On the Pairs of Orthogonal Ruled Surfaces

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1 EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 5 No ISSN On the Pirs of Orthogonl Ruled Surfces Filiz KANBAY Deprtment of Mthemtics Fculty of Arts nd Science Yıldız Technicl Uniersity Dutpṣ Istnbul Turkey Abstrct. In this work in three dimensionl Eucliden spce E 3 by using the ruled Bonnet surfces which he been known up to now the problem of finding some pirs of orthogonl ruled surfces is exmined nd only one pir of orthogonl ruled surfces cn be obtined Mthemtics Subject Clssifictions: 53A05 Key Words nd Phrses: Orthogonl Surfce Bonnet Surfce Ruled surfce Weingrten surfce Tngentil surfce 1. Introduction The orthogonl surfces re relted to the problem of isometric representtion. These surfces lso ply n importnt role in ll questions connected with the study of the infinitesiml bending problem. Becse the infinitesiml bending problem is reduced the problem of finding the orthogonl surfces[5 7 8]. The orthogonl ruled surfces re significnt to inestigte the isometric representtion of the ruled surfces or the infinitesiml bending of ruled surfces. For this im we cn use the Bonnet ruled surfces to exmine pirs of orthogonl ruled surfces. Generlly Bonnet surfces he been clssified into three ctegories: The surfces of constnt men curture other thn the plnes nd spheres. The isometric Weingrten surfces of non-constnt men curture which re isometric to surfce of reolution. The surfces of non-constnt men curture tht dmit single non-triil isometry. In[2] by using the method gien in[4] some specil Bonnet ruled surfces of non-constnt men curture re gien. It is shown tht the ruled surfces which re formed by the binormls two cures whose curtures nd bsolute lue of torsions re the sme re the Bonnet pirs. Moreoer the ruled miniml Bonnet surfces nd the ruled Weingrten surfces re Emil ddress: ÒÝÝÐÞºÙºØÖ c 2012 EJPAM All rights resered.

2 F. Knby/ Eur. J. Pure Appl. Mth 5 ( indicted (the deelopble surfce re not inestigted. The tngentil Bonnet surfces s the deelopble Bonnet surfces re exmined in[3]. In this work the min gol is to find some ruled orthogonl surfces by using the ruled Bonnet surfces. Using this method it is shown tht the pirs of orthogonl surfces except one pir cn not be obtined. 2. Ruled Bonnet Surfces nd Some Orthogonl Pirs 2.1. Some Ruled Bonnet Surfces A ruled surfce in three dimensionl Eucliden spce E 3 cn be gien by the ectoril eqution X(u =r(+ut( (1 where r=r( is directrix cure T=T( is unit ector with T 2 (= 2 ( 0 T ( r (= b( The first fundmentl form of the ruled surfce (1 is T( r (=cosθ( (0 θ π where f = d f d (2 ds 2 = du cosθdud+( 2 u 2 + 2bu+1d 2 (3 TThe lue of u for the centrl point; the prmeter of the distributionβ re expressed s u=α(= b( 2 ( β= 1 2(tTT (4 Here the centrl point is the limit of the point in which genertor g 1 is met by the common perpendiculr of g 1 nd neighboring genertor g 2 s g 2 pproches g 1 oer ruled surfce; the prmeter of distribution is the limit of the rtio of the shortest distnce between two genertors nd their included ngle. Now let n orthogonl trjectory to the genertors be tken s directrix nd T 2 (=1 T 2 (=1. Such ruled surfce cn be obtined from (1 under the conditions[6 1]: T = 1 β (T r T = α β β 2 +α 2 r + β 2 +α 2 r T r = αt +βt T (5 N = 1 βw [(β2 +α 2 uαt +(α ur ] w= (u α 2 +β 2 (β 0 We recll tht the director-cone is the cone formed by drwing through point lines prllel to the genertors nd it is determined by the function D(= r T =(TT T (6 β

3 F. Knby/ Eur. J. Pure Appl. Mth 5 ( Becseκ 2 = 1+D 2 ndτ= D 1+D 2 re the curture nd the torsion of the unit sphericl cure x = T( which determines the director-cone[2]. The miniml surfces nd the isothermic Weingrten ruled surfces re gien by the equtionsα =β = D= 0 ndα =β = D = 0 respectiely[2 6]. If isometric representtion between two surfces preseres the principl curtures of these surfce these surfces re sid to be Bonnet surfces. In[4] in order to find Bonnet surfce method is gien. In ccording to this A-net on surfce such tht when this net is prmetrized the conditions E=G F= 0 M= c= const. 0 re stisfied is clled n A-net where E F G re the coefficients of the first fundmentl form of the surfce nd L M N re the coefficients of the second fundmentl form. And necessry nd sufficient condition for surfce to be Bonnet surfce is tht the surfce cn he n A-net. In[2] on the ruled surfce the genertors nd orthogonl trjectories form n A-net if nd only if the prmeter of the distributionβ nd the bsciss of the centrl pointαre constnts. And the prmeter of distribution nd the bsciss of the centrl point of surfce re constnt then the surfce is Bonnet surfce. The torsion of the striction line of such Bonnet surfce is constnt nd is equl to the reciprocl of the prmeter of distribution. And the ruled surfces which re formed by the binormls two cures whose curtures nd bsolute lue of torsions re the sme re the Bonnet pirs. This mens tht the ruled surfces gien by the equtions X(u =r(+ β sinhut( Y(u = r(+ β sinhut( (7 respectiely re Bonnet pirs. Hereβ 0 nd D( is n rbitrry function. In[4] the cses D(=const. nd D(=0re indicted. In the cse D(=const. the Bonnet pirs re obtined tht X(u = cos sin + ǫ sinhusin ǫ sinhucos ǫ + ( sinhu (8 Y(u = cos sin + ǫ sinh usin ǫ sinh ucos ǫ + ( sinh u

4 F. Knby/ Eur. J. Pure Appl. Mth 5 ( whereǫ= sgn(b=sgn(β. In the cse D(=0 X(u = β sinhucos β sinhusin β Y(u = (9 β sinhucos β sinhusin β re found. These solutions re lid for the non-deelopble surfces(β 0[2]. In[3] the tngentil deelopble surfces of the circulr helices(β= 0 re gien s n exmple of the ruled Bonnet pirs. These surfces re clculted s X(u = Y(u = cos sin + b + sin cos bu 2( + 2 +b rctn( cos 2( + 2 +b rctn( sin 2 +b 2 2 +b 2 b(+ 2(2 +b rctn( 2 +b sin + cos bu 2( + 2 +b rctn( 2( + 2 +b rctn( All ruled Bonnet pirs which he been known up to now cn be tken the equtions (7 (8 (9 ( Some Orthogonl Ruled Surfces If two surfces S( nd S(b gien by the equtions =(u nd b=b(u mp upon ech other such tht their liner elements re orthogonl t corresponding points d db=0 then these surfces re clled to be orthogonl surfces. Now we recll the reltionship between the orthogonl surfces nd the isometric surfces: if two surfces S(x nd S(y gien by the equtions x=x(u nd y=y(u re isometric(dx 2 = dy 2 two surfces written s S(x+y nd S(x y re orthogonl surfces d(x+y d(x y=0. By using two surfces S(x nd S(y which cn be mp isometriclly on the ech other the orthogonl surfces S( nd S(b gien by the equtions =x+y b=x y (11 cn be written. By using this method we cn get the orthogonl surfces esily. For this im we cn use the ruled surfces gien by the equtions (7 (8 (9 (10 to find orthogonl 2 +b 2 2 +b 2 (10

5 F. Knby/ Eur. J. Pure Appl. Mth 5 ( ruled surfces. But the process of using the ruled surfces is not s esy s the generl process. Since the eqution of ruled surfce consists of two prts(x(u =r(+ut( mostly one of the first(r( or second(t( prt cn nish nd the surfce becomes cure. Becse of this only one pirs of orthogonl surfces gien by x y z = = = cos + cos + 2(2 +b rctn( + sin + sin 2 + b 2 sin sin + 2(2 +b rctn( + cos + cos 2 + b 2 2b( 2 +b rctn( 2 +b b (2 +b rctn( 2 +b (2 +b rctn( 2 +b 2 2 +b 2 2bu (12 x y = = cos cos + 2(2 +b rctn( sin + sin 2 + b 2 sin + sin 2 +b 2 + 2(2 +b rctn( + 2(2 +b rctn( + cos cos 2 + b 2 2 +b (2 +b rctn( 2 +b 2 2 +b 2 z = 2b+ 2b(2 +b rctn( 2 +b 2 cn be found. Theorem 1. By using the Bonnet Ruled surfces; only one pir of orthogonl surfces cn be found. This surfces re generted by the tngentil deelopble surfces of the circulr helices.

6 REFERENCES 210 The infinitesiml bending problem is reduced the finding them[5 7 8]: the eqution x t (u t=x(u + ty(u (13 gies the infinitesiml bending surfces of the surfce S(x under the condition dx t 2 (u t=(dx(u + tdy(u 2 (14 where t is rel infinitesiml prmeter. Here the condition (14 is equilent to the condition dx dy=0. Consequently we gie the following corollry: Corollry 1. One of the surfces gien by the eqution (12 cn be infinitesiml deformtion by using the other one. References [1] L Eisenhrt. "A Tretise On The Differntil Geometry Of Cures And Surfces" Doer Publictions Inc. New York p [2] F Knby. "Bonnet ruled surfces" Act Mthemtic Sinic English Series Vol 21 No: [3] I Rssos. "Tngentil deelopble surfces s Bonnet surfces". Act Mthemtic Sinic English Series 15( [4] Z Soyuc.ok. "The problem of non-triil isometries of surfces presering principl curtures" Journl of Geometry Vol [5] Z Soyuc. ok. "Infinitesiml Deformtions of surfces nd the stress distribution on some membrnes under constnt inner pressure" Int. J. Engng Sci. Vol 34 N0 9 pp [6] F Urs. "Diferensiyel Geometri II Dersleri" Yıldız Teknik Üniersitesi Fen-Edebiyt Fkultesi Mtemtik Bölümü Syı:261. Istnbul [7] F Urs. "On the net of the principl stresses relted with the infinitesiml bending of surfces" Bulletin of the Technicl Uniersity of Istnbul Vol 49 No: [8] I Veku. "Generlized Anliytic Functions. Pergmon Pres Oxford

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