2ξαξα)Ωα − 2 2 2 − 1 2 − 1 2 2 +(ξαξ˙α+ξα(ξα)2 (ξα)2 (ξα)3+(ξα)2ξα)B ]. 2 2 1 2 − 1 − 1 1 2 α The torsion of the curve β is 1 K2(s) τ = [Q (s)( ξ˙α+2(ξα)2 ξαξα) β 9 M2(s)+M2(s)+M2(s){ 3 − 2 2 − 1 2 − 1 2 3 Q (s)(ξα ξ˙α ξαξα)]ξα+[Q (s)( ξ˙α (ξα)2+ξαξα) − 2 1 − 2 − 1 2 1 3 − 1 − 1 1 2 Q (s)( ξ˙α+2(ξα)2 ξαξα)]ξα [Q (s)( ξα (ξα)2+ξαξα) − 1 − 2 2 − 1 2 2 − 2 − 1 − 1 1 2 Q (s)( ξα+2(ξα)2 ξαξα)](ξα ξα) , − 1 − 2 2 − 1 2 1 − 2 } where Q (s)= ξ˙α (ξα)2+ξαξα, 1 − 1 − 1 1 2 Q (s)= ξ˙αξα+2ξαξ˙α ξ¨α, 2 − 1 2 2 2 − 2 Q (s)=ξα(ξα)2 (ξα)2 (ξα)3+(ξα)2ξα+ξαξ˙α. 3 1 2 − 1 − 1 1 2 2 2 3.5 Example Example 3.1 Next, let us consider the following unit speed curve w =w(s) in E3 as follows 1 w(s)=(s,√2ln(sech(s)),√2arctan(sinh(s))). (3.25)