Special Involute-Evolute Partner D -Curves in E 3

Transcription

1 Special Ivolute-Evolute Parter D -Curves i E ÖZCAN BEKTAŞ SAL IM YÜCE Abstract I this paper, we take ito accout the opiio of ivolute-evolute curves which lie o fully surfaces ad by taki ito accout the Darboux frames of them we illustrate these curves as special ivolute-evolute parter D-curves i E. Besides, we d the relatios betwee the ormal curvatures, the eodesic curvatures ad the eodesic torsios of these curves. Fially, some cosequeces ad examples are ive. torsio. Keywor: Ivolute-evolute, Darboux Frame, ormal curvature, eodesic curvature, eodesic 000 AMS Subject Clas si catio: 5A0 1 Itroductio I di eretial eometry, there are may importat cosequeces ad properties of curves. Researchers follow labours about the curves. I the liht of the existi studies, authors always itroduce ew curves. Ivolute-evolute curves are oe of them. C. Hues discovered ivolutes while tryi to build a more accurate clock, [1]. Later, the relatios Freet apparatus of ivoluteevolute curve couple i the space E were ive i []. A. Turut examied ivolute-evolute curve couple i E, []. I this study, we cosider the otio of the ivolute-evolute curves lyi o the surfaces for a special situatio. We determie the special ivolute-evolute parter D-curves i E. By usi the Darboux frame of the curves we obtai the ecessary ad su ciet coditios betwee,, ad for a curve to be the special ivolute parter D-curve. ad of this special ivolute parter D-curve are foud. Fially, some special case ad examples are ive. Yildiz Techical Uiversity, Faculty of Arts Ad Scieces, Departmet of Mathematics, 10, Eseler, Istabul, Turkey, ozcabektas198@hotmail.com, sayuce@yildiz.edu.tr. 1

2 Prelimiaries I this sectio, we ive iformatios about Ivolute-evolute curves ad Darboux frame. Let (s) be a curve o a orieted surface M. Sice the curve (s) is also i space, there exists Freet frame ft; N; B at each poits of the curve where T is uit taet vector, N is pricipal ormal vector ad B is biormal vector, respectively. The Freet equatios of the curve (s) is ive by 8 >< >: T 0 = N N 0 = B 0 = T + B N where ad are curvature ad torsio of the curve (s), respectively. Sice the curve (s) lies o the surface M there exists aother frame of the curve (s) which is called Darboux frame ad deoted by ft; ;. I this frame T is the uit taet of the curve, is the uit ormal of the surface M ad is the uit vector ive by = T. Sice the uit taet T is commo i both Freet frame ad Darboux frame, the vectors N; B;; lie o the same plae. So that the relatios betwee these frames ca be ive as follows T 7 5 = T 0 cos ' si ' 7 5 N 0 si ' cos ' B 7 5 (.1) where ' is the ale betwee the vectors ad. The derivative formulae of the Darboux frame is T 7 5 = 0 T (.) 0 where, is the eodesic curvature, is the ormal curvature ad is the eodesic torsio of (s) : Here ad i the followi, we use dot to deote the derivative with respect to the arc leth parameter of a curve. The relatios betwee,, ad, are ive as follows = cos '; = si ' ; = + d' : (.) Furthermore, the eodesic curvature ad eodesic torsio of the curve (s) ca be calculated as follows

3 d = ; d d ; = ; d I the di eretial eometry of surfaces, for a curve (s) lyi o a surface M the followis are well-kow i) (s) is a eodesic curve, = 0; ii) (s) is a asymptotic lie, = 0; iii) (s) is a pricipal liepricipal lie, = 0; [7]. Let ad be two curves i the Euclidea space E. Let ft; N; B ad ft ; N ; B (.) be Freet frames of ad, respectively. The the curve is called the ivolute of the curve ; if the taet vector of the curve at the poits (s) passes throuh the taet vector of the curve at the poit (s) ad ht; T i = 0: Also, the curve is called the evolute of the curve :The pair f; is said to be a special ivolute-evolute pair. Special Ivolute-Evolute Parter D -Curves i E I this sectio, by cosideri the Darboux frame, we de e ivolute evolute parter D-curves ad ive the characterizatios of these curves. De itio 1. Let M ad N be orieted surfaces i three dimesioal Euclidea space E ad the arc leth parameter curves (s) ad (s ) lyi fully o M ad N, respectively. Deote the Darboux frames of (s) ad (s ) by ft; ; ad ft ; ;, respectively. If there exists a correspodi relatioship betwee the curves ad such that, at the correspodi poits of the curves, the Darboux frame elemet T of coicides with the Darboux frame elemet of, the is called a special evolute D-curve of ad is a special ivolute D -curve of : The, the pair f; is said to be a special ivolute evolute D -pair. Theorem 1. Let (s) ad (s ) be two curves i the Euclidea space E : If the pair f; is a special ivolute-evolute D-pair, the (s) = (s) + (c s) T (s) : Proof. Suppose that the pair f; is a special ivolute evolute D-pair. From de itio of

4 special ivolute-evolute D-pair, we kow (s) = (s) + (s) T (s) : (.1) Di eretiati both sides of the equatio (.1) with respect to s ad use the Darboux formulas, we obtai T (s ) = T (s) + (s)t (s) + (s) (s) (s) + (s) (s) (s) : Sice the directio of T coicides with the directio of, we et ad (s) = 1 (.) (s) = c s (.) where is c costat. Thus, the equality (.1) ca be writte as follows (s ) = (s) + (c s) T (s) : (.) Corollary 1. Let (s) ad (s ) be two curves i the Euclidea space E : If the pair f; is a special ivolute-evolute D-pair, the the distace betwee the curves (s) ad (s ) is costat. Theorem. Let M ad N be orieted surfaces i three dimesioal Euclidea space E ad the arc leth parameter curves (s) ad (s ) lyi fully o M ad N, respectively. (s ) is special ivolute D-curve of (s) if ad oly if the ormal curvature of (s ) ad the eodesic curvature, the ormal curvature ad the eodesic torsio of (s) satisfy the followi equatio = +! cos + : for some ozero costats, where is the ale betwee the vectors ad at the correspodi poits of (s) ad (s ). E Proof. Suppose that M ad N are orieted surfaces i three dimesioal Euclidea space ad the arc leth parameter curves (s) ad (s ) lyi fully o M ad N, respectively. Deote the Darboux frames of (s) ad (s ) by ft; ; ad ft ; ;, respectively. The by the de itio we ca assume that (s) = (s) + (s) T (s) (.5)

5 for some fuctio (s). By taki derivative of (.5) with respect to s ad applyi the Darboux formulas (.) we have From (.) we et O the other had we have Di eretiati (.8) with respect to s, we obtai + T = 1 + T+ + (.) T = + : (.7) T = cos si : (.8) = ( cos si ) T+ From the last equatio ad the fact that = si + cos si + cos we have + si + cos = ( si cos ) T+ si + cos: Sice the directio of T is coicidet with we have From (.) ad (.8) we obtai ad = = cos = : (.9) si (.10) By taki the derivative of this equatio ad applyi (.9) we et = ta (.11) that is desired. = +! cos + : (.1) Coversely, assume that the equatio (.1) hol for some ozero costats : The by usi (.10), (.11) ad (.1) ives us = + + (.1) 5

6 Let de e a curve (s) = (s) + (s) T (s) : By taki the derivative of the last equatio with respect to s twice, we et T = + (.1) ad + +T d s = + T+ + (.15) respectively. Taki the cross product of (.1) with (.15) we have = h By substituti (.1) i (.1) we et + + i T : (.1) = T : (.17) Taki the cross product of (.1) with (.17) we have = + T+ : (.18) From (.17) ad (.18) we have + = " # + + T+ (.19) ( + ) + ( + ) + : Furthermore, from (.1) ad (.17) we et 8 >< >: respectively. Substituti (.0) i (.19) we obtai + = ( = + (.0) = + ( ) ) + (.1) :

7 Equality (.1) ad (.1) shows that the vectors T ad lie o the plae Spf;. So, at the correspodi poits of the curves, the Darboux frame elemet T of coicides with the Darboux frame elemet of. Thus, the proof is completed. Special Case 1. Let (s ) be a asymptotic special ivolute D-curve of : i) Cosider that (s) is a asymptotic lie. The (s) is special evolute D-curve of (s ) if ad oly if the eodesic curvature, the eodesic ormal ad the eodesic torsio of (s) satisfy the followi equatio, = : ii) Cosider that (s) is a pricipal lie. The (s) is special evolute D-curve of (s ) if ad oly if he eodesic curvature ad the eodesic ormal of (s) satisfy the followi equatio, = : Theorem. Let the pair f; be a special ivolute evolute D-pair i the Euclidea space E :The the relatio betwee the eodesic curvature ive as follows ad the eodesic torsio of (s ) is + = 1 for some ozero costats, where is the ale betwee the vectors ad at the correspodi poits of (s) ad (s ). Proof. Let the pair f; be a special ivolute-evolute D-pair i the Euclidea space E :The from (.5) we ca write (s) = (s) + (s) T (s) for some costats. The last equatio is writte as follows (s) = (s) (s) T (s) : Sice the directio of T is coicidet with we have (s) = (s) (s) (s) : (.) By di eretiati (.) with respect to s ad sice the directio of T is coicidet with we have + = 1 : 7

8 Special Case. space E : Let the pair f; be a special ivolute-evolute D-pair i the Euclidea i) If is eodesic curve, the = 1 : ii) If is pricipal lie, the = 1 : Theorem. E : The the followi relatios hold: Let the pair f; be a special ivolute-evolute D-pair i the Euclidea space i) = d ii) = cos + si iii) = si + cos iv) = ( si cos ) Proof. i) By di eretiati the equatio h; i = cos with respect to s we have ( T ) ; + ; T = si d Usi the fact that the directio of T coicides with the directio of ad we easily et that T = cos si = si + cos Similarly, other choices are testi ed. = d Theorem 5. Let the pair f; be a special ivolute-evolute D-pair i the Euclidea space E : The eodesic curvature of (s ) is = ( cos + si ) 8

9 where is the ale betwee the vectors ad at the correspodi poits of (s) ad (s ). Proof. Suppose that the pair f; is a special ivolute-evolute D-pair i the Euclidea space E : From the rst equatio of (.) ad by usi the fact that T is coicidet with we have d = ; d = ( cos + si ) : space E : Special Case. Let the pair f; be a special ivolute-evolute D-pair i the Euclidea i) If is a eodesic curve, the the eodesic curvature of (s ) is = si : ii) If is a asymptotic lie, the the eodesic curvature of (s ) is = cos : Theorem. Let the pair f; be a special ivolute evolute D -pair i the Euclidea space E :The eodesic curvature of (s ) is = si cos + : where is the ale betwee the vectors ad at the correspodi poits of (s) ad (s ). Proof. Suppose that the pair f; is a special ivolute-evolute D-pair i the Euclidea space E : From the rst equatio of (.) ad by usi the fact that T is coicidet with we have = d ; d = si cos + : 9

10 Corollary. Let the pair f; be a special ivolute-evolute D-pair i the Euclidea space E : i) If is a eodesic curve, the the eodesic curvature of (s ) is = si cos : ii) If is a asymptotic lie, the the eodesic curvature of (s ) is = si cos : Example 1. Let (s) = si s; cos s; si s si s cos s be a curve. This curve lies o the surface z = x xy (mokey saddle). The special ivolute D-curve of the curve (s) ca be ive below (s) = si s + (c s) cos s; cos s + (s c) si s; si s si s cos s + (1 s)(9 si s cos s cos s) ; c R: For specially, c = 1 ad s [0; ]; we ca draw special ivolute-evolute D-pair f; with helpi the proramme of Mapple 1 as follow, F iure 1: Special Ivolute-Evolute Parter D-Curves Example. Let (s) = s si s; s cos s; s be a curve. This curve lies o the surface z = x + y. The special ivolute D-curve of the curve (s) ca be ive below (s) = s si s + (c s)(si s + s cos s); s cos s + (c s)(cos s s si s); s + (c s)s ; c R: This curve lies o the surface z = p x + y.for specially, c = 0 ad s [0; ]; we ca draw special ivolute-evolute D-pair f; with helpi the proramme of Mapple 1 as follow, 10

11 F iure : Special Ivolute-Evolute Parter D-Curves Refereces [1] Boyer, C., A history of Mathematics, New York: Wiley, (198). [] Hac saliho¼lu, H. H., Diferesiyel Geometri, Akara Üiversitesi Fe Fakültesi, (000). [] Turut, A. ad Erdo¼a, E., Ivolute Evolute Curve Couples of Hiher Order i R ad Their Horizotal Lifts i R, Commo. Fac. Sci. Uiv. Ak. Series A, 1 () (199) [] Do Carmo, M.P., Di eretial Geometry of Curves ad Surfaces, Pretice Hall, Elewood Cli s, NJ, 197. [5] Millma R.S., Parker G.D., H.H., Elemets of Di eretial Geometry, Pretice Hall Ic., Elewood Cli s,new Jersey, [] Kazaz M., U¼urlu H.H., Öder M., Kahrama T., Maheim Parter D Curves i Euclidea -space, arxiv:100.0 [math.dg]. [7] O Neill, B., Elematery Di eretial Geometry Academic Press Ic. New York,