Computation of Smarandache curves according to Darboux frame in Minkowski 3-space

2017·0.81 MB·

by  H.S. Abdel-Aziz

Journal of the Egyptian Mathematical Society 25 (2017) 382–390
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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
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[4] A.T. Ali , Special Smarandache curves in the Euclidean space, Int. J. Math.Comb.
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[8] M. Çetin , H. Kocayi g˘it , On the quaternionic Smarandache curves in Euclidean
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Acknowledgment [9] K.T. prü, M. Tosun , Smarandache curves according to Sabban frame on S 2 , Bo-
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