Balki Okullu, P., et al.: An Explicit Characterization of Spherical Curves According ... THERMAL SCIENCE: Year 2019, Vol. 23, Suppl. 1, pp. S361-S370 361
AN EXPLICIT CHARACTERIZATION OF SPHERICAL CURVES ACCORDING TO BISHOP FRAME AND AN APPROXIMATELY SOLUTION
by
Pinar BALKI OKULLU a*, Huseyin KOCAYIGIT a, and Tuba AGIRMAN AYDIN b a Faculty of Art and Science, Manisa Celal Bayar University, Manisa, Turkey b Faculty of Education, Bayburt University, Bayburt, Turkey
Original scientific paper https://doi.org/10.2298/TSCI181101049B
In this paper, spherical curves are studied by using Bishop frame. First, the dif- ferential equation characterizing the spherical curves is given. Then, we exhibit that the position vector of a curve which is lying on a sphere satisfies a third-order linear differential equation. Then we solve this linear differential equation by using Bernstein series solution method. Key words: spherical curve, Bishop frame, Bernstein series solution method
Introduction As is well known fundamental structure of differential geometry is the curves. With in the process, the best part of classical differential geometry topics have been expanded to space curves. There are many studies which implies different characterizations of these curves. Also spherical curves are the special space curves which lies on the sphere. There are also many studies on spherical curves. Firstly, Wong [1] give a universal formulation of the condition for a curve to be on sphere in 1963. Then, Breuer and Gottlieb [2] shown that the differential equa- tion characterizing a spherical curve can be solved explicitly to express the radius of curvature of the curve in terms of its torsion in 1971. Wong [3] proved that the explicit characterization of spherical curves recently obtained by Breuer and Gottlieb without any precondition on the curvature and torsion, a necessary and sufficient condition for a curve to be a spherical curve. The proof is based on an earlier result of the present author on spherical curves. Mehlum and Wimp [4] proof that the position vector of any 3-space curve lying on a sphere provides a third order linear differential equation in 1985. In 1988, Kose [5] give an explicit characterization of the dual spherical curve. Abdel Bakey [6] studied with dual spherical curves and obtained a differential equation which is characterizing the dual spherical curves. Then he give the explicit solution of this differential equation without the precondition on the dual torsion. A necessary and sufficient condition for a dual curve to be spherical curve is given in 2002. Ilarslan [7] present the spherical characterization of non-null regular curves in 3-D Lorentzian space is given. Furthermore, the differential equation which expresses the mentioned characterization is solved in 2003. Kocayigit [8] shown that the differential equation characterizing a spherical curve in n-dimensional Euclidean space n ≥ 3 can be solved explicitly to express nth curva- ture function of the curve in terms of its curvatures and its other curvature functions in 2003.
* Corresponding author, e-mail: [email protected] Balki Okullu, P., et al.: An Explicit Characterization of Spherical Curves According ... S362 THERMAL SCIENCE: Year 2019, Vol. 23, Suppl. 1, pp. S361-S370
Ayyildiz et al. [9] present the differential equation that is characterizing the dual Lorentzian spherical curves and then give an explicit solution of this differential equation are given in 2007. Camci et al. [10] studied with regular curves in 3-D Sasakian space and give the spherical characterizations of them. Furthermore, the differential equation which expresses the aforesaid characterization is solved in 2007. In this paper, we give the differential equation characterizing the spherical curves by using Bishop frame. Then, we present that the position vector of a curve lying on a sphere satisfies a third-order linear differential equation and the solution is given with Bernstein series method. Preliminaries In Euclidean 3-space scalar product is given:
222 , =++xxx1 23 3 3 1/2 here (,xxx , ) is an arbitrary vector in E and the norm of the vector xE∈ is x= (,)áñ xx . 123 Let α : I⊂→ IR E3 be arbitrary curve in the Euclidean space E3. If áñαα', '= 1, then the curve α is said to be of unit speed (parametrized by arc length parameter s). The {,TNB , }is named as Frenet frame of α and κ , τ are the curvature and the torsion of the curve α, respectively, [10]. Here, curvature functions are defined by κκ=()s = T '(), s τ () s= −áñ NB , '. Let 3 u= (, uuu123 , ) and v= (, vvv123 , ) be vectors in E . Cross product of u= (, uuu123 , ) and v= (, vvv123 , ) is defined:
eee123
u×= v u1 u 2 u 3 =( uv 23 − uv 32,, uv 13 − uv 32 uv 12 − uv 21)
vvv123 The Bishop frame in other words parallel transport frame is a different point of view for describing a moving frame which is well-define even when the curve has vanishing second derivative. The parallel transport of an orthonormal frame along a curve can be expressed sim- ply by parallel transporting each component of the frame. The tangent vector and any conve- nient arbitrary basis for the remainder of the frame are used. Consequently, the Bishop (frame) formulas is expressed: dT =kN11 + kN 2 2 ds dN 1 = −kT (1) ds 1 dN2 = −kT2 ds Here, we name the set {,TN12 , N } as Bishop frame and k1 and k2 are the first and second Bishop curvature, respectively, [11]. There is a relation between Frenet and Bishop frame elements and this relation is given: TT= NNN=cosθθ12 + sin B=−+sinθθ NN12 cos Balki Okullu, P., et al.: An Explicit Characterization of Spherical Curves According ... THERMAL SCIENCE: Year 2019, Vol. 23, Suppl. 1, pp. S361-S370 S363
2 2 1/2 where θ(s )= arctan( kk21 / ),τθ ( s )= '( s ) and κ()s= ( kk12 + ) . Bernstein series solution method Bernstein polynomials of nth-degree are defined:
nk− n xRxk ( − ) Bxkn, () = ,k= 0,1,..., n (2) k Rn The R is the maximum range of the interval [0,R ] over which the polynomials are de- fined to form a complete basis [12, 13]. Let us explain briefly the Bernstein series solution meth- od for the solution of the linear differential equations with variable coefficient defined:
m ()k ∑ Pk () s y () s= gs () 0 ≤≤ s R (3) k =0 Suppose that the eq. (2) has a solution, f . We wish to approximate f :
n pn() s= ∑ aBk kn, () s n≥ 1 (4) k =0 such that psn () satisfies eq. (3) on the nodes 0< We take second derivative of eq. (6) according to s, we have: '' kNx11,+= k 2 Nx 2 ,0 (8) From the equations below, we get: kk'' =−=21 Nx12,'' ,,Nx '' (9) kk12−− kk 12 kk12 kk 12 As a result, the expression of position vector xs() in terms of Bishop frame elements is: kk'' = 21+ xs( ) ''''N12N (10) kk12−− kk 12 kk 12 kk 12 3 3 Theorem 2. Let xs( ): I→ E is a curve in E with arc length parameter s. If x= xs() is spherical, then there is a correlation between the bishop curvatures given: ()kk'2+ () ' 2 12= 2 ' '2 r ()kk12− kk 12 Proof. Let x be a spherical curve with the arc length parameter s . If we take the inner product of eq. (10) by itself, we get: ()kk' 2 ()'2 = 21+ xs(),() xs ' '2 NN11, ' '2 NN2, 2 (11) ()kk12−− kk 12 ()kk12 kk 12 From eq. (11), we obtain: ()kk'2+ () ' 2 12= 2 ' '2 r (12) ()kk12− kk 12 Theorem 3. Position vector of a curve lying on a sphere satisfies a third-order linear differential equation. 2 3 Proof. Let áñxmxm−,0 −−= r be a sphere and x= xs() be a curve in E . Sphere 2 and curve have a contact of order 3 at point P if gs()=−áñ xmxm , −−= r 0 and ddggg23 d = = = 0 (13) ds ddss23 With the aid of eq. (1), the eq. (13), respectively, give: g',= Tx −= m 0 (14) g''= kNxm11 , −+ kNxm 2 2, −=− 1 (15) '' g'''= kNxm11 , −+ kNxm 2 2, −= 0 (16) From eq. (15) and eq. (16) we get: k ' k ' −= 2 −=− 1 Nx1, m '', Nxm2 , '' (17) kk12− kk 12 kk12− kk 21 Balki Okullu, P., et al.: An Explicit Characterization of Spherical Curves According ... THERMAL SCIENCE: Year 2019, Vol. 23, Suppl. 1, pp. S361-S370 S365 The vector xm− can be written as the linear combination of the Bishop frame ele- ments: xm−=αβ T + N12 + γ N (18) If we multiply both sides of the eq. (18), respectively, by TN,,12 N, we get: kk'' αβ= = 21 γ= 0, ' ', '' (19) kk12−− kk 12 kk 12 kk 21 If the center of the sphere is origin then the expression of the position vector of a curve lying on sphere in terms of Bishop frame elements: kk'' = 21− xN''''12 N (20) kk12−− kk 12 kk 12 kk 12 If we take the derivative of position vector three times, we, respectively, get: xT' = (21) x''= T ' = kN11 + kN 2 2 (22) 22 ' ' x''' =−++ k k T kN + kN (23) ( 1 2) 11 2 2 From eqs. (22) and (23), we get: ' 22 kx'''−+ kx '' k ( k + k ) x ' = 2 2 21 2 N1 '' (24) kk12− kk 12 ' 22 kx'''−+ kx '' k ( k + k ) x ' = 1 1 11 2 N2 '' (25) kk12− kk 12 If we put eqs. (24) and (25) in eq. (20), we get: ' ' ''22 22 ' ' kk++ kk k k (k1++ k 2)( kk 11 kk 2 2) =11 2 2 −+1 2 xx' '2''' ' '2 x'' ' '2 x' (26) ()()()kk12−− kk 12 kk12 kk 12 kk12 − kk 12 If the eq. (26) rearranged, we get: '' ''22 22'' ' '2 (kk11+ kk 2 2 ) x ''' −+ ( k 1 k 2 ) x '' ++ ( k 1 k 2 )( kk 11 + kk 2 2 ) x ' − ( kk 12 − kk 12 ) x = 0 (27) Equation (27) shows that the position vector of a curve lying on a sphere satisfies a third-order linear differential equation. 3 3 Corollary 1. Let xs( ): I→ E is a spherical curve in E with arc length parameter s. The centers of spheres with 3-contact points with xs() in the point xs() lie on straight line. kk'' = −+21 mx' 'N12 ''N kk12−− kk 12 kk12 kk 21 3 3 Corollary 2. Let xs( ): I→ E is a spherical curve in E with arc length parameter s. The radius of spheres with 3-contact points with xs() is: Balki Okullu, P., et al.: An Explicit Characterization of Spherical Curves According ... S366 THERMAL SCIENCE: Year 2019, Vol. 23, Suppl. 1, pp. S361-S370 22 kk'' = 21+ r ' ' '' kk12−− kk 12 kk 12 kk 21 The solution with Bernstein series method =−− ' ' =+ 22 ' + ' P0() s k 12 () sk () s k 12 () sk (), s P 1 () s k 1 () s k 2 () s k 11 () sk () s k 22 () sk () s 22 =−+='' '' + Ps2(){ ks 1 () ks 2 () } , Ps 3 () ksks 11 () () ksks 2 () 2 () Fs()= 0 and x()kk( s )= y () ( sk ), = 0,1,2,3 By using the previous equations the differential eq. (27) characterizing the spherical curves according to Bishop frame can be rewritten: m ()k ∑ Pk () sy () s= Fs () m = 3 0 ≤ s ≤π 2 (28) k =0 Let f be a solution of eq. (28). We wish to approximate f : n y= pn() s = ∑ aBk kn, () s n= 4 (29) k =0 such that psn () satisfies eq. (28) on the nodes 0≤ss01 < < ... < s 4 ≤π 2 . Here n = 4 is taken for simplicity. Putting psn () into eq. (28), we get the system of linear equations depending on aa01, ,,.., a 4 .Let us consider the eq. (28) and find the matrix forms of each term in the equation. First, we can convert the Bernstein series solution y= psn () defined by (29) and its derivatives ny= ()k () s to matrix forms, for n = 4 and k = 0,1,2,3. ()kk ys()= B4 () sA and y() s= B4 () sA (30) here Bs0,0() Bs 1,0 () Bs4,0 () Bs0,1() Bs 1,1 () Bs4,1 () Bs( ) = , A= [ a a a] 4 01 2 BsBs0,4() 1,4 () Bs4,4 () T TT On the other hand, it can be written [Bs4 ( )] as [B4 ( s )]= { DSs [ ( )]} or: T B4 () s= SsD () (31) for ji− (−1) n ni− ≤ j ,i j dij = R 1 ji− 0, ij> the matrix D is calculated: Balki Okullu, P., et al.: An Explicit Characterization of Spherical Curves According ... THERMAL SCIENCE: Year 2019, Vol. 23, Suppl. 1, pp. S361-S370 S367 1−π 2 3 2 π2 − 1 2 π 34 1 16 π 0 2π −π 323 32 π − 14 π 4 2 34 D = 0 0 32π−π 32 38 π π34 −π 0 0 0 12 14 4 0 0 0 0 1 16π It is clearly seen that the relation between the matrix Ss() and its derivative Ss' () is S' () s= SsB () . 01000 00200 B = 00030 and Ss()= 1 s s234 s s 00004 00000 To obtain the matrix Ss()k () in terms of the matrix Ss(), we can use the following procedure: S''() s= S ' () sB = SsB () 2 (32) S'''() s= S '' () sB = SsB () 3 0020 0 0006 0 0006 0 000024 B2 = 0 0 0 0 12 and B3 = 0000 0 0000 0 0000 0 0000 0 0000 0 Consequently, by substituting the matrix forms (31) and (32) into (30), we have the matrix relation: ys()= SsD () T A y' () s= S () s BDT A (33) y'' () s= SsBD () 2 T A y''' () s= SsBD () 3 T A Substituting the matrix relation (33) into (28) and then simplifying, we obtain the matrix equation: m kT ∑ Pk () sSsBD () A= f () s (34) k =0 By using the nodes {sii ,= 0,1,...,4;0 ≤s01 ≤ s < ... < s 4 ≤π 2 } in (34) we get the system of matrix equations: Balki Okullu, P., et al.: An Explicit Characterization of Spherical Curves According ... S368 THERMAL SCIENCE: Year 2019, Vol. 23, Suppl. 1, pp. S361-S370 m=3 kT ππ3 ∑ Pki( s ) Ss ( i ) BD A= f ( si ), ( i = 0,1,...,4) and s01= 0, s = , s 2 =π= , s 3 , s 4 =π 2 k =0 22 Pk (0) 0 0 0 0 0P (π /2) 0 0 0 k Pski()= 0 0Pk ()π 0 0 0 0 0Pk (3π /2) 0 0 0 0 0Pk (2π ) 10000 Ss()0 f (0) 0 234 1 (ππ /2) ( /2) ( π /2) ( π /2) Ss()1 f (π /2) 0 234 Ss()ii=Ss()2 = 1 ()ππ () () π () π,Fs ()= f ()π= 0 234 Ss()3 1 (3π2 / ) (3 π2 / ) (3 π2 / ) (3 π2 / ) f (3π /2) 0 Ss() 234 f (2π ) 0 4 1 (2ππ ) (2 ) (2 π ) (2 π ) The fundamental matrix equation can be written briefly: m kT ∑ Pk SB D A= F (35) k =0 Hence, the fundamental matrix eq. (35) corresponding to (29) can be written: WA= F or [WF;,] = A Ww= [ kh ], kh,= 0,1,...,4 (36) m kT Also W= ∑k =0 PSk D and the eq. (36) corresponds to a matrix of type (5×5). Now let us obtain the matrix equation of the conditions by means of the relation (33): kT S(0) BDA= [αk ] k = 0,1, 2 Firstly, the matrix forms for the conditions can be written: UAkk= [α ] or [Ukk;α ] k = 0,1, 2 (37) for T U0 = S(0) D = [ uu00 01 u 04 ] = [10000] 1 T U1 = S(0) BD =[ u10 u 11 u 14 ] =−π[ 2 2 π 000] =2T = = π−π2 22 π U2 S(0) BD[ u20 u 21 u 24 ] 12 6 3 0 0 Replacing the row matrices (37) by any m = 3 rows of the matrix (36), we get the augmented matrix [;]WF: wwwww00 01 02 03 04 ; f (0) uuuuu; α 00 01 02 03 04 0 WF; = uuuuu10 11 12 13 14 ; α1 uuuuu20 21 22 23 24 ; α1 wwwww50 51 52 53 54 ; f (2π ) Balki Okullu, P., et al.: An Explicit Characterization of Spherical Curves According ... THERMAL SCIENCE: Year 2019, Vol. 23, Suppl. 1, pp. S361-S370 S369 where wiij (= 0,1 j = 0,1,...,4) obtained: 23 3 26 9 wPPPPwPPP=−+−(0) (0) (0) (0), =++ (0) (0) (0) 00 0 ππ1 ππ232 3 01 1 ππ232 3 39 3 9 3 w= P(0) − Pw (0), = Pw(0), == 0, w Pw(2), =−π P(2 ) 02ππ23 2 3 03 π 3 3 04 50π2 2 51 π 33 3 9 269 w= P(2 π+ ) Pw (2 π ), =− P(2 π− ) P (2 π− ) P (2 π ) 52ππ23 2 3 53ππ 1 π22 3 23 3 wP=(2 π+ ) P (2 π+ ) P (2 π+ ) P (2 π ) 54 0 π 1 ππ232 3 As a result, we can write the following matrix: 01 0 0 00 π 0 1 /2 0 0 α0 −1 2 AW=( ) F = 0− 2 ππ /3 0 α1 RM T K Vα2 YZ Q L C0 where w (w++ w 2) ww − ( w ++ w 2) ww RM= 54 , = 50 51 52 04 00 01 02 54 ww03 54−− ww 04 53 ww03 54 ww 04 53 π (2w− w ) w −− (2 w w ) w ww− ww TL= 02 01 54 52 51 04 ,3= 02 53 03 52 2 ww03 54−− ww 04 53 ww03 54 ww 04 53 ww(+− w 2) w − ww ( +− w 2) ww w ZC= 53 00 01 02 03 50 51 52 54 , = 03 ww03 54−− ww 04 53 ww03 54 ww 04 53 π ww(+− 2) w ww ( + 2) w Q = 53 01 02 03 51 52 2 ww03 54− ww 04 53 and hence the elements aa01, ,..., a 4 of A are uniquely determined: ππ aa=α, = α + α ,2 a =− αα +π + α o 11123 2 2 1 2 3 aM3=αα 1 ++ T 2 K α 34, aZ =++ ααα 1 Q 2 T 3 If we put this ak unknowns in eq. (29), we obtain the Bernstein series solution y= psn () = m1 of the eq. (28). Conclusion In the present paper we dealt with spherical curves. The expression of the position vector of a spherical curve in terms of Bishop frame elements and the relation between the first and second Bishop curvatures for the spherical curve is given. We obtain a third order linear differential equation which is characterizing the curve that lies on a sphere. By using Bernstein series solution method, the solutions of the equation are approximately obtained. Balki Okullu, P., et al.: An Explicit Characterization of Spherical Curves According ... S370 THERMAL SCIENCE: Year 2019, Vol. 23, Suppl. 1, pp. S361-S370 References [1] Wong, Y. C., On an Explicit Characterization of Spherical Curves, Proc. Am. Math. Soc., 34 (1972), 1, pp. 239-242 [2] Breuer, S., Gottlieb, D., Explicit Characterization of Spherical Curves, Proc. Am. Math. Soc., 27 (1971), 1, pp. 126-127 [3] Wong, Y. C., A Global Formulation of the Condition for a Curve to Lie in a Sphere, Monatsh. Math., 67 (1963), 4, 363-365 [4] Mehlum, E., Wimp, J., Spherical Curves and Quadratic Relationships for Special Functions, Austral. Mat. Soc., 27 (1985), 1, pp. 111-124 [5] Kose, O., An Expilicit Characterization of Dual Spherical Curves, Doga Mat. 12 (1998), 3, pp. 105-113 [6] Abdel Bakey, R. A., An Explicit Characterization of Dual Spherical Curve, Commun. Fac. Sci. Univ. Ank. Series, 51 (2002), 2, pp. 1-9 [7] Ilarslan , K., et all., On the Explicit Characterization of Spherical Curves in 3-Dimensional Lorentzian Space, Journal of Inverse and Ill-Posed Problems, 11 (2003), 4, pp. 389-397 [8] Kocayigit, H., et all., On the Explicit Characterization of Spherical Curves in N-Dimensional Euclidean Space, Journal of Inverse and Ill-posed Problems, 11 (2003), 3, pp. 245-254 [9] Ayyilidiz, N., et all., A Characterization of Dual Lorentzian Spherical Curves in the Dual Lorentzian Space, Taiwanese Journal of Mathematics, 11 (2007), 4, pp. 999-1018 [10] Camci, C., et all., On the Characterization of Spherical Curves in 3-Dimensional Sasakian Spaces, J. Math. Anal. Appl., 342 (2008), 2, pp. 1151-1159 [11] Bishop, L. R., There is More Than one Way to Frame a Curve, Amer. Math. Monthly, 82 (1975), 3, pp. 246-251 [12] Bhatti, M. I., Brocken, B., Solutions of Differential Equations in a Bernstein Polynomial Basis, Bhatti, Journal of Computational And Applied Mathematics. 205 (2007), 1, pp. 272-280 [13] Isik, O. R., et all., A Rational Approximation Based on Bernstein Polynomials for High Order Initial and Boundary Values Problems, Applied Mathematics and Computation, 217 (2011), 22, pp. 9438-945 Paper submitted: November 1, 2018 © 2019 Society of Thermal Engineers of Serbia Paper revised: December 28, 2018 Published by the Vinča Institute of Nuclear Sciences, Belgrade, Serbia. Paper accepted: January 10, 2019 This is an open access article distributed under the CC BY-NC-ND 4.0 terms and conditions