New Representations of Spherical Indicatricies of Bertrand Curves in Minkowski 3-Space

Abstract

We obtained a new representation for timelike Bertrand curves and their Bertrand mate in 3-dimensional Minkowski space. By using this representation, we expressed new representations of spherical indicatricies of Bertrand curves and computed their curvatures and torsions. Furthermore in case the indicatricies of a Bertrand curve are slant helices, we investigated some new characteristic features of these curves.

1. Introduction

In the early 1900s, Bertrand had worked on special curves, which were then referred to by his name. He examined these curves and their applications to both differential equations in analytical mechanics and thermodynamics [1]. Bertrand curve is defined as a particular curve, which shares its principal normal vector with another special curve that called Bertrand pair. Also, there is a linear relation between the curvature and the torsion. In three-dimensional euclidean space, a helix is characterized as a curve whose tangent line makes a constant angle with a fixed direction. Lancret expressed the result, which describes helices, a curve is a helix if and only if the ratio is constant in 1802, and Saint Venant proved it in 1845. There are many highly interesting applications of helices such as helical structures in fractal geometry, DNA spirals, collagen triple helix and the helical ladder structures in architecture [2–5]. Izumiya and Takeuchi defined Slant helices. A Slant helix is a curve whose principal normal vector makes a constant angle with a fixed direction [6].

In three-dimensional Minkowski space, curves are classified into three groups: spacelike, timelike, and lightlike. In this paper, we examined timelike Bertrand curves and their timelike Bertrand mate with a geometric point of view. We obtained a new representation for timelike Bertrand curves and their Bertrand mate in 3-dimensional Minkowski space. By using this representation, we expressed new representations of spherical indicatricies of Bertrand curves and computed their curvatures and torsions. Furthermore, in case the indicatricies of a Bertrand curve are slant helices, we investigated some new characteristic features of these curves.

3-dimensional Minkowski space is the vector space endowed with Lorentzian inner product given by where and .

A vector is called a spacelike vector if , a timelike vector if , and a lightlike vector if and . Likewise, an arbitrary curve in is addressed as spacelike, timelike, or null, if its velocity vectors are respectively spacelike, timelike, or null, for every . For a vector , the norm of is defined by In addition, is called a unit vector if its norm is equal to . The Lorentzian sphere and hyperbolic sphere of radius in 3-dimensional Minkowski space are, respectively, given by For any and , the vectorial product of and is defined by

We denote the moving Frenet frame along the curve by where , and are the tangent, the principal normal and the binormal vector of the curve , respectively. Let be a unit speed timelike space curve with curvature , torsion and Frenet vector fields of be where is timelike and are spacelike vector fields. Then, Frenet formulas are

For given any two vectors , if and are positive (negative) timelike vectors then there is a unique nonnegative real number such that

Moreover, this unique nonnegative real number is called The Lorentzian timelike angle between and . If is a spacelike vector and be a positive timelike vector then there is a unique nonnegative real number such that

Therefore, this unique nonnegative real number is called The Lorentzian timelike angle between and (see [7]).

A unit speed curve is called a slant helix if there exists a constant vector field in such that is constant [8].

Theorem 1. Let be unit speed timelike curve in . Then, is a slant helix if and only if either one of the next two functions or is constant everywhere does not vanish [8].

2. New Representations of Timelike Bertrand Curves

Let be a unit speed timelike curve. If there exists a timelike curve whose principal normal vector coincides with that of , then is called a timelike Bertrand curve. The pair is said to be a timelike Bertrand pair. Let be a timelike Bertrand curve and let be a timelike Bertrand mate of and we denote the moving Frenet frame along the curve by and along the curve by . Thus, the tangent vector fields and the binormal vector fields are, respectively, related by for some functions . Furthermore, is the angle between the tangent vectors and . Similar relations can be given as follows for the curvatures and torsions: where , and , are the curvatures and torsions of and , respectively.

Furthermore, let be a nonhelical timelike Bertrand pair; and are arclength parameters of and , respectively. Hence, and . Let be a timelike Bertrand curve with a timelike Bertrand mate curve ; then, where and .

Moreover, we can give the relation between the arclength parameters and of and , respectively, as follows:

Theorem 2. Let be a timelike Bertrand curve and let be a nonhelical timelike Bertrand mate of . We denote the moving Frenet frame along the curve by and along the curve by . Then, where , , and , and , are the curvatures and torsions of and , respectively.

The geodesic curvature of the principal image of the principal normal indicatrix of is

Also, if we compute the derivative and put it in equation , we obtain

So we have .

Now, we can give the following result.

Corollary 3. Let be a timelike Bertrand curve and be a nonhelical timelike Bertrand mate of . Bertrand curve is a slant helix if and only if its mate curve is a slant helix.

Theorem 4. Let be a timelike Bertrand curve and be a nonhelical timelike Bertrand mate of and . Then, is a constant.

Proof. If we differentiate the equation with respect to , we obtain
By substituting and in the equation and choosing we can write then, if we differentiate the last equation again with respect to , we get
Since , , and are linear independent, and . Thus, and are constants. Therefore, is a constant.

3. New Representations of Spherical Indicatrix of Timelike Bertrand Curves

Definition 5. Let be a unit speed regular curve and let , , and be Frenet vectors in Minkowski 3-space. Thus, there exist unit tangent vectors along the curve . These unit tangent vectors generate a curve on the unit Lorentzian sphere (or hyperbolic sphere). The curve is called the Tangent indicatrix of the curve . Similarly, we can define the principal normal and binormal indicatrix of the curve .

3.1. The Tangent Indicatrix of Timelike Bertrand Curves

If we take a nonhelical timelike Bertnard pair in the above definition, we get the tangent indicatrix of a timelike Bertrand curve such that

Theorem 6. Let be a timelike Bertrand curve and . If the Frenet frame of the tangent indicatrix of the timelike Bertrand curve is then

From (22), we know . If we differentiate this equation with respect to and reorganize, we have where and . With a similar way, we can express following relation between the arclength parameters and of and , respectively:

Theorem 7. Let be a timelike Bertrand curve and let be a nonhelical timelike Bertrand mate of , and . Thus, curvature and torsion of the tangent indicatrix of the timelike Bertrand curve are respectively.

Corollary 8. Let be a nonplanar and nonhelical timelike Bertrand pair in . Thus, Bertrand curve is a slant helix if and only if the tangent indicatrix of the Bertrand curve is a helix.

Theorem 9. Let be a timelike Bertrand curve and let be a nonhelical timelike Bertrand mate of . Then, the geodesic curvature of the principal image of the principal normal indicatrix of is where and .

Corollary 10. Let be a timelike Bertrand curve and let be a nonhelical timelike Bertrand mate of . Then, the tangent indicatrix of the timelike Bertrand curve is a helix if and only if is satisfied.

3.2. The Principal Normal Indicatrix of Timelike Bertrand Curves

It is known that the principal normal indicatrix of a timelike Bertrand curve is

Theorem 11. Let be a timelike Bertrand curve and . If the Frenet frame of the principal normal indicatrix of the timelike Bertrand curve is then where .

If is a timelike Bertrand curve and is a nonhelical timelike Bertrand mate of , and , then

Furthermore, the curvature and the torsion of the principal normal indicatrix of the timelike Bertrand curve are respectively, where .

Theorem 12. Let be a timelike Bertrand curve and let be a nonhelical timelike Bertrand mate of . Then, the principal normal indicatrix of the timelike Bertrand curve is a helix if and only if is satisfied.

3.3. The Binormal Indicatrix of Timelike Bertrand Curves

It is known that the binormal indicatrix of a timelike Bertrand curve is

Theorem 13. Let be a timelike Bertrand curve and . If the Frenet frame of the binormal indicatrix of the timelike Bertrand curve is then

Theorem 14. Let be a timelike Bertrand curve and let be a nonhelical timelike Bertrand mate of , , and . Then, And the curvature and the torsion of the binormal indicatrix of the timelike Bertrand curve are respectively.

As a result of this theorem, we have the following.

Corollary 15. A Bertrand curve is a slant helix if and only if the binormal indicatrix of the Bertrand curve is a helix.

Theorem 16. Let be a timelike Bertrand curve and let be a nonhelical timelike Bertrand mate of . The geodesic curvature of the principal image of the principal normal indicatrix of is where and .

Corollary 17. Let be a timelike Bertrand curve and let be a nonhelical timelike Bertrand mate of . Then, the binormal indicatrix of the timelike Bertrand curve is a helix if and only if is satisfied.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.