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Journal of the Egyptian Mathematical Society 25 (2017) 268–271 Contents lists available at ScienceDirect Journal of the Egyptian Mathematical Society journal homepage: www.elsevier.com/locate/joems Original Article Smarandache curves in Euclidean 4- space E4 Mervat Elzawy Mathematics Departement, Faculty of Science, Tanta University, Tanta, Egypt a r t i c l e i n f o a b s t r a c t Article history: The purpose of this paper is to study Smarandache curves in the 4-dimensional Euclidean space E 4 , and Received 1 January 2017 to obtain the Frenet–Serret and Bishop invariants for the Smarandache curves in E 4 . The first, the second Revised 15 February 2017 and the third curvatures of Smarandache curves are calculated. These values depending upon the first, Accepted 5 March 2017 Available online 17 March 2017 the second and the third curvature of the given curve. ©2017 Egyptian Mathematical Society. Production and hosting by Elsevier B.V. 2010 MSC: This is an open access article under the CC BY-NC-ND license. 14H45 14H50 ( http://creativecommons.org/licenses/by-nc-nd/4.0/ ) 53A04 Keywords: Euclidean 4- space Smarandache curves Ferent frame Parallel transport frame 1. Introduction lel vector fields { M 1 ( s ), M 2 ( s ), M 3 ( s )} of the frame, such that they are perpendicular to T ( s ) at each point [7,8] . The parallel trans- It is well known that, if a curve is differentiable in an open in- port frame in four dimensional Euclidean space is studied in [9] . terval, at each point, a set of mutually orthogonal unit vectors can Smarandache curves were studied from deferent researchers in be constructed, and these vectors are called Frenet frame or mov- three dimensional Euclidean space [10–14] . Smarandache curves in ing frame vectors. The rates of these frame vectors along the curve 4-dimensional Galilean space are presented in [15] . In this paper define curvatures of the curve. The set, whose elements are frame we study Smarandache curves in 4-dimensional space according to vectors and curvatures of a curve is called Frenet apparatus of the Frenet frame and parallel transport frame. the curves. In recent years, the theory of degenerate submanifolds has been treated by researchers and some classical differential ge- ometry topics have been extended to Minkowski space [1–3] and 2. Preliminaries Galilean space [4] . For instance in [1,2] the authors extended and sctuurdviee din SEmuacrlaidnedaanc hsep accuer vEe 4s , winh oMsien kpoowsistkioi ns pvaecceto-trism ies. cAo mrepgousleadr Let L−→eat =α :( aR 1 ,→ a 2 ,E a 4 3 b, ae 4 a )n , −→ba r b=i t(rba 1r y, bc 2u , rbv 3e , bi 4n ) tahned E −→ucc l=id e(ac 1n , cs 2p ,a cc 3e , cE 4 4 ) . by Frenet frame vectors on another regular curve is called Smaran- be three vectors in E 4 , equipped with the standard inner product dache curve. Special Smarandache curves in three dimensional Eu- given by < −→a , −→b > = a 1 b(cid:2) 1 + (cid:2)a 2 b 2(cid:3) + a 3 b 3 + a 4 b 4 . The norm of a vec- clidean space studied in [5] . tor a ∈ E 4 is given by (cid:2)−→a (cid:2)= < −→a , −→a >. The curve α is said The Bishop frame [6] or parallel transport frame is an alter- to be of a unit speed or parametrized by arc length function s if native approach to defining a moving frame that is well defined < α(cid:4) , α(cid:4) > = 1 . The vector product of −→a , −→b , and −→c is defined by even when the curve has vanishing second derivative. We can par- the determinant allel transport an othonormal frame along a curve simply by par- (cid:4) (cid:4) (cid:4) (cid:4) allel transporting each component of the frame in Euclidean 4- (cid:4)e 1 e 2 e 3 e 4 (cid:4) space. The parallel transport frame is based on the observation −→a ×−→b ×−→c = (cid:4)(cid:4)a 1 a 2 a 4 a 4 (cid:4)(cid:4) that while T ( s ) for the given curve model is unique, we may chose (cid:4)(cid:4)b 1 b 2 b 3 b 4 (cid:4)(cid:4) any convenient arbitrary basis which consists of relatively paral- c 1 c 2 c 3 c 4 where e 1 ×e 2 ×e 3 = e 4 , e 2 ×e 3 ×e 4 = e 1 , e 3 ×e 4 ×e 1 = e 2 , e 4 × E-mail address: [email protected] e 1 ×e 2 = e 3 , e 3 ×e 2 ×e 1 = e 4 . http://dx.doi.org/10.1016/j.joems.2017.03.003 1110-256X/©2017EgyptianMathematicalSociety.ProductionandhostingbyElsevierB.V.ThisisanopenaccessarticleundertheCCBY-NC-NDlicense. ( http://creativecommons.org/licenses/by-nc-nd/4.0/ ) M. Elzawy / Journal of the Egyptian Mathematical Society 25 (2017) 268–271 269 The vectors t ( s ), n ( s ), b 1 ( s ), b 2 ( s ) are the moving Frenet frame Note that k 1 , k 2 , k 3 are the principal curvature functions accord- α along the unit speed curve . Then the Frenet formulas are given ing to Frenet frame and K 1 , K 2 , K 3 are the principal curvature func- α by tions according to the parallel transport frame of the curve . ⎡ ⎤ ⎡ ⎤⎡ ⎤ ⎢⎣ bbnt (cid:4)(cid:4) 12(cid:4) (cid:4) ⎥⎦ = ⎢⎣ −000k 1 −k00k 1 2 −k00k 2 3 k000 3 ⎥⎦ ⎢⎣ bbnt 12 ⎥⎦ 33..1 .M tabi 1n S rmeasrualntsd ache curves in E 4 according to the Frenet frame. t, n, b 1 , and b 2 are called, respectively, the tangent, the principal In this subsection we define tb 1 Smarandache curves and obtain normal, the first binormal and the second binormal vector fields of their Frenet apparatus. the curves. The functions k 1 ( s ), k 2 ( s ) and k 3 ( s ) are called respeαc- Definition 1. A regular curve in E 4 , whose position vector is ob- tively, the first, the second and the third curvature of the curve . The curve is called W −curv e if it has constant curvatures k 1 , k 2 tained by Frenet frame vectors on another regular curve, is called Smarandache curve. and k 3 . Let α= α(t) be an arbitrary curve in E 4 . The Frenet apparatus of the curve α can be calculated by the following equations. Definition 2. Let α= α(s ) be a unit-speed curve with constant α(cid:4) and nonzero curvatures k 1 , k 2 , k 3 and { t, n, b 1 , b 2 } be moving frame nt == (cid:5)(cid:2)(cid:2) α(cid:5)(cid:5) αα(cid:4) (cid:5)(cid:4)(cid:4) (cid:5)(cid:5) 22 αα(cid:4)(cid:4)(cid:4)(cid:4) −−<< αα(cid:4)(cid:4) ,, αα(cid:4)(cid:4)(cid:4)(cid:4) >> αα(cid:4)(cid:4) (cid:2)(cid:2) obT 1nh ( esio )t,r) e .t mb 1 1S.m Laerta αnd( sa )c hbee acu urvneits sapreee dd ecfiunrveed wbiyth βc(osn βs)ta =n t √1n 2 o (nt( sz )e r+o b 1 = ηb 2 ×t ×n cduerfivnaetdu rbeys kth 1e , kfr 2a , mke 3 vaencdto rβs (o s βf )α b( se ) t. bT 1h eSnm athraen Fdraecnheet cauprpvaersa tiuns Eo 4f b 2 = η(cid:2)(cid:2)(cid:5)tt ××nn ××αα(cid:4)(cid:4)(cid:4)(cid:4)(cid:4)(cid:4) (cid:5) (cid:2)(cid:2) βtu(s{ t o βf ,α n( β{, t ,b 1n β, ,b b 1 2 , βb, 2 k , 1k β 1, ,k k 2 β 2 ,, kk 3 3 β })}. ) can be formed by Frenet appara- (cid:2)(cid:5) α(cid:4) (cid:5) 2 α(cid:4)(cid:4) −< α(cid:4) , α(cid:4)(cid:4) > α(cid:4) (cid:2) k 1 = (cid:5) α(cid:4) (cid:5) 4 PThroeonf . Let β= β(s β) be tb 1 Smarandache curve of the curve α. k 2 = (cid:2)(cid:2)(cid:5) α(cid:4)(cid:5) (cid:5)t 2 ×α(cid:4)(cid:4)n − ×<α α(cid:4)(cid:4)(cid:4)(cid:4) (cid:5) , (cid:5) α α(cid:4)(cid:4) (cid:4) >(cid:5) α(cid:4) (cid:2)(cid:2) FBryo mdi fDfeerfienntiitaiotinn g(2 β) (w s βe) hwaviteh βre(ssp βe) c=t t√o1 2 s ( tw(se ) o+b bta 1i (ns )) k 3 = (cid:5)t ×< nα ×(IV α) ,(cid:4) b(cid:4)(cid:4) (cid:5) 2 (cid:5) > α (cid:4) (cid:5) dβd(ss β) = dβd(ss β β) ddss β = √12 ((k 1 −k 2 ) n+k 3 b 2 ) where ηis taken ±1 such that determinant of matrix [t,n,b1,b2] is The tangent vector of the curve β is given by equTahl et oB oisnheo. p frame or parallel transport frame is an alternative t β = A 1 n + A 2 b 2 (3.1) (cid:11) atahlplepe lrc outarrcavhne s thpoao sdr tev fiaonnf eisa hnai nmogro tsvheioncnogon frdrma dmaele ri fvrthaatmaitve ei s a[ 9wlo]en . lOgl ndaee ficcunarnevd ee xespviermensp swl yph aebrny- where ddss β = (k 1 −√k 2 2 ) 2 + k 23 , A 1 = (cid:11) (k( 1k − 1 −k 2k ) 2 2 ) + k 23 and A 2 = (cid:11) (k 1 −kk 3 2 ) 2 + k 23 β parallel transporting each component of the frame. The tangent Again differentiating the tangent vector of with respect to s β we can obtain β(cid:4) (cid:4) as follows vector and the convenient arbitrary basis for the remainder of the fp⎡⎢⎣winrrahg MMMeTm es (cid:4)t 123(cid:4)(cid:4)(cid:4)sr eo ⎤⎥⎦ee dap K =ra 1aer ,s ⎡⎢⎣a uKlls −−− 2ee ,0l KKKd a t 123.n r aTdnh KseK000p 3 1 op arartre aKf 000lrt 2la he mel teprK 000rao 3i n n⎤⎥⎦f sctp ⎡⎢⎣ihpoe MMMarT tl c 123cue ⎤⎥⎦uqrv ruveaa tαtαiuo. rneTs h fieun n scEet 4ti o{cn Ta,sn M abc 1ce , o Merdx 2-- , βnw(cid:11) β(cid:4)h (cid:4) eT==rh e Ae √ 3kp t2 1 r k[ +i 2 −n − cAkki 22 4 1p −A b(ak 3k 1l 23 1 =n −o ((cid:11)rkkm 1 k 2 21 a)− (tk l 1k+o − 2fk− )( 2tk 2k)h 1 2 1 (+ e+k k 1( 2ck k− 1u 23−k kr 2 2v ) −k ek 22 22 β− −ki 23ks ) 23 2 ) b 1 ] and (A3 4. 2=) M 3 }T hise ceaxllperde sthsieo npsa roafl letlh ter apnrsipnocript aflr acmuerv oaft ure[s6 ,a14re] . given as fol- k 21 (k 1 −k 2 ) 2 +(k 1 k 2 −k 22 −k 23 ) 2 lows: β(cid:4)(cid:4)(cid:4) = A 5 n + A 6 b 2 KK 12 == kk 11 (c−osc θosc φoss ψin, ψ + sin φsin θcos ψ) , where A 5 = 2(−k 21 (k (1( −kk 1 2 − )k − 2k ) 2 2 ( +k k 1 23 k ) 2 −32 k 22 −k 23 )) andA 6 =β (k( 3k ( 1k − 1 kk 2 2 −) 2k + 22 k− 23k ) 23 32 ) K 3 = (cid:3)k 1 ( sin φsin ψ + cos φsin θcos ψ) and θ(cid:4) folloTwhes second binormal vector of the curve is given easily as kφ 1(cid:4) c=o s θK+ 12 +θ(cid:4) Kc 2o2 t+ ψ K = 32 , 0k 2 = −ψ (cid:4) + φ(cid:4) sin θ, k 3 = sin ψ , b 2 β = (cid:3)(k ( 1k k 1 2 k − 2 −k 22k − 22 −k 23k ) 23t ) + 2 +k 1 k ( 21k ( 1k − 1 −k 2k ) 2 b ) 1 2 (3.3) (cid:3) β w(cid:3)hk e 23r −e (θ(cid:4) ) 2 θ(cid:4) = (cid:3)k 21k 3+ k 22 , ψ (cid:4) = −(k 2 + k 3 (cid:3)k 23k 21− +( θk(cid:4) 22 ) 2 ) , φ(cid:4) = b 1 βT=h e− fi(cid:3)kr 3s k nt 23 b++i n ((okkr 1 1m −−alkk 2v 2 )e) b 2c 2to r of the curve is (3.4) θ cos 270 M. Elzawy / Journal of the Egyptian Mathematical Society 25 (2017) 268–271 β β The first, second and third curvature of the of the curve are The third derivative of the curve reads k 1 β = 2[(k 1 k 2 −(kk 23 22 + − (kk 23 1 ) − 2 +k 2 k ) 21 2 ( )k 2 1 −k 2 ) 2 ] (3.5) β(cid:4)(cid:4)(cid:4) = (λ0 T α(cid:4) + λ1 M 1(cid:4) α+ λ2 M 2(cid:4) α + λ3 M 3(cid:4) α) (cid:11) 2 K 12 α +√ K 2 2 2 α + K 32 α k 2 β = (k 23 + (√k2 1 k − 3 [k k 2 1 ) ( 2k ) 1 (cid:3) k 2 ( −k 1 kk 222 −−kk 2322 )− +k k 23 21) 2( k+ 1 −k 21 k(k 2 1) ]− k 2 ) 2 (3.6) β(cid:4)(cid:4)(cid:4) = √ 2[ (−λ1 K 1 α −λ2 K 2 α −λ3(cid:11) K 3 2 α K) T 12 αα ++ Kλ 220 α K+ 1 α MK 32 1 α α+ λ0 K 2 αM 2 α+ λ0 K 3 αM 3 α] √ (3.11) k 3 β = (cid:3) k 23 + (k 1 −2 (k− 2 k) 2 1 (cid:3) A 4 ( Ak 5 1 −k 2 k− 2 Ak 3 22 A − 5 +k 23 k) 3 2 A+ 3 Ak 6 21 )( k 1 −k 2 ) 2 (3.7) T β×n β×β(cid:4)(cid:4)(cid:4) = C 1 M 1 α+ C 2 M 2 α + C 3 M 3 α where √ This completes the proof. (cid:2) C 1 = √ 2[ λ0 λ3 K 1 αK 2 α−(cid:3)λ0 λ λ202 + K 1 λ αK21 3+ α −λ22( λ+1 λ K23 1 α(2+ K 1λ2 α 2 +K 2 K α 22+ α λ+3 K K 32 3 α α)) (λ3 K 2 α−λ2 K 3 α)] 3tr.a2n. sTpMor 1t Sfrmamarea.n dache curves in E 4 according to the parallel C 2 = √ 2[ λ0 λ3 K 12 α −λ0(cid:3) λ λ1 K20 1+ αK λ 321 α +− λ(λ22 1+ K 1λ α23 +(2 λ K2 12 αK 2+ α K+ 22 α λ+3 K K 3 α 32 α ))( λ3 K 1 α −λ1 K 3 α)] tainI nt htehiirs psaurbasleleclt itornan wspeo rdte ffirnaem eT Ma n1 dS mthaer apnrdinaccihpea lc cuurrvveast uarneds . ob- C 3 = 2[ λ0 λ2 K 12 α −λ0(cid:3) λ λ1 K20 1+ αK λ 221 α +− λ(λ22 1+ K 1λ α23 +(2 λ K2 12 αK 2+ α K+ 22 α λ+3 K K 3 α 32 α ))( λ1 K 2 α −λ2 K 2 α)] Definition 3. Let α= α(s ) be a unit speed curve in E 4 and The second binormal of the curve β is given by the following { T α, M 1 α, M 2 α, M 3 α} be its moving parallel transport frame. TM 1 formula STmheaorarenmda c2h. eL ectu αrve=s αis( sd )e bfien ead u bnyit βsp(ese βd) c=u r√v1e 2 (wT αith+ cMon 1 αst)a . nt princi- b 2 β = C 1 M 1(cid:3) α +C 12 C + 2 M C 2 22 α ++ C C 32 3 M 3 α (3.12) pal curvatures K 1 α, K 2 α, K 3 α and β( s β) be TM 1 Smarandache curves in The first binormal of the curve β is given by the following for- E 4 defined by the parallel transport frame vectors of α= α(s ) . Then mula β the parallel transport frame of can be formed by the parallel transport frame of α and the principle curvatures of β (K 1 β, K 2 β, K 3 β) can b 1 β = b 2 β×T β×n β be obtained by the principal curvatures of α. = γ0 T α+ γ1 M 1 α + γ2 M 2 α + γ3 M 3 α (3.13) where the constants are given by Proof. To investigate the parallel transport frame of TM 1 Smaran- dMa 1c αh)e wcituhrvree spaceccotrtdoinsg to α( s ) differentiating β(s β) = √1 2 (T α+ γ0 = C 1 λ(cid:3)3 KC 2 2 α +− CC 12 λ+2 KC 3 2 α (cid:3)+ λ C2 2 +λ1 λ K2 3 α+− λC2 2 +λ3 λ K2 1 (cid:11) α+ 2 KC 3 2 λ 2+ K K 1 α 2 −+C 3K λ 21 K 2 α 1 2 3 0 1 2 3 1 α 2 α 3 α T βddss β = √12 (−K 1 αT α+ K 1 αM 1 α + K 2 αM 2 α + K 3 αM 3 α) γ1 = (cid:3) C 2 λ3 K 1 α(cid:3)+ C 2 λ0 K 3 α −C 3 λ2 K 1(cid:11) α − C 3 λ0 K 1 α The tangent vector of the curve β can be written as follows C 12 + C 22 + C 32 λ20 + λ21 + λ22 + λ23 2 K 12 α + K 22 α + K 32 α T β = −K 1 αT α+(cid:11) K 1 αM 1 α + K 2 αM 2 α + K 3 αM 3 α (3.8) γ2 = (cid:3) C 1 λ3 K 1 α(cid:3)+ C 1 λ0 K 3 α −C 3 λ1 K 1(cid:11) α − C 3 λ0 K 1 α 2 K 12 α + K 22 α + K 32 α C 12 + C 22 + C 32 λ20 + λ21 + λ22 + λ23 2 K 12 α + K 22 α + K 32 α where ddss β = √1 2 (cid:11)2 K 12 α + K 22 α + K 32 α γ3 = (cid:3)C 2 + CC 1 2 λ +2 K C 1 2 α (cid:3)+ λ C2 1 +λ0 λ K2 2 α+− λC2 2 + λ1 λ K2 1 (cid:11) α − 2 KC 2 2 λ+0 K K 1 α2 + K 2 Differentiating (3.8) with respect to s 1 2 3 0 1 2 3 1 α 2 α 3 α β T β(cid:4) = ddTs β β = λ0 T α+ λ1 M 1 α + λ2 M 2 α + λ3 M 3 α The (cid:12)paraλllel traθnsport ψframe for the curve h(cid:13)as the form wheλr2e =λ 0( 2= K 2−− √(√+2 2 2 K KK ( 12K 1 α2 α 1 2+ α K+ 2 +K αK 2 2K α2 2 2+ α ) ,+K 32K α 32) α λ) 3 ,= λ 1( =2 K ( −22 √ K+ 12 2 α K K− 1 +2 α√ KK 2+ 2 32K α α K 12 α + 2 K) 32 α ) M 1 β = + (cid:12)(cid:3)(cid:3) λ020λ λ c+120o cs+λo 21 sλβ +θ21cβ o+λcs22 o λ (cid:13)+s22 β ψ +λ β23 λ +23 +γ(cid:12)0 γ c1o cso θsβ θsβins iψn βψ βT α The first 1c αurvat2u αre of3 αthe c√ur v(cid:11)e β acco1r αding 2to α Fren3 αet frame is −(cid:3)C C 12 1 + si Cn 22 θ +β C 32 M 1 α+ (cid:3)λ λ220 c+o sλ(cid:13) θ21β +c oλs22 ψ+ β λ23 k 1 β = (cid:3) λ20 + λ21 + λ22 + λ23 = (cid:11)2 2 KK 12 α 1 2 α ++ K K 22 α 2 2 α ++ K K 32 α 3 2 α (3.9) + γ(cid:12)2 coλs θβsiθn ψ β−ψ(cid:3)C C 12 2 + si Cn 22 θ +β C 32 M 2 α The principal normal of the curve β is given by the following + (cid:3) λ32 c+o sλ 2β +c oλs2 + β λ2 + γ3 cos θβsin ψ β formula 0 1 (cid:13)2 3 n β = λ0 T α+(cid:3) λ λ120 M + 1 α λ+21 +λ2 λ M22 2+ α +λ23 λ 3 M 3 α (3.10) −(cid:3)C C 12 3 + si Cn 22 θ +β C 32 M 3 α (3.14) M. Elzawy / Journal of the Egyptian Mathematical Society 25 (2017) 268–271 271 (cid:14) M 2 β = λ0 (−cos φβ(cid:3)s λin2 ψ+ β λ+2 +si nλ φ2 β+s λin2 θβcos ψ β) is gTivhee(cid:17)n s beyco nd curvature of the curve β according to Frenet frame 0 1 2 3 (cid:15) + γ0 ( cos φβcos ψ β+ sin φβsin θβsin ψ β) T α k 2 β = (cid:3) λ2C 1+2 + λ 2C 22+ + λ C2 32+ λ2 (3.17) (cid:14) 0 1 2 3 + λ1 (−cos φβ(cid:3)s in ψ β+ sin φβsin θβcos ψ β) The third curvature of the curve β according to Frenet frame is λ2 + λ2 + λ2 + λ2 given by ++ γ(cid:12)1C ((cid:3) 1 c oCs is 2n φ+ φβ βCc 2co o+ss ψ Cθ β2β0 +(cid:13) (cid:15)siMn1 1φ αβs2i n θβ3s in ψ β) k 3 βT=h e− fi(rCs 1t Kc 1u αrv+a tCu 2r Ke 2 αof+ (cid:3)t hCeC 3 K12c 3u+ αr) vC(e 22λ β+1 K aC 1c 3 α2c +or dλi2n Kg 2 αto+ p λar3a Kl 3le αl) tran(s3p.1o8r)t (cid:14) 1 2 3 frame reads + λ2 (−cos φβ(cid:3)s λin2 ψ+ β λ+2 +si nλ φ2 β+s λin2 θβcos ψ β) K 1 β = (cid:3) λ20 + λ21 + λ22 + λ23 cos θβcos ψ β (3.19) 0 1 2 3 β + γ(cid:12)2 ( cos φβcos ψ β+(cid:13) (cid:15)sin φβsin θβsin ψ β) porTt hfrea smeceo rneda dcsu rvature of the curve according to parallel trans- + C(cid:3) 2 Cs i 2n + φ βC 2c o+s Cθ 2β M 2 α K 2 β = (cid:3) λ20 + λ21 + λ22 + λ23 [ −cos φβsin ψ β+ sin φβsin θβcos ψ β] (cid:14) 1 2 3 + λ3 (−cos φβ(cid:3)s λin2 ψ+ β λ+2 +si nλ φ2 β+s λin2 θβcos ψ β) The third curvature of the curve β according to parallel (t3ra.2n0s)- 0 1 2 3 port frame reads + γ(cid:12)3 ( cos φβcos ψ β+(cid:13) (cid:15)sin φβsin θβsin ψ β) (cid:3) + C(cid:3) 3 Cs i 2n + φ βC 2c o+s Cθ 2β M 3 α (3.15) K 3 β = λ20 + λ21 + λ22 + λ23 [ sin φβsin ψ β+ cos φβsin θβcos (ψ3 β.2]1 ) 1 2 3 (cid:14) The proof is complete. (cid:2) M 3 β = λ0 ( sin φβ(cid:3)sin λ ψ2 + β +λ2 c +os λ φ2β +s iλn2 θ βsin ψ β) Acknowledgement 0 1 2 3 (cid:15) + γ0 (−sin φβcos ψ β+ cos φβsin θβsin ψ β) T α I am grateful to the referee for his/her valuable comments and suggestions. (cid:14) + λ1 ( sin φβ(cid:3)sin ψ β+ cos φβsin θβsin ψ β) References λ2 + λ2 + λ2 + λ2 + γ1 (−sin φβcos0 ψ(cid:15) β+1 cos φ2 βsin3 θβsin ψ β) [1] MInt. . TJu.Mrgautth , . SCmoamrabnind.a Vchoel. b1r e(2a0d0th9 )p 4se6u–d4o9 . null curves in Minkowski space-time, φ θ [2] M. Turgut , S. Yilmaz , Smarandache curves in Minkowski space-time, Int. + C(cid:3) 1 cC o 2s + Cβ 2c o+s C 2β M 1 α [3] JU.M. Oatzhtu. rCkoamb ,b Ein.B. .VKo. lO. z3t u(2rk0a0b8 , )K 5. 1I˙–la5r5s .l anc , E. Ne sˆ ovi c´d , On Smarandache curves (cid:14) 1 2 3 lying in lightcone in Minkowski 3-space, J. Dyn. Syst. Geom. Theor. Vol. 12 + λ2 ( sin φβ(cid:3)sin ψ β+ cos φβsin θβsin ψ β) [4] (AN. uMmabgedre n1 ,) S(2. 0Y1ı4lm) a8z1 ,– O91n . the curves of constant breadth in four dimensional λ2 + λ2 + λ2 + λ2 Galilean space, Int. Math. Forum Vol. 9 (no. 25) (2014) 1229–1236 . 0 1 2 3 [5] A.T. Ali , Special Smarandache curves in the Euclidean space, Int. J.Math. Com- + γ2 (−sin φβcos ψ(cid:15) β+ cos φβsin θβsin ψ β) [6] bRi. nB. iVshool. p2 , (T2h0e1r0e) i3s 0m–3o6r .e than one way to frame a curve, in: The American + (cid:14)C(cid:3) 2 cC o 12 s + φ Cβ 22c o+s C θ 32β M 2 α [7] MSfr.a aBmtuheye uminka kEtuiuctcaullki d ,M eGoa.nn O t4hz-ltysup rMakc ,a eCr coEhn 4, ,s 1tNa9en7wt5 , r Taprtpeio.n 2dc4su 6rMv–e2ast5 h2a. .c Sccoir.d NinTgM tSoC Ip a3r a(Nlleol. t4r)a n(2s0p1o5rt) ++ γ3λ (3− ( ssiinn φφββ(cid:3)scion λs ψ20 ψ(cid:15) + β β +λ+21 c c+oos sλ φ φ22β β+ss iλinn23 θ θββssiinn ψ ψ β β)) [[89]] 1iT49Fn.7.1- d1–KcEi–e1r 4m1pl0 ,i 73ıkeBn8n , .o a .s Zlri. , .o SnEBo.ao clT.z ukePruuahrrcatal ,inn d ,.I e . BMaGinhao taks.r p ,m( a3Fcos.Ne.n) .i vcEE. 4 ck ,3 um1Cr vea(e2kspsc) ii( ,aa 2ncY0c .1o J.3Yr d)aM iy2nla1 ,g t3 hPt–.ao 2r S1apc7lail ..e r al( lClteJrMal ntSrs)ap no3sr pt( o1fr)rt a (mf2r0aem1 4ine) + C(cid:3) 3 cC o 2s + φ Cβ 2c o+s C θ 2β M 3 α (3.16) [10] iAAtry.Tt .i ocYlnea kHIuD 20t , , 1 J9M. 3.A5 Sp8ap6vl. .a GM , aTt.h T. aVmoilrucmi , eT h(2e0 S1m4)a r1a2n . dHaicnhdea wcuir vPeusb loisnh iSn 21g aCnodr pitosr adtuioanl-, 1 2 3 [11] M. Cetin , H. Kocayi g˘it , On the quaternionic Smarandache curves in Euclidean Note that 3-space, Int. J. Contemp. Math. Sciences Vol. 8 (no. 3) (2013) 139–150 . ⎡ (cid:11) ⎤ [12] M. Cetin , Y. Tuncer , M.K. Karacan , Smarandache curves according to Bishop θβ = (cid:16) (cid:11) k 3 β ds β, ψ β = −(cid:16) ⎣ k 2 β + k 3 β(cid:11) k 23 β −θβ(cid:4) 2 ⎦d s β, [13] f5Sr.0a –SmE6eN6 .Yi nU RETu ,c lAid. eCaAnL I3S-KsApNac , eN, GCe-n S. mMaartahn. dNaochtees cVuorlv. e2s 0o f( NMoa. n2n) h(eFiembr ucuarryv e2 c0o1u4-) k 2 + k 2 k 2 + k 2 ple according to Frenet frame, Int. J.Math. Combin. Vol.1 (2015) 1–13 . 1 β (cid:11)2 β 1 β 2 β [14] S. Senyurt , A. Calskan , An application according to spatial quaternionic and φβ = −(cid:16) k 23 β −θ θβ(cid:4) 2 ds β [15] MSMma. taEhrla.z naSwdoacyc. ,h 2Se5. cM(u2or0vs1ea7, ,) AS5mp3p–al5r. a6Mn . daathch. eS cci.u Vrvoel.s 9i n( ntoh.e 5 G) a(l2i0le1a5n) 421-s9p–a2c2e8 .G 4 , J. Egypt. cos β