AppliedMathematicalSciences,Vol. 9,2015,no. 5,219-228HIKARILtd, www.m-hikari.comhttp://dx.doi.org/10.12988/ams.2015.411961AnApplicationAccordingtoSpatialQuaternionicSmarandacheCurveS uleymanSenyurt1FacultyofArtsandSciences,DepartmentofMathematicsOrduUniversity,52100,Ordu,TurkeyAbdussametC alskanFacultyofArtsandSciences,DepartmentofMathematicsOrduUniversity,52100,Ordu,TurkeyCopyright c _ 2014 S uleyman Senyurt and Abdussamet C alskan. This is an open accessarticledistributedundertheCreativeCommonsAttributionLicense, whichpermitsunre-stricteduse, distribution, andreproductioninanymedium, providedtheoriginal workisproperlycited.AbstractInthispaper,wefoundtheDarbouxvectorofthespatialquaternioniccurve according to the Frenet frame. Then, the curvature and torsion ofthe spatial quaternionic smarandache curve formed by the unit Darbouxvectorwiththenormalvectorwascalculated. Finally;thesevaluesareexpresseddependinguponthespatialquaternioniccurve.MathematicsSubjectClassication: 11R52,53A04Keywords: Quaternionic curve, Quaternionic frame, Smarandache curves.1 IntroductionThequaternionwasintroducedbyHamilton. Hisinitialattempttogeneralisthe complex numbers by introducing a 3-dimensional object failed in the sensethatthealgebraheconstructedforthese3-dimensional objectsdidnothave1Correspondingauthor220 S uleymanSenyurtandAbdussametC alskanthedesiredproperties. In1987, Bharathi andNagaraj denedthequater-nioniccurvesin E3, E4andstudiedthedierential geometryofspacecurvesandintroducedFrenetframesandformulaebyusingquaternions,[2]. Follow-ing, quaternionicinclinedcurveshavebeendenedandharmoniccurvaturesstudied by Karada g and Sivrida g, [7]. In, Tuna and C oken have studied quater-nionvaluedfunctionsandquaternionicinclinedcurvesinthesemi-EuclideanspaceE42, [9]. Theyhave giventhe Serret-Frenet formulae for the quater-nionic curve in the semi-Euclidean space.Then they have dened quaternionicinclinedcurves andharmoniccurvatures for thequaternioniccurves inthesemi-Euclideanspace. Quaternionicrectifyingcurves havebeenstudiedbyG ungorandTosun, [5]. In[1], AlihasintroducedsomespecialSmarandachecurvesintheEuclideanspace. HehasstudiedFrenet-Serretinvariantsof aspecial case. In[4], ErisirandG ungorhaveobtainedsomecharacterizationsof semi-real spatial quaternionicrectifyingcurvesinIR31. Moreover, bytheaidof thesecharacterizations, theyhaveinvestigatedsemi real quaternionicrectifyingcurvesinsemiquaternionicspace.2 PreliminaryNotesInthissection, wegivethebasicelementsof thetheoryof quaternionsandquaternioniccurves. Amorecompleteelementarytreatmentof quaternionsand quaternionic curves can be found in [2] and [6], respectively. A real quater-nionqisanexpressionoftheformq= d + ae1 + be2 + ce3(2.1)wherea, b, c IRandei, 1 i 3arequaternionicunitswhichsatisfythenon-commutativemultiplicationrules_e12= e22= e32= e1e2e3= 1, e1, e2, e3 IR3e1e2= e3, e2e3= e1, e2e3= e1.(2.2)ThealgebraofthequaternionsisdenotedQbyanditsnaturalbasisisgivenby e1, e2, e3. Arealquaternioncanbegivenbytheformq= Sq + Vq(2.3)whereSq=disscalarpartandVq=ae1 + be2 + ce3isvectorpartofq. TheHamiltonconjugateof q =Sq+ Vqisdenedby q =Sq Vq. Summationoftwoquaternionsq1=Sq1+ Vq1andq2=Sq2+ Vq2isdenedasq1 q2=(Sq1+ Sq2) + (Vq1+ Vq2). Multiplicationof aquaternionq=Sq+ Vqwithascalar R is identied as q= Sq+Vq. These expression the symmetricreal-valued,non-degenerate,bilinearformasfollows:AnapplicationaccordingtospatialquaternionicSmarandachecurve 221, |Q: QQ R, q1, q2[Q=12(q1 q2 + q2 q1) (2.4)whichiscalledthequaternioninnerproduct. ThenthenormofqisN(q) =q q=a2+ b2+ c2+ d2. (2.5)If q=1, thenqiscalledunitquaternion. Letq1=Sq1+ Vq1=d1 + a1e1 +b1e2 +c1e3andq2= Sq2 +Vq2= d2 +a2e1 +b2e2 +c2e3betwoquaternionsin,Q,thenthequaternionproductofq1andq2isgivenbyq1q2= d1d2(a1a2 + b1b2 + c1c2) + (d1a2 + a1d2 + b1c2c1b2)e1+ (d1b2 + b1d2 + b1a2a1b2)e2 + +(d1c2 + c1d2 + a1b2b1a2)e3orq1q2= Sq1Sq2 Vq1, Vq2 + Sq1Vq2 + Sq2Vq1 + Vq1 Vq1(2.6)where , and denotetheinnerproductandvectorproductinEuclidean3-space. q is calledaspatial quaternionwhenever q+ q =0andcalledatemporal quaternion whenever q q= 0. A general quaternion qcan be givenas q=12(q + q) +12(q q). The three-dimensional Euclidean space is identiedwiththespaceofspatialquaternions,[2].QH= q Q [ q + q= 0inanobviousmanner. LetI= [0, 1]beanintervalinthereal line IRands I bethearc-lengthparameteralongthesmoothcurve: [0, 1] QH, (s) =3

i=1i(s)ei. (2.7)Thetangentvector

(s) = t(s)hasunitlength |t(s)| = 1foralls. Itfollowst

t + t + (t)

= 0whichimplies t

is orthogonal tot andt

t is aspatial quaternion. Let: [0, 1] QHbeadierentiablespatial quaternionscurvewitharc-lengthparameter s and t(s), n1(s), n2(s) be the Frenet frame of at the point (s),where___t(s) =

(s)n1(s) =

(s)N(

(s))n2(s) = t(s) n1(s),(2.8)222 S uleymanSenyurtandAbdussametC alskanandthecurve(s)isnonunitspeedcurvethenwesaythat___t(s) =

(s)(s) , (s) = N(

(s))n1(s) = n2(s) t(s)n2(s) =

(s)

(s) + (s)

(s)N_

(s)

(s) + (s)

(s)_(2.9)Let t(s), n1(s), n2(s)be the Frenet frame of (s). ThenFrenet formula,curvatureandthetorsionaregivenby___t

(s) = k(s)n1(s)n1

(s) = k(s)t(s) + r(s)n2(s)n2

(s) = r(s)n1(s),(2.10)and___k(s) =N_

(s)

(s) + (s)

(s)_(s)3r(s) =

(s)

(s),

(s)|Q_N_

(s)

(s) + (s)

(s)__2,(2.11)wheret(s), n1(s), n2(s)aretheunittangent,theunitprincipalnormalandtheunitbinormalvectorofaquaternioniccurve,respectively([2],[8]). Thefunc-tions k, r are called the principal curvature and the torsion, respectively. Thesearet(s) t(s) = n1(s) n1(s) = n2(s) n2(s) = 1t(s) n1(s) = n1(s) t(s) = n2(s)n1(s) n2(s) = n2(s) n1(s) = t(s)n2(s) t(s) = t(s) n2(s) = n1(s).Let: [0, 1] QHbeaunitspeedregularcurveand t(s), n1(s), n2(s)beitsmovingSerret-Frenet frame. Inthiscasetn1, n1n2, tn1n2QuaternionicSmarandachecurvescanbedenedbyAnapplicationaccordingtospatialquaternionicSmarandachecurve 223tn1=12(t(s) + n1(s))n1n2=12(n1(s) + n2(s))tn1n2=13(t(s) + n1(s) + n2(s)), (see[1], [3], [8]).3 An Application According to Spatial Quater-nionicSmarandacheCurve: [0, 1] QHspatial quaternionscurve, t(s), n1(s), n2(s)movingframemoves with a certain angular velocity around each axis s instantly. This axis iscalled instantaneous rotation axis of the spatial quaternionic curve. If DarbouxaxisvectorinthedirectionindicatedbyDD = xt + yn1 + zn2.FromDarbouxequations,t

, n1

, n2

QHderivativevectorst

= D t = zn1yn2n1

= D n1= zt + xn2n2

= D n2= yt xn1andfrom(2.10)z= k, y= 0, x = r. IfwewritethesevaluesD = rt + kn2. (3.1)ThenormisN(D) =_D D =k2+ r2. (3.2)LetDisinstantaneospfavectorofcurve. IftheanglebetweenDandn2is,fromFig.1,itisobtainedthatcos =D, kn2|QN(D)N(kn2)=D kn2 + kn2D2kk2+ r2,sin =D, rt|QN(D)N(rt)=D rt + rt D2rk2+ r2and224 S uleymanSenyurtandAbdussametC alskanFigure1: QuaternionicDarbouxV ectorcos =kk2+ r2, sin =rk2+ r2(3.3)Iftheunitvectorofquaternionicdarbouxvectorindicatedbyww =DN(D)= sin t + cos n2.Conclusion3.1Let : [0, 1] QHbe a unit speed regular curve and t(s), n1(s), n2(s)beitsmovingSerret-Frenet frame. Foranarbitrarycurve, withcurvatureandtorsion,k(s)andr(s)respectively. DarbouxvectorinthedirectionoftheaxisofthequaternioniccurveDD = rt + kn2andcos =kN(D),sin =rN(D)including,unitDarbouxvectorisw = sin t + cos n2.Let:[0, 1] QHbeaunitspeedregularcurveand t(s), n1(s), n2(s)beitsmovingSerret-Frenetframe. Quaternionicn1wSmarandachecurvescanbedenedby(s) =12(n1(s) + w(s)). (3.4)Now, we can investigate Serret-Frenet invariants of quaternionic n1wSmaran-dachecurvesaccordingto=(s). Dierentiating(3.4)withrespecttos,weget

= t(s)dsds=12_(

cos k)t + (r

sin )n2(3.5)wheredsds=_(

)2+ N(D)22

N(D)2

AnapplicationaccordingtospatialquaternionicSmarandachecurve 225Thetangentvectorofcurvecanbewrittenasfollowt(s) =(

cos k)t + (r

sin )n2_(

)2+ N(D)22

N(D), (3.6)dierentiating(3.6)withrespecttos,weobtaint

(s) =_2(1t + 2n1 + 3n2)_(

)2+ N(D)22

N(D)_2, (3.7)where___1= r2

cos k

cos2 r

sin cos (

)4sin k2(

)2sin r2(

)2sin + 2k(

)3sin cos + 2r(

)3sin2 k

(

)2r2k

2k

k

cos 2k

r

sin rr

cos + k2(

)2cos2 + r(

)2sin cos +k

+ krr

kr

sin

r

k sin 2= k(

)3cos + 3k3

cos + 3r2k

cos 2k2(

)2cos2 4kr(

)2sin cos k2(

)2k42k2r2+ 3k2r

sin r2(

)2+ 3r3

sin + r(

)3sin 2r2

2sin23= r

(

)2+ k2r

2kr

cos k2

sin + k

sin cos + r

sin2(

)4cos k2(

)2cos k2(

)2cos + 2k(

)3cos2 + 2r(

)3sin cos r

rkk

+ rk

cos + rk

cos + kk

sin k

(

)2sin cos r

(

)2sin2Theprincipal curvatureandprincipal normal vectoreldof curvearere-spectively,=_2(21 + 22 + 23)_(

)2+ N(D)22

N(D)_2(3.8)226 S uleymanSenyurtandAbdussametC alskanandn=1t + 2n1 + 3n2_21 + 22 + 23 (3.9)Ontheotherhand,weexpressb= t n. So,thebinormalvectorofcurveb=2(

sin r)t +_1(r

sin ) 3(

cos k)_n1__(

)2+ N(D)22

N(D)_(21 + 22 + 23)+2(

cos k)n2__(

)2+ N(D)22

N(D)_(21 + 22 + 23)(3.10)Wedierentiate(3.5)withrespecttosinordertocalculatethetorsion

=(

cos (

)2sin k

)t + (k

cos + r

sin k2r2)n12+(r

sin (

)2cos )n22(3.11)andsimilarly

=1t + 2n1 + 3n22where___1=

cos 3

sin (

)3cos k

k2

cos kr

sin + k3+ kr22= 2k

cos 2k(

)2sin 3kk

+ k

cos + r

sin + 2r

sin +2(

)2cos 3rr

3= (kr

3

) cos + (r2

+ (

)3) sin k2r r3+ r

Thetorsionofcurveis=2(11 + 22 + 33)

21 + 22 + 23(3.12)AnapplicationaccordingtospatialquaternionicSmarandachecurve 227where___

1= (k(

)2sin ) cos + (r(

)2sin k2

2r2

) sin + k2r + r3

2= (r

k(

)2) cos + (k

r(

)2k

) sin rk

+ (

)3+ kr

3= (k(

)2cos + r(

)2sin 2k2

r2

) cos kr

sin + k3+ kr2.Example: Letbespatialquaternioniccurve(s) =_12coss5+12sins5_e12s5e2+_12coss5+12sins5_e3.Intermsofdenition, weobtainspecial n1wsmarandachecurveaccordingtoFrenetframeofspatialquaternioniccurve,(seeFigure3).(s) =_12 coss512 sins5_e112e2 +_12 coss512 sins5_e3.Figure2: SpatialQuaternionicCurve Figure3: SmarandacheCurveReferences[1] A.T. Ali, Special SmarandacheCurvesintheEuclideanSpace, Interna-tional Journal ofMathematical Combinatorics,Vol.2,(2010),30-36.[2] K. Bharath and M. Nagaraj, Quaternion Valued Function of a Real Vari-ableSerret-FrenetFormulae,IndianJ.PureAppl.Math.,18(6),(1987),507-511.228 S uleymanSenyurtandAbdussametC alskan[3] M. C etinandH. Kocayi git, OntheQuaternionicSmarandacheCurvesintheEuclidean3-Space, Int. J. Contemp. Math. Sciences, 8, (2013),139-150.[4] T. Erisir and M.A. G ung or, Some Characterizations of Quaternionic Rec-tifying Curves inthe Semi-EuclideanSpace, HonamMathematical J.,Vol.36,1,(2014),67-83.http://dx.doi.org/10.5831/hmj.2014.36.1.67[5] M.A. G ung or and M. Tosun, Some characterizations of quaternionic recti-fyingcurves,Dierential Geometry-Dynamical Systems,Vol.13,BalkanSocietyofGeometers,GeometryBalkanPress,(2011),89-100.[6] H.H. Hacsalio glu, Hareket Geometrisi ve Kuaterniyonlar Teorisi, Univer-sityofGaziPress,1983.[7] M.Karada g,A.I.Sivridag,TekDe giskenliKuaterniyonDegerliFonksiy-onlarveEgilimC izgileri,Erc.Univ.FenBil.Derg.,13,(1997),23-36.[8] H.Parlatc,KuaterniyonikSmarandacheE grileri,Masterthesis,Univer-sityofSakarya,2013.[9] A. Tuna, A.C. C oken, Onthequaternionicinclinedcurvesinthesemi-Euclidean space, Applied Mathematics and Computation, Vol.155, 2, 2014,373-389.http://dx.doi.org/10.1016/s0096-3003(03)00783-5Received: December1,2014;Published: January1,2015