LOCUS OF THE CENTERS OF MEUSNIER SPHERES IN EUCLIDEAN 3-SPACE. 1. Introduction

Transcription

1 LOCUS OF THE CENTERS OF MEUSNIER SPHERES IN EUCLIDEAN -SPACE Beyha UZUNOGLU, Yusuf YAYLI ad Ismail GOK Abstact I this study, we ivestigate the locus of the cetes of the Meusie sphees Just as focal cuve is the locus of the cetes of the osculatig sphees, we ivestigate the geometical itepetatio o the locus of the cetes of the Meusie sphees We poved that if the cuve is a picipal lie, the locus of the cetes of the Meusie sphees of the cuve is a evolute cuve The, we give the elatios betwee this evolute cuve ad the focal cuve Also, we give some elatios betwee helices, slat helices ad the locus of the cetes of the Meusie sphees of the cuve Key wods Osculatig cicle, Osculatig sphee, Meusie sphee, Evolute cuve [000] 4H45, 4H50, 5A04,5A55 Itoductio The osculatig sphee of space cuve at a poit P is the sphee havig cotact of ode with at P The osculatig sphee ca also be defied as the limit of a vaiable sphee passig though fou poits of as the poits ted to P [9] It is ow that the focal cuve of a immesed smooth cuve : i dimesioal Euclidea space E is locus of the cetes of its osculatig sphees The study of the focal suface of a cuve ca povide useful geometic ifomatio about that cuve ad vice vesa [4] With the help of cotact of cuves with sufaces, we itoduced osculatig cicle ad osculatig sphee As the poit moves o the cuve, the cetes of osculatig cicle ad osculatig sphee tace out cuves amely the locus of cete of cuvatue ad the locus of the cete of spheical cuvatue Now istead of taig poits o cuves, let us coside tagets at diffeet poits of the cuve These tagets will geeate a suface ad we coside cuves o this suface These otios lead to the defiitios of ivolutes ad cosequetly evolutes of give cuve [] Daboux foud how to detemie the evolutes of a cuve, that is, the cuves whose tagets ae omals of Moeove, Vagas [4] poved that the focal suface of is foliated by the evolutes, ad all of them lie o the focal suface Classical texts i diffeetial geomety cotai a defiitio of a evolute of spatial cuve It is eough to say that, accodig to that defiitio, spatial cuve has a oe paamete family of evolutes W Blasche, K Leichweiss [] have itoduced evolute of the secod type R Uibe-Vagas uses the tem focal cuve fo the cuve called evolute [] Let S be a oieted suface i E All cuves o a suface passig though a give ay poit P ad havig the same taget lie at P also have the same omal cuvatue at P ad thei osculatig cicles fom a sphee This sphee is called Meusie sphee [4] I diffeetial geomety; a cuve of costat slope o geeal helix i Euclidea -space E, is defied by the popety that the taget vecto field maes a costat agle with a fixed staight lie (the axis of the geeal helix) A classical esult stated by M A Lacet i 80 ad fist

2 poved by B de Sait Veat i 845 ( fo details see [7,] ) is: A ecessay ad sufficiet coditio that a cuve be a geeal helix is that the atio of cuvatue to tosio be costat Izumiya ad Taeuchi, i [5], have itoduced the cocept of slat helix i Euclidea -space A slat helix i Euclidea space E was defied by the popety that its picipal omal vecto field maes a costat agle with a fixed diectio Moeove, Izumiya ad Taeuchi showed that is a slat helix i Euclidea -space if ad oly if the geodesic cuvatue of the spheical image of picipal omal idicatix N of a space cuve is a costat fuctio I the peset pape, we ivestigate the locus of the cetes of the Meusie sphee Just as focal cuve is the locus of the cete of the osculatig sphee, we ivestigate the geometical itepetatio o the locus of the cetes of the Meusie sphee We poved that if the cuve is a picipal lie, the locus of the cete of the Meusie sphee of the cuve is a evolute cuve The, we give the elatios betwee this evolute cuve ad the focal cuve Also, we give some elatios betwee helices, slat helices ad the locus of the cete of the Meusie sphee of the cuve Pelimiaies Let : I E be a abitay cuve i E Recall that the cuve is said to be a uit speed cuve (o paametized by ac legth fuctios) if,, whee, is the stadad ie poduct of E Let V ( s), V ( s), V ( s ) be the movig Feet fame alog the uit speed cuve, whee V( s), V( s ) ad V () s deote espectively the taget, the picipal omal ad the biomal vecto fields The the Feet fomulas ae give by The fuctios s ad cuve [0] Let S be a oieted suface i V 0 0 V V 0 V V 0 0V s ae called espectively the fist ad the secod cuvatue of the E ad : I S be a uit speed cuve o the suface S The cuve () s exists Feet fame ( ), ( ), ( ) İstead of the fame at the poit P s tihedo) T s s, Y sn s T s The aalogues of Feet's fomulas fo the tihedo,, by V s V s V s o each poits of it coside the thee uit vectos (the Daboux N s is the omal vecto field to S at P ad T Y N ae give

3 T 0 g T Y g 0 t Y N t 0 N I these fomulas g, ad t ae called the geodesic cuvatue, the omal cuvatue ad the goedesic tosio, espectively The elatios betwee geodesic cuvatue, omal cuvatue, geodesic tosio ad, ae give as follows cos, si, t g whee is measue of the agle fctio betwee Y ad V It is well ow that, if 0, the the cuve is called geodesic, if 0, the the cuve is called asymptotic ad if g t 0, the the cuve is called picipal cuve [] Theoem The adius of the osculatig cicle at the poit P the ecipocal of cuvatue of the cuve at P ad the positio vecto of its cete of the osculatig cicle is [] C s s V s Defiitio A sphee havig fou poit cotact with the cuve at a poit P is called the osculatig sphee at P o cuve The cete of the osculatig sphee is called the cete of spheical cuvatue [] Popositio Let S be a oieted suface ad all lyig o a suface S ad havig at a give ay poit P S the same taget lie have at this poit the same omal cuvatues The, thei osculatig cicles ae o a sphee called Meusie sphee [8] Defiitio A cuve o a give suface whose taget at each poit is alog a picipal diectio is called a lie of cuvatue [] Defiitio Let : I E be a ac leghted paamete cuve with ozeo cuvatues ad i E ad V ( s), V ( s), V ( s ) deote the Feet fame of the cuve The cuve C ( s) ( s) V( s) V( s) ( s) ( s) ( s) cosistig of the cetes of the osculatig sphees of the cuve The locus of these cetes is called the focal cuve of [4]

4 Defiitio 4 Let : I E be a uit speed cuve with Seet-Feet appaatus,, V, V, V ad be a evolute cuve of The the followig equality holds [] ( s ) ( s ) ( ) cot( ( ) ) ( ) ( s) V s ( s) s ds V s I this sectio we ivestigate the locus of cetes of Meusie sphees fo give ay cuve all its poit P LOCUS OF THE CENTERS OF MEUSNIER SPHERES IN EUCLIDEAN -SPACE Theoem Let : I S poit of the cuve The the cuve be a uit speed Feet cuve ad P s be ay s is a geodesic if ad oly if the cetes of the s ad the osculatig cicles coicides with the cetes of the Meusie sphees of the cuve osculatig cicles ae geodesics o the Meusie sphees Figue Meusie sphee of a cuve The pictue of the locus of the cetes of Meusie sphees i Euclidea -space is edeed i Figue whee i i,

5 Poof Assume that the poits C, M ad M ae the cete of osculatig cicle, the cete of the osculatig sphee ad the cete of Meusie sphee, espectively Let uit omal vecto field of S ad,, Ns be the V s V s V s is the Feet fame of the cuve By usig the Figue, the locus of the poit M which is cete of the Meusie sphee is the give by ss N s Sice cos cos si g s V s V s s V s V s s si V s cos V s s is a geodesic cuve, we get ss V s So, s is the locus of the cete of the osculatig cicle Covesely, we assume that the cete of the osculatig cicle coicides with the cete of the Meusie sphee of the cuve We ca easily see that g 0, that is, s is a geodesic cuve This completes the poof Theoem Mai Theoem Let : I R S be a ac leghted paamete cuve o S with cuvatues ad The the cuve is a picipal lie o S if ad oly if the locus of the cetes of the Meusie sphees of the cuve s coicides with evolute cuve of the cuve s Poof Assume that : I S be a ac leghted paamete cuve lyig o a oieted suface ad the cuve of the cuve Sice the cuve s is s is a picipal lie The locus of the cetes of the Meusie sphees s s s ds c holds ssvs cot svs 4 s is the picipal lie o the suface S, t 0 ad If this equalities ae witte i the Eq (4), the the cuve s is s s s V s s ds c V s So, fom the Defiitio 4, the cuve is the evolute of the cuve s cot s 0 s

6 Covesely, assume that the locus of the cetes of the Meusie sphees of the cuve s s We ca easily see that the cuve s is a coicides with evolute cuve of the cuve picipal lie o S This completes the poof Theoem Let : I S be a ac leghted paamete cuve with ozeo cuvatues, i E ad V ( s), V ( s), V ( s ) deote the Feet fame of the cuve Let C s be the focal cuve of the cuve s ad s The the focal cuve C s the Meusie sphees of the cuve s cuve s if ad oly if Poof Assume that the focal cuve cot sds s ce C be the locus of cetes of s coicides with the cuve s defiitio of focal cuve (Defiitio ) ad the Eq (), we obtai coicides with the By usig the V s V s V s cotv s cot cot sds s ce cot sds Covesely, we assume that the followig equality s ce holds It is obvious that the focal cuve C s coicides with the cuve s This completes the poof Coollay 4 Let : I S be a picipal lie o S The locus of the cetes of the Meusie sphees of the cuve s coicides with the locus of the cetes of the osculatig sphees of the cuve s if ad oly if s acsi c s Poof Assume that the locus of the cetes of the Meusie sphees of the cuve s coicides with the locus of the cetes of the osculatig sphees of the cuve s So, s V s cotsv s s V s V s s s s s s s s scot s Sice the cuve s is a picipal lie, s s ds c, the we have

7 s cot s ds c l si s ds l s l c Covesely, we assume that c s cetes of the Meusie sphees of the cuve osculatig sphees of the cuve s s s c s ds acsi s c acsi s s acsi holds It is obvious that the locus of the s coicides with the locus of the cetes of the s This complete the poof Theoem 5 Let S be o oieted suface ad : I S be a ac leghted paamete cuve o S with ozeo cuvatues ad Let the cetes of the osculatig sphees coicides with the cetes of the Meusie sphees of the cuve The the cuve is geodesic if ad oly if is the costat fuctio Poof Assume that the cetes of the osculatig sphees coicides with the cetes of the Meusie sphees of the cuve s, thee ae the followig equatio Sice the cuve g s is geodesic cuve, we have 0 ad tha is the costat fuctio Covesely, we assume that is the costat fuctio It is obvious that the cetes of the osculatig sphees coicides with the cetes of the Meusie sphees of the cuve s This completes the poof Theoem 6 Let : I S be a ac leghted paamete cuve o S with cuvatues, g, t ad S be the paallel suface of S We assume that the cuve s is the locus of the cetes of the Meusie sphees lyig o suface S with cuvatues, t ad the asymptotic cuvatue of cuve s is equal to g, cuvatue fuctios of the cuve have the followig elatios The the s ad the cuvatue of the fuctios of the cuve s

8 i) ii) iii) t g t t g 5 Poof Usig the asymptotic cuvatue of the cuve Theoem s is equal to pp0 i 6, we get the equatios (5) This completes the poof i the elatio Coollay 7 Let : I S be a ac leghted paamete cuve o S with cuvatues, g, t ad S be the paallel suface of S We assume that the cuve s is the locus of the cetes of the Meusie sphees lyig o suface S with cuvatues, t ad the asymptotic cuvatue of cuve s is equal to The the cuve g, s o S is a geodesic cuve if ad oly if the cuve Poof Usig Theoem 6, it is obvious s o S is geodesic cuve Defiitio A cuve : I with ( s) 0 is called a cylidical helix if the taget lies of mae a costat agle with a fixed diectio It has bee ow that the cuve () s is a cylidical helix if ad oly if ( )( s) costat If both of ( s) 0 ad () s ae costat, it is, of couse, a cylidical helix We call such a cuve a cicula helix [5] Defiitio Let : I be a uit speed space cuve with ( s) 0 A cuve is called a slat helix if the picipal omal lies of mae a costat agle with a fixed diectio [5] Theoem 8 Let : I S be a ac leghted paamete cuve lyig o a oieted suface S which is a costat agle suface If the cuve suface S, the the cuve s is a helix o E Poof The locus of the cetes of the Meusie sphees of the cuve If we diffeatiate the Eq (6), we have s is a picipal lie o the s is give s s N 6

9 Usig the cuve ds V T N T ty ds t N Y s is the picipal lie, we obtai ds V N ds ad calculatig the om of the last equatio we get If we wite the Eq (8) i Eq (7), ds ds V N Sice S is a costat agle suface, NU, is a costat fuctio The usig the Eq (9), V U is a costat fuctio, that is, the cuve s is a helix This completes the poof, Theoem 9 Let : I S be a picipal lie o S with Daboux fame T, Y, N ad the cuve cuve with Feet fame V, V, V ad s be the locus of the cetes of the Meusie sphees of the The thee exist the followig elatios V N, V T, V Y g, Poof Fom the Eq (9), we have V N The, if we diffeetiate the Eq (9) with espect to s, we get dv ds dn ds ds ds o usig the Eq (8) ad Feet-Seet fomulas V T 0 Calculatig the om of the last equatio ad usig this equality i the Eq (0), we get

10 Sice V V V, usig the Eq (9) ad Eq () we obtai V T V NT Y The if we diffeetiate the Eq () with espect to s, we get dv ds dv ds dt ds ds ds gy N İf we coside the Eq (9) ad Eq () we have V V V V g So, usig the equality the poof i the last equatio we obtai g which completes Coollay 0 Let : I S be a picipal lie o S ad the cuve s is the locus of the cetes of the Meusie sphees o suface S Thee is the followig elatios amog cuvatues of these cuves g Poof Usig Theoem 9, it is obvious Theoem Let : I S be a picipal lie o S ad the cuve s the locus of the cetes of the Meusie sphees of the cuve The cuve -dimesioal Euclidea space -dimesioal Euclidea space Poof Assume that the cuve E E if ad oly if the cuve be s is a helix i s is a slat helix i s is a helix i -dimesioal Euclidea space TU, is a costat fuctio Usig the Eq (9), we obtai V, U So, the cuve s is a slat helix i -dimesioal Euclidea space Covesely, assume that the cuve E It is obvious that the cuve s This completes the poof E So, is a costat fuctio E s is a slat helix i -dimesioal Euclidea space is a helix i -dimesioal Euclidea space E

11 Coollay Let : I S be a picipal lie o S ad the cuve s s The the cuve s s is a plae cuve the locus of the cetes of the Meusie sphees of the cuve geodesic if ad oly if the cuve be is a Poof Let : I S be a ac leghted paamete cuve o S ad the cuve s be the locus of the cetes of the Meusie sphees of the cuve s s is a geodesic, the the geodesic cuvatue of the cuve s tosio of the cuve s is 0 So, the cuve s is a plae cuve Covesely, assume that the cuve s is a plae cuve It is obvious that the cuve s is a geodesic This completes the poof Theoem Let : I S the locus of the cetes of the Meusie sphees o suface S The cuve -dimesioal Euclidea space Poof Let : I S If the cuve is equal to zeo The be a picipal lie o S ad the cuve s is s is a helix i E if ad oly if the cuve s is a plae cuve be a picipal lie o S If the cuve s g E, So, cot s -dimesioal Euclidea space s 0 Sice the cuve s s is a plae cuve Covesely, assume that the cuve is a helix i -dimesioal Euclidea space is a helix i is a costat fuctio ad is a picipal lie o S, ss 0 So the cuve s is a plae cuve It is obvious that the cuve s E This completes the poof Refeeces Blashe, W ad Leichweiss, Elematae Diffeetial geometie I, fifth editio, Spige-Velag, Beli, 97 Camc, Ç, Kula, L ad İlasla, K Chaacteizatios of the positio vecto of a suface cuve i Euclidie -Space, A Şt Uiv Ovidius Costata, Vol 9(), 0, Fuchs D, Evolutes ad Ivolutes of Spatial Cuves, Ameica Mathematical Mothly, Volume 0, Numbe, Mach 0, pp 7-(5) 4 Ia R Poteous, Geometic Diffeetiatio, pp 5--5, Cambidge Uivesity Pess ISBN , 00 5 Izumiya S, Taeuchi N, 004, New special cuves ad developable sufaces, Tu J Math 8, 5-6

12 6 Kzltug S, Taac O, ad Yaylı Y, O the cuves lyig o paallel sufaces i the Euclidea -space E, Joual of Advaced Reseach i Dyamical ad Cotol Systems, Vol 5, Issue, 0, pp 6-5 Olie ISSN: 94-0X 7 Lacet, M A, Mémoie su les coubes à double coubue, Mémoies pésetés à l'istitut (806) Mafedo P do Camo, Diffeetial Geomety of Cuves ad Sufaces, Petice-Hall, Millma RS, Pae GD, Elemets of diffeetial geomety, Petice-Hall, O'Neill, B, Elematey Diffeetial Geomety, Academic Pess Ic, New Yo, 996 Sabucuoglu A, Difeesiyel Geometi, Nobel Yayla, No:58, 006 Somasudaam D, Diffeetial Geomety A Fist Couse, Naosa Publishig House Pvt Ltd, Idia, 005 D J Stui, Lectues o Classical Diffeetial Geomety, Dove, New-Yo, Uibe-Vagas R, O Vetices, focal cuvatues ad diffeetial geomety of space cuves, Bull Baz Math Soc, New Seies 6(), 85-07, 005 Beyha UZUNOĞLU Depatmet of Mathematics, Faculty of Sciece, Uivesity of Aaa, Tadoğa, Aaa, TURKEY buzuoglu@aaaedut Yusuf YAYLI Depatmet of Mathematics, Faculty of Sciece, Uivesity of Aaa, Tadoğa, Aaa, TURKEY yusufyayli@scieceaaaedut Ismail GÖK Depatmet of Mathematics, Faculty of Sciece, Uivesity of Aaa, Tadoğa, Aaa, TURKEY igo@scieceaaaedut