Journal of the Egyptian Mathematical Society 25 (2017) 382–390 Contents lists available at ScienceDirect Journal of the Egyptian Mathematical Society journal homepage: www.elsevier.com/locate/joems Original Article Computation of Smarandache curves according to Darboux frame in Minkowski 3-space H.S. Abdel-Aziz a, M. Khalifa Saad a , b , ∗ a Math. Dept., Faculty of Science, Sohag University, 82524 Sohag, Egypt b Math. Dept., Faculty of Science, Islamic University in Madinah, Saudi Arabia a r t i c l e i n f o a b s t r a c t Article history: In this paper, we study Smarandache curves according to Darboux frame in the three-dimensional RReevceisievedd 9 6 M Aapyr i2l 021071 7 Mgaitnek soowmske i spspeacciael SEm 13 . aUrasnindga cthhee cuusruvaels tforarn as fgoirvmena ttiiomn ebliektew ceuernv eF rlyeinnegt faunlldy oDna rab otiumx eflrikaem seusr, fawcee . iFnivneasltlyi-, Accepted 28 May 2017 we defray a computational example to confirm our main results. Available online 7 June 2017 ©2017EgyptianMathematicalSociety.ProductionandhostingbyElsevierB.V. MSC: This is an open access article under the CC BY-NC-ND license. 53A04 ( http://creativecommons.org/licenses/by-nc-nd/4.0/ ) 53A35 53C50 Keywords: Smarandache curves Timelike curves Timelike surfaces Darboux frame Minkowski 3-space 1. Introduction (Frenet frame vectors) are respectively, the tangent, the principal normal and the binormal vector fields. In the study of the fundamental theory and the characteri- In the light of the existing studies in the field of geometry, zations of space curves, the corresponding relations between the many interesting results on Smarandache curves have been ob- curves are very interesting and important problem. tained by many mathematicians [3–9] . Turgut and Yilmaz have in- Among all space curves, Smarandache curves have special em- troduced a particular circumstance of such curves, they entitled it placement regarding their properties, because of this they deserve Smarandache TB 2 curves in the space E 41 [2] . They studied special especial attention in Euclidean geometry as well as in other ge- Smarandache curves which are defined by the tangent and second ometries. It is known that Smarandache geometry is a geometry binormal vector fields. In [4] , the author has illustrated certain spe- which has at least one Smarandache denied axiom [1] . An axiom cial Smarandache curves in the Euclidean space. is said to be Smarandache denied, if it behaves in at least two dif- Special Smarandache curves in such a manner that Smaran- ferent ways within the same space. Smarandache geometries are dache curves TN 1 , TN 2 , N 1 N 2 and TN 1 N 2 with respect to Bishop connected with the theory of relativity and the parallel universes. frame in Euclidean 3-space have been seeked for by Çetin et al. [6] . Smarandache curves are the objects of Smarandache geometry. By Furthermore, they worked differential geometric properties of δ definition, if the position vector of a curve is composed by Frenet these special curves and checked out first and second curvatures β δ frame’s vectors of another curve , then the curve is called a of these curves. Also, they get the centers of the curvature spheres Smarandache curve [2] . In three-dimensional curve theory, each and osculating spheres of Smarandache curves. unit speed curve with at least four continuous derivatives, one can Recently, H.S. Abdel-Aziz and M. Khalifa Saad have studied spe- associate three mutually orthogonal unit vector fields T, N and B cial Smarandache curves of an arbitrary curve such as TN, TB and TNB with respect to Frenet frame in the three-dimensional Galilean and pseudo-Galilean spaces [3,7] . The main goal of this article is to introduce and describe some ∗ EC-omrraeisl paodnddreinssge sa:u hthaobrd. [email protected] (H.S. Abdel-Aziz), special Smarandache curves in E 13 for a given timelike surface and a timelike curve lying fully on it with reference to its Darboux [email protected] (M.KhalifaSaad). http://dx.doi.org/10.1016/j.joems.2017.05.004 1110-256X/©2017EgyptianMathematicalSociety.ProductionandhostingbyElsevierB.V.ThisisanopenaccessarticleundertheCCBY-NC-NDlicense. ( http://creativecommons.org/licenses/by-nc-nd/4.0/ ) H.S. Abdel-Aziz, M. Khalifa Saad / Journal of the Egyptian Mathematical Society 25 (2017) 382–390 383 θ frame. We looking forward to that our results will be helpful to where is the angle between the vectors P and N . Therefore, the researchers who are specialized on mathematical modeling and Darboux frame of r ( s ) is given as follows (cid:4) (cid:5) (cid:4) (cid:5)(cid:4) (cid:5) other applications. T (cid:9) 0 κg κN T P (cid:9) = κg 0 −τg P , (3) 2. Basic concepts U (cid:9) κN τg 0 U κ κ τ Let us first recall the basic notions from Lorentz geometry [10] . where N , g and g are the normal curvature, geodesic curvature spaLceet, aRn 3d = le {t ( xx 1 = , x ( 2x , 1x , 3 x ) 2 | , xx 3 1 ,) xa 2n , dx 3 y ∈ = R ( y}b 1 ,e y 2a , y3 3- d) ibmee tnwsioo nvaelc tvoercst oinr asunrdf agceeos,d feosri ca tcourrsvioen r, =re rs(pse )c tlyivienlgy .o Inn at hsue rfdaicffee rMen tthiael fogelloomweintrgy a oref R 3 . The Lorentz scalar product of x and y is defined by well-known [11] (cid:4) x , y (cid:5) L = x 1 y 1 + x 2 y 2 −x 3 y 3 . (i) r ( s ) is a geodesic curve if and only if κg = 0 . E 13 = (R 3 , (cid:4) x , y (cid:5) L ) is called 3-dimensional Lorentzian space, ((iiiii)) rr (( ss )) iiss aa np raisnycmippatlo ltiinc el iinf ea nifd a nondl yo nifl yτ gif =κ N0 .= 0 . Minkowski 3-space or 3-dimensional Semi-Euclidean space. The ar- bitrary vector x in E 13 can have one of three Lorentzian causa(cid:4)l cha(cid:5)r- Definition 2. A regular curve in Minkowski 3-space, whose posi- acters; it can be spacelike, timelike and lightlike (null) if x, x L tion vector is composed by Frenet frame vectors on another regular > 0 or x = 0 , (cid:4) x , x (cid:5) L < 0 and (cid:4) x , x (cid:5) L = 0 and x (cid:6) = 0, respectively. curve, is called a Smarandache curve [2] . Similarly, a curve r , locally parameterized by r = r(s ) : I ⊂R −→ E 13 In the following, we investigate Smarandache curves TP , TU , PU where s is pseudo arclength parameter, is called a spacelike curve if (cid:4) r (cid:9) ( s ), r (cid:9) ( s ) (cid:5) L > 0, timelike if (cid:4) r (cid:9) ( s ), r (cid:9) ( s ) (cid:5) L < 0 and lightlike if and TPU and study some of their properties which represent the (cid:4) r (cid:9) (s ) , r (cid:9) (s ) (cid:5) L = 0 and r (cid:9) ( s ) (cid:6) = 0 for all s ∈ I . The two vectors main results. x(cid:4) x = , y(x (cid:5) 1 L , x= 2 ,0 x . 3A ) ,ls oy, =fo (r ya 1n , yy 2x , ,y y 3 )∈ ∈ E E 13 13 , Laorere notrzthiaong ovnecatlo irf parnodd uocnt loyf ixf 3.1. TP −Smarandache curves and y is defined by (cid:2)(cid:2)(cid:2)e 1 e 2 −e 3 (cid:2)(cid:2)(cid:2) defiFnreodm b tyh e Definition 2 , the TP − Smarandache curve of r can be x ×L y = (cid:2)(cid:2)x 1 x 2 x 3 (cid:2)(cid:2). y 1 y 2 y 3 α( s¯ ) = √1 ( T + P ). (4) (cid:3) 2 The norm of a vector x ∈ E 13 is given by (cid:10) x (cid:10) L = |(cid:4) x , x (cid:5) L |. We Differentiating Eq. (4) with respect to s , we obtain denote by { T, N, B } the moving Frenet frame along the curve r ( s ) itnh et hvee cMtoirnsk oowf sthkei stpaancgee nEt 13, ,p wrinhceirpea lt hneo rvmecatlo rasn dT , tNhe, Bb inaroer mcaalll eodf α(cid:9) = ddα s¯ dd ss¯ = √12 ( κg T +κ g P + ( κN −τg ) U ), r , respectively. Consider now the following definition that we are with the parameterization needed throughout this study. Definition 1. A surface M in the Minkowski 3-space E 13 is said to dd ss¯ = √12 ( κN −τg ), (5) be spacelike, timelike surface if, respectively the induced metric on the surface is a positive definite Riemannian metric, Lorentz met- and then rsiucr. faInce o itsh ear t iwmoerldikse, t(hsep ancoerlimkea)l vveeccttoorr [o1n0 ] .t he spacelike (timelike) T¯ α = ( κN 1− τg ) ( κg T +κ g P + ( κN −τg ) U ). (6) Differentiating Eq. (6) with respect to s and using Eq. (5) , we get 3. Smarandache curves of a timelike curve on a timelike √ surCfaocnes ider a timelike curve r = r(s ) in E 13 , parameterized by its dd T¯ s¯ α = ( κN− −τ2g ) 3 ( ω 1 T + ω 2 P + ω 3 U ), arc length s and lying fully on an oriented timelike surface M . Let where (cid:6) (cid:6) (cid:7)(cid:7) Ta,l oNn,g B r (b se ). tThhee nta, nFgreennte,t pfrrianmciep aisl ngiovremn abl ya nd binormal vector fields ω 1 = τg(cid:9) κg −κN(cid:9) κg + ( κN −τg ) κg(cid:9) + κg 2 + κN 2 −κN τg , (cid:4) (cid:5) (cid:4) (cid:5)(cid:4) (cid:5) NT (cid:9) (cid:9) = κ0 κ0 τ0 NT , (1) ω 2 = (cid:6)τg(cid:9) κg −κN(cid:9) κg + ( κN −τg )(cid:6) κg(cid:9) + κg 2 + ( κN −τg )τ g (cid:7)(cid:7), B (cid:9) 0 −τ 0 B where a prime denotes differentiation with respect to s . For this ω 3 = κg ( κN −τg )2 . frame the following are satisfying α The curvature and torsion of are given as follows (cid:4) T(cid:4) T , N , T (cid:5) (cid:5) == −(cid:4) T1 , ,B (cid:5)(cid:4) B= , B(cid:4) N (cid:5) ,= B (cid:4) (cid:5) =N , 0 N . (cid:5) = 1 , κ¯α = (cid:8)(cid:8)(cid:8)(cid:8)dd T¯ s¯ α (cid:8)(cid:8)(cid:8)(cid:8)= ( κN −1 τg ) 3 (cid:9) 2 (cid:6)−ω 12 + ω 22 −ω 32 (cid:7), (7) κ τ The coefficients and are the curve’s curvature and torsion. and genLte tv e{c Tt,o rP , oUf }r baen dth Ue Disa rtbhoeu xu nfirta mnoer mofa lr (t so ) wthieth s uTr faasc et hMe taannd- N¯ α = ω(cid:3) 1 T + ω 2 P + ω 3 U . P = T ×U, then the usual transformation between Frenet and Dar- −ω 12 + ω 22 −ω 32 boux frames takes the form [10,11] : (cid:4) (cid:5) (cid:4) (cid:5)(cid:4) (cid:5) On the other hand, we express PT = 10 cos0h θ sin0h θ NT , (2) B¯ α = −(cid:6)T¯ ×N¯(cid:7) = (cid:3) −1 U 0 sinh θ cosh θ B ( κN −τg ) −ω 12 + ω 22 −ω 32 384 H.S. Abdel-Aziz, M. Khalifa Saad / Journal of the Egyptian Mathematical Society 25 (2017) 382–390 (cid:2) (cid:2) So, the(cid:2)(cid:2)(cid:2)(cid:2) ωκbTg 1i n ormωκPg 2a l ve(c κtNo−ω r− U 3oτ fg α)(cid:2)(cid:2)(cid:2)(cid:2) .i s 3. Ifκ¯ αα =is (cid:16)a p 4r κinN(cid:9) c iκpga l− li2n κe,N t (cid:6)hκ2eN κ f3g(cid:9)o +llo 3w κing 2g +ho κldN 2 (cid:7), (cid:17) (cid:6) (cid:7) (cid:18) B¯ α = ( κN −τg ) (cid:3)− −ω1 1 2 + ω 22 −ω 32 ( ω¯ 1 T + ω¯ 2 P + ω¯ 3 U ), τ¯α = κN4 −κN √(cid:6)κ N(cid:6) (cid:6)κ3g (cid:6) κ5N(cid:9) 2 κ κg(cid:9) g+ + 2 κ (cid:6)Nκ(cid:9) (cid:6)g 23 κ+g(cid:9) κ+N 52 (cid:7) κ(cid:7)g 2+ κκNg(cid:9) (cid:9) (cid:7)+(cid:7)(cid:7) κg κN(cid:9)(cid:9) (cid:7) . where 2 2 4 κN(cid:9) κg −2 κN 2 κg(cid:9) + 3 κg 2 + κN 2 ω¯ 1 = ( τg −κN ) ω 2 + κg ω 3 , ω¯ 2 = ( κN −τg ) ω 1 −κg ω 3 , 3.2. TU −Smarandache curves ω¯ 3 = κg ω 1 −κg ω 2 . Let r = r(s ) be a timelike curve lying on an oriented timelike We consider the derivatives α(cid:9) (cid:9) , α(cid:9) (cid:9) (cid:9) with respect to s as follows surface M in Minkowski 3-space E 13 . Using Definition 2 , the TU − (cid:10) Smarandache curve of r is given by α(cid:9)(cid:9) = √12 (κg(cid:9) −τg κN + κN2 + κg2 ) T +(cid:11) (κg(cid:9) −τg2 + τg κN + κg2 ) P β( s¯ ) = √1 ( T + U ), (9) +(−τg(cid:9) + κN(cid:9) −κg τg + κg κN ) U , 2 it leads to α(cid:9)(cid:9)(cid:9) = √1 2 ( λ1 T + λ2 P + λ3 U ), T¯ β = (cid:3) −2 κN 2 +1 ( κg + τg )2 ( κN T + ( κg + τg )P +κ N U ). (10) where (cid:6) (cid:7) By taking the derivative of Eq. (10) with respect to s , we have λλ12 == −τg(cid:9)2 ( τ κgN(cid:9) κ−N 3+ τ gκ )N(cid:9) + ( 3 2 κ κNN(cid:9) −τg τ+g ) κ +g (cid:6) κ3g κ g(cid:9)3 + κg (cid:9)κ +g 2 κ +g 2 κ +N 2 κ −N 2τ −g 2 (cid:7)τg+ 2 κ+g(cid:9)(cid:9) ,κ g(cid:9)(cid:9) , dd T¯ s¯ β = (cid:6)−2 κN 2 +√ ( κ2 g + τg )2 (cid:7)2 ( (cid:11)1 T + (cid:11)2 P + (cid:11)3 U ), where⎛ (cid:6) (cid:7)⎞ λ3 = −−ττgg(cid:9)(cid:9) (cid:9) κ +g + κ N(cid:9)κ(cid:9) .N(cid:9) κg + ( κN −τg )(cid:6) 2 κg(cid:9) + κg 2 + κN 2 −τg 2 (cid:7) (cid:11)1 = ⎝ κg 4 + +3 κ κgN 3 (cid:6) τ+−g κ 2+g κ (cid:6) κN− N3(cid:9) ( (−κ κg(cid:9)g( +κ+g (cid:9) τ τ+g(cid:9)g ) )τ κ2 g (cid:9)N −) τ+κg τg+ 2g 3κ κ(cid:7)NN τ 2g 2− (cid:7)3 τg 2 ⎠ , α (cid:6) (cid:6) (cid:7)(cid:7) Iwτ¯nαh t=ehre e− li(g2 κh √Nt 2 −o fκτ ¯tαg2h ) e− (cid:12) 3aχχb τ(cid:6)12og(cid:9) 2(cid:13)v κe, g, t−he3 κtoN(cid:9) 2r κsigo −n κofN(cid:9) ( κNis −giτv(cid:7)ge )n by (8) (cid:11)(cid:11)23 == κ(cid:17)N− κ−−g(cid:9)2 (cid:6)κ(−κNκ g((cid:9) N κ+ 2g τ++g(cid:9) )ττ κgg ( N) κ −+g +τ(g κ(cid:9) τgκg + )N(cid:7) ( τ(cid:6) κ−gg ) 2 + κ2 τN κ g2N(cid:9) ) + −+ ( 2κ κ κN(cid:9)g N (+ 2 κ g+τ +g () κ2 τ (cid:7)gg )+2 (cid:18) τg. )2 , χχ12 == (cid:14)+5 κ τgg(cid:9)(cid:9) κ (cid:6)+g6 ( κ κ κgN(cid:9)N (cid:6) κ 2−g ( ×+ κτN g(3 ) κ− κ +Ng(cid:9)τ +−2g ) κ +5τ2 g (g κ 3 ( κ) ( κg(cid:6) κN2N3 N ++ κ− − g (cid:9)(τ τ+ κgτg N) g )5 ) −− κ +gττ 2 g(g(cid:9) (cid:9)+ κ) + τN κg κ −NN (cid:9)2(cid:9)τ (cid:7)−g ) κκgN(cid:9)(cid:9) τ(cid:15)g. (cid:7) (cid:7) , κTκN¯¯¯hββ βe,= r=τe¯ β f(cid:8)(cid:8)(cid:8)(cid:8)o(cid:3)(cid:11)adr1der T T¯e, s ¯ β+ fer(cid:8)(cid:8)(cid:8)(cid:8) ox(cid:11)pm=2r P ea (cid:6)s+f−so e(cid:11)r2de3 κ mUaNs e2. nf+oti lo(l1o κn wge ds+ eτqgu )2 a (cid:7)ti2o (cid:9)ns 2, (cid:6)t−he(cid:11) 12c u+r v(cid:11)a22t u−re(cid:11) 32f u (cid:7)n, ctions Thus we can state the following Corollary. −(cid:11)2 + (cid:11)2 −(cid:11)2 1 2 3 Corollary 1. Let α( s¯ ) be a timelike curve lies on a timelike surface M β Besides, the binormal vector of is in1. MIfi nαkoisw (cid:16)ask g i e(cid:6)3o-dsepsaicc ec uE 1r3v ,(cid:7) et, htehne n the equations B¯ β = (cid:3) −2 κN 2 + ( κg + −τg1 ) 2 (cid:3) −(cid:11)12 + (cid:11)22 −(cid:11)32 ( (cid:11)¯1 T + (cid:11)¯2 P + (cid:11)¯3 U ), κ¯α = 2( κτgN 2 − −τκg(cid:6) N) 2 2 , (cid:7) w(cid:11)¯1h =er eκ N (cid:11)3 + ( κg + τg )(cid:11) 1 , τ¯α = −( κN −4 √τg2 )(cid:6) 5 τ gτ 2g (cid:9)− κNκ −N 2κ (cid:7)N(cid:9) τg , (cid:11)(cid:11)¯¯23 == κ( κNg (cid:11) +1 − τgκ )N(cid:11) (cid:11)1 3− , κN (cid:11)2 . β After differentiate with respect to s , we get are hold. (cid:14) (cid:15) 2. Ifκ¯ αα =is (cid:16)an 2 a (cid:6)s2y τmg(cid:9) pκtogt −ic τlign (cid:6)e2,τ κgt h3g(cid:9) e+ f o3l κlogw 2 i−ngτ gh 2o (cid:7)l(cid:7)d , βsim(cid:9)(cid:9) =ila r√l1y2 , (κN(cid:9) + τg κg + +κN(2 κ+N(cid:9) κ−g2 κ) Tg τ +g −(κτg(cid:9)g 2+ + τ κg(cid:9) N+2 ) Uτg κN + κN κg ) P , (cid:17) (cid:6) (cid:7) (cid:18) τ4 (cid:6) (cid:6)−(cid:6)3 τg(cid:9) 2 κg −τg(cid:9) 3 κg(cid:9) + 5(cid:7) κg 2 τ(cid:7)g (cid:7) β(cid:9)(cid:9)(cid:9) = √12 ( μ1 T + μ2 P + μ3 U ), τ¯α = g + τg4 √τg2 (cid:6) κ2g τ g5(cid:9) κκg(cid:9)g +− 2τ κg (cid:6)g 22 κ−g(cid:9) 2+ τ 3g 2 κ g +2 − κgτ(cid:9)(cid:9) g 2+ (cid:7)(cid:7) κg τg(cid:9) (cid:9) . wμh1 e=re 2 τg(cid:9) κg + κg(cid:9) ( 3 κg + τg ) + κN (cid:6)3 κN(cid:9) + κg 2 + κN 2 −τg 2 (cid:7)+ κN(cid:9)(cid:9) , H.S. Abdel-Aziz, M. Khalifa Saad / Journal of the Egyptian Mathematical Society 25 (2017) 382–390 385 (cid:6) (cid:7) where μμ23 == −(κ2g(cid:9) κ +g(cid:9) τ τgg (cid:9)− ) κτNg(cid:9) (+ κ g( κ+g 3+ τ τg )g )+ 2 κ κNN(cid:9) (cid:6) 3+ κ κN(cid:9) g + 2 + κ gκ 2N + 2 −κNτ 2g 2− τ+g 2 κ (cid:7)g(cid:9)(cid:9)+ + κ τN(cid:9)(cid:9)g (cid:9). (cid:9) , (cid:13)1 = ⎛⎝ −−κκNg (cid:6) (cid:6)4κ κgN (cid:9)2 2 κ−g 2+ κ N7 2 κ (cid:7)N(cid:9) κ κN(cid:9)g(cid:9) 3+ + κ 2g(cid:9) (cid:6) κκg N3(cid:9) 3 (cid:6)(cid:6) κκ3gg κ(cid:9) 2 2g κ 2+g +κ κN N4 2 κ (cid:7)N +2 (cid:7) (cid:6)+2 κκN(cid:9)N (cid:6)+7 κκgg 22 κ+N 2+ κ 2N κ2 (cid:7)N(cid:9)(cid:9)κ (cid:7)g(cid:9)(cid:7)(cid:9) (cid:7)⎠⎞ . β Following that, the torsion of is obtained as wτ¯βhe=r e2 √12 κ ¯β2 ( ξ1 + ξ2(cid:6) −ξ3 ), (cid:7) (11) 3Le.3t. A {P Tss,U uP −m, SUem }t ahbraaetn Ddγaarc=bh oeγu c(xu s¯ )rf vriaesms ae t iomf eγli.k Teh ceunr vbey lDyienfign fiutilolyn o2n , tMhe i nP UE 1 −3 . (cid:6) −κN 2 κN(cid:9) + κg 2(cid:6) + 2 κN 2 −τg 2 (cid:7) Smarandache curve of γ is identified by ξ1 = (κ(cid:6)g(cid:9) + τg(cid:9) ) κN + ( κg ++ κ τg(cid:9)g(cid:9) )+ 2τ κg(cid:9)(cid:6) (cid:9)N(cid:9) (cid:7) ,+ κg 2 + κN 2 −τg 2 (cid:7)(cid:7) γ( s¯ ) = √12 ( P + U ). ξ2 = (cid:6)2 τg(cid:9) κ(κg g(cid:6)(cid:9)+ + κ τg(cid:9) g((cid:9) ) 3 κ κNg ++ τ( κg )g ++ κτNg ) (cid:6) 3κ κN(cid:9)N (cid:9) ++(cid:6) 2 κ κgN 2 2+ − κτNg 2 ( −κg τ+g 2 τ (cid:7)(cid:7)g+ (cid:7)) κN(cid:9)(cid:9) (cid:7), Ta¯n γd= ( ( κg (cid:3)+ κN )T +τ g P −τg U ) . (12) ξ3 = (cid:6)−2 κg(cid:9) τg− −(κτgg(cid:9)(cid:9) (+ κ gτ g+(cid:9) ) κ3N τ g+ ) +( κ κg N+ (cid:6) 3τ κg )N(cid:9) +κN(cid:9) κ +g 2 κ+g ( κ κNg 2+ − ττg )g 2 (cid:7)+ κN(cid:9)(cid:9) (cid:7). Differentiati√n−g (E κqg. +(1 κ2N) )w2 ith respect to s , we have CorTohlluasr yw e2 . caLent gβiv(e s¯ )t hbee foa lltoimwienligk eC ocruorlvlea rlyi.e s on M in Minkowski dd T¯ s¯ γ = ( κg + 2κ N )3 ( ζ1 T + ζ2 P + ζ3 U ), 3-space E 13 , then where (cid:6) (cid:7) 1. If β is (cid:16)a geodesic curve, the following are satisfied ζ1 = τg κN2 −κg2 , κ¯β = (cid:6)τg 2 −2 χ2¯1 κ N 2 (cid:7)3 , ζ2 = (cid:6)(κg(cid:9) + κN(cid:9) ) τg −( κg + κN )(cid:6) τg(cid:9) + κg ( κg + κN ) −τg 2 (cid:7)(cid:7), (cid:6) (cid:6) (cid:7)(cid:7) wχ¯1h e=re 4 κN 6 +−52 κ τNg(cid:9) 2 2 τ κgN 4 2 − −τ2g 6 κ N+(cid:9) 2 τ 2g κ2 N(cid:9)− τg8 (cid:6) κ2N τ 4g(cid:9) τ κgN 2 +− τ2g τ 3g(cid:9) (cid:7) κN τg 3 , ζT3h e= c u(cid:8)−rv(aκtgu(cid:9) (cid:8)r+e κoN(cid:9)f )γ τg i+s d(e κtge +rm(cid:9) κinN e)d τ bg(cid:9) y− κN ( κg + κN ) + τg 2 . (cid:8) (cid:8) (cid:6) (cid:7) τ¯β = 4 √χ2 ¯2 χ¯3 , κ¯γ = (cid:8)(cid:8)dd T¯ s¯ γ (cid:8)(cid:8)= ( κg +1 κN )3 2 −ζ12 + ζ22 −ζ32 . where ⎛ (cid:6) (cid:7) ⎞ Further, we define the principal normal and the binormal vectors χ¯2 = ⎜⎜⎜⎜⎜⎜⎝ −(cid:6)κN(cid:9) τ(cid:6)g τ−+g(cid:9) κ(cid:6) (cid:6)τκNτg(cid:9) gNκ⎛(cid:9)+κ κ (cid:6)NN N3 (cid:7)τ κ(cid:6)g+τ (cid:6)−N(cid:9)g 2 τ 23+ κ g τ− (cid:6)N (cid:9)κg +(cid:9)κ τ+2NN(cid:9) κg κ2 Nκ +(cid:9)+N(cid:9)(cid:9)− (cid:7)N −2κ 2τ κNg−2 (cid:6)N2 κ 3(cid:7)2τN κg+− 2 2N(cid:9) (cid:7) κτ++gN(cid:9)(cid:9) 2 (cid:7)κ (cid:7)τN(cid:7)g(cid:9) 2 (cid:9)⎞ (cid:7) −τg 2 (cid:7)⎟⎟⎟⎟⎟⎟⎠ , aBN¯¯s γ γ f=o=l l o(cid:3)(cid:3)ζw1 − − Ts ( ζ+ κ12 g ζ ++2 P ζκ 2+2N − )ζ2 3− (cid:3)ζ U312 − , ζ12 + ζ22 −ζ32 (cid:6)ζ¯1 T + ζ¯2 P + ζ¯3 U (cid:7), χ¯3β = (cid:6)τg 2 −12 κN 2 (cid:7)3 ⎜⎝ 4 κ+−N 286 +κ κ−NN(cid:9)5 2τ4 κ τg τN (cid:6)gg(cid:9) 222 2 τ κ τ−ggN(cid:9) 4 κ 22 − N τ− g+(cid:9)τ κ2g N τ κ6 τgN(cid:9) 3 2g 3(cid:7)τ g 2 ⎟⎠ . wζζζ¯¯¯123h ===er eτ−τ ggτ ζζg31 ζ +−1 − τ( κg( ζg κ 2+g , +τ gτ )gζ )2ζ . 3 , 2. If is(cid:16) a n a(cid:6)symptotic line, then (cid:7) If we differentiate γ(cid:9) to get γ(cid:9) (cid:9) , γ(cid:9) (cid:9) (cid:9) , then we obtain the torsion of τ¯βκ¯β= =− 4 √−2 2(cid:6) κκgg 6 6 ++ 4 4 κ κgg 5 5 τ τgg ++( 7 κ7 κg κ g+g 4 4 τ ττggg 2 2 ) (+9 + κ (cid:6) g 8τ8 g+ κ(cid:9) κ κgg τg3 3 gτ τ− )gg6 3 3κ +g+(cid:9) τ 7g7 (cid:7) κ κgg 2 2 ττgg 44 ++ 44 κ κgg ττgg 55 ++ ττgg 66 (cid:7),. γγ(cid:9)(cid:9)a s= f o√+l1l(o2 −w (cid:10)τs(g κ(cid:9) g+(cid:9) + κ Nκ κN(cid:9) g+ + τ κg κN2 g− −ττg g2 κ ) UN ) (cid:11) T, + (τg(cid:9) + κg2 + κN κg −τg 2 ) P β 3. Ifκ¯ β =is (cid:25)(cid:26)(cid:26)(cid:26)(cid:27)a p 2ri (cid:28)nci4p κalN 6li n−+e,2 κt κgh 4N(cid:9)e 2 κ κfNog 2l 2(cid:6)l o−κ−wg 22iκ n κ−gg N6(cid:9) 2κa+ κgr e(cid:6)2N κ 2 κcg (cid:7)lg(cid:9) 3a 3 κ r −gifi 3 2eκd κN g(cid:9)− κN2 (cid:7) κg(cid:9) 2 κN 2 (cid:29), γwν(cid:9)h(cid:9)(cid:9)1e =r=e √21 τ2 g(cid:9) (( ν κ1g T− +κ νN 2) P + + ( κνg(cid:9)3 U− )κ, N(cid:9) ) τg + ( κg + κN )(cid:6) κg 2 + κN 2 −τg 2 (cid:7) τ¯β = 4 √2 (cid:17) 4 κN 6 −+2 κ κg 4N(cid:9)(cid:6) 2 κ κ κNgg 2 2 2 −−−22κ κ κg N6(cid:9)N κ +2 g(cid:7) (cid:6)32κ κ(cid:13)gg(cid:9) 3 κ1 − g 3 2κ κN g(cid:9)− κN2 (cid:7) κg(cid:9) 2 κN 2 (cid:18), ν2+ = κg (cid:9)2(cid:9) + κN(cid:9) κ κN(cid:9)g(cid:9) ,+ κg(cid:9) ( 3 κg + κN ) + τg (cid:6)−3 τg(cid:9) + κg 2 + κN 2 −τg 2 (cid:7)+ τg(cid:9) (cid:9) , 386 H.S. Abdel-Aziz, M. Khalifa Saad / Journal of the Egyptian Mathematical Society 25 (2017) 382–390 (cid:6) (cid:7) 3.4. TPU −Smarandache curves ν3 = 2 κg(cid:9) κN + κN(cid:9) ( κg + 3 κN ) − 3 τg(cid:9) + κg 2 + κN 2 τg + τg 3 −τg(cid:9) (cid:9) , Let r = r(s ) be a timelike curve lying on a timelike surface M τ¯γ = 2 √12 κ ¯γ2 ( (cid:17)1 + (cid:17)2 + (cid:17)3 ), (13) irn ( s )M. Ainckcoowrdsiknig 3t-os ptahcee dEe 13fi naintido n{ To, fP S, mUa } rabned tahceh eD caurbrvoeu,x t hfrea mTPeU o −f Smarandache curve of r is expressed as where⎛ (cid:6) ⎞ (cid:17)(cid:17)12 == ⎜⎜⎝⎛⎜⎜⎝⎛ ++− ττ−gg (cid:6)(cid:6)(cid:6)(cid:6)−+−3τ−−τ(cid:6)g τ κ(cid:9)32 2g(g(cid:9)g(cid:9)( τ ( κ κ κ(cid:6)κ+g κ(cid:9)gg(N(cid:9) (cid:9)g(cid:9) κg + κ +κκ(cid:6) +N+g gg κ κ 2+− κ+κg N+ N2 (cid:9)N κ κ) κ ) + ) NκN−g(cid:9) τ(cid:9) +)( N(gκ2 3 κ (2− Nκ κκ g− −2 Ngg(cid:9)+2 −2(++τ κ κ τ3gg τ gg2κκ τ κ 2g (cid:7)N+gNN(cid:9) 2 (cid:7) 2 (cid:7) + ))(cid:7))(cid:7) (cid:7)κ τ + gτN g )(cid:9)τ (cid:9)2 (cid:7)g(cid:9) ⎞⎟⎟⎠ (cid:9) (cid:7)⎟⎟⎠ , , ⎞ TδTD¯ddh δ(i T¯f ssi¯¯f δ=s )e ri==me( n (p4√ κt1li (gia3 κe+t s(eg T (cid:3)+κtE o+Nq − κ ). PNT 2(√ )1 + (+2 4 κ3 ( ( )gU κ κ w+gN ) . +i −κth Nττ )grg ()e ) Pκ s2 pN + ( e ξ− (c1 κt T τNt g o+− ) sξτ 2ag P n)U d+ )u ξ.s 3e U E )q, . (3) , we obta(i1n4 ) (cid:17)3 F=ro ⎝m +th (e κ ga +b(cid:6)o2 κv τNeg−(cid:9) ()c (cid:6) κa2κgl τcg− 2ug(cid:9) l++κaNt κκi )ogN n+ 2 2 s − ,−( wκκτg(cid:9)eN g− 22c (cid:7)aκτn+N(cid:9)g )iκn τg(cid:9)tg(cid:9) r +od κuN(cid:9)c(cid:9) e(cid:7) ⎠th .e following re- wξ1h =er e(− κ(g κ +N κ−N τ)g(cid:6) )−(cid:6) τκgg(cid:9)(cid:9) ( + κg 2 + (cid:6)κ κgN 2 )+ + κ κNN (cid:9)2 (+ κ g( + κg τ−g )κ N )τ g (cid:7)(cid:7)(cid:7), sult: ⎛ ⎞ Corollary 3. Let γ( s¯ ) be a timelike curve lies on M in Minkowski 3- τg(cid:9) ( κg + κN ) ( κg + 2 κN −τg ) s1p.a cIsfep eγEc 13t i i,vs ethalye nge odesic curve, the curvature and torsion of γ are, re- ξ2 = −( κg + κN )⎜⎝ −κ+N(cid:9) 2 ( ( κ κg g+ + 2 (cid:6)κ κκNNg(cid:9) ) (− ( κ κgτg +g+ ) (2 κ κ κNgN +− − τττgg g )) ) (+ κ g( κ+N τ−g )τ )g ) ⎟⎠ , (cid:16) (cid:6) (cid:7) (cid:6) (cid:7) (cid:6) τ¯κ¯γγ== 2 √2 κ2 1N (cid:17)3 42 ⎛⎜⎜⎜⎝ τg(cid:9) +−+ τκ κ−gN++N(cid:9) (cid:6) 2 2τ− τ κ τggN(cid:9)− κ(cid:6)g (cid:6)(cid:9) 2 κ2Nκ4 (cid:6) κ N3N κ(cid:9)− N 2 N+(cid:6)(cid:9) ττκ κ3 ggNN2 κ −2−τ κN τg (cid:9)N2g (cid:9)κ23 τ−+N+ τ κg 5g N 2 2 (cid:9) τ32 2 κ −g τ+ κN κg N2 N2(cid:9)24(cid:9) τ (cid:7) κ (cid:7) τ τgNg(cid:9) g(cid:9)(cid:9)(cid:9) (cid:9)(cid:7)− (cid:7)(cid:7)κ⎞⎟⎟⎟⎠N 2 , τg 2 + 8 κN(cid:9) τg 3 , ξTtκ¯i3hδv ee==nly ,(cid:8)(cid:8)(cid:8)(cid:8) (−t κdh(dN eT κ¯ s¯ − δN c (cid:8)(cid:8)(cid:8)(cid:8)u−τrg=v τ)a g4t τ)u (cid:6) (g(cid:9)r κ−(e κg κ ga+g (cid:9)n + +κd κ N2p N) (1r 2 ) κi ( n−g κ c+Niκp −N(cid:9)κa (lN τ κ )ngg ( o) κ+2 r gm(cid:9) τ+ag 3 l)κ (cid:6) vN−e +ξc1t2 o τ +rg ) oξ(cid:7) f2(cid:7)2 . δ −aξr32e (cid:7), ,r esp(1e5c-) where and (cid:17)4 = (cid:4) 2 κN 3 (cid:6)2 τg(cid:9) −κN 2 (cid:7)−4 κN(cid:9) κN 2 τgκ −N 52 κN (cid:6)4 τg(cid:9) −κN 2 (cid:7)τg 2 + 8 κN(cid:9) τg 3 (cid:5) . N¯ δ = (cid:3)ξ1 − T ξ+12 ξ+2 P ξ 2+2 − ξ3ξ U32 . δ γ Also, the binormal vector of is given by 2. If is an asymptotic line, we get κ¯γ = (cid:16) 2 κg 3 (cid:6)2 τg(cid:9) + κg 2 (cid:7)−4 κg(cid:9) κg 2 τg κ−g 52 κg (cid:6)4 τg(cid:9) + 3 κg 2 (cid:7)τg 2 + 8 κg(cid:9) τg 3 , B¯ δ = (cid:3) −2 ( κg + κN ) ( κN −−1τ g )(cid:3) −ξ12 + ξ22 −ξ32 (cid:6)ξ¯1 T + ξ¯2 P + ξ¯3 U (cid:7), (cid:12) (cid:13) (cid:17) where τ¯γ = 2 √1 2 (cid:17)56 . ξ¯1 = ( κg + τg )ξ 3 −( κN −τg )ξ 2 , w(cid:17)5h e=r e⎛⎜⎜⎜⎝(cid:4) + τ(cid:6)g (cid:14)−κg− (cid:17)κg− (cid:6)(cid:7)1κ20g (cid:6) +2 ττ g−(cid:9) τg(cid:9) 2 g+ 4(cid:6)+7 τ3 7 κg κ 2 τg g(cid:7) 2g (cid:9)23τ τκ (cid:7) κgg(cid:9)gτ (cid:9) g 2(cid:9)−g+ 2 2+ κ 2κ+g τ2(cid:6) τg(cid:9) g κ(cid:6)4g 3 τg τ 4g+g(cid:9) (cid:6) 4 3 2 κ τ(cid:18)gg(cid:9) (cid:9)2 (cid:7)− −(cid:7)(cid:7)(cid:6)22 τ τgg(cid:9) 2 + (cid:7) κg 2 (cid:7)κg(cid:9)(cid:9)(cid:5) (cid:15) ⎟⎟⎟⎠⎞ , ξξTδ¯¯23h(cid:9)(cid:9) e=== d ((e√ κκ1ri Ng v (cid:30)+a−t iτvτ+geg( ) (s)κξ κ ξ gδ1(cid:9)1g (cid:9) (cid:9) +− (cid:9)+− , κ δ(τ( κN(cid:9)(cid:9) κg (cid:9)(cid:9) g(cid:9)g++ o ++f κκ δ κκNg22 N Na++ ) r)ξ e ξ κκ2 3 gN.2 , κ+g τ+g κκgN −τg τ−g κτNg 2) T) P (cid:31) , (cid:17)6 = 2 κg 3 2 τg(cid:9) + κg 2 −4 κg(cid:9) κg 2 τg κ−g 52 κg 4 τg(cid:9) + 3 κg 2 τg 2 + 8 κg(cid:9) τg 3 δ(cid:9)(cid:9)(cid:9) = √13 ( η+1( T− +τ g(cid:9)η +2 P κ +N(cid:9) +η3 κ UN ) κ, g + τg κg + κN2 −τg 2 ) U γ 3 3. If is a principal line, the following hold (cid:16) (cid:6) (cid:7) where κ¯γ = 2( κκgg 2+ − κ(cid:6)κN N) 2 2 , (cid:7) η1 = 2+ τ (g (cid:9)κ (g κ +g − κNκ )N(cid:6) )κ +g 2 3+ κ Nκ(cid:9) κN N2 −−τκgN(cid:9) 2 τ (cid:7)g+ + κ κg(cid:9)g(cid:9) (cid:9) (+ 3 κκgN(cid:9) (cid:9) + , τg ) τ¯γ = ( κg +2 √ κ2 N (cid:12) )3 2 ((κ κ κgNg(cid:9) +2 κ − κgκN N)− 2 2 )κ (cid:13)g(cid:9) κN . η2 = κg(cid:9) ( 3 κg + κN ) + τg(cid:9) ( κN −3 τg ) H.S. Abdel-Aziz, M. Khalifa Saad / Journal of the Egyptian Mathematical Society 25 (2017) 382–390 387 Fig. 1. The timelike curve r ( u ) on the timelike surface M . Fig. 2. TP and TU −Smarandache curves. (cid:6) (cid:7) + ( κg + τg ) 2 κN(cid:9) + κg 2 + κN 2 −τg 2 + κg(cid:9)(cid:9) + τg(cid:9) (cid:9) , ⎛ 2 κg(cid:6) 4 ( κN −τg ) + 2 κg 3 ( κN −τg )τ g (cid:7) ⎞ ηIn3 t=h eκ+ Nl(cid:9) ( i ( κg κhNg t − +o τf3 g κ t)hN(cid:6) e 2)s κ−eg(cid:9) τ+dge(cid:9) (κr κigv g2a ++tiv 3κe τNsg ,2 ) t− heτ gc 2u (cid:7)r−veτ’sg(cid:9) (cid:9) t+or κsiN(cid:9)o(cid:9) .n of δ can be (cid:17)10 = ⎜⎜⎜⎝ + κg (cid:14) +2+2 κ κN κ g3N 2 τ (cid:6) g− (− κκNN2 + (cid:6)− κκ2Ngτ (cid:9) τ2(cid:9) g τg+ ) (cid:6)g2 τ2 τ( g+ κg 3(cid:9) (cid:9) N (cid:7) −κ ++Nτ (cid:6) gττ(cid:9)− (cid:9)gg + (cid:6))2 κ τ− κgg(cid:9)(cid:9) N3(cid:9) τ(cid:9)+ (cid:7)g−(cid:9) (cid:9) κ+τN(cid:9)g(cid:9) (cid:9) κ(cid:7) (cid:9) (cid:7)+N(cid:9)(cid:9) κN(cid:9)(cid:9) (cid:7) (cid:15) ⎟⎟⎟⎠ . computed as follows τ¯δ = 3 √3 κ ¯δ2 ( ( κN(cid:17) −7 +τg )(cid:17) (8 (cid:17) 9 + 2 (cid:17)10 ) ) , (16) Cspoarcoel lEa 13r ,y t4h.e nL et δ( s¯ ) be a timelike curve lies on M in Minkowski 3- δ where 1. If is a geodesic curve, the curvature and torsion can be expressed (cid:17)(cid:17)78 == 2⎛⎜⎜⎜⎜⎝ τ −g(cid:9) ⎛⎜⎝2 κ +N(cid:9) ( ( κ κNN− −−τgτ(cid:9)τ 2gg (cid:6) ) )3−(cid:30) ⎧⎨⎩ κ κκN (cid:9)g N(cid:6) (cid:9)2+ 3 2+− (( κ ( 3κg4 κ κg2 κg g+ +g ++ κ κ2 Nτ 4N κ κg+ κ) N)g(cid:9) ( (cid:12) N(κ2 κ 2 5 g+(cid:9)+−N κ κ( ++ 2κg 5 g6τ g κ−(cid:9)+ κ (−g κ(cid:9) 3 g)g2(cid:9)+g (cid:9) τ2 ( κ2 +τ κ+ κg (g κgκg ) 5 g τ (N+N+ ((cid:9) τ κ(cid:9) κg+ κ g + (cid:9))ggκ τ(cid:9)g τ +g−N−3 g2 ) τ )(cid:7)κ2τ κ gNg κ )N )N− +τ )gτ (cid:7)τg g2 ) (cid:13)(cid:31)⎞⎟⎠ ⎫⎬⎭ ,⎞⎟⎟⎟⎟⎠ , aτw¯κsδ¯h δf=eo=r(cid:4)l el3 o ( w √(cid:19)(cid:6) s3 8 2 κ κ¯δN2 3,3 ( κ (cid:19)N 1− τg )3 ,τ (cid:7)g(cid:9) 2 κN 2 −2 τg(cid:9) κN (cid:5) (cid:4) (cid:5) (cid:19)1 = κN(cid:9) −2 ( κN −τg )2 τg + κN(cid:9) 2 τg 2 −4 κN(cid:9) ( κN −τg )2 τg 2 , (cid:17)9 = +2 κg(cid:9) & −( κN −τg )(cid:12) −2κ κg(cid:9) 2g+ ( 2 κ τ +gN ( κ− κgN (τ 4+g ) κ τNg −) τg ) (cid:13)−τg(cid:9) (cid:9) + κN(cid:9)(cid:9) ’ , −4 κN 2 ( κN −τg ) 3 ( κN + τg ) 388 H.S. Abdel-Aziz, M. Khalifa Saad / Journal of the Egyptian Mathematical Society 25 (2017) 382–390 ⎛ ⎞ (cid:19)2 = ⎜⎜⎝ (cid:6)κN(cid:9) (cid:6)2 κ+κN−N (cid:9)22 2 κ2+ τN κg τ(N (cid:9)( κg −( 2 Nκ (cid:7)4 N− κ+ −N τ (+gτ κ )g N(cid:6) τ) −(cid:6) g− )(κ κτ−NNg τ )τ−g(cid:6) (cid:9)g (cid:9)(cid:9) κ 2+τ κNg N( τ) κ (τ g κκNg 2NN(cid:9)− (cid:9) +(cid:7)+τ+ τg2 g)(cid:9)2 τ (cid:9) τ (cid:7)g τg )g (cid:9)+ κN(cid:9)(cid:9) (cid:7)(cid:7) ⎟⎟⎠ . AA 12 == 2v √+2 c (o1s√ +h ( vu ) ) c +os (h1 ( u+ ) v+ ) scionshh ( (u2 ) u (cid:6) )√ −2 c−o2s√h c ( o3s uh ) ( u ) + 2 v sinh (u ) (cid:7), 2. If δ is an asymptotic line, we have √ 2 2 ( ( − 2v sinh (u ) + sinh (2 u ) , κ¯δ = 12 32 κg(cid:19) 3 τ3g 3 , A 3 = 1 + 2 v + cosh (2 u ) + sin√h2 (u ) + (1 + v ) sinh (2 u ) − sin√h (2 3 u ) , (cid:19) τ¯δ = 3 √3 4κ ¯ δ2 , and ( B 1 , B 2 , B 3 ) where (cid:17) (cid:6) (cid:7) (cid:18) U = (cid:20) , (cid:19)3 = + τg 2 τ(cid:6)κg(cid:9) 2g(cid:9) κ 2 g+ 2 −4 κ2g τ 2 g((cid:9) κ κgg τ +g τκgg(cid:9) ) +2 + 2 κ4g κ 2g (cid:9) − (cid:6)κ2g 2 τ g− 2 τg 2 (cid:7)(cid:7) , where 1 (cid:19)4 = ⎛⎝ +− ττg (cid:6)g(cid:9) 2κ κg(cid:9) g+ ( 3 2 κ κgg + ( κ 1g+ 0+ τ κ gτg(cid:9) ) (g 5−)(cid:7) κ(cid:6)2gτ τ+gg(cid:9) (cid:6) τ τκgg g((cid:9) ) κ+ +g (2 2κ κ κg(cid:9)g(cid:9) g (cid:7)2 −−τ2g τ )g ( 2 κ (cid:7)g ++ 2 τ τgg )(cid:9) (cid:9) (cid:7) ⎠⎞. BBB 123 === −1− √1+ − 22 ( v√1 + 2+ v c v os )isn hcho ( (s2uh u )( ) u +− ) s+√in 2 2h ss (ii2nn uhh ) ( ( .uu ) ) , 2 , 3. If δ is (a p ri(nc ipal line, we obtain Accorκding to Eq. (3) , the geodesicτ curvature κg , the normal curva- τ¯κδ¯δ== 3 √1 3 38 (cid:4) +κ2N κ 3 N((cid:9) κ κ(cid:19)gN +(cid:6)5 − κ5N κ )g(cid:9)3 −κ ,g κ −N(cid:9) 22 κ+ κg (cid:19)g( 3 3 κ+gκ 2+ (cid:6) κ4g κ(cid:9) +N ) 2 κg 2 (cid:7)κN −κg(cid:9)(cid:9) (cid:7)(cid:5) , tκugr e= (cid:4)+N T (cid:9)ca ,o nPsd(cid:5) h = t(hu e )(cid:3) (cid:6)g21 (cid:20)e o+2d (cid:6)4e−s vi 2c− st2ion √rhsi2 (ouv n )s (cid:6)in2gh v o (+fu )t√h (cid:7)(cid:7)2 e, sciunrhv (eu r ) ( (cid:7) u ) are 6 N where ⎛ (cid:6)(cid:6) (cid:7) ⎞ (cid:19)5 = ⎝ κN(cid:9) 2 κg− 2 8+ κ gκ 2N κ 2N 2(cid:6) κ−g(cid:9) 8+ κ 2g κ κNg 32 −2 +4 κ 4N κ 4g (cid:7)(cid:9) κ(cid:7)g κN ⎠ , κN = (cid:4) U (cid:9) , T(cid:5) ⎛ 4 √ 2 (cid:6)−4 + v (cid:6)−2 + 3 v + 6 v 2 (cid:7)(cid:7) ⎞ 4. C(cid:19)o6m =pu ⎛⎝ta +tκ+i4oN2 κn (cid:6)− (cid:6)gaκ κ22lg(cid:9) (cid:6) g κ2(cid:9)e κ N+(cid:9)x+ g κa 2 κ4mg + κg κ (pN g(cid:9)κ 2 l κeN κκg 2g (g(cid:9) (cid:7) κ++g− κ−22N κ κ2)g(cid:7) N κ(κ κκNN g(cid:9)g)(cid:9)(cid:9) (cid:9) +(cid:7)⎞⎠ κ .N ) = 2 ( (cid:20)12 )3 / 2 ⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝ +1+2 v2− ( √5√ 2 + 2 (8v + 4− c+3 v−o2 √ ) s14()h+2 11 (s 6(v v1−i46n (0 ) √+27 u hs s+(i 2)i (v+n4n 2 (− (3h√ h 4 u+44 v ( ( 4 )3 v+ u( 43 ( (7 − )+ u v553 ++ ) ) 2 v+2+ +c (cid:6) )44√ o 932c v v2s 2 o v v+ ))h( (s ( )))− ( 1h2 )2cc57 3 (c2oo+ u 4 os vs+ ) uhshv + h ) (((2 35(u v62 u ) )+ v u )s 2 ) i4(cid:7) n vshi )n ()5h) u (3 ) u ) ⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠, + 2 sinh (6 u ) In this example, we construct some special Smarandache curves ( TP, TU, PU and TPU ) of a timelike curve which lies on a timelike surface. Moreover, using Mathematica program, we compute their ⎛ √ (cid:6) (cid:7) ⎞ differential geometric properties ( Figs. 1–3 ). 1 −2 v − 2 −4 + v + 4 v 2 M(uSu, vp )p o=s er (wue ) +ar ev Qg(ivue )n , a timelike ruled surface represented as τg = (cid:4) P (cid:9) , U(cid:5) = (cid:20)12 ⎜⎜⎜⎜⎜⎝ + √c−2 o (s√−h2 (2 u( 7+ ) + +v )42 c v(o3 (s2 +h + ( 53 v v u ) ) )) c+ soi snchoh (s (2hu u ( ) )4 u ) ⎟⎟⎟⎟⎟⎠ . wr(hue )r =e t(cid:6)hue, √tim2 ecloikshe ub,a √se2 c suinrvhe u i (cid:7)s, given by −√2 + (23( +2 +4 v v ) ( s3in +h 2(3 v u ) )) s−invh s (i2n uh ) ( 4 u ) and (cid:6)√ √ (cid:7) where Q(u ) = 2 cosh u, 1 , 2 sinh u , ⎛ √ ⎞ 5 + 12 v (1 + v ) + 2 2√ c osh (u ) is thT(cid:6)he er ut√rlii npgle sv eocft oDr√a or bf oMu .x fra(cid:7)me can be computed as follows (cid:20)2 = ⎜⎝ + cos+h (44( u1 ) + − 32 v √ ) c2 o (s√1h + (2 6 u v ) ) − sin2 h (2u c )o +sh 4 ( s3i un )h (2 u ) ⎟⎠ T = 1 , 2 sinh u, 2 cosh u , −2 2 sinh (3 u ) P = ( A 1 , (cid:20)A 2 , A 3 ) , If we (choose u = 0 a(nd v = 0 , the curvatures ( κg , κN and τg ) are w(cid:20)1h =er (cid:25)(cid:26)(cid:26)(cid:27)e (cid:6)1 +1 2 v + cosh (2 u ) −−(cid:6)√1 2 + s i√nh 2v (u s )in (cid:7)2h +(u (cid:6) )√ −2 s (i1n h+ ( 2v u) )c (cid:7)o2s h (u ) −2 sinh 2 (u ) (cid:7)2 , κTPg − = Sm 25a r, aκnNd a=ch −e2 c ur 25v e, τg = 110 (cid:6)8 + 2 √2 (cid:7) . H.S. Abdel-Aziz, M. Khalifa Saad / Journal of the Egyptian Mathematical Society 25 (2017) 382–390 389 Fig. 3. PU and TPU −Smarandache curves. The TP− Smarandache curve can be computed as PU− Smarandache curve The PU− Smarandache curve can be computed as γPU = (cid:4)( γ √1 , γ2 , γ3 ), (cid:6)√ (cid:7)(cid:5) 2 (1 + v ) cosh (u ) + sinh (u ) 2 −2 cosh (u ) + 2(1 + v ) sinh (u ) γ1 = √ (cid:20) , 2 1 (cid:17) √ √ √ (cid:18) γ2 = 2 + 6 v + 2 cosh (u ) + 2(2 + v ) cosh (2 u ) −√ 2(cid:20) cosh (3 u ) −2 2 (1 + v ) sinh (u ) + 2 sinh (2 u ) , (cid:17) √ 2 2 1 √ (cid:18) γ3 = 4 v + 2 cosh (2 u ) + 2 (1 −2 v ) sinh√ ( u )(cid:20) + 2(2 + v ) sinh (2 u ) − 2 sinh (3 u ) . 2 2 1 αTP = ( α1 , α2 , α3 ), In this case, we get ( u = 0 and v = 0 ) where (cid:30) √ (cid:6)√ (cid:7)(cid:31) α1 = √1 1 + 2 2 (1 + v ) cosh (u ) + sinh (u(cid:20) ) 2 −2 cosh (u ) + 2 v sinh (u ) , 2 1 (cid:30) (cid:31) √ α2 = √ 2 sinh (u ) + √1 v + cos√h 2 ( u ) + (1 + v ) cosh (2 u ) −(cid:20)cos√h ( 23 u ) − 2v sinh (u ) + sinh (2 u ) , 2 1 (cid:30) (cid:31) α3 = √1 cosh (u ) + 1 + 2 v + cosh (2 u ) + sin√h 2 (u )(cid:20) + (1 + v ) sinh (2 u ) − sin√h ( 23 u ) . 2 1 Therefore, we get ( u = 0 and v = 0 ) κ¯γ = 2 . 204 , τ¯γ = 0 . 113 . κ¯α = 0 . 793 , τ¯α = 0 . 455 . TPU− Smarandache curve TU− Smarandache curve The TPU− Smarandache curve is given by βTUF o=r (t βhi1s , βcu2r ,v βe3, )w, e get δTPU = ((cid:6) δ21 √ , δ3 2(cid:20) , δ13 (cid:7) ) , wβ1h e=r e√ 1 (cid:17)1 + −√2 (1 + v ) cos(cid:20)h (u ) + 2 sinh2 (u ) (cid:18), δ1 = (cid:17) 2 (cid:6)√ 2 (1 + v ) cosh (u ) ++ 2√(cid:20) 2 1s i−nh2 ( sui )n h+ ( 22 us i)n h (u ) 2 + 2 v sinh (u ) 2 (cid:7) (cid:18), β2 = √122 (cid:17)√2 sinh (u ) + 1 + 2 v +1 cosh ((cid:20)2 u1 ) −√2 sinh (u ) (cid:18), δ2 = ⎛⎝ √ 2 c+os2h (cid:17) (u )1 + + 2 3(− v2 +√+ 2sv i( )n1 ch +o (s2 vh u ) ( ) 2+ + u (cid:20) ) s −i2n h√ (2 u c )o s(cid:18)h (3 u ) ⎠⎞, β3 = √12 (cid:17)√2 cosh (u ) + −1 −√2 v sin(cid:20)h (1u ) + sinh (2 u ) (cid:18), δ3 = (cid:17) 4 v + 2 cosh (2 u− ) √+ 2 √ s2 in (h1 ( −3 u2 ) v + ) s 2in(cid:20)h2 ( uco ) s+h (2u( )2 + v ) sinh (2 u ) (cid:18), after some calculations, we obtain ( u = 0 and v = 0 ) it follows that ( u = 0 and v = 0 ) κ¯β = 2 . 965 , τ¯β = 0 . 199 . κ¯δ = 1 . 319 , τ¯δ = 0 . 549 . 390 H.S. Abdel-Aziz, M. Khalifa Saad / Journal of the Egyptian Mathematical Society 25 (2017) 382–390 5. Conclusion [3] H.S. Abdel-Aziz , M.K. Saad , Smarandache curves of some special curves in the Galilean 3-space, Honam Math. J. 37 (2) (2015) 253–264 . [4] A.T. Ali , Special Smarandache curves in the Euclidean space, Int. J. Math.Comb. In the present paper, we have studied special curves called 2 (2010) 30–36 . Smarandache curves according to Darboux frame in the three- [5] O. Bektas , S. Yuce , Smarandache curves according to Darboux frame in Eu- dimensional Minkowski space E 13 . These curves are composed us- [6] cMli.d eÇaenti ns ,p aYc. eT, uRnoçmer. ,J .M M.Ka.t hK. aCroamcapnu , tS.Smcia. r3a n(d1a) c(h2e0 1c3u) r4ve8s– 5a9c .c ording to Bishop ing Frenet frame vectors of another curve. Moreover, some results frame in Euclidean space, Gen. Math. Notes 20 (2) (2014) 50–66 . for the meaning curves are obtained. Finally, for confirming our re- [7] M. 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