Characterizations of Slant Helices. According to Quaternionic Frame

Transcription

1 Appled Mathematcal Scence, Vol. 7, 0, no. 75, HIKARI Ltd, Chaactezaton of Slant Helce Accodng to Quatenonc Fame Hüeyn KOCAYİĞİT and Beyza Betül PEKACAR Celal Baya Unvety, Faculty of At and Scence, Depatment of Mathematc 507 Muadye Campu, Muadye, Mana, Tukey Copyght 0 Hüeyn KOCAYİĞİT and Beyza Betül PEKACAR. Th an open acce atcle dtbuted unde the Ceatve Common Attbuton Lcene, whch pemt unetcted ue, dtbuton, and epoducton n any medum, povded the ognal wok popely cted. Abtact. In th tudy, we tuded ome ntegal chaactezaton of lant helce accodng to quatenonc fame n - and -dmenonal Eucldean pace E and E. Futhemoe, we obtan ome neceay and uffcent condton fo a pace cuve to be a lant helx accodng to quatenonc fame. Mathematc Subject Clafcaton : 0G0, H50, H5 Keywod : Quatenonc cuve, quatenonc fame, lant helx.. Intoducton The quatenon wee ft defned by Hamlton n 8. They ae actually mult-dmenonal complex numbe and magnay pat of a quatenon an magnay vecto baed on thee magnay othogonal axe. Some cuve ae pecal n dffeental geomety. They atfy ome elatonhp between the cuvatue and toon and have an mpotant ole. One of thee cuve geneal helx whch defned by the popety that the tangent of the cuve make a contant angle wth a fxed taght lne called the ax of the geneal helx. Futhemoe, ecently new pecal cuve have been defned and tuded by Izumya and Takeuch. They have defned lant helx whch a pecal cuve whoe pncpal nomal vecto make a contant angle wth a fxed decton [7]. Kula and et al. tuded ome chaactezaton of lant helce n the Eucldean - pace [9]. Afte that, Önde and et al. have egaded

2 70 Hüeyn KOCAYİĞİT and Beyza Betül PEKACAR the noton of lant helx n E and defned thee new cuve a B -lant helx [0]. Bahaath and Nagaaj defned the quatenonc cuve n E and E and tuded the dffeental geomety of pace cuve and ntoduced Fenet fame and fomulae by ung quatenon []. Followng, quatenonc nclned cuve have been defned and tuded by Kaadağ and Svdağ [8]. In [], Çöken and Tuna have tuded thee cuve n the em-eucldean pace E. Quatenonc ectfyng cuve have been tuded by Güngö and Toun [5]. Gök and et al. have condeed the defnton gven by Önde and et al. fo patal quatenonc cuve and have defned quatenonc B -lant helx. They have gven new chaactezaton fo thee cuve n E and E [, ]. In th tudy, we defne quatenonc lant helce and obtan ome neceay and uffcent condton fo a pace cuve to be a lant helx accodng to quatenonc fame n the Eucldean pace E and E.. Pelmnae In [6,] and [], a moe complete elementay teatment of quatenon and quatenonc cuve can be found epectvely. A eal quatenon q defned by q= ae + ae+ ae+ ae () whee a, ( ) ae odnay numbe, and e,( ), e =+ ae quatenonc unt whch atfy the non-commutatve multplcaton ule e e = e,( ) () e ej = ej e = ek,(, j, k ) whee ( jk ) an even pemutaton of () n the Eucldean pace. The algeba of the quatenon denoted by Q and t natual ba gven by { e, e, e, e }. We can expe a eal quatenon n () by the fom q= q + vq, () whee q = a cala pat and vq = ae + ae+ ae vecto pat of q. So the conjugate of q = q + vq defned by q = v. () q Ung thee bac poduct we can wte the ymmetc eal-valued, non-degeneate, blnea fom a follow: h: Q Q IR, ( q, p) h( q, p) = ( q p+ p q) (5) whch called the quatenon nne poduct [6]. The nom of q q

3 Chaactezaton of lant helce 7 (, ) q = h q q = q q = q q= a + a + a + a. (6) If q =, then q called unt quatenon. Then, nvee of the quatenon q gven by q =. (7) q q Let q= q + vq = ae + ae+ ae+ ae and p = p + vp = be + be + be + be be two quatenon n Q. Then the quatenon poduct of q and p gven by q p= v, v + v + v + v v (8) q p q p q p p q q p whee, and denote the nne poduct and vecto poduct n Eucldean -pace E, epectvely. And then f q+ q = 0, q called a patal quatenon and f q q = 0, q called a tempoal quatenon []. Theoem.. The thee-dmenonal Eucldean pace E dentfed wth the q Q: q+ q = 0 n an obvou manne. Let pace of patal quatenon { } I = [ 0,] be an nteval n eal lne IR and let α : I IR Q, α() = α () e (9) be an ac-lengthed cuve wth nonzeo cuvatue { k, } and { t(), n(), n() } = denote the Fenet fame of the cuve α (). Then the Fenet fomulae of the quatenonc cuve α () ae gven by t 0 k 0 t n = k 0 n n 0 0 n (0) whee k() pncpal cuvatue and () toon of α () []. Theoem.. The fou-dmenonal Eucldean pace E dentfed wth the I = 0, be an nteval n eal lne R and let pace of unt quatenon. Let [ ] γ : I IR Q, γ() α () e =, = e = + () be a mooth cuve n K, k, K and { (), (), (), T N () N N } denote the Fenet fame of the cuve γ (). Then the Fenet fomulae of the quatenonc cuve γ () ae gven by E wth nonzeo cuvatue { }

4 7 Hüeyn KOCAYİĞİT and Beyza Betül PEKACAR T 0 K 0 0 T N K 0 k 0 N = () N 0 k 0 K N N 0 0 ( K) 0 N whee K() pncpal cuvatue, k() toon and ( K)( ) btoon of γ () [].. Chaactezaton of Slant Helce n E In th ecton, we gve the defnton and chaactezaton of lant helce n accodng to quatenonc fame. Ft, we gve the followng defnton: E α : I E be a egula patal quatenonc cuve wth the Defnton.. Let quatenonc fame { t, n, n }. If the unt vecto n () make a contant angle θ wth ome fxed unt vecto u, that h( n, u) = co θ = cont. fo all I, then α called a patal quatenonc lant helx accodng to quatenonc fame. Then we can gve the followng: Theoem.. Let α = α () be a unt peed patal quatenonc cuve n E wth non-zeo cuvatue. Then α a quatenonc lant helx f and only f the functon k () k + k contant eveywhee. ( ) Poof. We aume that α a quatenonc lant helx and u be the contant vecto feld uch that the functon hn ( ( ), u) = c contant. Then fo the mooth functon a = a () and a = a () we have the followng fom of the vecto feld u u = a()() t + cn() + a() n(), I. () By dffeentatng () t follow a ck = 0, ak a = 0, (5) c + a = 0.

5 Chaactezaton of lant helce 7 Fom the econd equaton n (5) we have a = a k. (6) Moeove, nce the vecto u contant we have h( u, u ) = a + c + a = cont. (7) Fom (6) and (7), we obtan a + = m = cont. k If m = 0 then a = 0 and fom (5) we have a = c= 0. Th mean that u = 0 whch a contadcton. Thu t mut be aumed that m 0. Then we have, m a = m. ( k ) + The thd equaton of (5) yeld d m m = c. d ( ) k + Th can be wtten a k c cont. = m = ( + k ) k m whch gve the deed. Conveely, aume that the condton () atfed. In ode to mplfy the computaton, we aume that the equaton () a contant, namely c. We defne a vecto feld u a follow k u = t + cn + n. (8) + k + k du By dffeentatng (8) t follow 0 d =, that, u a contant vecto. On the othe hand, h ( n (), u ) = 0 and th mean that α a lant helx.. The Slant Helce Accodng to Quatenonc Fame n E Defnton.. A unt peed quatenonc cuve α : I E called a lant helx f t unt pncpal nomal N make a contant angle wth a fxed decton u. Theoem.. Let α : I E be a unt peed quatenonc cuve n E. Defne the functon K G0 = K() d, G =, G = G0, G = [ kg+ G ]. (9) k K

6 7 Hüeyn KOCAYİĞİT and Beyza Betül PEKACAR Then α a lant helx f and only f the functon G = c (0) = 0 contant and non-zeo. Moeove, the contant c = ec θ, beng θ the angle between N and a fxed decton u. Poof. Let α be a unt peed quatenonc cuve n E. Aume that α a quatenonc lant helx cuve. Let u be the decton wth whch N make a contant angle θ and, wthout lot of genealty, we uppoe that huu (, ) =. Conde the dffeentable functon a, 0, u = a () T() + a () N (), I () 0 that, a0 = h( T, u), a = h( N, u), () fo any. Becaue the vecto feld u contant, a dffeentaton n () togethe (9) gve the followng ytem of odnay dffeental equaton: a 0 Ka = 0, ak 0 ka = 0, () ak + a ( K) a = 0, a( K) + a = 0. Let u defne the functon G = G( ) a follow a() = G() a, 0. () We pont out that a 0 : on the contay, () gve a = 0, fo 0 and o, u = 0, whch a contadcton. Snce, () lead to G0 = K() d, G =, K G = G0, (5) k G = [ kg+ G ]. K The lat equaton () lead to followng condton: G + ( K) G = 0. (6) We do the change of vaable: dt t ( ) = ( K)( udu ), = ( K)( ). (7) d In patcula, and fom the lat equaton of (5), we have kt () G () t = G() t G(). t ( K)( t) =

7 Chaactezaton of lant helce 75 A a conequence, f α a lant helx, ubttutng the equaton (6) to the lat equaton, we expe kt () G () t + G() t = G(). t ( K)( t) By the method of vaaton of paamete, the geneal oluton of th equaton obtaned kt () G() t = A G()n t t dt cot ( K)( t) (8) kt () + B + G ()co t t dt nt ( K)( t) whee A and B ae abtay contant. Then (8) take the followng fom G() = ( A k() G()n ( K)() d d) co ( K)() d (9) + ( B + kg () ()co ( K)() d d) n ( K)() d. Fom (6), the functon G gven by ( ) ( ) G() = A k() G()n ( K)() d d n ( K)() d (0) B + kg () ()co ( K)() d dco ( K)() d. By ung equaton (5) we have G0 + G = cont. () Ung equaton (9) and (0), we have ( () ()n ( )() ) ( B kg K d d) G + G = A k G K d d + + () ()co ( )(). It follow fom () and () that = 0 G = c. Moeove th contant c can be calculated a follow. Fom () togethe the equaton (5), we have c= G = ec a = = θ = 0 a = 0 a whee we have ued (0) and the fact that u a unt vecto feld. We do the convee of theoem. Aume that the condton (5) atfed fo a cuve α. Let θ be o that c = ec θ. Defne the unt vecto u by u = co θ GT 0 + GN. = ()

8 76 Hüeyn KOCAYİĞİT and Beyza Betül PEKACAR By takng account (5), a dffeentaton of u du gve that 0 d =, whch t mean that u a contant vecto feld. On the othe hand, the quatenon cala poduct between the unt tangent vecto feld N wth u hn ( ( ), u) = co θ. Thu α a quatenonc lant helx n the pace E. Theoem.. Thee ae no quatenonc lant helce wth contant and non-zeo cuvatue n the pace E. Poof. Let u uppoe a quatenonc lant helx wth contant and non-zeo cuvatue. Then the equaton n () and () hold. Then, we have K () (()) k G = + ( K( ) d) + + = 0 k () ( K)() and t eay to ay that G nowhee contant. By the theoem (.) we = 0 ave that thee doe not ext a quatenonc lant helx wth contant and non-zeo cuvatue n the pace E. Theoem.. Let α : I E be a unt peed quatenonc cuve n E. Then α a quatenonc lant helx f and only f thee ext a c -functon G () uch that dg G = [ kg+ G ], = k( ) G( ) () ( K) d whee K G0 = K() d, G =, G = G0, G = [ kg+ G ]. k K Poof. Let aume that α a lant helx cuve. By ung Theoem. and by dffeentaton the (contant) functon gven n (9), we obtan G( G + ( K) G) = 0. Th how (). Conveely, f () hold, we defne a vecto feld u by u = co θ GT 0 + GN. = du By the quatenon equaton, 0 d =, and o, u contant. On the othe hand, hn ( ( ), u) = coθ contant, and th mean that α a quatenonc lant helx.

9 Chaactezaton of lant helce 77 Theoem.. Let α : I E be a unt peed quatenonc cuve n E. Then α a quatenonc lant helx f and only f the followng condton atfed ( ) G() = A kgn ( K) d d n ( K)() u du fo ome contant A and B. ( ) B + kg co ( K) d d co ( K)( u) du Poof. Suppoe that α a lant helx. Let defne m ( ) and n ( ) by φ () = ( K)() u du, m () = G()co φ + G()n φ+ kgn φd, (5) n () G()n φ G()co φ kgco φ = d. If we dffeentate equaton (5) wth epect to and takng nto account of dm () and (5), we obtan 0 d = and dn d = 0. Theefoe, thee ext contant A and B uch that m () = A and n () = B. By ubttutng nto (5) and olvng the eultng equaton fo G (), we get G( ) = ( A kgn φd)n φ ( B+ kgco φd)co φ. Conveely, uppoe that () hold. In ode to apply Theoem., we defne G () by G( ) = ( A kgn φd)co φ+ ( B+ kgco φd)nφ wth φ ( ) = ( K)( u) du. A dect dffeentaton of () gve () G = ( K) G kg. Th how the left condton n (5). Moeove, a taght fowad computaton lead to G () = ( K) G, whch fnhe the poof. Refeence [] Bahaath, K., Nagaaj, M., Quatenon valued functon of a eal Seet-Fenet fomulae, Indan J. Pue Appl. Math. 6 (985), [] Çöken, A.C., Tuna, A., On the quatenonc nclned cuve n the em-eucldean pace E, Appled Math. and Comp., 55 (00), 7-89.

10 78 Hüeyn KOCAYİĞİT and Beyza Betül PEKACAR [] Gök, İ., Okuyucu, O.Z., Kahaman, F., Hacalhoğlu, H.H., On the Quatenonc B -lant helx n the Eucldean pace E, Adv. Appl. Clffod Algeba, (0), [] Gök, İ., Kahaman, F., Hacalhoğlu, H.H., On the Quatenonc B -lant helce n the em Eucldean Space E, Appled Math. and Comp., Vol. 8(0), [5] Güngo, M.A., Toun, M., Some chaactezaton of quatenonc ectfyng cuve, Dffeental Geomety-Dynamcal Sytem, Vol., 0, pp [6] Hacalhoglu, H.H., Haeket Geomet ve Kuatenyonla Teo, Gaz Ünvete, Fen-Edebyat Fakülte Yayınlaı Mat. No:, 98. [7] Izumya, S., Takeuch, N., New pecal cuve and developable uface,tuk.j. Math. Vol. 8 (00), 5-6. [8] Kaadağ, M., Svdağ, A.I., Kuatenyonk Eğlm Czgle çn kaaktezayonla. Ec. Ünv. Fen Bl. Deg. - (997), 7-5. [9] Kula, L., Ekmekç, N., Yaylı, Y., İlalan, K., Chaactezaton of Slant Helce n Eucldean - pace, Tuk.J. Math. Vol. (00), 6-7. [0] Önde, M., Kazaz, M., Kocayğt, H., Kılıç, B -lant helx n Eucldean -pace E, Int. J. Cont. Math. Sc. vol., no.9 (008), -0. [] Wad, J.P., Quatenon and Cayley Numbe, Kluwe Academc Publhe, Boton/London, 997. Receved: Mach, 0