THE DARBOUX TRIHEDRONS OF REGULAR CURVES ON A REGULAR SURFACE

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1 International lectronic Journal of eometry Volume 7 No 2 pp (2014) c IJ TH DARBOUX TRIHDRONS OF RULAR CURVS ON A RULAR SURFAC MRAH TUNÇ AND MİN OZYILMAZ (Communicated by Levent KULA) Abstract In this work we study Differential eometry of the curves on a regular surface in uclidean space R 3 by using parameter curves which are not perpendicular to each other The aim of this study is to investigate the formulas between the Darboux Vectors of the curve (c) the parameter curve (c 1 ) and the parameter curve (c 2 ) which are not intersect perpendicularly 1 Introduction Roughly saying geometry set out with the measurement of distances and angles The geometry of curves in R 3 can be investigated by the topics which are general helices involute-evolute curve couples spherical curves and Bertrand curves Such special curves are used many areas as seen computer graphics topological combinatorics especially the modelling of cars in physics and in some of real world problems like mechanical design or robotics In recent years the theory of degenerate submanifol has been treated by researchers and some of the classical differential geometry topics have been extended to Lorentzian manifol [6] In [6] authors had assumed that the parameter curves of any surface are orthognal Moreover some authors have aimed to determine the Frenet-Serret invariants in higher dimensions Another orthonormal frame on surface is the Darboux trihedrons We use it to examine the special curves on surface as geodesic curves and asymptotic curves By using the Darboux vector various well-known formulas of differential geometry had been produced by [ ] In this work we study to investigate the formulas between the Darboux Vectors of the curve (c) the parameter curves (c 1 ) and (c 2 ) which are not intersecting perpendicularly Thus we will find an opportunity to investigate regular surface by taking the parameter curves which are intersect under the angle θ Date: Received: September and Accepted: June Mathematics Subject Classification 53A04 Key wor and phrases Classical Differential eometrycurvessurfaces 61

2 62 MRAH TUNÇ AND MİN OZYILMAZ 2 Preliminary To meet the requirements in the next sections here the basic elements of the theory of curves in the space R 3 are briefly presented (A more complete elementary treatment can be found in [1]) The uclidean space R 3 provided with the standard flat metric given by (21) < >= dx dx dx 2 3 where (x 1 x 2 x 3 ) is a rectangular coordinate system of R 3 Recall that the norm of an arbitrary vector a R 3 is given by a = < a a > Let φ = φ(s) be a regular curve in R 3 φ is called an unit speed curve if the velocity vector v of φ satisfies v = 1 For the vectors u w R 3 it is said to be orthogonal if and only if < u w >= 0 On the other hand the vector w is called angular velocity vector of motion If we consider any orthogonal trihedron as { e 1 e 2 e 3 } we can write their derivative formulas as follows: (22) d e 1 dt = w e i i = where is vectoral product[1] Let we take a surface as y = y(u v) Denote by { t n b} the Frenet-Serret frame along the curve (c) on y = y(u v) Another orthogonal frame on y = y(u v) is the Darboux trihedron as { t N g} For an arbitrary curve (c) on surface the orientation of the Darboux trihedron is written as (23) t N = g g t = N N g = t and the Darboux vector of this trihedron is written as (24) w = t T g + N R g + g R n where < t t >= 1 < N N >= 1 < g g >= 1 < t g >= 0 < t N >= 0 < g N >= 0 where 1 T g 1 1 R n R g are geodesic torsion normal curvature geodesic curvature and T g R n R g are radius of geodesic torsion radius of normal curvature radius of geodesic curvature respectively Alsothe Darboux derivative formulas can be written as follows: (25) = w t d N = w N d g = w g

3 TH DARBOUX TRIHDRONS OF RULAR CURVS ON A RULAR SURFAC 63 3 The Darboux Vector For The Darboux Trihedron Of A Regular Curve Let us express the parameter curves u = const as (c 1 ) and v = const as (c 2 ) which are constant on a regular surface y = y(u v) But these curves are intersect under the angle θ (not perpendicular) Let any curve that is passing through a point P on the surface be (c) Let us take curves which are passing through the same point P as (c 1 ) and (c 2 ) Let the unit tangent vectors of curves (c) (c 1 ) and (c 2 ) at the point P be t t 1 and t 2 respectively From [1] the edges of the Darboux trihedrons of parameter curves are (31) t i N = g i g i t i = N N gi = t i (i = 1 2) Here the three Darboux trihedrons are written as below: [ t N g] [ t 1 N g 1 ] [ t 2 N g 2 ] Let the arc-lenght parameter of the curves (c)(c 1 ) and (c 2 ) be s s 1 and s 2 respectively Thus we can write t 1 = r u r u = r u (32) t 2 = r v r v = r v du t = r u + r dv v Moreoverthe parameter curves are intersect under the angle θthus we have (33) < t 1 t 2 >= cosθ Then the normal vector of surface is (34) N = t 1 t 2 t 1 t 2 = t 1 t 2 By considering the first two formulas of (32) in the third term (35) du t = r u + r dv v = du t 1 + dv t 2 is written[1] On the other hand let us consider the real angle between t and t 1 as α and if we take inner product both side of (35) with t 1 and t 2 then (36) (37) < t t 1 >= cosα = du + cos θ dv < t t 2 >= cos(θ α) = cosθ du + dv

4 64 MRAH TUNÇ AND MİN OZYILMAZ are obtained Thus from (36) and (37) (38) sinα = du = dv are written Finally if we put (38) into (35) we have the following equation between the tangent vectors of the curves (c) (c 1 ) and (c 2 ): (39) t = t 1 + sinα t 2 Here we shall denote the arc-elements 1 and 2 of the parameter curves which are belongs to regular surface y = y(u v) and then we express as follows: (310) 2 = du 2 + 2F dudv + dv 2 Thus considering (38) and (310) we have 2 1 = du = dv 2 (311) = du = 1 sinα = dv = 2 Corollary 31 The third elements g g 1 and g 2 of the Darboux trihedrons [ t N g] [ t 1 N g 1 ] and [ t 2 N g 2 ] are linear dependent Proof If we substitute the equation (39) in the first equality of (23) and consider the Darboux trihedrons of (c 1 ) and (c 2 ) we have just obtain (312) g = Thus we get the expression g 1 + sinα g 2 Theorem 31 The Darboux trihedrons [ t 1 N g 1 ] and [ t 2 N g 2 ]of the parameters curves (c 1 ) and (c 2 ) of the surface y = y(u v) are written by Darboux instantaneous vectors as follows: (313) t i = w i t i N = w i N g i = w i g i (i j = 1 2) Proof If we consider the Darboux trihedrons [ t 1 N g 1 ] and [ t 2 N g 2 ] of parameter curves we see that the normal vector N is coincide Then considering (34)

5 TH DARBOUX TRIHDRONS OF RULAR CURVS ON A RULAR SURFAC 65 (314) g 1 = t 1 N t = t 1 ( 1 t 2 ) = t 2 (< t 1 t 1 >) + t 1 (< t 1 t 2 >) = t 1 cosθ t 2 (315) g 2 = t 2 N t = t 2 ( 1 t 2 ) = t 2 < ( t 1 t 2 ) > + t 1 (< t 2 t 2 >) are obtained From (22) we write = t 2 cosθ + t 1 (316) t 1 = w 1 t 1 N = w 1 N 1 g 1 = w 1 g (317) t 2 2 = w 2 t 2 N 2 = w 2 N g 2 2 = w 2 g 2 If (314) is substituted in the third equality (316)we get (318) g 1 = [ 1 sin θ (cos θ t 1 t 2 )] = 1 s 1 s 1 sin θ ( t 1 cos θ t 2 ) = s 1 s 1 1 sin θ [( w 1 t 1 ) cos θ t 2 1 ] (319) g 1 1 = w 1 g 1 = w 1 [ s 1 sin θ (cos θ t 1 t 2 )] = 1 sin θ (( w 1 t 1 ) cos θ ( w 1 t 2 )) Then from (318) and (319) we have (320) t 2 1 = w 1 t 2 Thus the derivative of t 2 with respect to s 1 is written by the vectoral product of w 1 and t 2 Similarly it is easy to see that the other vectors can be written by the same metod Corollary 32 By using the vectors t 1 t 2 and Nwe can express w w 1 and w 2 as follows: (321) t w = 1 α) cos(θ α) t [sin(θ + ] + 2 sin θ T g R n sin θ [sin α T g cos α N ] + R n R g

6 66 MRAH TUNÇ AND MİN OZYILMAZ (322) (323) 1 w 1 = t 1 [ + cos θ ] (T g ) 1 sin θ(r n ) 1 t 2 sin θ(r n ) 1 + N (R g ) 1 t 1 1 w 2 = + [ cos θ N ] t 2 + sin θ(r n ) 2 (T g ) 2 sin θ(r n ) 2 (R g ) 2 Proof From (24) we can write the Darboux vectors of the [ t N g] [ t 1 N g 1 ] and [ t 2 N g 2 ] as w = t T g + N R g + g R n (324) w 1 = t 1 (T g ) 1 + N (R g ) 1 + g 1 (R n ) 1 w 2 = t 2 (T g ) 2 + N (R g ) 2 + g 2 (R n ) 2 Then if we consider the equation(39)(314) and (315) according to the vectors t 1 and t 2 and substitute in (324) we get (321)(322) and (323) Theorem 32 If we consider the tangent vectors t 1 and t 2 of the parameter curves (c 1 ) and (c 2 ) on the surface y = y(u v) then we obtain the following relations: (325) (326) i) < t 1 t 2 >= < t 2 t 1 >= ( )v cos θ( ) u s 1 s 1 ii) < t 2 t 1 >= < t 1 t 2 >= ( )u cos θ( ) v s 2 s 2 Proof From (32) t 2 = ru and t 2 = rv are written And also we know that (327) < t 1 t 2 >= cosθ < r u r v >= cosθ (328) = ( ) 2 = r 2 u ( ) v = r uv r u (329) By taking differential from t 1 = = ( ) 2 = r 2 v ( ) u = r vu r v t 1 v = ru and t 2 = rv we obtain r u v ( ) ( ) v r u Thus we write t 2 u = r u v ( ( ) u r v

7 TH DARBOUX TRIHDRONS OF RULAR CURVS ON A RULAR SURFAC 67 (330) < t 2 t 1 v >= r v ( r uv ( )v r u ) = ( ) u cosθ( ) v (331) < t 1 t 2 u >= r u ( On the other hand we have r u v ( ) ( ) u r v ) = ( ) v cosθ( ) u (332) t 1 = t 1 dv = 1 t 1 s 2 v 2 v (333) t 2 = t 2 du = 1 t 2 s 1 u 1 u Thus taking inner product of (332) and (333) by the vector t 2 and the vector t 1 and considering (330)and(331) we have (326) and(325) The other cases can be seen easily Corollary 33 If we take differential from < t 1 t 2 >= cos θ with respect to u and v we get (334) The same hol for < t 1 t 2 v >= < t t 2 1 )u cos θ( ) >= [( v ] v (335) Thus we have (336) < t 1 t 2 u >= ( )v cos θ( ) u v (< t 1 t 2 u > u (< t 1 t 2 v >) = v (( )v cos θ( ) u ) + u (( )u cos θ( ) v ) Theorem 33 The following geodesic curvature equalities are satisfied for the parameter curves (c 1 ) and (c 2 ) (337) (338) 1 1 = (R g ) 1 (cosθ( ) u ( ) v ) 1 1 = (R g ) 2 (( u ) cosθ( ) v )

8 68 MRAH TUNÇ AND MİN OZYILMAZ Proof i) From (325) and (320) we write < t 1 t 2 >= ( ( )v cosθ( ) u ) s 1 It follows that we have t 2 s 1 = w 1 t 2 ( ) v cos θ( ) u =< t 1 w 1 t 2 >=< w 1 t 2 t 1 >= < N w 1 > Then from (324) if we take inner product both of side w 1 with N we obtain Thus < N w 1 >= < N N > (R g ) 1 N w 1 = 1 (R g ) = (R g ) 1 (cosθ( ) u ( ) v ) is obtained Similarly (ii) can be proven Theorem 34 Let us consider any curve (c) on the surface and the arc elements of curves (c) (c 1 ) and (c 2 ) is 1 and 2 respectively Let the Darboux instantaneous rotation vectors of (c 1 ) and (c 2 ) be w 1 w 2 and if the real angle between the tangent t of curve (c) and t 1 is α then (339) and ( w 1 + sinα w 2) t 1 = A t 1 (340) are satisfied 1 = A t 1 2 = A t 2 Proof If we consider (311) and (313) then d N = A N t 1 s = t 1 1 s 1 + t 1 2 s 2 = ( w 1 t 1 ) + sinα ( w 2 t 1 ) = ( w 1 + sinα w 2) t 1 = A t 1 is obtained Similarlythe others are satisfied Corollary 34 The following equality (341) is hold < t 2 d t 1 >= < t 1 d t 2 >= < A N >

9 TH DARBOUX TRIHDRONS OF RULAR CURVS ON A RULAR SURFAC 69 Theorem 35 Let us consider the curves (c) (c 1 ) and (c 2 ) which are intersect a point P on the surface y = y(u v) Let the Darboux instantaneous rotation vectors of these curves at the point P be w w 1 and w 2 respectively We obtain following equality between the Darboux vectors w w 1 and w 2 : (342) Proof From (39) w = w 1 + sinα w 2 + N dα (343) t = t 1 + sinα t 2 can be written Then by taking derivatives with respect to s from equation (343) we obtain (344) 1 = + sinα dt 2 α) + [ cos(θ t 1 + cosα t 2 ] dα On the other hand considering the Darboux trihedrons [ t 1 N g 1 ] and [ t 2 N g 2 ] we write (345) t 1 = N g 1 t 2 = N g 2 From (314) and (315) if g 1 and g 2 are substituted in (345) we obtain (346) t 1 = N 1 (cosθ t 1 t 2 ) t 2 = N 1 ( cosθ t 2 + t 1 ) By substituting the equations (346) in (344) we have (347) 1 = According to the theorem 4 + sinα dt 2 α) + [ cos(θ sin 2 N (cosθ t 1 t 2 ) θ + cosα sin 2 θ N ( t 1 t 2 cosθ)] dα 1 = A t 2 t 1 = A t 2 A = w 1 + sinα w 2 are known And by using the trigonometric expression we find = A t 1 + sinα A t 2 + N [ t 1 + sinα t 2 ] dα = A [ t 1 + sinα t 2 ] + N [ t 1 + sinα t 2 ] dα = A t + ( N t) dα = [ A + N dα ] t

10 70 MRAH TUNÇ AND MİN OZYILMAZ Thus (348) where = k t (349) After that k = [ A + N dα ] (350) A = k N dα is obtained By writing (349) in the third expression of (340) we obtain (351) d N = A N = ( k dα N) N = k N Since W is the Darboux vector we have = w dn (352) t = w N Then considering (25) (347) (350) and (351) = w t = k t 0 = k t w t = ( k w) t (353) and ( k w) = λ t (354) d N = w N = k N 0 = k t w N = ( k w) N k w = µ N are written At the end if we make equal (352) to (353) we have (355) k w = λ t = µ N λ = µ = 0 Finally k w = 0 can be written Thus we get the theorem References [1] Akbulut F Bir Yuzey Uzerindeki grilerin Darboux Vektorleri ge Univ Faculty of Science Izmir 1983 [2] Do Carmo MP Differential eometry of Curves and Surfaces Prentice-Hall nglewood Cliffs 1976 [3] KlingenbergW A Course in Differential eometry Springer-Verlag Berlin and New York 178 p 1978 [4] KuhnelW Differential eometry Curves-Surfaces-Manifol iesbaden Braunchweig 1999

11 TH DARBOUX TRIHDRONS OF RULAR CURVS ON A RULAR SURFAC 71 [5] MilmanMS and ParkerD lements of Differential eometryprentice-hall Incnglewood Cliffs New Jersey 1977 [6] Ugurlu H and Caliskan A The Space-like and Time-like Surface eometry by the Darboux Vector Celal Bayar Univ Manisa 2012 Department of Mathematics Faculty of Science University of ge Bornova Izmir TURKY -mail address: emrahtunc@hotmailcom Department of Mathematics Faculty of Science University of ge Bornova Izmir TURKY -mail address: ozyilmaz@egeedutr