surfaces family with common smarandache asymptotic curve according to bishop frame in euclidean...

Bol. Soc. Paran. Mat. (3s.) v. 341(2016): 187202.c SPMISSN-2175-1188online ISSN-00378712inpressSPM:www.spm.uem.br/bspmSurfacesFamilyWithCommonSmarandacheAsymptoticCurveAccordingToBishopFrameInEuclideanSpaceGlnuraakAtalayandEmin Kasapabstract: Inthispaper, weanalyzedtheproblemof consructingafamilyofsurfaces from a given some special Smarandachecurves in Euclidean 3-space. Usingthe Bishop frame of the curve in Euclidean 3-space, we express the family of surfacesasalinearcombinationofthecomponentsofthisframe, andderivethenecessaryandsucientconditionsforcoecentstosatisfyboththeasymptoticandisopara-metric requirements. Finally, examples are given to show the family of surfaces withcommonSmarandacheasymptoticcurve.KeyWords: Smarandache AsymptoticCurve, BishopFrame.Contents1 Introduction 1872 Preliminaries 1883 SurfaceswithcommonSmarandacheasymptoticcurve 1904 ExamplesofgeneratingsimplesurfaceswithcommonSmarandacheasymptoticcurve 1971. IntroductionIn dierential geometry, there are many important consequences and propertiesof curves[1], [2], [3]. Researches follow labours about the curves. In the light of theexisting studies, authors always introduce new curves. Special Smarandache curvesareoneofthem. SpecialSmarandachecurveshavebeeninvestigatedbysomedif-ferential geometers [4,5,6,7,8,9,10]. This curveis denedas, aregular curveinMinkowski space-time,whoseposition vectoriscomposedbyFrenetframevectorson another regular curve, is Smarandache curve[4]. A.T.Alihasintroduced somespecial Smarandache curves in the Euclidean space[5]. Special Smarandache curvesaccording to Bishop Frame in Euclidean 3-space have been investigated by etin etal[6]. In addition, Special Smarandache curves according to Darboux Frame in Eu-clidean 3-space hasintroducedin[7]. Theyfoundsomepropertiesof thesespecialcurvesandcalculatednormal curvature, geodesiccurvatureandgeodesictorsionof thesecurves. Also, theyinvestigatespecial SmarandachecurvesinMinkowski3-space, [8]. Furthermore, theyndsomepropertiesof thesespecial curvesandtheycalculatecurvatureandtorsionofthesecurves. SpecialSmarandachecurves2000MathematicsSubjectClassication: 53A04,53A05187TypesetbyBSPMstyle.c Soc. Paran. deMat.188 GlnuraffakAtalayandEminKasapsuchas-SmarandachecurvesaccordingtoSabbanframeinEuclideanunitspherehas introduced in[9]. Also, they give some characterization of Smarandache curvesandillustrateexamplesoftheirresults.OntheQuaternionicSmarandacheCurvesinEuclidean3-Spacehavebeeninvestigatedin[10].One of themost signicant curve on a surface is asymptotic curve. Asymptoticcurveonasurfacehasbeenalong-termresearchtopicinDierential Geometry,[3,11,12]. Acurveonasurfaceiscalledanasymptoticcurveprovideditsvelocityalways points in an asymptotic direction,that is thedirection in which thenormalcurvatureiszero. AnothercriterionforacurveinasurfaceMtobeasymptoticis that its accelerationalways be tangent toM, [2].Asymptoticcurves arealsoencounteredinastronomy, astrophysicsandCADinarchitecture.Theconceptoffamilyofsurfaceshavingagivencharacteristiccurvewasrstintroduced by Wang et.al. [12]in Euclidean 3-space. Kasap et.al. [13]generalizedthe work of Wang by introducing new types of marching-scale functions, coecientsoftheFrenetframeappearingintheparametricrepresentationofsurfaces. Withthe inspiration of work of Wang, Li et.al. [14] changed the characteristic curve fromgeodesictolineofcurvatureanddenedthesurfacepencilwithacommonlineofcurvature. Recently, in[15] Bayramet.al. dened the surface pencil with a commonasymptoticcurve. Theyintroducedthreetypes of marching-scalefunctions andderived thenecessary andsucientconditionson themtosatisfybothparametricandasymptoticrequirements.Bishopframe, whichisalsocalledalternativeorparallel frameof thecurves,wasintroducedbyL. R. Bishopin1975bymeansof parallel vectorelds, [16].Recently, manyresearchpapersrelatedtothisconcepthavebeentreatedintheEuclidean space, see[17,18]. And,recently, this special frame is extended to studyof canal and tubular surfaces, we refer to[19,20]. Bishop and FrenetSerret frameshaveacommonvectoreld, namelythetangentvectoreldoftheFrenetSerretframe. ApracticalapplicationofBishopframesisthattheyareusedintheareaofBiologyandComputerGraphics. Forexample, itmaybepossibletocomputeinformationabout theshopeof sequencesof DNAusingacurvedenedbytheBishopframe. TheBishopframemayalsoprovideanewwaytocontrol virtualcamerasincomputeranimatons, [21].In this paper, we study the problem: given a curve (with Bishop frame), how tocharacterizethosesurfacesthatposessthiscurveasacommonisoasymptoticandSmarandachecurveinEuclidean3-space. Insection2, wegivesomepreliminaryinformation aboutSmarandache curvesinEuclidean3-space anddeneisoasymp-toticcurve. Weexpresssurfacesasalinearcombinationof theBishopframeofthegivencurveandderivenecessaryandsucientconditionsonmarching-scalefunctionstosatisfybothisoasymptoticandSmarandacherequirementsinSection3. Weillustratethemethodbygivingsomeexamples.2. PreliminariesThe Bishopframe or parallel transport frame is analternativeapproachtodeningamovingframethatiswell denedevenwhenthecurvehasvanishingsecondderivative. Onecanexpress parallel transport of anorthonormal frameSurfacesFamilyWithCommonSmarandacheAsymptoticCurve 189alongacurvesimplybyparallel transportingeachcomponentof theframe [16].Thetangentvectorandanyconvenientarbitrarybasisfortheremainderof theframeareused(fordetails,see [17]). TheBishop frameisexpressed as [16,18].dds__T(s)N1(s)N2(s)__=__0 k1(s) k2(s)k1(s) 0 0k2(s) 0 0____T(s)N1(s)N2(s)__(2.1)Here,weshallcalltheset {T(s), N1(s), N2(s)}asBishopFrameandk1andk2asBishop curvatures. The relation between Bishop Frame and Frenet Frame of curve(s)isgivenasfollows;__T(s)N1(s)N2(s)__=__1 0 00 cos (s) sin (s)0 sin (s) cos (s)____T(s)N(s)B(s)__(2.2)where___ (s) = arctan_k2(s)k1(s)_ (s) = d(s)ds(s) =_k21(s) + k22(s)(2.3)HereBishopcurvaturesaredenedby_k1(s) = (s) cos (s)k2(s) = (s) sin (s)(2.4)LetrbearegularcurveinasurfacePpassingthroughsP, thecurvatureofratsandcos =n.N,whereNisthenormalvectortorandisnnormalvectortoPatsand . denotesthestandard innerproduct. Thenumberkn= cos isthencalledthenormalcurvatureofats [2].Let sbeapoint inasurface P.Anasymptoticdirection ofPat sisadirectionofthetangentplaneofPforwhichthenormalcurvatureiszero. Anasymptoticcurve of Pisaregular curver Psuch thatfor eachsrthetangent lineof ratsisanasymptoticdirection[2].Anisoparametriccurve(s)isacurveonasurface=(s, t)isthathasaconstantsort-parametervalue. Inotherwords, thereexistaparameterorsuchthat(s) = (s, t0)or(t) = (s0, t).Givenaparametriccurve(s),wecall(s)anisoasymptoticofasurfaceifitisbothaasymptoticandanisoparametriccurveon.Let=(s)beaunitspeedregularcurveinE3and {T(s), N1(s), N2(s)}beitsmovingBishopframe. SmarandacheTN1curvesaredenedby= (s) =12 (T(s) + N1(s)) ,SmarandacheTN2curvesaredenedby= (s) =12 (T(s) + N2(s)),SmarandacheN1N2curvesaredenedby= (s) =12 (N1(s) + N2(s)),SmarandacheTN1N2curvesaredenedby= (s) =13 (T(s) + N1(s) + N2(s)), [5].190 GlnuraffakAtalayandEminKasap3. SurfaceswithcommonSmarandacheasymptoticcurveLet = (s, v) be a parametric surface. The surface is dened by a given curve = (s)asfollows:(s, v) = (s)+[x(s, v)T(s)+y(s, v)N1(s)+z(s, v)N2(s)], L1 s L2, T1 v T2(3.1)wherex(s, v) , y(s, v)andz(s, v)areC1functions. Thevaluesof themarching-scale functionsx(s, v) , y(s, v) and z(s, v) indicate, respectively; the extension-like,exion-like and retortion-like eects, by the point unit through the time v, startingfrom(s)and {T(s), N1(s), N2(s)}istheBishopframeassociatedwiththecurve(s).Ourgoalistondthenecessaryandsucientconditionsforwhichthesomespecial Smarandachecurvesof theunitspacecurve(s) isanparametriccurveandanasymptoticcurveonthesurface(s, v).Firstly,sinceSmarandachecurveof(s)isanparametriccurveonthesurface(s, v),thereexistsaparameterv0[T1, T2]suchthatx(s, v0) = y(s, v0) = z(s, v0) = 0, L1 s L2, T1 v T2. (3.2)Secondly, accordingtotheabovedenitions, thecurve(s)isanasymptoticcurveonthesurface(s, v)ifandonlyifthenormalcurvaturekn=cos = 0,whereistheanglebetweenthesurfacenormaln(s, v0)andtheprincipalnormalN(s) of the curve(s). Since n(s, v0).T(s) =0, L1s L2, byderivatingthisequationwithrespecttothearclengthparameters, wehavetheequivalentconstraintdnds(s, v0).T(s) = 0 (3.3)for thecurve(s) tobeanasymptoticcurveonthesurface(s, v), where .denotesthestandardinnerproduct.Theorem3.1. : SmarandacheTN1curveofthecurve(s)isisoasymptoticonasurface(s, v)ifandonlyif thefollowingconditionsaresatised:_x(s, v0) = y(s, v0) = z(s, v0) = 0,zv(s, v0) = tan (s)yv(s, v0).Proof: Let (s) be aSmarandache TN1curve onsurface (s, v).From(3.1) ,(s, v),parametricsurfaceisdenedbyagivenSmarandacheTN1curveofcurve(s)asfollows:(s, v) =12(T(s) + N1(s)) + [x(s, v)T(s) + y(s, v)N1(s) + z(s, v)N2(s)].IfSmarandacheTN1curveisanparametriccurveonthissurface,thenthereexistaparameterv = v0suchthat,12 (T(s) + N1(s)) = (s, v0),thatis,x(s, v0) = y(s, v0) = z(s, v0) = 0 (3.4)SurfacesFamilyWithCommonSmarandacheAsymptoticCurve 191Thenormal vectorof(s, v)canbewrittenasn(s, v) =(s, v)ds(s, v)vSinceThenormalvectorcanbeexpressed asn(s, v) = [z(s, v)dv(k1(s)2+y(s, v)ds+ k1(s)x(s, v))y(s, v)dv(k2(s)2+z(s, v)ds+ k2(s)x(s, v))]T(s)+[x(s, v)dv(k2(s)2+z(s, v)ds+ k2(s)x(s, v))z(s, v)dv(k1(s)2+x(s, v)dsk1(s)y(s, v) k2(s)z(s, v))]N1(s)+[y(s, v)dv(k1(s)2+x(s, v)dsk1(s)y(s, v) k2(s)z(s, v))x(s, v)dv(k1(s)2+y(s, v)ds+ k1(s)x(s, v))]N2(s) (3.5)Using(3.4),ifwelet___1(s, v0) =k1(s)2zdv(s, v0) k2(s)2ydv(s, v0),2(s, v0) =k1(s)2zdv(s, v0) +k2(s)2xdv(s, v0),3(s, v0) = k1(s)2ydv(s, v0) k1(s)2xdv(s, v0).(3.6)weobtainn(s, v0) = 1(s, v0)T(s) + 2(s, v0)N1(s) + 3(s, v0)N2(s).FromEqn. (2.2),wegetn(s, v0) = 1(s, v0)T(s) + cos (s) 2(s, v0) + sin (s) 3(s, v0))N(s)+(cos (s) 3(s, v0) sin (s) 2(s, v0))B(s).Fromtheeqn. (3.2),weshouldhavens(s, v0).T(s) = 0(cos (s)2(s,v0)+sin(s)3(s,v0))N(s)+(cos (s)3(s,v0)sin (s)2(s,v0))B(s))s.T(s) = 01s(s, v0) (s) (cos (s) 2(s, v0) + sin (s) 3(s, v0)) = 0. (3.7)192 GlnuraffakAtalayandEminKasapFrom (3.4), wehave1s(s, v0) = 0.Weusing Eqn. (3.6) and Eqn.(2.4) in Eqn.(3.7),since(s) = 0,wegetz(s, v0)dv= tan (s) y(s, v0)dv(3.8)whichcompletestheproof. Combiningtheconditions (3.4) and(3.8), wehavefoundthenecessaryandsucientconditionsforthe (s, v) tohavetheSmarandacheTN1curveof thecurveisanisoasymptotic.Theorem3.2. : SmarandacheTN2curveofthecurve(s)isisoasymptoticonasurface(s, v)ifandonlyif thefollowingconditionsaresatised:_x(s, v0) = y(s, v0) = z(s, v0) = 0,zv(s, v0) = tan (s)yv(s, v0).Proof: Let (s) be aSmarandache TN2curve onsurface (s, v).From(3.1) ,(s, v)parametricsurfaceisdenedbyagivenSmarandacheTN2curveofcurve(s)asfollows:(s, v) =12(T(s) + N2(s)) + [x(s, v)T(s) + y(s, v)N1(s) + z(s, v)N2(s)].If Smarandache TN2curve is an parametric curve on this surface , then there existaparameterv= v0suchthat,12 (T(s) + N2(s)) = (s, v0),thatis,x(s, v0) = y(s, v0) = z(s, v0) = 0 (3.9)Thenormal vectorcanbeexpressed asn(s, v) = [z(s, v)dv(k1(s)2+y(s, v)ds+ k1(s)x(s, v))y(s, v)dv(k2(s)2+z(s, v)ds+ k2(s)x(s, v))]T(s)+[x(s, v)dv(k2(s)2+z(s, v)ds+ k2(s)x(s, v))z(s, v)dv(k2(s)2+x(s, v)dsk1(s)y(s, v) k2(s)z(s, v))]N1(s)+[y(s, v)dv(k2(s)2+x(s, v)dsk1(s)y(s, v) k2(s)z(s, v))x(s, v)dv(k1(s)2+y(s, v)ds+ k1(s)x(s, v))]N2(s) (3.10)SurfacesFamilyWithCommonSmarandacheAsymptoticCurve 193Using(3.9),ifwelet___1(s, v0) =k1(s)2zdv(s, v0) k2(s)2ydv(s, v0),2(s, v0) =k2(s)2zdv(s, v0) +k2(s)2xdv(s, v0),3(s, v0) = k2(s)2ydv(s, v0) k1(s)2xdv(s, v0).(3.11)weobtainn(s, v0) = 1(s, v0)T(s) + 2(s, v0)N1(s) + 3(s, v0)N2(s).FromEqn. (2.2),wegetn(s, v0) = 1(s, v0)T(s) + cos (s) 2(s, v0) + sin (s) 3(s, v0))N(s)+(cos (s) 3(s, v0) sin (s) 2(s, v0))B(s).Fromtheeqn. (3.2),weshouldhavens(s, v0).T(s) = 0(cos (s)2(s,v0)+sin(s)3(s,v0))N(s)+(cos (s)3(s,v0)sin (s)2(s,v0))B(s))s.T(s) = 01s(s, v0) (s) (cos (s) 2(s, v0) + sin (s) 3(s, v0)) = 0. (3.12)From(3.9), wehave1s(s, v0) =0.WeusingEqn. (3.11) andEqn.(2.4) inEqn. (3.12),since(s) = 0,wegetz(s, v0)dv= tan (s) y(s, v0)dv(3.13)whichcompletestheproof. Theorem3.3. : SmarandacheN1N2curveof thecurve(s)isisoasymptoticonasurface(s, v)if andonlyif thefollowingconditionsaresatised:___x(s, v0) = y(s, v0) = z(s, v0) = 0,k1 (s) + k2 (s) = 0,zv(s, v0) = tan (s)yv(s, v0).Proof: Let (s) beaSmarandacheN1N2curveonsurface(s, v).From(3.1) ,(s, v)parametric surfaceisdenedbyagiven SmarandacheN1N2curve ofcurve(s)asfollows:(s, v) =12(N1(s) + N2(s)) + [x(s, v)T(s) + y(s, v)N1(s) + z(s, v)N2(s)].194 GlnuraffakAtalayandEminKasapIf SmarandacheN1N2curveisanparametriccurveonthis surface, thenthereexistaparameterv = v0suchthat,12 (N1(s) + N2(s)) = (s, v0),thatis,x(s, v0) = y(s, v0) = z(s, v0) = 0 (3.14)Thenormal vectorcanbeexpressed asn(s, v) = [z(s, v)dv(y(s, v)ds+ k1(s)x(s, v)) y(s, v)dv(z(s, v)ds+k2(s)x(s, v))]T(s) + [x(s, v)dv(z(s, v)ds+ k2(s)x(s, v))z(s, v)dv(k1(s) + k2(s)2+x(s, v)dsk1(s)y(s, v)k2(s)z(s, v))]N1(s) + [y(s, v)dv(k1(s) + k2(s)2+x(s, v)dsk1(s)y(s, v) k2(s)z(s, v))x(s, v)dv(y(s, v)ds+ k1(s)x(s, v))]N2(s) (3.15)Using(3.14),ifwelet___1(s, v0) = 0,2(s, v0) =_k1(s)+k2(s)2_zdv(s, v0),3(s, v0) = _k1(s)+k2(s)2_ydv(s, v0).(3.16)weobtainn(s, v0) = 2(s, v0)N1(s) + 3(s, v0)N2(s).FromEqn. (2.2),wegetn(s, v0) = cos (s) 2(s, v0) + sin (s) 3(s, v0))N(s)+(cos (s) 3(s, v0) sin (s) 2(s, v0))B(s).Fromtheeqn. (3.2),weshouldhavens(s, v0).T(s) = 0(cos (s)2(s,v0)+sin(s)3(s,v0))N(s)+(cos (s)3(s,v0)sin (s)2(s,v0))B(s))s.T(s) = 0(s) (cos (s) 2(s, v0) + sin (s) 3(s, v0)) = 0. (3.17)WeusingEqn. (3.16) andEqn.(2.4)inEqn. (3.17), since(s) = 0,wegetcos (s)_k1(s) + k2(s)2_z(s, v0)dv= sin (s)_k1(s) + k2(s)2_y(s, v0)dvSurfacesFamilyWithCommonSmarandacheAsymptoticCurve 195For,k1 (s) + k2 (s) = 0weobtainz(s, v0)dv= tan (s) y(s, v0)dv(3.18)whichcompletestheproof. Theorem3.4. SmarandacheTN1N2curveof thecurve(s)isisoasymptoticonasurface(s, v)if andonlyif thefollowingconditionsaresatised:___x(s, v0) = y(s, v0) = z(s, v0) = 0,k1 (s) + k2 (s) = 0,zv(s, v0) = tan (s)yv(s, v0).Proof: Let(s)beaSmarandacheTN1N2curveonsurface(s, v).From(3.1),(s, v) parametric surface is dened by a given Smarandache TN1N2 curve of curve(s)asfollows:(s, v) =12(T(s) + N1(s) + N2(s)) +[x(s, v)T(s) +y(s, v)N1(s) +z(s, v)N2(s)].IfSmarandacheTN1N2curveisanparametriccurveonthissurface, thenthereexistaparameterv=v0suchthat,12 (T(s) + N1(s) + N2(s))=(s, v0), thatis,x(s, v0) = y(s, v0) = z(s, v0) = 0 (3.19)Thenormal vectorcanbeexpressed asn(s, v) = [z(s, v)dv(k1(s)3+y(s, v)ds+ k1(s)x(s, v))y(s, v)dv(k2(s)3+z(s, v)ds+ k2(s)x(s, v))]T(s)+[x(s, v)dv(z(s, v)ds+ k2(s)x(s, v) +k2(s)3)z(s, v)dv(k1(s) + k2(s)3+x(s, v)dsk1(s)y(s, v)k2(s)z(s, v))]N1(s)+[y(s, v)dv(k1(s) + k2(s)3+x(s, v)dsk1(s)y(s, v) k2(s)z(s, v))x(s, v)dv(k1(s)3+y(s, v)ds+ k1(s)x(s, v))]N2(s) (3.20)Using(3.19),ifwelet___1(s, v0) =k1(s)3zdv(s, v0) k2(s)3ydv(s, v0),2(s, v0) =k2(s)3xdv(s, v0) +_k1(s)+k2(s)3_zdv(s, v0),3(s, v0) = k1(s)3xdv(s, v0) _k1(s)+k2(s)3_ydv(s, v0).(3.21)196 GlnuraffakAtalayandEminKasapweobtainn(s, v0) = 1(s, v0)T(s) + 2(s, v0)N1(s) + 3(s, v0)N2(s).FromEqn. (2.2),wegetn(s, v0) = 1(s, v0)T(s) + cos (s) 2(s, v0) + sin (s) 3(s, v0))N(s)+(cos (s) 3(s, v0) sin (s) 2(s, v0))B(s).FromtheEqn. (3.2),weshouldhavens(s, v0).T(s) = 0(cos (s)2(s,v0)+sin(s)3(s,v0))N(s)+(cos (s)3(s,v0)sin (s)2(s,v0))B(s))s.T(s) = 0(s) (cos (s) 2(s, v0) + sin (s) 3(s, v0)) = 0. (3.22)From(3.19),wehave1s(s, v0) = 0.WeusingEqn. (3.21)andEqn.(2.4)inEqn.(3.22),since(s) = 0,wegetcos (s)_k1(s) + k2(s)3_z(s, v0)dv= sin (s)_k1(s) + k2(s)3_y(s, v0)dvFor,k1 (s) + k2 (s) = 0weobtainz(s, v0)dv= tan (s) y(s, v0)dv(3.23)whichcompletestheproof. Nowletusconsiderothertypesof themarching-scalefunctions. IntheEqn.(3.1) marching-scalefunctions x(s, v) , y(s, v) andz(s, v) canbechoosenintwodierentforms:1) Ifwechoose___x(s, v) =p

k=1a1kl(s)kx(v)k,y(s, v) =p

k=1a2km(s)ky(v)k,z(s, v) =p

k=1a3kn(s)kz(v)k.thenwecansimplyexpressthesucientconditionforwhichthecurve(s)isanSmarandacheasymptoticcurveonthesurface(s, v)as_x(v0) = y(v0) = z(v0) = 0,a31n(s)dz(v0)dv= tan (s) a21m(s)dy(v0)dv.(3.24)SurfacesFamilyWithCommonSmarandacheAsymptoticCurve 197wherel(s), m(s), n(s), x(v), y(v)andz(v)areC1functions, aijIR, i =1, 2, 3, j= 1, 2, ..., p.2)Ifwechoose___x(s, v) = f_p

k=1a1kl(s)kx(v)k_,y(s, v) = g_p

k=1a2km(s)ky(v)k_,z(s, v) = h_p

k=1a3kn(s)kz(v)k_.then we can write the sucient condition for which the curve (s) is an Smaran-dacheasymptoticcurveonthesurface(s, v)as_x(v0) = y(v0) = z(v0) = f(0) = g(0) = h(0) = 0,a31n(s)dz(v0)dvh(0) = tan (s) a21m(s)dy(v0)dvg(0).(3.25)wherel(s), m(s), n(s), x(v), y(v), z(v), f, gandhareC1functions.Alsoconditionsfordierenttypesofmarching-scale functionscanbeobtainedbyusingtheEqn. (3.4)and(3.9).4. ExamplesofgeneratingsimplesurfaceswithcommonSmarandacheasymptoticcurveExample4.1. Let (s)=_45 cos(s), 35 cos(s), 1 sin(s)_beaunit speedcurve.ThenitiseasytoshowthatT(s) =_45 sin(s),35 sin(s), cos(s)_, = 1, = 0.FromEqn.(2.3), (s) = d(s)ds (s) = c, c =cons tant. Herec =4canbetaken.FromEqn. (2.4),k1=12, k2=32.FromEqn. (2.1),N1= _k1T,N2= _k2T.N1(s) =_25 cos(s), 35 cos(s),12 sin(s)_,N2(s) =_235cos(s),3310cos(s),32sin(s)_.If wetakex(s, v) =0, y(s, v) =ev 1, z(s, v) =3 (ev1) ,weobtainamemberofthesurfacewithcommoncurve(s)as1(s, v) =

45cos(s) (1 + 2(ev1)) , 35cos(s)

1 12 (ev1)

, 1 sin(s) (1 + 2 (ev1))

where0 s 2, 1 v 1(Fig. 1).198 GlnuraffakAtalayandEminKasapFigure1: 1(s, v)asamemberofsurfacesandcurve(s).Ifwetakex(s, v)=0, y(s, v)=ev 1, z(s, v)=3 (ev1)andv0=0thentheEqns.(3.4)and(3.7)aresatised. Thus, weobtainamemberof thesurfacewithcommonSmarandacheTN1asymptoticcurveas2(s, v)=

252(2 sin(s)+cos(s))+85 cos(s)(ev1),352(sin(s)cos(s))+310cos(s)(ev1),12(12sin(s) cos(s)) + 2 sin(s)(ev1)

where0 s 2, 1 v 1(Fig. 2).A member of thesurface withcommon SmarandacheTN2asymptotic curve as3(s, v) =___252(2 sin(s) 3 cos(s)) +85 cos(s)(ev1),352(sin(s) +32cos(s)) +310 cos(s)(ev1),12(32sin(s) cos(s)) + 2 sin(s)(ev1)___where0 s 2, 1 v 1(Fig. 3).AmemberofthesurfacewithcommonSmarandacheN1N2asymptoticcurveas4(s, v) =

252 cos(s)(1+3)+85 cos(s)(ev1), 352 cos(s)(1+32)+310cos(s)(ev1),122 sin(s)(1 +3) + 2 sin(s)(ev1)

where0 s 2, 1 v 1(Fig. 4).AmemberofthesurfaceanditsSmarandacheTN1N2asymptoticcurveas5(s, v) =___253(2 sin(s) + cos(s) +3 cos(s)) +85 cos(s)(ev1),353(sin(s) cos(s) +32) +310 cos(s)(ev1),13(cos(s) +12 sin(s)(1 +3)) + 2 sin(s)(ev1)___where0 s 2, 1 v 1(Fig. 5).SurfacesFamilyWithCommonSmarandacheAsymptoticCurve 199Figure2: 2(s, v)asamemberofsurfacesanditsSmarandacheTN1asymptoticcurveof(s).Figure3: 3(s, v)asamemberofsurfacesanditsSmarandacheTN2asymptoticcurveof(s).200 GlnuraffakAtalayandEminKasapFigure4: 4(s, v)asamemberofsurfaces anditsSmarandacheN1N2asymptoticcurveof(s).Figure 5: 5(s, v) as a member of surfaces and its Smarandache TN1N2 asymptoticcurveof(s).SurfacesFamilyWithCommonSmarandacheAsymptoticCurve 201References1. B.ONeill, ElementaryDierentialGeometry,AcademicPressInc.,NewYork,1966.2. M.P. do Carmo, Dierential Geometry of Curves and Surfaces, Prentice Hall, Inc., EnglewoodClis, New Jersey,1976.3. B.ONeill, Semi-RiemannianGeometry,AcademicPress,New York,1983.4. M. Turgut, andS. Yilmaz, Smarandache Curves inMinkowski Space-time, InternationalJournalofMathematicalCombinatorics,Vol.3,pp.51-55.5. Ali, A.T. ,SpecialSmarandacheCurves in Euclidean Space,InternationalJournalof Mathe-maticalCombinatorics,Vol.2,pp.30-36,2010.6. etin,M. ,TunerY. ,Karacan,M.K. , SmarandacheCurvesAccordingto Bishop FrameinEuclideanSpace.arxiv: 1106.3202v1[math.DG],16Jun2011.7. Bekta, . andYce, S. , SmarandacheCurvesAccordingtoDarbouxFrameinEuclideanSpace.arxiv: 1203.4830v1[math.DG],20Mar2012.8. Turgut, M. andYlmaz, S. , Smarandache Curves inMinkowski Space-time. InternationalJ.Math.Combin.Vol.3 (2008),51-55.9. Takpr, K. ,and Tosun. M. , Smarandache Curves According to Sabban Frame on . BoletimdaSociedadeParaneansedeMatematica,vol,32,no.1,pp.51-59,2014.10. etin, M. , andKocayiit, H. , OntheQuaternionicSmarandacheCurvesinEuclidean3-Space.Int.J.Contemp.Math.Sciences,Vol.8,2013,no.3,139150.11. Deng,B.,2011.SpecialCurvePatternsforFreeformArchitecturePh.D.thesis,EingereichtanderTechnischenUniversitatWien,FakultatfrMathematikund Geoinformationvon.12. G. J. Wang, K. Tang, C. L. Tai, Parametric representation of a surface pencil with a commonspatialgeodesic,Comput.Aided Des.36(5)(2004)447-459.13. E. Kasap,F.T. Akyildiz,K. Orbay,A generalizationofsurfacesfamily with commonspatialgeodesic,Applied MathematicsandComputation,201(2008)781-789.14. C.Y. Li, R.H. Wang, C.G. Zhu, Parametricrepresentationof a surface pencil with a commonline ofcurvature, Comput.AidedDes.43(9)(2011)1110-1117.15. Bayram E. , Gler F. , Kasap E. Parametric representation of a surface pencil with a commonasymptoticcurve.Comput.AidedDes.(2012),doi:10.1016/j.cad.2012.02.007.16. Atalay. G. , KasapE. SurfacesFamilyWithCommonSmarandacheAsymptoticCurve.Boletim daSociedadaParanaensedeMatematica(InPress),doi:10.5269/bsmp.v34i1.24392.17. L.R.Bishop,Thereis morethanonewaytoFrameaCurve,Amer.Math.Monthly82(3)(1975)246-251.18. B. Bukcu, M.K. Karacan, Special BishopmotionandBishopDarbouxrotationaxisofthespacecurve,J.Dyn.Syst.Geom.Theor.6(1)(2008)2734.19. B. Bukcu, M.K. Karacan, The slant helices according to Bishop frame, Int. J. Math. Comput.Sci.3(2)(2009)6770.20. M.K.Karacan,B.Bukcu,Analternativemovingframefortubularsurfacesaroundtimelikecurvesin theMinkowski3-space,BalkanJ.Geom.Appl.12(2)(2007)7380.21. M.K. Karacan,B. Bukcu,An alternativemoving frame for a tubular surface around a space-likecurvewithaspacelikenormal inMinkowski 3-space, Rend. Circ. Mat. Palermo57(2)(2008)193201.22. M. Petrovic,J. Vertraelenand L. Verstraelen,2000. Principal Normal SpectralVariations ofSpaceCurves.Proyecciones19(2),141-155.202 GlnuraffakAtalayandEminKasapGlnuraakAtalayandEminKasapOndokuzMayisUniversity,ArtsandScienceFaculty,Department of Mathematics, Samsun55139, TurkeyE-mail address: gulnur.saffak@omu.edu.trE-mail address: kasape@omu.tr