Bol. Soc. Paran. Mat. (3s.) v. 342(2016): 5362.c SPMISSN-2175-1188online ISSN-00378712inpressSPM:www.spm.uem.br/bspmSmarandacheCurvesInTermsofSabbanFrameofFixedPoleCurveSleymanenyurtandAbdussametalikanabstract: Inthispaper, westudythespecial SmarandachecurveintermsofSabban frame of Fixed Pole curve and we give some characterization of Smarandachecurves. Besides,weillustrateexamplesofourresults.Key Words: Smarandache Curves, Sabban Frame, Geodesic Curvature, FixedPole CurveContents1 Introduction 532 Preliminaries 543 SmarandacheCurvesAccordingtoSabbanFrameofFixedPoleCurve 553.1 CTC-Smarandache Curves . . . . . . . . . . . . . . . . . . . . . . . . 553.2 TC(C TC)-Smarandache Curves. . . . . . . . . . . . . . . . . . . . 573.3 CTC(C TC)-Smarandache Curves. . . . . . . . . . . . . . . . . . . 583.4 Example. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 601. IntroductionAregularcurveinMinkowski space-time, whosepositionvectoriscomposedbyFrenetframevectorsonanotherregularcurve,iscalledaSmarandachecurve[12]. Special Smarandachecurveshavebeenstudiedbysomeauthors. AhmadT.Ali studiedsomespecial SmarandachecurvesintheEuclideanspace.Hestud-iedFrenet-Serretinvariantsof aspecial case [1]. M. etin, Y. TunerandK.Karacaninvestigatedspecial smarandachecurves accordingtoBishopframeinEuclidean 3-Space and they gave some dierential goematric properties of Smaran-dachecurves [5]. enyurtandalkaninvestigatedspecial Smarandachecurvesintermsof Sabbanframeofspherical indicatrixcurves andtheygave somechar-acterizationofSmarandachecurves, [3].Also, intheirotherwork, whentheunitDarbouxvectorofthepartnercurveofMannheimcurveweretakenastheposi-tion vectors, the curvature and the torsion of Smarandache curve were calculated.Thesevalueswereexpressed dependingupontheMannheimcurve, [4]. Theyde-nedNC-Smarandachecurve, thentheycalculatedthecurvatureandtorsionofNB and TNB- Smarandache curves together with NC-Smarandache curve, [10]. .2000MathematicsSubjectClassication: 53A04.53TypesetbyBSPMstyle.c Soc. Paran. deMat.54 SleymanenyurtandAbdussametalikanBektaandS. Ycestudiedsomespecial smarandachecurvesaccordingtoDar-bouxFrameinE3[2]. M. TurgutandS. Ylmazstudiedaspecial caseof suchcurvesandcalleditsmarandacheTB2curvesinthespaceE41[12]. K.Tas.kpr, M. Tosun studied special Smarandache curves according to Sabban frame onS2[11].In this paper, the special smarandache curves such as CTC, TC(CTC), CTC(CTC) created by Sabban frame , {C, TC, C TC} , that belongs to xed pole of acurve are dened. Besides, we have found some results.2. PreliminariesThe Euclidean 3-spaceE3be inner product given by,= x21 +x32 +x23where (x1, x2, x3) E3. Let : I E3be a unit speed curve denote by {T, N, B}themovingFrenetframe. Foranarbitrarycurve E3, withrstandsecondcurvature, andrespectively, the Frenet formulae is given by[6]T= NN= T+BB= N.(2.1)Accordingly, the spherical indicatrix curves of Frenet vectors are (T), (N) and (B)respectively.These equations of curves are given by[7], [9]T(s) = T(s)N(s) = N(s)B(s) = B(s)(2.2)Let: I S2beaunitspeedspherical curve. Wedenotesasthearc-lengthparameter of. Let us denote by(s) = (s)t(s) = (s)d(s) = (s) t(s).(2.3)Wecall t(s) aunit tangent vector of. {, t, d}frame iscalled theSabban frameofonS2. Then we have the following spherical Frenet formulae of:= tt= +gdd= gt(2.4)where is called the geodesic curvature ofgonS2andg= t, d, [8]. (2.5)SmarandacheCurvesInTermsofSabbanFrameofFixedPoleCurve 553. SmarandacheCurvesAccordingtoSabbanFrameofFixedPoleCurveInthis section, weinvestigateSmarandachecurvesaccordingtotheSabbanframe of xed pole curve (C). LetC(s) =C(s)bea unitspeed regular sphericalcurves onS2.We denotesCas the arc-lenght parameter of xed pole curve(C)C(s) = C(s) (3.1)Dierentiating(3.1) , we havedCdsCdsCds= C(s)andTCdsCds= cos T sin B (3.2)From the equation (3.2)TC= cos T sinBandC TC= NFrom the equation (2.3)C(s) = C(s)TC(s) = cos T sin B(C TC)(s) = N(s)is called the Sabban frame of xed pole curve (C) .From the equation (2.5)g= TC, C TC = g= WThenfromtheequation(2.4)wehavethefollowingsphericalFrenetformulaeof(C):C= TCTC= C +W(C TC)(C TC)= WTC(3.3)3.1. CTC-SmarandacheCurvesDenition3.1. Let S2beaunit sphereinE3andsuppose that theunit speedregularcurveC(s)=C(s) lyingfullyonS2. Inthiscase, CTC- Smarandachecurvecanbedenedby(s) =12(C +TC). (3.4)56 SleymanenyurtandAbdussametalikanNow we can compute Sabban invariants ofCTC- Smarandache curves. Dier-entiating (3.4) , we haveTdsds=12
(cos sin )T+ WN (cos + sin )B
,wheredsds=
2 +
W
22. (3.5)Thus, the tangent vector of curveis to beT=1
2 +
W
2
(cos sin)T+ WN (cos + sin)B
. (3.6)Dierentiating(3.6), we getTdsds=1(2 + (W)2)32(1T+2N+3B) (3.7)where1= 2(sin + cos )
2W+
W
3
W2(sin + cos ) W
W
(cos sin)2=
2 +
W
2
(cos sin) +(cos + sin )
+ 2
W
3= W
2 +
W
2
2(cos sin ) W2(cos sin)+W
W
(cos + sin).Substituting the equation (3.5) into equation (3.7) , we reachT=2
2 +
W
22(1T+2N+ 3B). (3.8)Considering the equations (3.4) and (3.6) , it easily seen that(C TC)=1
4 + 2
W
2
W(cos sin )T+ (3.9)+2N+ (W(cos + sin ))B
SmarandacheCurvesInTermsofSabbanFrameofFixedPoleCurve 57From the equation (3.8) and (3.9) , the geodesic curvature of(s) isg= T, (C TC)
=1
2+
W
232
W(cos sin )1 + 22 +W(cos + sin )3
.3.2. TC(C TC)-SmarandacheCurvesDenition3.2. LetS2beaunitsphereinE3andsupposethattheunitspeedreg-ularcurveC(s) = C(s)lyingfullyonS2. Inthiscase,TC(CTC)-Smarandachecurvecanbedenedby(s) =12(TC+C TC). (3.10)Now we can compute Sabban invariants of TC(C TC) - Smarandache curves.Dierentiating(3.10), we haveTdsds=12
(sin Wcos )T+ WN+ (Wsin cos )B
wheredsds=
1 + 2
W
22. (3.11)In that case, the tangent vector of curveis as followsT=1
1 + 2
W
2
(sin Wcos )T+ (3.12)+WN+ (Wsin cos )B
Dierentiating(3.12), it is obtained thatTdsds=1
1 + 2
W
2
32(1T+2N+3B) (3.13)where58 SleymanenyurtandAbdussametalikan1= cos +W(sin + 2Wcos + 2
W
2sin 2W23)
W
(cos 2Wsin )2= (sin Wcos + 2
W
2(sin Wcos ))(W(sin cos ) 2
W
3(sin cos )) +
W
3= sin +W(+cos + 2
W
2+ 2
W
2cos + 2W sin) +
W
(sin + 2Wcos ).Substituting the equation (3.11) into equation (3.13) , we getT=2
1 + 2
W
22(1T+2N+3B) (3.14)Using the equations (3.10) and (3.12) , we easily nd(C TC)=1
2 + 4
W
2
(2Wsin cos )T+ (3.15)+N+ (2Wcos + sin)B
So, the geodesic curvature of(s) is as followsg= T, (C TC)
=1
1+2
W
232((2Wsin cos )1 +2 + (2Wcos + sin )3).3.3. CTC(C TC)-SmarandacheCurvesDenition 3.3. Let S2bea unit sphereinE3and supposethatthe unit speed regu-larcurve C(s) = C(s)lyingfullyon S2. In this case,CTC(CTC)- Smarandachecurvecanbedenedby(s) =13(C +TC+C TC). (3.16)SmarandacheCurvesInTermsofSabbanFrameofFixedPoleCurve 59Let us calculate Sabbaninvariants of CTC(C TC) - Smarandachecurves.Dierentiating(3.16), we haveTdsds=13
(cos sin Wcos )T++WN+ (sin cos + Wsin)B
wheredsds=
2(1 W+
W
2)3. (3.17)Thus, the tangent vector of curveisT=1
2
1 W+
W
2
(cos sin Wcos )T (3.18)+WN+ (sin cos + Wsin )B
Dierentiating(3.18), it is obtained thatTdsds=122
1 W+
W
2
32(1T+2N+3B) (3.19)where1=
1 W+
W
2
2(sin + cos ) + 2W sin 2W
+W(cos sin ) + 2
W
(cos +Wsin)2=
1 W+
W
2
2
cos sin Wcos
+2
sin + cos Wcos
+ 2
W
1 W
+
W
23=
1 W+
W
2
2W+ 2(sin cos ) + 2W cos
+
W
2sin + 2
W
(sin +Wcos ) W(sin + cos ).Substituting the equation (3.17) into equation (3.19) , we reachT=34
1 W+
W
2
2(1T+2N+3B). (3.20)60 SleymanenyurtandAbdussametalikanUsing the equations (3.16) and (3.18) , we have(C TC)=16
1 W+
W
2
(2Wsin Wcos (3.21)cos )T+ (1 W)N+ (sin + 2Wcos +Wsin )B
From the equation (3.20) and (3.21), the geodesic curvature of(s) isg= T, (C TC) =142(1 W+ (W)2)32[1(2Wsin Wcos cos ) +2(1 W) +3(sin + 2Wcos + Wsin)].3.4. ExampleLet us consider the unit speed spherical curve:(s) = {9208 sin16s 1117 sin 36s, 9208 cos 16s +1117 cos 36s,665 sin 10s}.It is rendered in Figure 1.Figure 1: FixedPolecurve(T)In terms of denitions, we obtain Smarandache curves according to Sabban frameonS2, see Figures 2 - 4.SmarandacheCurvesInTermsofSabbanFrameofFixedPoleCurve 61Figure 2: CTC- Smarandache CurveFigure 3: TC(C TC) - Smarandache CurveFigure 4: CTC(C TC) - Smarandache Curve62 SleymanenyurtandAbdussametalikanReferences1. Ali A. T., Special Smarandache Curves inthe EuclideanSpace, International Journal ofMathematicalCombinatorics,Vol.2,pp.30-36,2010.2. Bekta. andYceS., Special Smarandache Curves According toDarbouxFrameinEu-clidean3- Space,RomanianJournal of MathematicsandComputerScience, Vol.3, Issue1,pp.48-59,2013.3. alkanA. andenyurt, S., SmarandacheCurvesInTermsof SabbanFrameof SphericalIndicatrixCurves,Gen.Math.Notes,inpress,2015.4. alkanA.andenyurt, S., NCSmarandacheCurvesof MannheimCurveCoupleAc-cordingtoFrenet Frame, International Journal of Mathematical Combinatorics, Vol:1, pp.13-15,2015.5. etinM., TunerY. andKaracanM. K., SmarandacheCurvesAccordingtoBishopFrameinEuclidean3-Space,Gen.Math.Notes,Vol.20,pp.50-66,2014.6. DoCarmo, M.P., Dierential Geometryof CurvesandSurfaces, PrenticeHall,EnglewoodClis, NJ,1976.7. Hacsaliholu H.H.,Dierantial Geometry,AcademicPressInc.Ankara,1994.8. KoenderinkJ.,SolidShape,MIT Press,Cambridge,MA,1990.9. ONeill B.,ElemantaryDierential Geometry,AcademicPressInc.NewYork,1966.10. SenyurtS.andSivasS.,AnApplicationof SmarandacheCurve,UniversityofOrdujournalofscienceandtechnology,Vol:3(1),pp.46-60,2013.11. TakprK.andTosunM.,SmarandacheCurvesonS2,BoletimdaSociedadeParanaensedeMatemtica3Srie.Vol.32,no:1,pp.51-59,2014.12. TurgutM. andYlmazS., SmarandacheCurvesinMinkowski Space-time, International J.Math.Combin.,Vol.3,pp.51-55,2008.Sleymanenyurt(Correspondingauthor)Facultyof ArtsandSciences,Departmentof Mathematics,OrduUniversity,52000, Ordu/TurkeyE-mail address: [email protected] alikanFacultyof ArtsandSciences,Departmentof Mathematics,OrduUniversity,52000, Ordu/TurkeyE-mail address: [email protected]
