On k-type pseudo null Darboux helices in Minkowski

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On k-type pseudo null Darboux helices in Minkowski 3-space Emilija Nešović a,∗ , Ufuk Öztürk b , Esra Betül Koç Öztürk b a

Department of Mathematics and Informatics, Faculty of Science, University of Kragujevac, Radoja Domanovića 12, Kragujevac 34000, Serbia b Department of Mathematics, Faculty of Science, University of Çankırı Karatekin, Çankırı 18100, Turkey

a r t i c l e

i n f o

Article history: Received 25 December 2015 Available online xxxx Submitted by W. Sarlet Keywords: Serret–Frenet formulae Pseudo null curve Darboux vector Minkowski 3-space Curve of constant slope

a b s t r a c t In this paper, we introduce k-type pseudo null Darboux helices in Minkowski 3-space, for k ∈ {1, 2, 3}. We obtain the relationship between 1-type and 2-type pseudo null Darboux helices and show that all 3-type pseudo null Darboux helices are also 1-type and 2-type helices, but the converse does not hold. Finally, we give some examples. © 2016 Elsevier Inc. All rights reserved.

1. Introduction In science and nature, helix is one of the most fascinating curves (see for example, [3,9,15]). Also, the helix curve or helical structures can be found in fractal geometry, [11,13]. Moreover, helix is more convenient in use within most of computer aided geometric design systems, since a helix segment can be represented accurately with a combination of trigonometric functions and polynomials, [8,16]. In classical differential geometry, a curve in Euclidean 3-space is said to be a curve of constant slope (or a general helix) if its tangent vector field encloses constant angle with a fixed direction (called an axis of the curve) in each point. The ratio of the torsion τ and the curvature κ of such curve is constant, which is the necessary and sufficient condition for a curve to be a curve of constant slope, [4,12]. The above characterization of general helices in 3-dimensional space forms is generalized by Barros in [2]. The special curves of constant slope are circular helices, having both curvature functions κ and τ constant. Every curve of constant slope is a slant helix, a curve whose principal normal vector makes a constant angle with some fixed direction, [5]. Characterizations of the slant helices are studied in [1,6,7]. * Corresponding author. E-mail addresses: [email protected] (E. Nešović), [email protected], [email protected] (U. Öztürk), [email protected], [email protected] (E.B.K. Öztürk). http://dx.doi.org/10.1016/j.jmaa.2016.03.014 0022-247X/© 2016 Elsevier Inc. All rights reserved.

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Darboux helices in Euclidean space E3 are introduced in [17] as the curves whose Darboux vector D = τ T +κB makes constant angle with some fixed direction. In Minkowski 3-space, pseudo null Darboux helices are defined in [10] as the curves whose Darboux vector D and some fixed direction U satisfy the relation D, U  = constant. The relationship between pseudo null Darboux helices and pseudo null curves of constant precession is also obtained in [10]. If α is a general helix in Euclidean 3-space, its unit Darboux vector D0 is constant vector, so an angle between D0 and any fixed direction is trivially constant. The same property holds for timelike and spacelike generalized helices with non-null principal normal in Minkowski 3-space. In particular, if α is a slant helix in Euclidean 3-space, then its principal normal vector and Darboux vector make different constant angles with the same fixed direction [17]. Hence the mentioned classes of curves have only one fixed axis, making constant angle with their tangent, principal normal, or Darboux vector. In this paper, we show that pseudo null Darboux helix α with non-constant torsion in Minkowski 3-space has completely different property. Namely, its Darboux vector D is not constant vector, so there exists fixed direction V ∈ E31 satisfying the condition V, D = constant. Moreover, also there exist fixed directions U1 , U2 and U3 non-collinear with V , satisfying the conditions Uk , Wk  = constant, where Wk is some Frenet vector of α and k ∈ {1, 2, 3}. The mentioned property allows us to introduce k-type pseudo null Darboux helices in Minkowski 3-space. We obtain the relationship between 1-type and 2-type pseudo null Darboux helices and show that all 3-type pseudo null Darboux helices are also 1-type and 2-type helices, but the converse does not hold. Finally, we deduce that fixed directions U1 , U2 and U3 are not unique and give some examples. 2. Preliminaries The Minkowski 3-space E31 is the real vector space E3 equipped with the standard indefinite flat metric ·, · defined by x, y = −x1 y1 + x2 y2 + x3 y3 ,

(1)

for any two vectors x = (x1 , x2 , x3 ) and y = (y1 , y2 , y3 ) in E31 . Since ·, · is an indefinite metric, an arbitrary vector x ∈ E31 \ {0} can have one of three causal characters: it can be spacelike, timelike or null (lightlike), if x, x is positive, negative or zero, respectively.In particular, the vector x = 0 is a spacelike. The norm (length) of a vector x ∈ E31 is given by x = |x, x|. An arbitrary curve α : I → E31 can locally be spacelike, timelike or null (lightlike), if all of its velocity vectors α (s) satisfy α (s), α (s) > 0, α (s), α (s) < 0 or α (s), α (s) = 0 and α (s) = 0, respectively. A spacelike curve α : I → E31 is called a pseudo null curve, if its principal normal vector N (s) and its binormal vector B(s) are linearly independent null vectors. The Frenet formulae of a pseudo null curve α have the form [14] ⎡

⎤ ⎡ T 0 ⎢ ⎥ ⎢ = ⎣N ⎦ ⎣ 0 B −κ

⎤⎡ ⎤ κ 0 T ⎥⎢ ⎥ τ 0 ⎦⎣N ⎦, 0 −τ B

(2)

where the first curvature κ can take only two values: κ = 0 when α is a straight line, or κ = 1 in all other cases. The second curvature (torsion) τ (s) is an arbitrary function of the arclength parameter s of α. The Frenet frame vectors of α satisfy the equations T, T  = 1, N, N  = B, B = 0, T, N  = T, B = 0, N, B = 1,

(3)

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and T × N = N, N × B = T, B × T = B.

(4)

When the Frenet frame {T, N, B} of a non-geodesic pseudo null curve α makes an instantaneous helix motion in E31 , there exists an axis of the frame’s rotation. The direction of such axis is given by the vector D(s) = τ (s)T (s) − N (s),

(5)

which is called a Darboux vector (centrode). The Darboux vector D satisfies the Darboux equations T  (s) = D(s) × T (s) , N  (s) = D(s) × N (s) , B  (s) = D(s) × B (s) . The relations (2) and (5) imply D (s) = τ  (s)T (s) − τ (s)N (s) + τ (s)N (s) = τ  (s)T (s).

(6)

Therefore, if a pseudo null curve has the torsion τ (s) = constant, then it has non-constant Darboux vector, i.e. D(s) = constant. Recall that a pseudo null curve α is called a pseudo null Darboux helix [10], if there exists a fixed direction V = 0 in E31 such that D, V  = c,

c ∈ R.

(7)

The fixed direction V is called an axis of the pseudo null Darboux helix and it is given by V (s) = λe−



τ (s)ds

N (s),

λ ∈ R+ .

(8)

3. k-type pseudo null Darboux helices in E 31 In this section we introduce k-type pseudo null Darboux helices and give a classification of such curves in E31 . We consider only non-geodesic k-type pseudo null Darboux helices, having the first curvature κ = 1. Throughout this section let R0 denote R\{0}. If α is a pseudo null curve in E31 with Frenet frame {T, N, B}, let us set T (s) = W1 (s),

N (s) = W2 (s),

B(s) = W3 (s).

(9)

Definition 1. A pseudo null Darboux helix α with Frenet frame {W1 , W2 , W3 } and axis V in Minkowski 3-space E31 is called a k-type pseudo null Darboux helix for k ∈ {1, 2, 3}, if there exists a fixed direction Uk = 0 such that Wk , Uk  = ck ,

ck ∈ R,

(10)

whereby Uk and V are not collinear directions. We call the fixed directions V and Uk as the first axis and the second axis of the k-type pseudo null Darboux helix, respectively.

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If a pseudo-null curve α has constant torsion τ , according to relation (6) its Darboux vector D is a constant vector, so every fixed direction V in E31 trivially satisfies the condition D, V  = constant. To avoid this situation, throughout this section we will assume that α has non-constant torsion τ (s) = constant for each s. Theorem 1. Let α be a pseudo null curve in E31 with the torsion τ = constant. If α is a 1-type pseudo null Darboux helix in E31 whose second axis U1 satisfies T, U1  = a,

a ∈ R,

(11)

then U1 is a spacelike axis given by U = aT + bN,

(12)

where a = 0 and b = 0 has the form −



b (s) = e



 c1 − a

τ (s)ds



e

τ (s)ds

ds ,

c1 ∈ R.

(13)

Proof. Assume that α is a 1-type pseudo null Darboux helix, parameterized by the arc-length function s, whose second axis U1 satisfies T, U1  = a,

a ∈ R.

(14)

By using (14), the fixed direction U1 can be decomposed as U1 = aT + bN + cB,

(15)

where b = b(s) and c = c(s) are some differentiable functions of s. Differentiating the equation (15) with respect to s and using the Frenet equations (2), we obtain the system of differential equations ⎧ ⎪ ⎨

c = 0, b + τ b + a = 0, ⎪ ⎩ c − cτ = 0. 

(16)

The second equation of the relation (16) implies b (s) = e−



τ (s)ds

 c1 − a



e

τ (s)ds

ds ,

c1 ∈ R.

(17)

Substituting (17) in (15), we obtain that the second axis of α is a spacelike axis of the form U1 = aT + bN,

a ∈ R0 ,

(18)

where b is given by the relation (17). If a = 0, then axis U1 given by (18) and the axis V given by (8) are parallel, which is a contradiction. Hence a = 0, which completes the proof. 2 Remark 1. Note that the second axis U1 is not unique. Namely, depending on the choice of arbitrary constant c1 ∈ R in the relation (17), we obtain infinity many different axes U1 satisfying the condition T, U1  = a. All of them lie in a lightlike plane spanned by {T, N } and have the same tangential component.

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Corollary 1. There are no 1-type pseudo null Darboux helix in E31 with the torsion τ = constant whose second axis U1 satisfies T, U1  = 0.

(19)

Corollary 2. If α is 1-type pseudo null Darboux helices in E31 with torsion τ = constant, then its first axis V is a lightlike direction and its second axis U1 is a spacelike direction. If α is a 1-type pseudo null Darboux helix, then its second axis U1 satisfies the relation T, U1  = constant.

(20)

Differentiating the previous relation with respect to s we get N, U1  = 0.

(21)

This means that every 1-type pseudo null Darboux helix is a 2-type pseudo null Darboux helix having the same axis U1 . Now we can ask the following question: “Is every 2-type pseudo null Darboux helix 1-type pseudo null Darboux helix having the same axis U2 ?” The answer is given in the next theorem. Theorem 2. Every 2-type pseudo null Darboux helix in E31 with the torsion τ = constant is a 1-type pseudo null Darboux helix. Proof. Assume that α is a 2-type pseudo null Darboux helix parameterized by the arc-length function s. Then there exists a fixed direction U2 = 0 in E31 , such that N, U2  = a,

a ∈ R.

(22)

By using (22), the fixed direction U2 can be decomposed as U2 = u1 T + u2 N + aB,

(23)

where u1 = u1 (s) and u2 = u2 (s) are some differentiable functions of s. Differentiating the equation (23) with respect to s and using the Frenet equations (2), we obtain the system of differential equations ⎧ ⎪ ⎨ ⎪ ⎩

u2

u1 − a = 0, + u2 τ + u1 = 0, aτ = 0.

(24)

From the first and the third equation of (24) we get a = 0,

u1 = c = constant.

(25)

Substituting (25) in (23) we find that the second axis U2 reads U2 = cT + u2 N.

(26)

In particular, substituting u1 = c in the second equation of (24), we obtain the first order linear differential equation u2 + u2 τ + c = 0,

(27)

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whose general solution reads u2 (s) = e−



τ (s)ds

 c2 − c



e

τ (s)ds

ds ,

c2 ∈ R.

(28)

Therefore, U2 is a spacelike axis if c = 0, or a lightlike axis if c = 0. If c = 0, then axes U2 and V are collinear, which is a contradiction. Thus c = 0. It can be easily verified that T, U2  = c, which means that every 2-type pseudo null Darboux helix is a 1-type Darboux helix having the same axis U2 . 2 Remark 2. Note that the second axis U2 is not unique. Namely, depending on the choice of real constant c2 in the relation (28), we obtain infinity many different axes U2 satisfying the condition N, U2  = 0. All of them lie in a lightlike plane spanned by {T, N } and have the same tangential component. Corollary 3. There are no 2-type pseudo null Darboux helix in E31 with the torsion τ = constant whose second axis U2 satisfies T, U2  = 0.

(29)

Corollary 4. If α is 2-type pseudo null Darboux helices in E31 with torsion τ = constant, then its first axis V is a lightlike direction and its second axis U2 is a spacelike direction. Theorem 3. Let α be a pseudo null curve in E31 with the torsion τ = constant. Then α is a 3-type pseudo null Darboux helix if only if its torsion is given by τ (s) =

√

√ 2c1 tan

2c1 (s + c2 ) , 2

(30)

where c1 ∈ R+ 0 , c2 ∈ R. Proof. Assume that α is a 3-type pseudo null Darboux helix parameterized by the arc-length function s. Then its second axis U3 = 0 satisfies the relation B, U3  = a,

a ∈ R.

(31)

By using (31), the second axis U3 can be decomposed as U3 = u1 T + aN + u3 B,

(32)

where u1 = u1 (s) and u3 = u3 (s) are some differentiable functions of s. Differentiating the equation (32) with respect to s and using the Frenet equations (2), we obtain the system of differential equations ⎧  ⎪ ⎨ u1 − u3 = 0, u1 + aτ = 0, ⎪ ⎩ u − u τ = 0. 3 3

(33)

If a = 0, then U3 = 0, which is a contradiction. Hence a = 0. From the first and the second equation of (33) we get 

u1 = −aτ, u3 = −aτ  .

(34)

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Substituting (34) in the third equation of (33), we obtain the second order differential equation τ  − τ τ  = 0.

(35)

Consequently, the torsion of the pseudo null 3-type Darboux helix, i.e. the general solution of the last equation, is given by the relation (30). Conversely, assume that the torsion function τ is given by (30). Consider the vector U3 given by U3 = a (−τ T + N − τ  B) ,

(36)

where a ∈ R0 . Differentiating the previous equation with respect to s and using the Frenet equations (2), we find U3 = 0. Hence U3 is a fixed direction. It can be easily checked that B, U3  = a.

(37)

According to the Definition 1, the curve α is a 3-type pseudo null Darboux helix. 2 Corollary 5. Let α be a 3-type pseudo null Darboux helix in E31 with the torsion τ = constant. Then its first axis V is a lightlike direction given by (8) and its second axis U3 is a timelike direction given by U3 = a (−τ T + N − τ  B) ,

a = 0.

(38)

Corollary 6. There are 1-type and 2-type pseudo-null Darboux helices in Minkowski 3-space which are not 3-type pseudo null Darboux helices. Corollary 7. Every 3-type pseudo null Darboux helix is a 1-type and a 2-type pseudo-null Darboux helix, with respect to the different axes U1 , U2 and U3 . 4. Examples Example 1. Let us consider a pseudo null curve α (Fig. 1) in E31 with the parameter equation   3 3 s√ −s α(s) = s3 , s√+s , . 2 2

(39)

The Frenet frame of α reads   2 2 3s√ −1 T (s) = 3s2 , 3s√+1 , , 2 2   N (s) = 6s 1, √12 , √12 ,  2  2 2   4 3s +1 −2 3s2 −1 −2 √ √ , , B(s) = −1−9s . 12s 12 2s 12 2s

(40) (41) (42)

The curvature functions of α are given by κ(s) = 1,

τ (s) =

1 . s

(43)

The relations (5), (40), (41) and (43) imply that the Darboux vector of α reads   2 2 √ , − 1+3s √ D(s) = τ (s)T (s) − N (s) = −3s, 1−3s . s 2 s 2

(44)

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Fig. 1. The 1-type pseudo null Darboux helix α.

According to the Theorem 1, if α is a 1-type pseudo null Darboux helix, then its axes V and U1 are respectively given by V = λe−



τ (s)ds

λ ∈ R+ ,

N,

a ∈ R0 ,

U1 = aT + bN,

(45)

where b (s) = e−





τ (s)ds

 c1 − a



e

τ (s)ds

ds ,

c1 ∈ R.

(46)

By using the relations (40), (41), (43), (45) and (46), we get   V = λ 6, √62 , √62 ,

  1 −a √ 1 , 6c√ U1 = 6c1 , a+6c . 2 2

(47)

The relations (40), (44) and (47) imply D, V  = 0,

T, U1  = a.

(48)

It can be easily checked that N, U1  = 0, which means that α is also a 2-type pseudo null Darboux helix. Moreover, by choosing arbitrary constant c1 ∈ R in the relation (47), we obtain infinity many mutually non-collinear axes U1 , having the same tangential component T, U1  = a. All of them are not collinear with the first axis V . Example 2. Let us consider a pseudo null curve α (Fig. 2) in E31 with the parameter equation  α(s) =

s3 +3s2 , 9

√

2(s3 +3s2 +9s) , 18

√

2(s3 +3s2 −9s) 18

 .

(49)

The Frenet’s frame vector fields of α read  T (s) =

s2 +2s 3 ,

√

2(s2 +2s+3) , 6

√

2(s2 +2s−3) 6



,  s+1   √ √  2, 2, 2 , N (s) = 3   2 +2s)2 +9 9−(s2 +2s)2 −6(s2 +2s) 9−(s2 +2s)2 +6(s2 +2s) √ √ B(s) = − (s12(s+1) , , . 12 2(s+1) 12 2(s+1)

(50) (51) (52)

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Fig. 2. The 2-type pseudo null Darboux helix α.

The first and the second curvature of α have the form κ(s) = 1,

τ (s) =

1 . 1+s

(53)

From the relations (5), (50), (51) and (53) we obtain that the Darboux vector of α reads  2  √ √ 2 +2s−1) 2(s2 +2s+5) +2s+2 D(s) = − s3(s+1) , − 2(s , − . 6(s+1) 6(s+1)

(54)

According to the Theorem 2, if α is a 2-type pseudo null Darboux helix, then its axes V and U2 are respectively given by relations (8) and (26). Substituting (51) and (53) in (8), we obtain that first axis is given by V =

λ √ √ (2, 2, 2), 3

λ ∈ R+ .

(55)

Next, substituting (50), (51) and (53) in (26), it follows that the second axis has the form 2c1 , U2 = ( 3

√

2(3c + 2c1 ) , 6

√

2(2c1 − 3c) ), 6

c ∈ R0 ,

c1 ∈ R.

(56)

It can be easily verified that D, V  = 0 and U2 , N  = 0. Since T, U2  = c, α is also a 1-type pseudo null Darboux helix. In particular, by choosing arbitrary real constant c1 in the relation (56), we obtain infinity many mutually non-collinear axes U2 , having the same normal component N, U2  = c. All of them are not collinear with the first axis V . Example 3. Let α be a 3-type pseudo null Darboux helix, parameterized by the arc-length parameter s. According to the Theorem 3, the torsion of α is given by (30). From the Frenet equations (2), it follows that the principal normal vector N of α satisfies the first order linear differential equation N  = τ N.

(57)

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Fig. 3. A 3-type pseudo null Darboux helix α for c1 = 2 and c2 = 0.

By using (30), we find that the general solution of the previous differential equation reads N (s) = α (s) = A1 sec2

√

c √1 2

 (s + c2 ) ,

(58)

where A1 is a constant null vector, c1 ∈ R+ 0 and c2 ∈ R. Integrating the last relation, we obtain  √  c α (s) = A3 − A1 c21 log cos √21 (s + c2 ) + A2 s,

(59)

where A1 , A2 and A3 are the constant vectors in E31 . Taking A1 = (1, 1, 0), A2 = (0, 0, 1) and A3 = (0, 0, 0), we get that the curve α is given by (Fig. 3)   √   √   c c α (s) = − c21 log cos √21 (s + c2 ) , − c21 log cos √21 (s + c2 ) , s .

(60)

In particular, the Frenet frame vectors of α read √

 √ √ √ c c tan( √21 (s + c2 )), √c21 tan( √21 (s + c2 )), 1 ,     √ √ c c N (s) = sec2 √21 (s + c2 ) , sec2 √21 (s + c2 ) , 0 , ⎛  c  1  c  ⎞ 2 1 1 − c11 sin2 2 (s + c2 ) − 2 cos 2 (s + c2 ) , ⎜  1  ⎟  c  c 2 2 ⎜ 1 1 1 (s + c ) + cos (s + c ) ,⎟ 2 2 B (s) = ⎜ − c1 sin ⎟. 2 2 2  ⎝ ⎠     c1 c1 − c21 sin (s + c ) cos (s + c ) 2 2 2 2 √2 c1

T (s) =

(61) (62)

(63)

The relations (5), (30), (61) and (62) imply that the Darboux vector of α has the form √

c

√ c

D(s) = (tan2 ( √21 (s + c2 )) − 1, tan2 ( √21 (s + c2 )) − 1,

√

√

c

2c1 tan( √21 (s + c2 ))).

(64)

According to the Corollary 5, if α is a 3-type pseudo null Darboux helix, then its the first axis V and the second axis U3 of α are respectively given by V = λ(1, 1, 0),

λ ∈ R+ ,

(65)

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and + 1 2−c1 U3 = a (−τ T + N − τ  B) = a( 2+c 2 , 2 , 0), a ∈ R0 , c1 ∈ R0 .

(66)

From the relations (63)–(66) we find D, V  = 0 and B, U3  = a. By choosing positive real constant c1 in the relation (66), we obtain infinity many mutually non-collinear axes U3 , having the same binormal component B, U2  = a. All of them are not collinear with the first axis V . Finally, consider two fixed directions U1 = (b, b, c), c = 0 and U2 = (d, d, e), e = 0. It can be easily checked that T, U1  = c and N, U2  = 0, which means that the 3-type pseudo null Darboux helix α is also a 1-type and a 2-type pseudo null Darboux helix. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17]

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