S, we call the base curve and the director curve. The straight lines

Transcription

1 Developable Ruled Sufaces wih Daboux Fame i iowsi -Space Sezai KIZILTUĞ, Ali ÇAKAK ahemaics Depame, Faculy of As ad Sciece, Ezica Uivesiy, Ezica, Tuey ahemaics Depame, Faculy of Sciece, Aau Uivesiy, Ezuum, Tuey Sizilu4@homailcom, alicama5@homailcom Absac: I his pape, we obai he disibuio paamee of a uled suface eeaed by a saih lie i Daboux ihedo movi alo wo diffee cuves wih he same paamee Besides, we ive ecessay ad suficie codiios fo his uled suface o become developable i iowsi -space [Kızıluğ S, Çama A Developable Ruled Sufaces wih Daboux Fame i iowsi -Space Life Sci J 0;0(4):906-94] (ISSN:097-85) hp://wwwlifesciesiecom 5 Keywods Daboux Fame, Asymoic cuve, Geodesic cuve, icipal lie, Ruled suface, iowsi space SC: 5A5 Ioducio Ruled sufaces ae he sufaces eeaed by a coiuously movi of a saih lie i he space These sufaces ae oe of he mos impoa opics of diffeeial eomey Ruled sufaces, paiculaly developable sufaces, have bee widely sudied ad applied i mahemaics ad eieei The eeaio ad machii of uled sufaces play a impoa ole i desi ad maufacui of poducs ad may ohe aeas Because of his posiio of uled sufaces, may eomees have sudied o hese sufaces i boh Euclidea space ad iowsi space [6, 7, 8, 0, ] A uled suface i iowsi -space defied by E is (locally) he map, : I E, s, v s v s : I E whee : I E \ 0, S, we call he base cuve ad he dieco cuve The saih lies v s v s ae smooh mappis ad I is a ope ieval o he ui cicle ae called ullis The uled suface, is called developable if he Gaussia cuvaue of he eula pa of, vaishes This is equivale o he fac ha, is developable if ad oly if he disiuished paamee de,, 0, I[], S Izumiya ad N Taeuchi[8] sudied a special ype of uled suface wih Daboux veco They called he uled suface ecifyi developable suface of he space cuve Tuu ad Hac saliholu[] have sudied imelie uled sufaces i iowsi -space ad ive some popeies of hese sufaces ayl ad S Saacolu[0] sudy developable uled sufaces i iowsi -space ad ive ecessay ad sufficie codiio uled suface o become developable Besides, Kızıluğ ad aylı [9] ivesiaed cuves o ubula suface by usi Daboux fame I his pape, mai use of mehod i pape of ayl ad S Saacolu, we sudy developable uled sufaces wih Daboux Fame elimiaies Le whee abiay veco E be a iowsi -space wih aual Loez eic, dx dx dx, ( x, x, x) v ( v, v, v) is a ecaula coodiae sysem of E Accodi o his meic, i E ca have oe of hee Loezia causal chaaces; i ca be spacelie if a 906

2 v,v > 0 o v 0 v,v < 0 v,v 0, imelie if ad ull (lihlie) if ad v 0 Similaly, a ( s) ' abiay cuve is spacelie, imelie o ull (lihlie), if all of is velociy vecos (s) ae spacelie, imelie o ull (lihlie), especively[] We say ha a imelie veco is fuue poii o pas x ( x,, ) poii if he fis compoud of he veco is posiive o eaive, especively Fo ay vecos x x y ( y,, ) ad y y i E, he veco poduc of x y ad is defied by x y ( x y x y, x y x y, x y x ) y E A suface i he iowsi -space is called a imelie suface if he iduced meic o he suface is a Loez meic ad is called a spacelie suface if he iduced meic o he suface is a posiive defiie Riemaia meic, ie, he omal veco o he spacelie (imelie) suface is a imelie (spacelie) veco[] Le S E be a oieed suface i ad le coside a o-ull cuve (s) lyi fully o S Sice (s) he cuve is also i he space, hee exiss a Fee fame T, N, B alo he cuve whee T is ui ae veco, N is picipal omal veco ad B is biomal veco, especively oeove, sice he cuve (s) lies o he suface S (s) hee exiss aohe fame alo he cuve This fame is called Daboux T, Z fame ad deoed by, which ives us a oppouiy o ivesiae he popeies of he cuve accodi o he suface I his fame T is he ui ae of he cuve, Z is he ui omal of he suface S alo (s) ad is a ui veco ive by Z T Sice he ui ae T is commo i boh Fee fame N, B, ad Daboux fame, he vecos ad Z lie o he same plae So ha he elaios bewee hese fames ca be ive as follows: he suface is a oieed imelie suface, he elaios bewee he fames ca be ive as follows he cuve (s) is imelie T 0 Z 0 0 cos si 0 T si N cos B he cuve (s) is spacelie T 0 0 T 0 cosh sih N Z 0 sih cosh B he suface is a oieed spacelie suface, he he cuve (s) lyi o is a spacelie cuve So, he elaios bewee he fames ca be ive as follows T 0 0 T 0 cosh sih N Z 0 sih cosh B I all cases, is he ale bewee he vecos ad N [, 5] Accodi o he Loezia causal chaaces of he suface ad he cuve (s) lyi o, he deivaive fomulae of he Daboux fame ca be chaed as follows: 907

3 i) he suface is a imelie suface, he he cuve imelie cuve Thus, he deivaive fomulae of he Daboux fame of T 0 T 0, Z 0 Z T, T,,, Z, Z ii) he suface is a spacelie suface, he he cuve Thus, he deivaive fomulae of he Daboux fame of (s) is ive by T 0 T 0, Z 0 Z T, T,,, Z, Z I hese fomulae, ad (s) lyi o ca be a spacelie o a (s) is ive by (s) lyi o is a spacelie cuve ae called he eodesic cuvaue, he omal cuvaue ad he eodesic osio, especively[] The elaios bewee eodesic cuvaue, omal cuvaue,eodesic osio ad, ae ive as follows if boh ad if imelie ad s ae imelie o spacelie Z s is spacelie cos, si,, cosh, sih, whee N, he ale fucio is bewee he ui omal ad biomal o s I he diffeeial eomey of sufaces, fo a cuve well-ow[], (s) 0 i) is a eodesic cuve, ii) iii) (s) is a asympoic lie (s) is a picipal lie 0 0, (s) lyi o a suface he followis ae Developable Ruled Sufaces wih Daboux Fame i iowsi -Space : I E Le be a cuve i iowsi -space ad T,, Z be a Daboux fame, whee I his fame T is he ui ae of he cuve, Z is he ui omal of he suface alo (s) ad is a ui veco ive by Z T As we have said above, wih he assisace of, we ca defie cuve : I E wih he same paamee of he cuve s ad such ha at b cz Ad also, we ca e he uled suface ha poduced dui he cuve he movi space H as: s wih each fixed lie of () () () (4) (5) 908

4 Le be a fixed ui veco Thus, s, v s v s T,,Zad T Z Sp The he disibuio paamee of he uled suface fo he cuve ca be ive as: de,,, (7) We ca obai he disibuio paamee of he uled suface eeaed by lie of he movi space H Diffee cases ca be ivesiaed as followi: The Ruled Suface, ad he cuve is imelie Le he uled suface be imelie ad be imelie By ai deivaive of (6) wih espec o s ad by usi Daboux fomulas () we have T Z (8) Besides,, So, Fom (5), (8) ad (9) we obai a b c Fo a, c 0, Fo b, c 0, b he disibuio paamee is ive as follows a he disibuio paamee is ive as follows Fo c, a b 0, he disibuio paamee is ive as follows Now we ivesiae some diffee cases fo c, a b 0 Fisly, fom () we ca ive he followi lemma: Lemma : The uled suface, ha poduced dui he cuve movi space H is developable if ad oly if cos si (6) (9) (0) () () s wih each fixed lie of he 909

5 oof: The uled suface fom (), we e we eplace, is developable,the cos si ad i Eq (4), we e cos si 0 () (4) Special Cases The Case: T holds: I his case,, 0, he 0 0 Thus, Thus, fom Eq () (s) ad also (s) eodesic cuvehece he followi poposiio s wih each oposiio : Le T The he uled suface, ha poduced dui he cuve fixed lie of he movi space H is developable if ad oly if (s) ad also (s) eodesic cuve The Case: I his case, 0 poposiio holds,, he 0 0 Thus, oposiio : Le The he uled suface fixed Thus, fom Eq () (s) ad also, lie of he movi space H is developable if ad oly if (s) eodesic cuvehece he followi ha poduced dui he cuve (s) ad also (s) eodesic cuve s wih each The Case: Z I his case,, 0 Thus, fom Eq () 0 Thus we ca easily see he uled suface, is developable 90

6 oposiio 4: Le Z The he uled suface fixed lie of he movi space H is developable, ha poduced dui he cuve s wih each 4 The Case: is i he T plae 0 I his case, Thus, fom Eq () 0, poposiio holds he 0, Thus, (s) ad also oposiio 5: The uled suface ha poduced dui he cuve T plae of he movi space H (s) is developable if ad oly if ad also 5 The Case: is i he T Z plae I his case, 0 Thus, fom Eq () 0, he Thus, liehece he followi poposiio holds: 0, (s) ad also oposiio 6: The uled suface ha poduced dui he cuve T Z plae of he movi space H is developable if ad oly if (s) ad also ad picipal lie 6 The Case: is i he Z plae I his case, 0 Thus, fom Eq () 0, he Thus, cuve Hece he followi poposiio holds 0, (s) ad also oposiio 7: The uled suface ha poduced dui he cuve Z plae of he movi space H is developable if ad oly if (s) ad also ad asympoic cuve (s) eodesic cuvehece he followi s wih each fixed lie i he (s) eodesic (s) boh eodesic cuve ad picipal s wih each fixed lie i he (s) boh eodesic cuve (s) boh eodesic cuve ad asympoic s wih each fixed lie i he (s) boh eodesic cuve, The Ruled Suface ad he cuve is spacelie Le he Ruled suface be spacelie ad be spacelie By ai deivaive of (6) wih espec o s ad by usi Daboux fomulas () we have 9

7 9 T Z (5) Besides,, (6) So, fom (5), (5) ad (6) we obai c b a Fo, a 0, c b he disibuio paamee is ive as follows (7) Fo, b 0, c a he disibuio paamee is ive as follows (8) Fo, c 0, b a he disibuio paamee is ive as follows (9) Now we ivesiae some diffee cases fo, c 0 b a Fisly, fom (9) we ca ive he followi lemma Lemma 8: The uled suface, ha poduced dui he cuve s wih each fixed lie of he movi space H is developable if ad oly if cos si oof: The uled suface, is developable, The 0 (0) Fom (0), we e () we eplace cos ad si i Eq (), we e cos si Special Cases The Case: T I his case,, 0 Thus fom Eq (9),

8 0, poposiio holds: he 0 Thus, (s) ad also (s) asympoic cuve Hece he followi s wih each oposiio 9: Le T The he uled suface, ha poduced dui he cuve fixed lie of he movi space H is developable if ad oly if (s) ad also (s) asympoic cuve The Case: I his case,, 0 Thus, fom Eq (9) 0, Thus, we ca easily see he uled suface is developable The Case: Z, I his case, 0 Thus, fom Eq (9) 0, poposiio holds: he 0 Thus, (s) ad also (s) asympoic cuve Hece he followi s wih each oposiio 0: Le Z The he uled suface, ha poduced dui he cuve fixed lie of he movi space H is developable if ad oly if (s) ad also (s) asympoic cuve 4 The Case: is i he T plae I his case, 0, 0 Thus, fom Eq (9) he Thus, liehece he followi poposiio holds 0, (s) ad also oposiio : The uled suface ha poduced dui he cuve T plae of he movi space H is developable if ad oly if (s) ad also picipal lie 5 The Case: is i he T Z plae I his case, 0 Thus, fom Eq (9) (s) boh asympoic cuve ad picipal s wih each fixed lie i he (s) asympoic cuve ad 9

9 0, poposiio holds: he 0 Thus, (s) ad also (s) asympoic cuvehece he followi, oposiio : The uled suface ha poduced dui he cuve T plae of he movi space H is developable if ad oly if (s) ad also 6 The Case: is i he Z plae I his case, 0 Thus, fom Eq (9) 0, 0 he Thus, cuvehece he followi poposiio holds:, (s) ad also oposiio : The uled suface ha poduced dui he cuve Z plae of he movi space H (s) is developable if ad oly if ad also ad asympoic cuve s wih each fixed lie i he (s) asympoic cuve (s) boh eodesic cuve ad asympoic s wih each fixed lie i he (s) boh eodesic cuve Refeeces [] Gay A, ode Diffeeial Geomey of Cuves ad Sufaces wih ahemaica, secod ed, Ccess, USA, 999 [] Tuu A, Hacisalihoğlu HH, Time-lie Ruled Sufaces i he iowsi -Space Fa Eas J ah Sci Vol 5 No (997), 8-90 [] O Neill B, Semi-Riemaia eomey wih applicaios o elaiviy,academic pess, New o, 98 [4] Hacısalihoğlu HH, Diffeeial Geomei-II AÜ, Fe Fa, 994 [5] Uğulu HH, Kocayiği H, The Fee ad Daboux Isaaeous Roai Vecos of Cuves o Time-Lie Sufaces, ahemaical ad Compuaioal Applicaios, Vol, No, pp -4 (996) [6] Abdel-Bay RA, Abd-Ellah HN, Hamdoo F, Ruled sufaces wih imelie ulis, App ah ad Comp, 47 (004) 4 5 [7] Abdel Bay RA, The elaio amo Daboux vecos of Ruled Sufaces i a lie Coueces, Riv a, Uiv ama 6(997), 0- [8] Izumiya S ad Taeuchi N, Special Cuves ad Ruled Sufaces, Beiaezu Aleba ud Geomeie Coibuios o Aleba ad Geomey, 44,No, 00, 0- [9] Kızıluğ S, aylı, Timelie ubes wih Daboux fame i iowsi -space, Ieaioal Joual of hysical Scieces, Vol 8(), (0), pp-6 [0] ayli ad Saacolu S, O Developable Ruled Sufaces i iowsi Space Adv Appl Cliffod Alebas (0), [] aylı, O The oio of he Fee Vecos ad Spacelie Ruled Sufaces i he iowsi -Space ahemaical ad Compuaioal Applicaios 5(000), /7/0 94