Selçuk J. Appl. Math. Selçuk Journal of Vol. 11. No.1. pp. 15-25 , 2010 Applied Mathematics
Bishop Frame of The Spacelike Curve with a Spacelike Binormal in Minkowski 3-Space
Bahaddin Bukcu1, Murat Kemal Karacan2
1Gazi Osman Pasa University, Faculty of Sciences and Arts, Department of
Mathe-matics, Tokat, Türkiye e-mail:bbukcu@ yaho o.com
2Usak University, Faculty of Sciences and Arts, Department of Mathematics, 1 Eylul
Campus, 64200, Usak, Türkiye e-mail:murat.karacan@ usak.edu.tr
Received Date: July 31, 2008 Accepted Date: March 12, 2010
Abstract. In this study, we have generalized for a spacelike curve with a space-like binormal which is studied by Bishop [1] to Minkowski 3-Space. In addition, the Darboux vector (matrix) for spacelike curve with a spacelike binormal was found. Furthermore, using the derivative of the tangent vector T of the spacelike curve with a spacelike binormal, the relations between the curvature functions
; and k1; k2 was found.
Key words: Bishop Frame; Adapted Frame; Spacelike Curve; Minkowski 3-Space.
2000 Mathematics Subject Classi…cation: 83A05; 53B30; 51B20. 1. Preliminaries
Let R3= f(x
1; x2; x3)jx1; x2; x3 2 Rg be a 3-dimensional vector space, and let x = (x1; x2; x3) and y = (y1; y2; y3) be two vectors in R3. The Lorentz scalar product of x and y is de…ned by
hx; yiL= x1y1 x2y2+ x3y3: E3
1= R3; hx; yiL is called 3-dimensional Lorentzian space, Minkowski 3-Space or 3- dimensional Semi-Euclidean space. The vector x in E3
1is called a spacelike vector, null vector or a timelike vector if hx; xiL > 0 or x = 0, hx; xiL = 0 , hx; xiL < 0, respectively. For x 2 E31, the norm of the vector x de…ned by
kxkL= q
and x is called a unit vector if kxkL= 1. For any x; y 2 E31, Lorentzian vectoral product of x and y is de…ned by
x ^Ly = (x2y3 x3y2; x1y3 x3y1; x1y2 x2y1) :
The Lorentzian sphere of center m = (m1; m2; m3) and radius r 2 R+ in the Minkowski 3-space is de…ned by
S12= a = (a1; a2; a3) 2 E31 ha m; a miL= r2 :
Denote by fT; N; Bg the moving Frenet frame along the curve . Then T; N and B are the tangent, the principal normal and vector binormal of the curve respectively. If is a spacelike curve with a spacelike binormal, then this set of orthogonal unit vectors, known as the Frenet-Serret frame, has the following properties [2]:
T0 = N; N0 = T + B; B0= N; hT; T; iL = 1; hN; NiL= 1; hB; BiL= 1:
2. Introduction
The Frenet frame of a 3-times continuously di¤erentiable non-degenerate space-like curve with a spacespace-like binormal invariant in semi-euclidean space has long been the standard vehicle for analyzing properties of the spacelike curve with a spacelike binormal invariant under semi-euclidean motions. For arbitrary mov-ing frames that is, orthonormal basis …elds, we can express the derivatives of the frame with respect to the spacelike curve with a spacelike binormal parameter in term of the frame itself, and due to semi-ortonormality the coe¢ cient matrix is always semi- skew symmetric. Thus it generally has three nonzero entries. The Frenet frame gains part of its special signi…cance from the fact that one of the three derivatives is zero. Another feature of the Frenet frame is that it is adapted to the spacelike curve with a spacelike binormal : the members are either tangent to or perpendicular to the spacelike curve with a spacelike binormal . It is the purpose of this paper to show that there are other frames which have these same advantages and to compare them with the Frenet frame. 3. Parallel Fields
3.1. Relatively Parallel Fields
We say that a normal vector …eld N along a spacelike curve with a spacelike binormal is relatively parallel if its derivative tangential. Such a vector …eld turns only whatever amount is necessary for it to remain normal, so it is as close to being parallel as possible without losing normality. Since its derivative is perpendicular to it, a relatively parallel normal vector …eld has constant length. Such …elds occur classically in the discussion of curves which are said
to be parallel to given spacelike curve with a spacelike binormal. Indeed, if a spacelike curve with a spacelike binormal considered as a displacement vector function of a parameter t, then if N is relatively parallel, the spacelike curve with a spacelike binormal with displacement vector + N has velocity
( + N )0 = (v + f ) T;
where T is the unit tangential vector …eld of ; v is the speed of ;and N0 = f T: Thus the segment between two spacelike curves is locally perpendicular to both. Whether or not this segment is locally a segment of minimum length between the two spacelike curves depends on the curvature and the length of N: It is easily veri…ed that the segment local minimizes length if N is short enough. Conversely, a spacelike curve with a spacelike binormal which runs at constant distance from must be given + N; where N relatively parallel. A single normal vector …eld N0 at a point (t0) generates a unique relatively parallel …eld N such that
N (t0) = N0.
The uniqueness is trivial: the di¤erence of two relatively parallel …elds is obvi-ously relatively parallel, so if two such coincide at one point, their di¤erence has constant length 0. To show existence one takes auxiliary adapted frames; the Frenet frame would do if it exists, but we want existence even for degenerate spacelike curves with a spacelike binormal, that is, those which have curvature vanishing at some points. such frames can be constructed locally by applying the Gram-Schmidt process to T two parallel …elds.
Theorem 3.1. Let be a spacelike curve with a spacelike binormal with unit speed. If T; N1; N2is adapted frame, then we have
(3.1) 2 4 T 0 N10 N20 3 5 = 2 4 001 001 0212 02 12 0 3 5 2 4 NT1 N2 3 5 .
Proof: If T , N2 are spacelike vectors but N1timelike vector then we can write
(3.2) T0 = 00T + 01N1+ 02N2, N10 = 10T + 11N1+ 12N2, N20 = 20T + 21N1+ 21N2: 9 = ;
for some the functions 00, 01, 02, 10, 11, 12, 20, 21and 22. Then we have equalities below
hT; N1i = 0 D T0; N1 E | {z } 01 +DT; N10E | {z } 10 = 0 01 = 10;
hT; N2i = 0 D T0; N2 E | {z } 02 +DT; N20E | {z } 20 = 0 02+ 20 = 0; hN1; N2i = 0 D N10; N2 E | {z } 12 +DN1; N 0 2 E | {z } 21 = 0 12 = 21:
From the last equalities we get eq. (3.1). Moreover K is semi skew-matrix for satisfying KT = K , where is a diag(1; 1 1) matrix. Now we …nd the condition for a normal …eld of constant L to relatively parallel. There is a smooth function such that
N = L [N1cosh + N2sinh ] : Di¤erentiating, we have
N0= L 0 12 (N1sinh + N2cosh ) + ( 01cosh + 02sinh ) T : From this we see that N is relatively parallel if and only if
0= 12:
Since there is a solution for satisfying any initial condition, this shows that locally relatively parallel normal …elds exist. To get global existence we can patch to gether local ones, which exist on a covering by interval. Smoothness at the points where they link together is consequence of the uniqueness part. We de…ne a tangential …eld to be relatively parallel if it is a constant multiple of the unit tangent …eld T . An arbitrary …eld is relatively parallel if its tangential and normal components are relatively parallel. We spell out the complete hypotheses for the existence and uniqueness of these …elds as follows.
Theorem 3.2. Let be a Ck spacelike curve with a spacelike binormal in Minkowski 3-space which is regular; that is the velocity never vanishes (k 2). Then for any vector X0 at (t0) there is a unique Ck 1relatively parallel …eld X along such that X(t0) = X0 and the scalar product of two relatively is constant.
Proof: To prove that the scalar product hX; Y iLof two relatively parallel …elds X; Y is constant, we observe that it is trivial for tangential ones and maybe veri…ed for the tangential and normal parts separately. Thus we assume X and Y are normal, with derivatives f T and gT . Then the derivative of hX; Y iL is
d dthX; Y iL = D X0; YE L+ D X; Y0E L = hfT; Y iL+ hX; gT iL = f hT; Y iL+ g hX; T iL = f:0 + g:0 = 0
as desired. Thus, hX; Y iLis constant. 3.2. Special Adapted Frames
It should be clear that the relatively parallel …elds on C2 regular form a 3-dimensional vector over R with distinguished subspaces consisting of an oriented 1-dimensional tangential part and a 2-dimensional normal part, and there is a Lorentzian scalar product inherited from the pointwise scalar product on the ambient Semi-Euclidean. We call an Semi-Orthonormal basis of this vector space which …ts the two subspaces a relatively parallel adapted frame or RPAF. If we assume that the ambient Semi-Euclidean space has a preferred orientation, then so does the normal space of the spacelike curve with a spacelike binormal , and we may refer to properly oriented RPAF. The totality of RPAF’S are in the form of two circles(in the Lorentzian mean),one in each orientation class, since they can be parametrized by the 2-dimensional Semi-Orthogonal group, according to the following obvious result.
Theorem 3.3. If fT; N1; N2g is a relatively parallel adapted frame, then RPAF’s consists of frames the form fT; aN1+ bN2; cN1+ dN2g, where
a b c d
runs through semi-orthogonal matrices having entries.
Proof: Now if fT; N1; N2g is a RPAF, denoting derivatives with respect to arc length by a dot, we have,
(3.2) 2 4 T0 N10 N20 3 5 = 2 4 k01 k01 0k2 k2 0 0 3 5 2 4 NT1 N2 3 5 :
This shows that we accomplished our original goal showing that there are other adapted frames which have only two nonzero entries in their Cartan matrices .
In fact, given some one such RPAF, Theorem 3.3 tells us that possible Cartan matrices for RPAF’s are
K = 2 4 0 ak1+ bk0 2 ck1+ dk0 2 0 0 3 5
where denotes an entry which can be determined by using semi skew-symmetry. The Frenet frame has Semi-Cartan matrix
2
4 0 0 0
0 0
3 5 ;
and is unique once the orientation of the ambient space and a convention on the sign of torsion have been chosen. The only other possibilities for Cartan matrix with one entry vanishing would be
2 4 00 00 gf f g 0 3 5 ; 2 4 f0 f0 0g g 0 0 3 5 :
It is simple to relate the entries of the various Cartan matrices. Indeed, = T0
L= kk1N1+ k2N2kL= q
jk2 2 k21j : Writing the principal normal as
(3.3) N = N1cosh + N2sinh = k1
N1+ k2
N2; and di¤erentiating we obtain
N0 = T + B = T + 0(N1sinh + N2cosh ) :
fT; N1; N2g is properly oriented, we conclude that B = N1sinh + N2cosh and hence 0= : Thus and inde…ned integralR (s)ds are polar coordinates for the curve (k1; k2):
4. The Normal Development of The Spacelike Curve with a Spacelike Binormal
We want to view (k1; k2) as a sort of invariant of spacelike curve with a space-like binormal. This slightly more di¢ cult to conceive than in a case of ( ; ), since the RPAF is not unique. However, we have spelled out what degree of freedom there is Theorem 3.3. (k1; k2) is determined up to an Semi-Orthogonal transformation in the non oriented case and up to a semi-rotation about the ori-gin in the oriented case. Thus we must think of (k1; k2) as a parametrized (by an
arc- length for ) continuous spacelike curve in a Centro Semi-Euclidean plane, that is a Semi-Euclidean plane having distinguished point. When conceived of in this way we call (k1; k2) the normal development of spacelike curve with a spacelike binormal . The situation is not really so di¤erent from the case of the Frenet invariants ( ; ), because in the non oriented case ( ; ) and the Frenet frame are determined only up to an action by the two-element group, with the non identity changing the sign of and B. That is, cannot be distinguished from ( ; ): The standard facts about the relation (k1; k2) and spacelike curve with a spacelike binormal as an object of Semi-Euclidean geometry correspond to similar facts about (k1; k2) and : The proofs are identical with the Frenet case, and in fact are partly given in uni…ed form in [3] .
Theorem 4.1. Two C2 regularly spacelike curve with a spacelike binormal in Semi-Euclidean space are congruent if and only if they have the same normal development. For any parametrized continuous spacelike curve with a spacelike binormal in Centro Semi-Euclidean plane there is a C2 regular spacelike curve with a spacelike binormal in Semi - Euclidean space having the given spacelike curve with a spacelike binormal as its normal development, i.e., Two spacelike curves with a spacelike binormal are congruent if and only if they have the same arc-length parametrization of their curvature and torsion.
The modi…cations for the oriented case are clear: make both Semi-Euclidean space and the Centro Semi-Euclidean plane be oriented and congruencies be proper.
Theorem 4.2. Let be a spacelike curve with a spacelike binormal . A C2 regular spacelike curve if and only if its normal development lies on a line not the origin. The distance of this line from the origin and the radius of the Lorentzian Sphere or Semi-Sphere are reciprocals.
Proof: )If lies on the Lorentzian sphere of center p and radius r, then
(4.1) h p; piL= r2:
Di¤erentiating with respect to arc length gives
(4.2) hT; piL= 0;
so
(4.3) p = f N1+ g N2;
for some functions f; g. From equation (4:3), we get (4.4) f = h p; N1iL; g = h p; N2iL:
On deriving for equation (4:4), we have f0 = d dsh p; N1iL = hT; N1iL D p; N10E L = 0 h p; k1T iL = 0:
Thus f is constant. Similarly, g is constant, as well. Then di¤erentiating equa-tion (4:2), we get hT; piL = 0 D T0; pE L+ hT; T iL = 0 hk1N1 k2N2; piL+ 1 = 0 k1h p; N1iL | {z } f k2h p; N2iL | {z } g + 1 = 0 f k1+ gk2 = 1: That is, (k1; k2) is on the line
(4.5) f x + g y = 1:
Moreover, distance of line l from the origin is 1
g2 f2 = 1
r2 = d; g
2 f2> 0:
(: Vice versa, suppose that
f x + g y = 1; where f and g are constant. Let
! p = f N1+ g N2; then p0 = 0+ f N10+ gN20 p0 = T + f k1T + gk2T p0 = (f k1+ gk2 1) T p0 = 0;
so p is constant. Moreover, let
then
d
dsh p; piL= hT; piL = 0:
So h p; piL = r2 is constant. Thus spacelike curve with a spacelike binormal lies on the Lorentzian sphere of center p and radius r :
De…nition 4.1. If a rigid body moves along a spacelike curve with a spacelike binormal (which we suppose is unit speed), then the motion of body consists of translation along spacelike curve with a spacelike binormal and rotation about spacelike curve with a spacelike binormal . The rotation is determined by an angular velocity vector ! which satis…es
T0 = ! ^LT; N
0
1= ! ^LN1; N
0
2= ! ^LN2: The vector ! is called the Darboux vector.
Theorem 4.3. If T is tangent vector of a space curve with a spacelike binormal; then the following the formulas hold:
(a) T0^LT 00 = 2! + (k0 2k1 k2k10) T ; k16= 0 (b) det T; T0; T00 = k02k1 k2k10 (c) det T;T 0 ;T00 kT ^LT0k 2 L = 0 or
where ! is the Darboux vector of spacelike curve with a spacelike binormal . Proof: (a) First of all let us …nd Darboux vector !: Then we write
(4.6) ! = aT + bN1+ cN2
and take cross products with T; N1and N2 to determine a; b and c. So we get ! ^LT = c (N2^LT ) + b (N1^LT )
k1N1 k2N2 = cN1 bN2) c = k1 and b = k2: Thus we can write Darboux vector as follows,
! = k2N1 k1N2= (0; k2; k1)fT; N1; N2g: Since T0 = k1N1 k2N2, we get T00 = k10N1 k 0 2N2+ k1(k1T ) + k2(k2T ) = k21+ k22 T k 0 1N1+ k 0 2N2 = 2T k10N1+ k 0 2N2: Moreover one can easily …nd
From de…nition of Darboux vector !; we write T0 = ! ^LT: So we get T0^LT00 = T 00 ^L(! ^LT ) = f hT00; T iL! + hT00; !iLT g = 2! hT00; k2N1 k1N2iLT = 2! k2hT00; N1iLT + k1hT00; N2iLT = 2! k2k10T + k1k20T = 2! + (k02k1 k2k10) T: (b) From the last equality we get
hT; T0^LT00iL = T; 2! + (k20k1 k2k01) T L det(T; T0; T00) = 2hT; !iL+ (k02k1 k2k10) hT; T iL
= 2hT; k2N1 k1N2i + (k02k1 k2k10) (+1) = k20k1 k2k01:
(c) From the equality T0 = ! ^LT , we can easily write the vector (T ^LT0) the following as:
T ^LT0 = T ^L(! ^LT )
= hT; T iL! + hT; !iLT ; (T is spacelike vector.) = ! + 0:T
= !:
And then if we take the norm of the equality T ^LT0= !; we …nd kT ^LT0k2L= k22 k21:
From the equality (3.3) we immediately see that
(4.7) tanh = k2 k1 or = arg tanh(k2 k1 ): Di¤erentiating from the equality (4.7), we …nd
0 = ( k2 k1) 0 1 (k2 k1) 2 = k0 2k1 k10k2 k2 1 k2 1 k22 k2 1 = k1k02 k2k10 (k2 2 k21) :
Thus we have 0 = det(T; T0; T00) kT ^LT0k2L or : References
1. Bishop, L. R., ”There is more than one way to frame a curve”, Amer. Math. Monthly, Volume 82, Issue 3, 246-251, 1975.
2. Petrovic-Torgasev, M., Sucurovic, E., ”Some Characterizations of The Lorentzian Spherical Timelike and Null Curves ”, Matematiqki Vesnik, Vol:53, pp: 21-27, 2001. 3. O’Neill, B., ”Elemantary Di¤erential Geometry”, Academic Press, Newyork, 1966.
