Special equiform Smarandache curves in Minkowski space-time

2017·1.1 MB·

by  E.M. Solouma

Journal of the Egyptian Mathematical Society 25 (2017) 319–325
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Special equiform Smarandache curves in Minkowski space-time

E.M. Solouma a , b, ∗
a Al Imam Mohammad Ibn Saud Islamic University, College of Science, Department of Mathematics and Statistics, KSA, P. O. Box 90950, Riyadh 11623, Saudi
Arabia
b Department of Mathematics, Faculty of Science, Beni-Suef University, Egypt
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 introduce special equiform Smarandache curves reference to the equiform Frenet frame
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Available online 23 May 2017
illustrate these curves.
MSC: ©2017 Egyptian Mathematical Society. Production and hosting by Elsevier B.V.
53B30 This is an open access article under the CC BY-NC-ND license.
53C40
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Keywords:
Smarandache curve
Equiform Frenet frame
Minkowski space-time
1. Introduction 2. Preliminaries
A regular non-null curve in Minkowski space-time, whose posi- The Minkowski 3-space E 31 is the Euclidean 3-space E 3 provided
tion vector is composed by Frenet frame vectors on another regular with the metric
curve, is called a Smarandache curve [1] . Recently special Smaran-
dache curves have been studied by some authors [2–5] . G = −d z 12 + d z 22 + d z 32 ,
wscplaiatrIhcinfe y lri tekhtfehies er se wunbrcoafears kicct,e o wc Motehn ecisn eteup qtMdiuyoii nnfsokspr omeowcf i saFkMlri e ienn3qke-usto ipwffaorscrakmemi eE3S 31-m o.s pfaI naraac enSc deuEcar 31tcvi heoae nn ζ dc2u o , rgnvwiv eeeas wbpνiihtc=rtea s0rr;e ,y i t(av zne 1cd a,c tnzol 2 ir gb, hezν 3t l t)∈ii km iEes 31 ea ilf ic kraGeenc ( tiνfha, na Gνvg()euν =l,ao νrn0 )e c a <oono f0dr d , t νihsnrpae(cid:4) =ateec e0Lsl.oyi krSseetimen imtfiz l iaGaor(nflνy E,,c 31νla a. n)uA y>sn e ay 0r d baoeri--r-
of equiform Fremet frame that will be used during this work. trary curve ζ = ζ(s ) can be timelike, spacelike or lightlike if all of
Section 3 is delicate tηo thξe ηstξudy of ηthξe special four equiform its velocity vectors ζ(cid:5) ( s ) are timelike, spacelike or lightlike, respec-
Smarandache curves, T , T , and T -equiform Smarandache
tively.
ccFuuurrrvvthaeetsu rbmreyo kbre 1e (,i θnw)g,e ta hpnerde cskoe 2nn (n θt e)s cootmfio tenh epw reiotqphue itrfhtoiere msfi rossnpt aatchneedl i kcsuee crcvouenrsdv wee qhζueiinnf o trEhm 31e . lengLκteht ζin =Eτ 31ζ (asn )d b {e t , an ,r ebg, uκl,a rτ }n obne- intus lFl rceunrevte inpvaarraimanettsr iwzehde rbey {a t,r cn-,
ζ b }, and are the moving Frenet frame and the natural curvature
curve has constant curvature or it is a circular helix. Finally, we ζ
functions respectively. If is a spacelike curve with spacelike prin-
give an example to clarify these curves. We hope these results will ζ
cipal normal vector, then the Frenet formulas of the curve can be
be helpful to mathematicians who are specialized on mathematical
given as [6–8] :
modeling. (cid:2) (cid:3) (cid:2) (cid:3)(cid:2) (cid:3)
t˙ (s ) 0 κ(s ) 0 t(s )
n˙ (s ) = −κ(s ) 0 τ(s ) n (s ) , (1)
b˙ ( s ) 0 τ(s ) 0 b(s )
∗ Corresponding author at: Al Imam Mohammad Ibn Saud Islamic University, Col- (cid:4) (cid:5)
lege of Science, Department of Mathematics and Statistics, KSA, P. O. Box 90950, where ·= d , G(t , t ) = G (n, n ) = −G (b, b) = 1 , and G(t, n ) =
Riyadh 11623, Saudi Arabia. ds
E-mail address: [email protected] G(t, b) = G(n, b) = 0 .
http://dx.doi.org/10.1016/j.joems.2017.04.003
1110-256X/©2017EgyptianMathematicalSociety.ProductionandhostingbyElsevierB.V.ThisisanopenaccessarticleundertheCCBY-NC-NDlicense.
( http://creativecommons.org/licenses/by-nc-nd/4.0/ )
320 E.M. Solouma / Journal of the Egyptian Mathematical Society 25 (2017) 319–325
Definition 2.1. A surface M in the Minkowski 3-space E 31 is said to 3.1. T η-equiform Smarandache curves in E 13
be timelike, spacelike surface if, respectively the induced metric on
the surface is a Lorentz metrica, positive definite Riemannian met-
ric. In other words, the normal vector on the timelike(spacelike) Definition 3.2. Let ζ = ζ(θ) be a regular equiform spacelike curve
surface is a spacelike(timelike) vector [8] . lying completely on a spacelike surface M in E 31 with moving
η ξ η
equiform Frenet frame { T, , }. Then T -equiform Smarandache
defiLneet ζth:e I e→qu Eif 31o rbme ap asrpaamceeltiekre ocfu rζveb yin θM=in ∫k κowdss .k Ti hsepna,c ew Ee 31 .h aWvee curves are defined by
ρ= ddθs , where ρ= κ1 is the radius of curvature of the curve ζ. Let (cid:8) = (cid:8) (θ∗) = √12 (T (θ) + η(θ)) . (3)
F be a homothety with the center in the origin and the coefficient
μ. If we put ζ¯ = F(ζ) , then it follows Theorem 3.1. Let ζ = ζ(s ) be a spacelike curve with spacelike prin-
s¯ = μs and ρ¯ = μρ, cTi ηp-aelq nuoifromrmal Svmecatroarn dina cEh 31e . cIuf rζve iiss aa lscoir ccuirlcaur lahre lhixe liwx iathn dκ it>s th0e, tnhaetn-
wvetahetrue rroee f so¯ ζfi st [ht9hi]se . cFaurrrocvm-ele. ntThghatehtr eppfaoorirnaemt, , θewteiesr aroenfc ζae¯lqla un{if Tdo, rρηm¯, thiξne,v} arbarieda inutths epo afmr acoumvr--- uκr(cid:8)a (lθ c∗u)r v=a tuρr(ek √f 2u 2−n c2ti)o n:s arke 2 s(cid:4) =at i±sfi√ed 2 , t he following equation,
iξpna(gθl )en qo=ur miρfoabrl(m sv ) e Fcarteroenr e atth nfedr a meeqqeuu iiwffoohrremmre btTai nn(θogre)m n=ta lρv vete(csct o)to ,r ,r η e(reqθsu)p i=feoc rρtmivn e (lpsy )r. i naAcndid-- τ(cid:8) (θ∗) = √ρ22 k2 2 (2 k 2 + 1) −(2k( 22 k− 3 +1) 2 (cid:6)k) 2 + k 22 (k 2 + 2) (cid:7) :
dζmito=ivo ζinn(agθl l)ey ,q autrhiefeo drmfiefir snFtre edann ebdty f srkea 1cm (oθen )do =fe ζρq˙ u==if oζdrd(mρsθ ) acinus drgv iakvt 2eu (nrθ ea) so= f[ 1κτt0h.] :e S oc,u trhvee Prookf. 2 L(cid:4) =et −(cid:8)√ 3= 2 (cid:8). (θ∗) be a T η-equ2i form Smarandache curves ref(e4r)-
(cid:2) T (cid:5) (θ) (cid:3) (cid:2) k 1 (θ) 1 0 (cid:3)(cid:2) T (θ) (cid:3) eunsicneg tEoq t. h(e2 ) ,e qwuei fogermt spacelike curve ζ = ζ(θ) . From Eq. (3) and
ηξ(cid:5)(cid:5) ((θθ)) (cid:4)= −01 (cid:5) kk 12 ((θθ)) kk 21 ((θθ)) ηξ((θθ)) , (2) (cid:8) (cid:5) (θ∗) = ddθ(cid:8)∗ ddθθ∗= √12 ((k 1 −1) T (θ) + (k 1 + 1) η(θ) + k 2 ξ(θ)) ,
where (cid:5) = dθ , G(T , T ) = G(η, η) = −G(ξ, ξ) = ρ2 , and (5)
d
G(T , η) = G(T , ξ) = G(η, ξ) = 0 .
hence
The pseudo-Riemannian sphere with center at the origin and of
rSa 12d =iu s{ u(cid:2)r ∈= E1 31 i:n − tuh 21e +M uin 22k +ow us 23k =i 3 1- .s }p ace E 31 is a quadric defined by T (cid:8) (θ∗) = ρ(cid:8) 2 k 21 1− k 22 −2 ((k 1 −1) T (θ) + (k 1 +1 ) η(θ) + k 2 ξ(θ)) ,
Let ζ = ζ(θ) be a regular non-null curve parametrized by arc- (6)
lferanmgteh {i Tn, ηM, inξk,}o. wTshkein 3 T-s ηp, aTc ξe, Eη 31ξ waintdh Tit ηs ξm-eoqvuinifgo remq uSimfoarmra nFdraecnheet where (cid:8)
curves of ζ are defined, respectively as follows [11] : dθθ∗ = ρ 2 k 21√ − k 22 −2 . (7)
d 2
(cid:8) = (cid:8) (θ∗) = √1 (T (θ) + η(θ)) ,
2 Now
√
(cid:8) = (cid:8) (θ∗) = √12 (T (θ) + ξ(θ)) , ddθT (cid:8)∗ = ρ2 (cid:6)2 k 2 −2k 2 −2 (cid:7)2 (λ1 T (θ) + λ2 η(θ) + λ3 ξ(θ)) ,
(cid:8) = (cid:8) (θ∗) = √1 (η(θ) + ξ(θ)) , 1 2
2 where
(cid:8) = (cid:8) (θ∗) = √13 (T (θ) + η(θ) + ξ(θ)) . ⎧⎨⎩λλ 12 == ((kk 11 −+ 11))((22 k k 11 kk (cid:5)1 (cid:5)1 −−kk 2 2 kk (cid:5)2 (cid:5)2 )) ++ ((22 k k 2121 −−kk 22 22 −−22))((kk (cid:5)1 (cid:5)1 ++ kk 21 21 −+ 3k 22 k 1+ ) ,k 1 −2) ,
3. Special equiform Smarandache curves in E 13 λ3 = k 2 (2 k 1 k (cid:5)1 −k 2 k (cid:5)2 ) + (2 k 21 −k 22 −2)(k (cid:5)2 + 2 k 1 k 2 ) .
Then
In this section, we define the special equiform Smarandache (cid:13) (cid:14) (cid:15)
curves reference to the equiform Frenet frame of a curve ζ (cid:12)(cid:12) (cid:12)(cid:12) 2 λ2 + λ2 −λ2
ienq uMifoinrmko wcusrkvia t3u-rsep afucne ctEio 31 .n sF uorf tthheer meqoureif,o rwme Sombatraainn dathceh e ncautruvreasl κ(cid:8) (θ∗) = (cid:12)(cid:12)ddθT (cid:8)∗ (cid:12)(cid:12)= ρ(cid:6)2 k 21 −k 22 −2 (cid:7)32 , (8)
ltyhien gc ucrovmesp lwetheelyn othne p ceusrdvoe- sζphhearse cSo 12n astnadn tg icvuer vsaotmuree porro pite rist ieas coirn- and 1 2
cDuelfiarn ihteiolinx 3.1. A curve in Minkowski space-time, whose position N (cid:8) (θ∗) = λ1 T (θρ)(cid:8) + λ λ212 + η (λθ22) − +λ λ233 ξ(θ) .
vector is composed by Frenet frame vectors on another curve, is
Also
called a Smarandache curve.
cialA fso rcmon osef qtuheen ecqeu wifoitrhm t hSem aabraonvdea dchefie nciutirovne,s wine Ein 31 tirno dtuhcee foal lsopwe-- B (cid:8) (θ∗) = p1 1 { m 1 T (θ) + m 2 η(θ) + m 3 ξ(θ) } ,
ing subsection where
E.M. Solouma / Journal of the Egyptian Mathematical Society 25 (2017) 319–325 321
mmm 123 === λλλ222 kk(k 22 1(cid:8)−− −λλ133 )((kk− 11 λ+−1 1 1(k)) 1 ,,(cid:8) +1) NddoθTw (cid:8)∗ = ρ2 (k√ 2 2 + 1) 3 (ε 1 T (θ) + ε 2 η(θ) + ε 3 ξ(θ)) ,

andN po 1w =, f rρom2 (cid:16)E kq 21 .− (5k) 22 −2 λ21 + λ22 −λ23 . w⎧⎨h εe 1r =e (k 2 + 1)(k (cid:5)1 −k 2 −1)−k 1 k 2 ,
(cid:8) (cid:5)(cid:5) (θ∗) = √1 2 [ k (cid:5)1 + k 21 −2 k 1 −1] T (θ) + [ k (cid:5)1 + k 21 (cid:17) ⎩ εε 23 == k(k 1 2( k+ 2 +1) 1[ k) (cid:5)1 2 ,+ k 2 (k 2 + 1) + k 22 −k 21 ] −k 1 k (cid:5)2 .
+ k 22 + 2 k 1 −1] η(θ) + [ k (cid:5)2 + 2 k 1 k 2 + k 2 ] ξ(θ) , Then √ (cid:8)
2 ε 2 + ε 2 −ε 2
a(cid:8)n (cid:5)(cid:5)d(cid:5) ( θth∗u) s= √12 (β1 T (θ) + β2 η(θ) + β3 ξ(θ)) , κan(cid:8) d(θ ∗) = ρ(k1 2 + 12) 3 3 , (15)
w⎧⎨h βe1r e= k (cid:5)1(cid:5) + 3 k (cid:5)1 (k 1 −1) + k 21 (k 1 −3) −k 2 (k 2 + 2) , N (cid:8) (θ∗) = ε 1 T (θρ)(cid:8) +ε ε 12 2 + η( εθ 22) − +ε ε 32 3 ξ(θ) .
⎩Hτ(cid:8)e ββ (nθ23c ∗e==), w=kk (cid:5)1(cid:5)2 e(cid:5)(cid:5) √ ρ++h22 akk v(cid:18) (cid:5)1(cid:5)2 e ++w (cid:12) 133 2 +((+kk w 11(cid:12) kk 2 2 (cid:5)1(cid:5)1 − +++ (cid:12) wkk 2 22 3 kk (cid:19) (cid:5)2(cid:5)2 )), ++ 33 kk 11 k(k 2 1( k− 1 +1) 1 +) −k 21 k(3 2 . k 1 + 1) −(91) , AB (cid:8)ls (oθ ∗) = +p1 2 [ ε(cid:16) 2 [ k ε 1 2 k− 1 ε− 1(cid:8) (εk 3 2 ( k+ 2 1+) ]1 ξ)(] Tθ ()θ (cid:17)), + k 1 (ε 1 −ε 3 ) η(θ)
1 2 3 where p 2 = ρ(k 2 + 1) ε 12 + ε 22 −ε 32 .
where Now, from Eq. (12) we have
(cid:16)
ww 12 == ((kk (cid:5)1(cid:5)2 ++ k2 21 k 1+ k 2k 22+ + k 22 ) k[ 1 β −1 (k1 1) [+ β 31 ()k − 1 −β21 ()k − 1 −β11 k) 2] ] , , (cid:8) (cid:5)(cid:5) (θ∗) = √1 2 [ k (cid:5)1 + k 21 −k 2 + 1] T (θ(cid:17) ) + [ k (cid:5)2 + 2 k 1 (k 2 + 2)] η(θ)
w(cid:12) 31 == k(k (cid:5)1 (cid:5)1k 2+ − k k21 (cid:5)2− (k2 1 k + 1 −1)1 +)[ βk 22 ( kk 2 2 − −βk3 21 ( −k 1 k+ 1 −1)2] ,) , + [ k (cid:5)1 + k 2 (2 k 2 + 1)] ξ(θ) ,
(cid:12) 2 = k (cid:5)1 k 2 −k (cid:5)2 (k 1 −1) −k 1 k 2 (k 1 + 1) , and
(cid:12) 3 = k 1 (2 k 1 + 1) −2 k (cid:5)1 + k 22 (k 1 + 1) + 2 . (cid:8) (cid:5)(cid:5)(cid:5) (θ∗) = √1 (δ1 T (θ) + δ2 η(θ) + δ3 ξ(θ)) ,
2
κ τ
Now, if and are non-zero constants, then the natural curva-
tEuqr.e ( 4fu) nwcthioicnhs mκe(cid:8) a, nτs(cid:8) thaaret tahlseo Tn ηo-enq-zueifroor mconSmstaanratsn daancdh es actuisrfvyeinigs w⎧⎪⎪⎨h δe1 r=e k (cid:5)1(cid:5) −k (cid:5)2 + 3 k 1 (k (cid:5)1 −k 2 ) + k 1 (k 21 −1) ,
circular heilx. (cid:2) δ2 = k (cid:5)2(cid:5) + k (cid:5)1 + 3(k (cid:5)1 k 2 + k 1 k (cid:5)2 ) + k 21 (2 k 2 + 3)
3.2. T ξ-equiform Smarandache curves in E 13 ⎪⎪⎩ δ3 =+ k (cid:5)1k(cid:5) 2+ ( 2k k (cid:5)2 22 + + k k 1 2 ( k− (cid:5)1 1+) 3+ k 21 ) , + k 2 (5 k (cid:5)2 + 4 k 1 k 2 ) .
Definition 3.3. Let ζ = ζ(θ) be a regular equiform spacelike curve Hence, we have
lec(cid:8)Tcyiuξqh pi=runeavigo lfe(cid:8) orsn (erc oθmmaorr∗mm e )3F p ad=r.ll2e ee .nvfit √eeeL1nlct2 ey e tt o d(f rrζT oa b n(mi=ynθ ea)ζE +( 31{ss T. p ),ξ I af(ηb cθeζ,e )lξai)ik }s .se . p aTa shccueeirrlnficak uceTle a ξc r-u Merhqve eulii inxwf o iwrEtmh 31it h swSp κmaictahe>rl ia kmn0ed, o pavt(rhc1iinhe0nnge-) τ(cid:8) (θ∗)= √ρ2 2 ⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎩ ((kk (cid:5)1(cid:5)2 ++(cid:21)+− k22 (cid:21)(cid:21) 1 kkkk k 221 11 (cid:5)2 k k[+ 2 k+ (cid:5)2 21 k+ + ( 2− k 2)k 2 (cid:6)2 k 22 δ+ 1 k−1 ) 2 ( 1(+kkk) 21 2 2(( 2−+k+ k (cid:5)1 k1 211+ (cid:5)1) −) ( −k+kk 21 2 δ (cid:5)1 1 −+ 2)] k (cid:22) −k1 12 2 (cid:7)) k +++ 2 ( 12kk) 22 1 k (cid:6) (( 22δkδ +2 23 k +−3 1 k −1δ 21 )+δ ) −3 1(1k) (cid:22) 2 (cid:22) 22+ 1) (cid:7)(⎫⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎭16 , )
T -equiform Smarandache curve is contained in a palne and its cur-
vature is satisfied the following equation, So, if κ and τ are non-zero constants, then κ(cid:8) is non-zero constant and sat-
√ (cid:8) isfying Eq. (11) , also τ(cid:8) = 0 which means that the T ξ-equiform Smarandache
κP(cid:8)ro (θof∗.) L=e t (cid:8)2 = (cid:8)( (1θ −∗) kb 22e )ρ (ak( 2 kT 2 ξ+ -+ e1 q1)u 2) i 2−f orkm 22 ( 3Sm k 22a −ran2d) a:c he kc 2u (cid:4) =rv e−s1 o . f (ζ11=) c3u.3rv. eη iξs -ceoqnutaifionremd iSnm aa prlaannde.a ch(cid:2)e curves in E 13
ζ(θ) . Then from Eq. (10) , we have
Definition 3.4. Let ζ = ζ(θ) be a regular equiform spacelike curve
(cid:8) (cid:5) (θ∗) = √1 (k 1 T (θ) + (k 2 + 1) η(θ) + k 1 ξ(θ)) . (12) lying completely on a spacelike surface M in E 31 with moving
2 equiform Frenet frame { T, η, ξ}. Then ηξ-equiform Smarandache
T (cid:8) (θ∗) = ρ(k 21 + 1) (k 1 T (θ) + (k 2 + 1) η(θ) + k 1 ξ(θ)) , (13) c(cid:8)u =rv e(cid:8)s ( θar∗e) d=e fi√1n ed(cid:4) ηb(yθ ) + ξ(θ) (cid:5). (17)
where 2
ddθθ∗ = ρ(k√ 2 2 + 1) . (14) Tcihpeaol rneomrm 3a.l3 .v eLcetto rζ i=n ζE( 31s . )I fb eζ ai ss paa cceirlickuel acru rhveeli xw iwthit hsp κace>li k0e, ptrhienn-
322 E.M. Solouma / Journal of the Egyptian Mathematical Society 25 (2017) 319–325
ηξ-equiform Smarandache curve is also circular helix and its the nat- 3.4. T ηξ-equiform Smarandache curves in E 13
ural curvature functions are satisfied the following equation,
√
κ(cid:8) (θ∗) = √2 (kρ 22 −1) , (18) Dlyeinfign itcioomn p3le.5t.e lLye to ζn =a ζs(pθa)c ebleik ae rseugruflaacre e qMu ifionr mE 31s pwaciethli kme couvrinvge
τ(cid:8) (θ∗) = ρ2 k2 2 : k 2 (cid:4) = 0 . ecuqruvifeosr mar eF rdeenfiente fdr abmy e { T, η, ξ}. Then T ηξ-equiform Smarandache
Pcurorvoef . ζL=et ζ(cid:8)( θ=) (cid:8) . F(θro∗m) bEeq a. (η17ξ)- ,e wquei fgoertm Smarandache curves of the (cid:8) = (cid:8) (θ∗) = √13 (T (θ) + η(θ) + ξ(θ)) . (24)
(cid:8) (cid:5) (θ∗) = √1 2 (−T (θ) + (k 1 + k 2 ) η(θ) + (k 1 + k 2 ) ξ(θ)) , (19) Tcihpeaol rneomrm 3a.l4 .v eLcetto rζ i=n ζE( 31s . )I fb eζ ai ss paa cceirlickuel acru rhveeli xw iwthit hsp κace>li k0e, ptrhienn-
hence T ηξ-equiform Smarandache curve is also circular helix and its the
T (cid:8) (θ∗) = ρ1 (−T (θ) + (k 1 + k 2 ) η(θ) + (k 1 + k 2 ) ξ(θ)) , (20) natural curvatur√e fu(cid:8)n ctions are satisfied the following equation,
wddθhθ∗er=e √ρ 2 . (21) κτ(cid:8)(cid:8) ((θθ∗∗)) == 3√ ρ33 2 (2k2 2( ρ 1k− 2− (3kk) 22( 2 k +) 2 :1+ ) 1|) k 2 1 |: < 1k , 2 (cid:4) = −1 , 3 . (25)
Now
√ Proof. Let (cid:8) = (cid:8) (θ∗) be a T ηξ-equiform Smarandache curves of
ddθT (cid:8)∗ = ρ22 (γ1 T (θ) + γ2 η(θ) + γ3 ξ(θ)) , the curve ζ = ζ(θ) . Then from Eq. (24) , we get
wγ1h e=r e− (k 1 + k 2 ) , (cid:8) (cid:5) (θ∗) = √13 ((k 1 −1) T (θ) + (k 1 + k 2 + 1) η(θ) + (k 1 + k 2 ) ξ(θ)) .
γ2 = k (cid:5)1 + k (cid:5)2 + k 2 (k 1 + k 2 ) −1 , (26)
γ3 = k (cid:5)1 + k (cid:5)2 + k 2 (k 1 + k 2 ) .
Then (cid:13) 2 (cid:14)γ2 + γ2 −γ2 (cid:15) T (cid:8) (θ∗) = ρ(cid:8) k 21 +1 2 k 2 + 2 ((k 1 −1) T (θ) + (k 1 + k 2 + 1) η(θ)
κ(cid:8) (θ∗) = 1 ρ 2 3 , (22) + (k 1 + k 2 ) ξ(θ)) , (27)
where
and (cid:8)

N (cid:8) (θ∗) = γ1 T (θρ)(cid:8) +γ 1γ2 2+ η (γθ22) −+ γγ332 ξ(θ) . ddθθ∗ = ρ k 21 +√ 3 2 k 2 + 2 . (28)
Also Now
√
B (cid:8) (θ∗) = +ρ (cid:8)[ γ γ3 1+2 + γ1 1γ (2k2 1− +γ k32 2 { ) []( ηγ(2θ −) −γ3[ ) γ(2k + 1 + γ 1k ( 2k ) 1] T+ ( θk 2) )] ξ(θ) } . ddθT (cid:8)∗ = ρ2 (cid:6)k 21 + 23 k 2 + 2 (cid:7)2 (χ1 T (θ) + χ2 η(θ) + χ3 ξ(θ)) ,
where
(cid:8) (cid:5)(cid:5) (Fθro∗)m = E q√.1 2 ( 1 {9−) ,[ 2w ke 1 h+a vke 2 ] T (θ) + [ k (cid:5)1 + k (cid:5)2 + (k 1 + k 2 ) 2 −1] η(θ) ⎧⎪⎪⎪⎪⎨ χχ12 == ((kk 121 −+ 12 k)( 2k + 1 k 2 (cid:5)1 ) +[ k k (cid:5)1 (cid:5)2+ ) −k (cid:5)2 (+k 2 k + 1 + 1 )k( 2k ( 21k + 1 + 2 kk 2 2 )+ − 21) ,]
and +[ k (cid:5)1 + k (cid:5)2 + (k 1 + k 2 ) 2 ] ξ(θ) } , ⎪⎪⎪⎪⎩ χ3 =+ ( k(k 21 1+ k (cid:5)1 2 + k 2 k + (cid:5)2 ) 2(k)[ 1 k + (cid:5)1 +k 2 k + (cid:5)2 +1) k , 2 (k 1 + k 2 + 1)]
(cid:8) (cid:5)(cid:5)(cid:5) (θ∗) = √1 (ω 1 T (θ) + ω 2 η(θ) + ω 3 ξ(θ)) , + (k 1 + k 2 )(k 1 k (cid:5)1 + k (cid:5)2 ) .
2 Then
(cid:13)
w⎧here (cid:6) (cid:7) (cid:14) (cid:15)
⎨⎩ ωωω 123 === −kk (cid:5)1(cid:5)1(cid:5)(cid:5) 3++ k kk (cid:5)1 (cid:5)2(cid:5)2(cid:5)(cid:5)+ −+ 2 33 k k( (cid:5)2 k 1+ 1− +kk 1 k 2( 2 2+ ) k( 13k +( (cid:5)1 k + k1 2 +k ) (cid:5)2 +k ) 2 + ()k( (k 1k (cid:5)1+ 1 + +k k 2 k (cid:5)2) 2 2) ) +− 3 . 1(k 1, + k 2 ) 3 , κan(cid:8) d(θ ∗) = ρ3(cid:6) kχ 21 12+ + 2 kχ 222 + − 2χ (cid:7)322 , (29)
Hτ(cid:8)e (nθc∗e), w= e√ h2 a⎧⎨⎩ ve[−− k[( (cid:5)1 kω + (cid:5)1 3 + k− (cid:5)2 k ρω+ (cid:5)2 2 2+ (cid:21)( )k k(( 1 21kk 1+ 1− ++ kk 2 22kk ) 2 2+ 2 ) ) (− 222 ]([1 kk ω 1] (cid:5)1 [ 2 + ω+ + 3 kk 2ω+ (cid:5)2 ) ) 1 ω (cid:22) (k 1 1( k+ 1 k+ 2 k)] 2 )]⎫⎬⎭ . NAB (cid:8)l (cid:8)s ( (oθθ ∗∗)) == χρ1(cid:8)(cid:16) T k(θ 21ρ )+(cid:8) + 2 χ kχ12 2 2 + +η (χ 2θ12(cid:8) 2) −+ χ 1χ2χ 3+23 ξ χ(θ22 )− . χ32
(23)
Then, if κ and τ are non-zero constants, then the natural curva- × [ −χ3 + (χ2 −χ3 )(k 1 + k 2 )] T (θ)
κ τ
tEuqr.e ( 1f8u)n cwtihoinchs m(cid:8)e , ans(cid:8) tahraet athlseo ηnξo-ne-qzueirfoo rmco nSsmtaanrtasn daancdh esactuisrfvyei nigs + [ χ1 (k 1 + k 2 ) −χ1 (k 1 (cid:17)− 1)] η(θ) + [ χ2 (k 2 −1)
circular heilx. (cid:2) −χ1 (k 1 + k 2 + 1)] ξ(θ) ,
E.M. Solouma / Journal of the Egyptian Mathematical Society 25 (2017) 319–325 323
Fig. 2. Equiform spacelike curve ζ= ζ(θ) on S 12 .
Fig. 1. Spacelike curve ζ= ζ(s ) on S 12 .
From Eq. (26) , we have
(cid:8) (cid:5)(cid:5) (θ∗) = √1 3 { [ k 21 −k 1 −k 2 −1] T (θ)
+ [ k (cid:5)1 + k (cid:5)2 + 2 k 1 + (k 1 + k 2 ) 2 −1] η(θ)
+ [ k (cid:5)1 + k (cid:5)2 + k 2 + (k 1 + k 2 ) 2 ] ξ(θ) } ,
and thus
(cid:8) (cid:5)(cid:5)(cid:5) (θ∗) = √1 (φ1 T (θ) + φ2 η(θ) + φ3 ξ(θ)) ,
3
where
⎧
⎪⎪⎪⎨⎪⎪⎪⎩ φφφ123 ===++ 2kk k( (cid:5)1(cid:5)1 k(cid:5)(cid:5)k 1 (cid:5)1++ 1( ( 2k +kk k 1 (cid:5)2(cid:5)2 21(cid:5)(cid:5)k + ++ 2+ )1 2+ 3k) . k 1 k − (cid:5)1 (cid:5)2 − ++21 k kk) (cid:5)2 2 2+ +( (k 3 ( 2k kk 1− 1 1 ( −k+1 21) 1k− +) 2 )k +3 3 1 ( −3−k( 11kk + , 1 2 +−k 2 k2 ) 2() )k +( (cid:5)1 k +1 (cid:5)1 , + k (cid:5)2 k ) (cid:5)2 )
Hence, we have
τ(cid:8) (θ∗) = √ρ23 (cid:18)qv 2 1 ++ qv 2 2 +− vq 3 2 (cid:19), (30) Fig. 3. The T η-equiform Smarandache curve (cid:8) ( θ∗) on S 12 .
2 3 4. Example
where (cid:14)√ √ √ (cid:15)
Let ζ(s ) = 3 s, s sin ( 3 ln s ), s cos ( 3 ln s ) be a unit speed
v 1 = (k 21 −k 1 −k 2 −1)[ φ2 (k 1 + k 2 ) −φ3 (k 1 + k 2 −1)] , spacelike curve parametrized by arc-length s with spacelike princi-
qqqvv 23123 ===== [[−(( kkkk( (cid:5)1(cid:5)1 11 k +++− 1 + kk1k (cid:5)2(cid:5)2 2 ) k )[++ 2 k( ) 2 (cid:5)12k ( k+ 2 kk 1 1 1+ k k−+ (cid:5)2 2( kk++( 2 1k 2 11−+ k)+ 1 1 k− 2)k+ ) 2−( 2 )k( ] 2k 1[[ k 1 φ−− (cid:5)1+ 1 1 +(1 kk])[k 1 2( φ (cid:5)2)k+ 3 2 (cid:5)1+ (] k+ k+2 2 1 kk −k− 1 (cid:5)2 1 1 ++(12)) k( k −−k 2 1 1 )φ−φ ,+ 213 ( (kk)k 2 1 1 ) − 2+ ] , 1k ) 2] ) ,] , p⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨at nl ( (nsc−s o) o) r s=s=mi (n a( √ √( √l1 √3 2 v3 (le 3, 0nc sl , tsn ioc n) ros − ()si √ n −(√ √3 E√3 31l 3 n s3 l(i n ssnc )e o s (e+ s) √ (−F √3 √ig √l3 . 3n c3 1 lson ) s).s i)sT n , () h √ )(e , √ 3n 3 lin tl n si )ss, ) e, a sy to show that
NtEcuiqorr.cwe u(+ ,2l fa 5ui(rf)nk h κcw+teih iao1lixnn c+.dsh kκτm(cid:2) 2(cid:8) e +a, arτ ne1(cid:8) s) n[ta2ohrn(aek-t z 2at e+lhrsoeo 1 Tc)n ηo −on-nesk-qt 1zau (enikrftoo 1s r,+ mcto h1neS)snm]t a+atnhr t1aesn . dnaanactdhu ersaalct uicsrufvyreivnaigs- ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩ κbτ(==s√ )2 3 =32sc √,s o 3(s 2, ( , √ √2k 3 3 2 lsρ=nin =s √)(2 √ 3− 2 . √s 3 332 l ,ns i sn ) (+ √k 13 32 =lcno ss2 ) √1( ) √ 3 , , 3 ln s ),
324 E.M. Solouma / Journal of the Egyptian Mathematical Society 25 (2017) 319–325
Fig. 4. The T ξ-equiform Smarandache curve (cid:8) ( θ∗) on S 12 .
√
Hence, the equiform parameter is θ = ∫ κds = 2 3 s + c. Here
we take c = 0 , then we have s = e θ/ 2 √3 and ρ= e θ/√ 2 √ 3 . So the
ζ 2 3
equiform spacelike curve is define as (see Fig. 2 )
(cid:26) (cid:26) (cid:27) (cid:26) (cid:27)(cid:27)
√ √ √ θ √ θ
ζ(θ) = 3 e θ/ 2 3 , e θ/ 2 3 sin , e θ/ 2 3 cos .
2 2
It easy to show that
√ (cid:26) (cid:26) (cid:27) (cid:26) (cid:27) (cid:26) (cid:27) (cid:26) (cid:27)(cid:27)
T (θ) = e θ/ 2 3 1 , √1 sin θ + cos θ , √1 cos θ −sin θ .
2 3 2 2 3 2 2
It is clear that T is an equiform spacelike vector. Also
√ (cid:26) (cid:26) (cid:27) (cid:26) (cid:27) (cid:26) (cid:27) (cid:26) (cid:27)(cid:27)
η(θ) = e θ/ 2 3 0 , √1 cos θ −sin θ , √−1 sin θ −cos θ ,
4 3 2 2 3 2 2
and
√ (cid:26) (cid:26) (cid:27) (cid:26) (cid:27) (cid:26) (cid:27) (cid:26) (cid:27)(cid:27)
ξ(θ) = e θ/ 2 3 √4 , sin θ + √ 3 cos θ , cos θ −√ 3 sin θ .
4 3 2 2 2 2
η ξ
Then is an equiform spacelike vector and is an equiform time-
like vector.
The T η-equiform Smarandache curve (cid:8) ( θ∗) of the curve ζ( θ) is
given by √(s ee Fi√g . (cid:26)3 ) (cid:26) (cid:27) Fig. 5. The ηξ-equiform Smarandache curve (cid:8) ( θ∗) on S 12 .
(cid:8) (θ∗) = 6 e θ/ 2 3 2 √3 , (2 √ 3 + 1) cos θ
24 2
√ (cid:26)θ(cid:27) √ (cid:26)θ(cid:27) √ (cid:26)θ(cid:27)(cid:27) The ηξ-equiform Smarandache curve (cid:8) ( θ∗) of the curve ζ( θ) is
+ (2 − 3 ) sin , (2 − 3 ) cos −(2 3 + 1) sin . given by (see Fig. 5 )
2 2 2
giveTnh eb yT ξ(s-eeqe uFiifgo.r m4 ) Smarandache curve (cid:8) ( θ∗) of the curve ζ( θ) is (cid:8) (θ∗) = √6 e θ/ 2 √ 3 (cid:26)1 , cos (cid:26)θ(cid:27), −sin (cid:26)θ(cid:27)(cid:27).
√ √ (cid:26) (cid:26) (cid:27) 6 2 2
(cid:8) (θ∗) = 6 e θ/ 2 3 2(2 + √ 3 ) , (2 + √ 3 ) sin θ
24 2
√ (cid:26)θ(cid:27) √ (cid:26)θ(cid:27) √ (cid:26)θ(cid:27)(cid:27) The T ηξ-equiform Smarandache curve (cid:8) ( θ∗) of the curve ζ( θ) is
+ (2 3 + 3) cos , (2 + 3 ) cos −(2 3 + 3) sin . given by (see Fig. 6 )
2 2 2
E.M. Solouma / Journal of the Egyptian Mathematical Society 25 (2017) 319–325 325
Acknowledgment
The author is grateful to the Editor in Chief and the referees for
their efforts and constructive criticism which have improved the
manuscript
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18 2 (cid:27) 2 2
√ (cid:14) (cid:15)
−(2 + 3 ) sin θ .
2