Some associated curves of Frenet non-lightlike curves in E13

J. Math. Anal. Appl. 394 (2012) 712–723

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Some associated curves of Frenet non-lightlike curves in E31 Jin Ho Choi a , Young Ho Kim b , Ahmad T. Ali c,d,∗ a

Department of Mathematics, University of Ulsan, Ulsan 680-749, Republic of Korea

b

Department of Mathematics, Kyungpook National University, Taegu 702-701, Republic of Korea

c

King Abdul Aziz University, Faculty of Science, Department of Mathematics, PO Box 80203, Jeddah, 21589, Saudi Arabia

d

Mathematics Department, Faculty of Science, Al-Azhar University, Nasr City, 11884, Cairo, Egypt

article

info

Article history: Received 22 January 2012 Available online 27 April 2012 Submitted by H.R. Parks Keywords: Minkowski 3-space General helix Slant helix Associated curves

abstract In this paper, we introduce the notion of the principal (binormal)-directional curve and the principal (binormal)-donor curve of the Frenet curve in the Minkowski space E31 . Furthermore, we study some associated curves of a given Frenet curve to classify the general helices and the slant helices. Also, we introduce the natural representation of general helices and slant helices from the natural representation of plane curves and general helices, respectively. As an example, we give the natural representation of the closed slant helices constructed by circles. © 2012 Elsevier Inc. All rights reserved.

1. Introduction Lorentzian geometry helps to bridge the gab between modern differential geometry and mathematical physics of general relativity by giving an invariant treatment of Lorentzian geometry. The fact that relativity theory is expressed in terms of Lorentzian geometry is attractive for geometers who can penetrate surprisingly quickly into cosmology (redshift, expanding universe and big bang) and a topic no less interesting geometrically, the gravitation of a single star (perihelion procession, bending of light and black holes) [1]. Despite its long history, the theory of curves is still one of the most important interesting topics in differential geometry and it is being studied by many mathematicians until now; see for example [2–8]. Among the curves, a helix has been drawing attention of scientists as well as mathematicians because of its various applications, for example, explanation of DNA, carbon nano-tube, nano-springs, α -helices, etc.; see for example [9–14]. From the view of differential geometry, a helix is a geometric curve with a non-vanishing constant curvature and a non-vanishing constant torsion. Indeed a helix which is often called a circular helix and a W -curve is a special case of the general helix [15,16]. A general helix or a curve of constant slope is defined by the property that the tangent lines make a constant angle with a fixed direction. A necessary and sufficient condition for a curve to be general helix in the Euclidean space E3 or in the Minkowski space E31 is that the ratio of curvature to torsion be constant [17–19]. All helices (W -curves) in E31 are completely classified by Walrave in [20]. For instance, the only planar spacelike degenerate helices are circles and hyperbolas. In fact, a circular helix is the simplest three-dimensional spirals. One of the most interesting spiral examples is k-Fibonacci spirals. These curves appear naturally in the study of the k-Fibonacci numbers {Fk,n }∞ n=0 and the related hyperbolic k-Fibonacci function. Fibonacci numbers and the related Golden Mean or Golden Section appear very often in

∗

Corresponding author at: Mathematics Department, Faculty of Science, Al-Azhar University, Nasr City, 11884, Cairo, Egypt. E-mail addresses: [email protected] (J.H. Choi), [email protected] (Y.H. Kim), [email protected] (A.T. Ali).

0022-247X/$ – see front matter © 2012 Elsevier Inc. All rights reserved. doi:10.1016/j.jmaa.2012.04.063

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theoretical physics and physics of the high energy particles [21,22]. Three-dimensional k-Fibonacci spirals were studied from a geometric point of view in [23]. Ali and Lopez [24], in an analogous way as in E3 , studied the concept of a slant helix in Minkowski space E31 by saying that the principal normal lines make a constant angle with a fixed straight line. They characterize a slant helix if and only if either one of the next three functions

 ′  ′ τ τ κ2 κ2 or (τ 2 − κ 2 )3/2 κ (κ 2 ± τ 2 )3/2 κ is a constant. Also, they studied spherical images of tangent indicatrix and binormal indicatrix of a timelike slant helix and showed that the spherical images are spherical helices. In [25], Choi and Kim introduced the notions of the principal (binormal)-directional and the principal (binormal)-donor curves in E3 . By using this notion, they gave certain relation and the classifications of the general helices and slant helices. The aim of this paper, by the definition of W -directional and W -donor curves, to determine the position vector of a general helices from the plane curves. Also, we can determine the position vector of a slant helices from the general helices and so on. The outline of this work are as follows: In Section 2, we introduced the notion of the principal (binormal)-directional curve and principal (binormal)-donor curve of Frenet curve in Minkowski space E31 . The relationships between the curvatures of a Frenet curve in E31 and the curvatures of its principal-direction (principal-donor, respectively) according to the causal character of the Frenet frame are introduced in Section 3. In Section 4, we studied the characterization of general helices of its principal-donor curves. The main result in this section is summarized in Theorem 4.10, which stages that the principaldonor of general helices are plane curves. Characterizations of slant helices in terms of its principal-donor curves are studied in Section 5. The main result in this section is summarized in Theorem 5.7, which stages that the second principal-donor of slant helices are plane curves or in other words the principal-donor of slant helices are general helices. Finally, as an application, we give some examples of closed slant helices in E31 . 2. Preliminaries and definitions The Minkowski three-dimensional space E31 is the real vector space R3 endowed with the standard flat Lorentzian metric given by: g = −dx21 + dx22 + dx23 , where (x1 , x2 , x3 ) is a rectangular coordinate system of E31 . If u = (u1 , u2 , u3 ) and v = (v1 , v2 , v3 ) are arbitrary vectors in E31 , we define the (Lorentzian) vector product of u and v as the following:

 −i  u × v = − u1 v 1

j u2

v2



k   u3  .

v3 

Since g is an indefinite metric, recall that a vector v ∈ E31 can have one of three Lorentzian characters: it can be spacelike if g (v, v) > 0 or v = 0, timelike if g (v, v) < 0 and null if g (v, v) = 0 and v ̸= 0. Similarly, an arbitrary curve γ = γ (s) in E31 can locally be spacelike, timelike or null (lightlike) if all of its velocity vectors γ ′ (s) are respectively √ spacelike, timelike or null (lightlike) for every s ∈ I ⊂ R. The pseudo-norm of an arbitrary vector a ∈ E31 is given by ∥a∥ = |g (a, a)|. γ is called an unit speed curve if velocity vector v of γ satisfies ∥v∥ = 1. For vectors v, w ∈ E31 it is said to be orthogonal if and only if g (v, w) = 0. Minkowski space is originally from the relativity in physics. In fact, a timelike curve corresponds to the path of an observer moving at less than the speed of light, a null curve correspond to the moving at a speed of light and a spacelike curves to moving faster than light [26]. The importance of the study of null curve and its presence in the physical theories is clear from the fact that: the classical relativistic string is a surface or world-sheet in Minkowski which satisfies the Lorentzian analogue of minimal surface equation [27]. The string equations simplify to the wave equation and a couple of extra simple equations (by solving the 2-dimensional wave equation it turns out that string are equivalent to pairs of null curves or a single null curve in the case of an open string) [28]. Denote by {T, N, B} the moving Frenet frame along the curve γ in the space E31 . Then T, N and B are the tangent, the principal normal and the binormal vector of the curve γ . For an arbitrary curve γ (s) in the space E31 , the following Frenet formulas are given in [1,6,15,16,20,29,30]. If γ be a unit speed non-lightlike curve, then the Frenet formulas are as follows:

 ′

T N′  = B′



0

ϵB κ 0

κ 0

ϵT τ

0

τ

0

T N , B

 

(2.1)

where ϵX = g (X, X), κ is the curvature function and τ is the torsion function. A curve with given curvature and torsion functions is unique up to isometry of E31 . Let γ be a spacelike Frenet curve in E31 , i.e., ϵT = 1. Then, the principal normal vector field N along the curve γ is either spacelike or timelike. Then we have the following two cases:

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(1) If γ is a spacelike Frenet curve with a spacelike principal normal, then the Frenet formulas of γ are given by: T′ = κ N,

N′ = −κ T + τ B,

B′ = τ N.

(2.2)

B × T = −N.

(2.3)

For this trihedron, we write T × N = B,

N × B = −T,

(2) If γ is a spacelike Frenet curve with a timelike principal normal, then the Frenet formulas of γ are given by: T′ = κ N,

N′ = κ T + τ B,

B′ = τ N.

(2.4)

For this trihedron, we write T × N = −B,

N × B = −T,

B × T = N.

(2.5)

A spacelike Frenet curve γ satisfying (2.2) or (2.4) is said to be of type 1 or type 2, respectively. In other words, a spacelike Frenet curve in E31 is of type 1 or of type 2 if g (N, N) = 1 or g (N, N) = −1, respectively. Let γ be a timelike Frenet curve in E31 , i.e., ϵT = −1. In this case, the principal normal vector field N along the curve γ is a spacelike. Then we have: (3) If γ be a timelike Frenet curve, then the Frenet formulas of γ are given by: T′ = κ N,

N′ = κ T + τ B,

B′ = −τ N.

(2.6)

B × T = −N.

(2.7)

For this trihedron, we write T × N = −B,

N × B = T,

In [25], Choi and Kim introduced the definition of W -direction and W -donor in Euclidean space E3 . In the following, we will define this concepts in Minkowski space E31 . Definition 2.1. Let γ : I → E31 be a unit speed Frenet curve in Minkowski space E31 with the Frenet frame {T, N, B} and W a unit vector field along γ . The curve γ¯ : I → E31 is called a W -directional curve of γ if the tangent T¯ of the curve γ¯ is equal to W , i.e., T¯ = W . The curve γ whose W -directional curve is γ¯ is called the W -donor curve of γ¯ . Remark 2.2. For a given curve in E31 , its W -directional and W -donor curves are unique up to the translations (for the details, see [25]). Remark 2.3. If W = T, then the T-directional curve γ¯ of the curve γ is trivially γ . Remark 2.4. If W = N, Then T¯ = N. In this case, the N-directional curve γ¯ of γ is called the principal-directional curve of γ and the curve γ is called the principal-donor curve of γ¯ . When a curve γ¯¯ is the principal-directional curve of γ¯ , we call γ¯¯ the second principal-directional curve of γ and γ the second principal-donor curve of γ¯¯ . Definition 2.5. If W = B, Then T¯ = B. In this case, the B-directional curve γ¯ of the curve γ is called the binormal-directional of γ and the curve γ is called the binormal-donor curve of γ¯ . 3. Principal-directional and principal-donor curves of non-lightlike Frenet curves In this section, we study the principal-directional and the principal-donor curves of a non-lightlike Frenet curves in Minkowski space E31 . Firstly, we consider the principal-directional curve of a spacelike Frenet curve of type 1, spacelike Frenet curve of type 2 and timelike Frenet curve. Before this and from Lemmas 3.2, 3.3 and 3.4 in Ali [2] and from Lemmas 3.1 and 3.2 in Ali and Melih [31], we can write the following important lemma: Lemma 3.1. There is no timelike general helix or spacelike general helix of type 1 in Minkowski 3-space with condition | κτ | = 1. Then, we have the following three theorems: Theorem 3.2. Let γ be a spacelike Frenet curve of type 1 in E31 with the curvature function κ and the torsion function τ and γ¯ the principal-directional curve of γ . (a) If |κ| > |τ |, γ¯ is a spacelike curve of type 1 with curvature function κ¯ and torsion function τ¯ given by

κ¯ =



κ 2 − τ 2,

τ¯ =

κ 2  τ ′ . κ − τ2 κ

(3.1)

κ 2  τ ′ . τ 2 − κ2 κ

(3.2)

2

(b) If |κ| < |τ |, γ¯ is a spacelike curve of type 2 with curvature function κ¯ and torsion function τ¯ given by

κ¯ =



τ 2 − κ 2,

τ¯ =

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Proof. In case (a), the signature of the Frenet frame along γ is given by {+, +, −}. This means that the tangent vector field T and the principal normal vector field N are spacelike while the binormal vector field B is timelike. By the definition of the principal-directional curve of γ , we have γ¯ ′ (s) = T¯ = N and T¯ ′ = N′ = −κ T + τ B. From the assumption |κ| > |τ |, we ¯ have g (T¯ ′ , T¯ ′ ) = κ 2 − τ 2 > 0. Hence, the principal normal √ vector field N along γ¯ is spacelike, that is, γ¯ is a spacelike curve 2 2 of type 1. Moreover, the curvature of γ¯ is given by κ¯ = κ − τ . Then, the Frenet vectors along γ¯ are given by

−κ T + τ B

T¯ = N,

−τ T + κ B

. κ −τ κ2 − τ 2     ¯ ′ , N¯ ), we get τ¯ = κτ ′ − τ κ ′ / κ 2 − τ 2 = Since τ¯ = g (B ¯ = √ N

2

2

,

¯= √ B

The proof of the case (b) is similar to that of case (a).

(3.3) κ2 κ 2 −τ 2

 τ ′ κ

.



The similar works lead Theorems 3.3 and 3.4. Theorem 3.3. Let γ be a spacelike Frenet curve of type 2 in E31 with curvature function κ and torsion function τ and γ¯ the principal-directional curve of γ . Then, γ¯ is a timelike curve with curvature function and torsion function are

κ¯ =



κ 2 + τ 2,

τ¯ =

κ 2  τ ′ . τ + κ2 κ

(3.4)

2

Theorem 3.4. Let γ be a timelike Frenet curve in E31 with curvature function κ , torsion function τ and γ¯ the principal-directional curve of γ . (a) If |τ | > |κ|, γ¯ is a spacelike of type 1 and the curvature function and torsion function of the curve γ¯ are given by:

κ¯ =



τ 2 − κ2

τ¯ =

κ 2  τ ′ . τ − κ2 κ

(3.5)

2

(b) If |τ | < |κ|, γ¯ is a spacelike of type 2 and the curvature function and torsion function of the curve γ¯ are given by:

κ¯ =



κ 2 − τ 2,

τ¯ =

κ 2  τ ′ . κ2 − τ 2 κ

(3.6)

Secondly, we consider the principal-donor curve of a spacelike Frenet curve of type 1, spacelike Frenet curve of type 2 and timelike Frenet curve. Then, we have the following three theorems: Theorem 3.5. Let γ be a spacelike Frenet curve of type 1 in E31 and a principal-donor curve of a spacelike curve γ¯ with curvature κ( ¯ s) and torsion τ¯ (s). (a) If the spacelike γ¯ is of type 1, then the curvature and the torsion of the curve γ are given by     τ¯ (s)ds . (3.7) τ¯ (s)ds , τ (s) = κ( ¯ s) sinh κ(s) = κ( ¯ s) cosh (b) If the spacelike γ¯ is of type 2, then the curvature and the torsion of the curve γ are given by

κ(s) = κ( ¯ s) sinh



 τ¯ (s)ds ,



τ (s) = κ( ¯ s) cosh

 τ¯ (s)ds .

(3.8)

Proof. For the proof of (a) and (b), we must solve the differential equations of (3.1) and (3.2) with respect to κ(s) and τ (s), τ (s) respectively. Since these two equations are very similar, it suffice to provide the proof of (a). Firstly, we put the ratio κ(s) by f (s). Then, the second equation of (3.1) is rewritten as

τ¯ (s) =

f (s)′

(3.9)

1 − f (s)2

and f (s)2 is less than 1. This implies

τ (s) = κ(s) tanh





τ¯ (s)ds = tanh−1 [f (s)]. Hence, we have

 τ¯ (s)ds .

(3.10)

Substituting (3.10) into the first equation of (3.1), we get the curvature function κ(s) as

κ(s) = κ( ¯ s) cosh



 τ¯ (s)ds .

(3.11)

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J.H. Choi et al. / J. Math. Anal. Appl. 394 (2012) 712–723 Table 1 The Lorentzian character and type of the principal-directional curve.

γ

γ¯ (principal-direction)

Necessary condition

Note

Spacelike of type 1

Spacelike of type 1 Spacelike of type 2

|κ| > |τ | |κ| < |τ |

Theorem 3.2 (a) Theorem 3.2 (b)

Spacelike of type 2

Timelike

None

Theorem 3.3

Timelike

Spacelike of type 1 Spacelike of type 2

|κ| < |τ | |κ| > |τ |

Theorem 3.4 (a) Theorem 3.4 (b)

Table 2 The Lorentzian character and type of the principal-donor curve.

γ¯ Spacelike of type 1

γ (principal-donor)

How to construct

Spacelike of type 1

T = − cosh

   τ¯ ds N¯ + sinh τ¯ ds B¯    T = − sinh τ¯ ds N¯ + cosh τ¯ ds B¯     T = sinh τ¯ ds N¯ + cosh τ¯ ds B¯     T = cosh τ¯ ds N¯ + sinh τ¯ ds B¯     T = cos τ¯ ds N¯ + sin τ¯ ds B¯

Timelike Spacelike of type 1

Spacelike of type 2

Timelike Timelike





Spacelike of type 2

So that the torsion function τ (s) is obtained as

τ (s) = κ( ¯ s) sinh



 τ¯ (s)ds .

Thus, the proof is completed.

(3.12)



Similarly, we have Theorems 3.6 and 3.7. Theorem 3.6. Let γ be a spacelike curve of type 2 in E31 and a timelike principal-donor curve γ¯ with the curvature function κ¯ and torsion function τ¯ . Then, the curvature function and torsion function of the curve γ are given by

κ(s) = κ( ¯ s) cos



 τ¯ (s)ds ,

τ (s) = κ( ¯ s) sin



 τ¯ (s)ds .

(3.13)

Theorem 3.7. Assume that a timelike curve γ (s) in E31 is a principal-donor curve of a spacelike curve with curvature κ( ¯ s) and torsion τ¯ (s). (a) If γ¯ is of type 1, then the curvature and torsion of the curve γ are given by:

κ(s) = κ( ¯ s) sinh



 τ¯ (s)ds ,

τ (s) = κ( ¯ s) cosh



 τ¯ (s)ds .

(3.14)

(b) If γ¯ is of type 2, then the curvature and torsion of the curve γ are given by:

κ(s) = κ( ¯ s) cosh





τ¯ (s)ds ,

τ (s) = κ( ¯ s) sinh





τ¯ (s)ds .

(3.15)

Lastly, we summarize the relations of principal-directional and principal-donor curves in E31 . Remark 3.8. Let γ be a Frenet curve in E31 with curvature function κ and torsion function τ and γ¯ a principal-directional curve of γ . Then, the Lorentzian character and the type of γ¯ is determined by the simple inequality of |κ| and |τ | together with the Lorentzian character and the type of γ . From the Theorems 3.2–3.4, we have Table 1. Remark 3.9. Let γ¯ be a Frenet curve in E31 with curvature function κ¯ and torsion function τ¯ and γ a principal-donor curve of γ¯ . Then, the Lorentzian character and the type of γ is determined by constructing γ by γ¯ together with the Lorentzian ¯ , B¯ } and T by the Frenet frame along γ¯ and the unit tangent vector field along γ , character and the type of γ . Denote {T¯ , N respectively. From the Theorems 3.2–3.7, we have Table 2. In next two sections, we will construct the general helices and the slant helices in E31 . Last two Remarks 3.8 and 3.9 are very useful to construct them.

J.H. Choi et al. / J. Math. Anal. Appl. 394 (2012) 712–723

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4. Principal-donor curves of non-lightlike general helices In this section, we study the principal-donor curves of a non-light general helices in Minkowski space E31 . The following lemma gives a motivation of this section. Lemma 4.1. An non-lightlike Frenet curve γ in E31 is a general helix if and only if its principal-directional curve γ¯ is a plane curve. Proof. ‘‘if’’ part is followed by Theorems 3.2–3.4 and ‘‘only if’’ part is followed by Theorems 3.5–3.7.



To study the general helices, firstly, we state the following important two lemmas (see Section 4 in [2,31]). Lemma 4.2. The position vector γ of a spacelike plane curve in E31 with the curvature function κ(s) of type 1 and of type 2 are given by up to isometry:

γ (s) =

 

0, cos



   κ(s)ds , sin κ(s)ds ds,

(4.1)

and

γ (s) =

 

 sinh



κ(s)ds , cosh







κ(s)ds , 0 ds,

(4.2)

respectively. Lemma 4.3. The position vector γ of a timelike plane curve in E31 with the curvature function κ(s) is given by up to isometry:

γ (s) =

 

 cosh



κ(s)ds , sinh







κ(s)ds , 0 ds.

(4.3)

Note. The planar spacelike degenerate helices are circles (type 1) and hyperbolas (type 2) while the planar timelike degenerate helices are hyperbolas only. Above theorems for the principal-directional curve of a given curve in E31 give us various applications. Especially, it can be used to construct the general helices. Secondly, from Theorems 3.2–3.4, we can proved the following important three lemmas, respectively. Lemma 4.4. Let γ be a spacelike general helix with intrinsic equations (κ = κ(s) and τ = mκ(s)) in E31 of type 1. (a) If  κτ  = |m| < 1, then γ can be locally expressed by

 

γ (s) = √

 

1

1 − m2



m, sin

 1 − m2

    κ(s)ds , − cos 1 − m2 κ(s)ds ds

(4.4)

2 and  τits  principal-directional curve γ¯ (s) is a spacelike plane curve of type 1 in E .   (b) If κ = |m| > 1, then γ can be locally expressed by

γ (s) = √

−1

m2 − 1

 



cosh

 m2 − 1

     κ(s)ds , sinh m2 − 1 κ(s)ds , m ds

(4.5)

and its principal-directional curve γ¯ (s) is a spacelike plane curve of type 2 in E21 . Proof. In case (a), the position vector γ¯ (s) of a spacelike plane curve of type 1 with curvature function κ¯ = κ( ¯ s) in E2 ⊂ E31 is given by (4.1). Then, the curve γ¯ has the Frenet vectors as the following:

       ¯ (s) = 0, cos T κ( ¯ s ) ds , sin κ( ¯ s ) ds ,          N¯ (s) = 0, − sin      B¯ (s) = (1, 0, 0).

κ( ¯ s)ds , cos

κ( ¯ s)ds

,

From (3.1), we know that κτ is constant, say m, 0 < |m| < 1. Thus, the intrinsic equations of the curve γ are

κ(s) = √

κ( ¯ s) 1 − m2

,

τ (s) = mκ(s).

(4.6)

718

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√

If we put κ( ¯ s) =

1 − m2 κ(s) and solve the Eq. (3.3), we get the tangent vector field T as follows:

−N¯ + m B¯ T = √ 1 − m2 = √



1 1 − m2

m, sin



 1 − m2

    κ(s)ds , − cos 1 − m2 κ(s)ds .

(4.7)

Thus, the spacelike general helix γ of type 1 in E31 is given by (4.4). The proof of the case (b) is similar to that of the case (a).  Lemma 4.5. A spacelike general helix γ in E31 of type 2 with intrinsic equations (κ = κ(s) and τ = mκ(s)) is locally expressed by

γ (s) = √

 

1

1 + m2

sinh





1 + m2

     κ(s)ds , cosh 1 + m2 κ(s)ds , m ds

(4.8)

and its principal-directional curve γ¯ is a timelike plane curve in E21 . Lemma 4.6. Let γ be a timelike general helix with intrinsic equations (κ = κ(s) and τ = mκ(s)) in E31 . (a) If  κτ  = |m| > 1, then γ can be locally expressed by

 

γ (s) = √

1

 

m2 − 1

m, cos



 m2 − 1

    κ(s)ds , − sin m2 − 1 κ(s)ds ds

(4.9)

2 and  τits  principal-directional curve γ¯ (s) is a spacelike plane curve of type 1 in E . (b) If  κ  = |m| < 1, then γ can be locally expressed by

γ (s) = √

−1

1 − m2

  cosh



 1−

m2



κ(s)ds , sinh



1−

 m2





κ(s)ds , m ds

(4.10)

and its principal-directional curve γ¯ (s) is a spacelike plane curve of type 2 in E21 . Each proof of Lemmas 4.5 and 4.6 is similar to one of Lemma 4.4. Now, from the above three lemmas, we can write the following three corollaries: Corollary 4.7. Let γ be a spacelike Frenet curve of type 1 in E31 and γ¯ the spacelike principal-directional curve of γ . Then, the principal-directional curve γ¯ is a plane curve in E2 or E21 if and only if γ is a general helix with |κ| > |τ | or |κ| < |τ |, respectively. Moreover, γ¯ is a circle in E2 or a spacelike hyperbola in E21 if and only if γ is a helix with |κ| > |τ | or a helix with |κ| < |τ |, respectively. Corollary 4.8. Let γ be a timelike Frenet curve in E31 and γ¯ the spacelike principal-directional curve of γ . Then, the principaldirectional curve γ¯ is a plane curve if and only if γ is a general helix. Moreover, γ¯ is a hyperbola if and only if γ is a helix. Corollary 4.9. Let γ be a timelike Frenet curve in E31 and γ¯ the spacelike principal-directional curve of γ . Then, the principaldirectional curve γ¯ is a plane curve in E2 or E21 if and only if γ is a general helix with |κ| < |τ | or |κ| > |τ |, respectively. Moreover, γ¯ is a circle in E2 or a spacelike hyperbola in E21 if and only if γ is a helix with |κ| < |τ | or a helix with |κ| > |τ |, respectively. Consequently, we can characterize shortly the non-lightlike general helices in E31 in terms of the associated curve. Theorem 4.10. A non-lightlike general helix in E31 is the principal-directional curve of a plane curve in some planes. 5. Principal-donor curves of non-lightlike slant helices In this section, we construct the slant helices in Minkowski space E31 by using the notions of principal-directional and principal-donor curves. In fact, the notions of the principal-directional and principal-donor curves are very useful to study the relationship between the general helices and the slant helices in E31 . The following remarks explain why.

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719

Remark 5.1. Let γ = γ (s) be a curve in E31 and W = W (s) a unit vector field along the curve γ . If W has constant angle with a constant vector V of E31 along γ , then the tangent vector of the W -directional curve γ¯ of γ has also constant angle with the vector V along γ¯ . Conversely, for a curve γ¯ in E31 , if its tangent has constant angle with a constant vector V of E31 , then its W -donor curve γ is a curve in E31 satisfying the condition that W has constant angle with the vector V along γ¯ . Remark 5.2. In Remark 5.1, we take the principal normal vector field N instead of the vector field W along γ . Then, γ is a slant helix in E31 , that is, the vector field N has constant angle with a constant vector V in E31 if and only if its principaldirectional curve is a general helix in E31 , that is, its tangent has constant angle with the constant vector V of E31 . In the other word, a slant helix in E31 is a principal-donor curve of a general helix in E31 and a general helix in E31 is a principal-directional curve of a slant helix in E31 . Let γ¯ be a spacelike general helix in E31 with κτ¯¯ = m. Then, its spacelike principal-donor curve γ1 has the curvature       function κ1 = κ¯ cosh m κ( ¯ s)ds and the torsion function τ1 = κ¯ sinh m κ( ¯ s)ds and its timelike principal-donor curve

      γ2 has the curvature function κ2 = κ¯ sinh m κ( ¯ s)ds and the torsion function τ2 = κ¯ cosh m κ( ¯ s)ds . The curves γ1 and γ2 satisfy the equations of spacelike and timelike slant helices in E31 , respectively. In fact,      ′  ′ cosh2 m κ¯ ds κ12 τ1 = tanh m κ ¯ ds = m. (5.1) κ¯ (κ12 − τ12 )3/2 κ1 Similarly, γ2 can be checked. The principal-donor curves of non-lightlike general helices of other type satisfy the equations of slant helices in E31 . This is another proof of Remark 5.2. We call the value of equation of a slant helix by the slant helix constant. Thus, we have the following. Proposition 5.3. Let γ¯ be an non-lightlike general helix in E31 with the curvature function κ¯ and the torsion function τ¯ and γ its principal-donor curve. Then, γ is a non-lightlike slant helix with the slant helix constant κτ¯¯ . In the previous section, we constructed the general helices in E31 by the plane curves. Remark 5.2 gives an idea constructing the slant helices in E31 by the general helices. Using the method of the previous section, we can construct the slant helices in E31 from the general helices. Theorem 5.4. Let γ be a spacelike slant helix with the curvature function κ and the torsion function τ in E31 of type 1 and m a slant helix constant of γ . (a) If |κ| > |τ | and |m| < 1, γ can be locally expressed by

γ (s) = −

 

sinh [m K1 (s)]

− √

1 − m2

,

√   m sinh [m K1 (s)] sin 1 − m2 K1 (s) , × cosh [m K1 (s)] cos 1 − m2 K1 (s) − √ 1 − m2  √   m sinh [m K1 (s)] cos 1 − m2 K1 (s) × cosh [m K1 (s)] sin 1 − m2 K1 (s) + ds, √ 1 − m2 √ where K1 (s) = κ 2 − τ 2 ds. (b) If |κ| > |τ | and |m| > 1, γ can be locally expressed by √      m sinh [m K1 (s)] cosh m2 − 1K1 (s) γ (s) = cosh [m K1 (s)] sinh m2 − 1K1 (s) − , √ m2 − 1 √    m sinh [m K1 (s)] sinh m2 − 1K1 (s) , × cosh [m K1 (s)] cosh m2 − 1K1 (s) − √ m2 − 1  sinh [m K1 (s)] ×− √ ds, m2 − 1 √ where K1 (s) = κ 2 − τ 2 ds. 

(5.2)

(5.3)

720

J.H. Choi et al. / J. Math. Anal. Appl. 394 (2012) 712–723

(c) If |κ| < |τ |, γ can be locally expressed by

√   m cosh [m K2 (s)] sinh 1 + m2 K2 (s) γ (s) = sinh [m K2 (s)] cosh 1 + m2 K2 (s) − , √ 1 + m2 √    m cosh [m K2 (s)] cosh 1 + m2 K2 (s) × sinh [m K2 (s)] sinh 1 + m2 K2 (s) − , √ 1 + m2  cosh [m K2 (s)] ds, × √ 1 + m2 √ where K2 (s) = τ 2 − κ 2 ds.  



(5.4)

Theorem 5.5. Let γ be a spacelike slant helix with the curvature function κ and the torsion function τ in E31 of type 2 and m a slant helix constant of γ . (a) If |m| > 1, γ can be locally expressed by

γ (s) =

 

sin [m K3 (s)] m sin [m K3 (s)] cos

√

m2 − 1

,

√



m2 − 1K3 (s)

√

m2 − 1

− cos [m K3 (s)] sin + cos [m K3 (s)] cos



 m sin [m K3 (s)] sin m2 − 1K3 (s) , √

√



m2 − 1K3 (s)

m2 − 1





m2 − 1K3 (s)

 ds,

(5.5)

√

where K3 (s) = κ 2 + τ 2 ds. (b) If |m| < 1, γ can be locally expressed by

√   m sin [m K3 (s)] cosh 1 − m2 K3 (s) γ (s) = − cos [m K3 (s)] sinh 1 − m2 K3 (s) + , √ 1 − m2 √    m sin [m K3 (s)] sinh 1 − m2 K3 (s) × cos [m K3 (s)] cosh 1 − m2 K3 (s) + , √ 1 − m2  sin [m K3 (s)] ds, × √ 1 − m2 √ κ 2 + τ 2 ds. where K3 (s) =  



(5.6)

Theorem 5.6. Let γ be a timelike slant helix with the curvature function κ and the torsion function τ in E31 and m a slant helix constant of γ . (a) If |κ| < |τ | and |m| < 1, γ can be locally expressed by

γ (s) = −

  −

cosh [m K2 (s)]

√

1 − m2

m cosh [m K2 (s)] sin

−

, sinh [m K2 (s)] cos

√

1 − m2 K2 (s)

, sinh [m K2 (s)] sin

1 − m2

+ where K2 (s) =

√ τ 2 − κ 2 ds.

√

√

1 − m2



1 − m2 K2 (s)



√

m cosh [m K2 (s)] cos







1 − m2 K2 (s)



1 − m2 K2 (s)

ds,

(5.7)

J.H. Choi et al. / J. Math. Anal. Appl. 394 (2012) 712–723

721

(b) If |κ| < |τ | and |m| > 1, γ can be locally expressed by

√   m cosh [m K2 (s)] cosh m2 − 1K2 (s) γ (s) = m2 − 1K2 (s) − , sinh [m K2 (s)] sinh √ m2 − 1 √    m cosh [m K2 (s)] sinh m2 − 1K2 (s) × sinh [m K2 (s)] cosh m2 − 1K2 (s) − , √ m2 − 1  cosh [m K2 (s)] ×− √ ds, m2 − 1 √ where K2 (s) = τ 2 − κ 2 ds. (c) If |κ| < |τ |, γ can be locally expressed by  √     m sinh [m K1 (s)] sinh 1 + m2 K1 (s) γ (s) = cosh [m K1 (s)] cosh , 1 + m2 K1 (s) − √ 1 + m2 √    m sinh [m K1 (s)] cosh 1 + m2 K1 (s) × cosh [m K1 (s)] sinh 1 + m2 K1 (s) − , √ 1 + m2  sinh [m K1 (s)] × √ ds, 1 + m2 √ where K1 (s) = κ 2 − τ 2 ds.  



(5.8)

(5.9)

In Theorem 4.10, we characterized the non-lightlike general helices in E31 in terms of the associated curve. Similarly, we can give a characterization of the non-lightlike slant helices in E31 by the associated curve. Theorem 5.7. A non-lightlike slant helices in E31 is a second principal-donor curve of a plane curve in some planes. A non-lightlike Frenet curve γ in E31 is called a circular slant helix or hyperbolic slant helix if its second principal-directional curve is a circle in E2 or a hyperbola in E21 , respectively. These curves are said to be simple. Remark 5.8. The expressions of the simple slant helices in E31 are more simpler than that of other slant helices. Moreover, these curves have global expressions. In fact, the functions K1 (s), K2 (s) and K3 (s) are linear functions and hence the Eqs. (5.2)–(5.9) are globally integrable. We now study the closed simple slant helix in E31 . From Eqs. (5.2)–(5.9) and Remark 5.8, we know that there are no closed simple slant helices whose the expressions (5.2)–(5.4) and (5.6)–(5.9), but there exists the closed simple slant helix whose the expression (5.5). Remark 5.9. Let γ be a spacelike circular slant helix of (5.5) and γ¯¯ and γ¯ a circle in E2 with the radius r and a helix with the intrinsic equation (| κτ¯¯ | = |m| > 1) as its principal-directional curve and its second principal-directional curve, respectively. Since the curvature κ¯¯ of γ¯¯ equals to 1/r , κ¯ is given by √ 1

K3 ( s ) =

√

m2 −1

r

κ 2 + τ 2 ds =



κ¯ ds = √ s 2

m −1

r

. Thus the function K3 in (5.5) is given by

. Thus, by the simple integration, we get that γ is closed if and only if √ m

m2 −1

is

rational. From the similar work, we can check that other simple slant helices are not closed. Examples. In fact, a spacelike circular slant helix of (5.5) can be expressed by

 γ (s) = −r

1 m

 cos

√ r



ms

m2 − 1





+ 2m m2 − 1 sin  − 2m m2 − 1 sin

√ r

ms



√

ms

m2 − 1

 √ r

ms

m2 − 1

 cos

s r

 s   ms sin , 2m2 − 1 cos √ r

m2 − 1

 r

  , 2m2 − 1 cos

 cos

s r

r

m2 − 1

 sin

s r

 .

(5.10)

722

J.H. Choi et al. / J. Math. Anal. Appl. 394 (2012) 712–723

Fig. 1. (a) A closed simple slant helix. (b) A non-closed simple slant helix. 3 We put m = √ and r = 1. Then, we have the closed condition √ m 2 2

m2 −1

= 3 and an example of a closed spacelike circular

slant helix of type 2 which is expressed by (Fig. 1(a))

 √

2 2

γ1 (s) = −

3

4

cos[3s], 2 cos [s],

1 4

 sin[4s] − sin[2s] .

Putting by m = 2 and r = 1, we have the closed condition √ m

m2 −1

=

√2 3

and an example of a non-closed spacelike circular

slant helix of type 2 which is expressed by (Fig. 1(b))

 γ2 (s) = −

1







2s 2s cos √ , 7 cos[s] cos √ 2 3 3



2s

× 7 sin[s] cos √

3





  √ 2s + 4 3 sin[s] sin √ , 3

√ 2s − 4 3 cos[s] sin √ 

3



.

From Remark 5.9 we get a characterization of closed simple slant helix in E31 . Theorem 5.10. A closed simple slant helix γ in E31 is a spacelike circular slant helix of type 2 expressed by (5.10) with the slant is rational. helix constant m satisfying the condition that √ m m 2 −1

Acknowledgments The authors would like to express their sincere gratitude to the referee for the valuable suggestions which improve the paper. The second author was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2010-0007184). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11]

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