Abstract
The concept of the natural mate and the conjugate curves associated to a smooth curve in Euclidian 3-space were introduced initially by Dashmukh and others. In this paper, we give some extra results that add more properties of the natural mate and the conjugate curves associated with a smooth space curve in . The position vectors of the natural mate and the conjugate of a given smooth curve are investigated. Also, using the position vector of the natural mate, the necessary and sufficient condition for a smooth given curve to be a Bertrand curve is introduced. Moreover, a new characterization of a general helix is introduced.
1. Introduction
The differential geometry of curves and surfaces is an ancient topic in differential geometry, but it is still an active area of research. This is because of its applications in several fields such as computer graphics, computer vision, medical imaging, physics, and aerospace. A helix plays a crucial role in many applications in engineering and also, in DNA structures. In fact, a DNA molecule can be described by double helix. Also, it has been observed that in a molecular model of the DNA there are two side-by-side in opposing direction helices linked by hydrogen bonds (cf. [1]). The rectifying curves are used to analyze joint kinematics (cf. [2, 3]). The Salkowski curves are useful in constructing closed curves with constant curvature and continuous torsion such as knotted curves (cf. [4]). The Salkowski curves are examples of slant helice with constant curvatures.
In differential geometry, curves and their Frenet frames play central roles for creating special surfaces (c.f [5–10]). The Frenet frame associated with a regular curve in , which is a moving frame along the curve, forms an orthonormal basis for the Euclidean space at each point of the given curve. This allows geometers to analyze a curve and to study the position vector of the given curve and other curves. The terminologies of natural mate and conjugate associated with a smooth curve were introduced and studied in [11]. Mainly, some relationships between a given curve and its natural mates were investigated in [11] as well as the necessary and sufficient conditions for the natural mate associated with a given Frenet curve to be a spherical curve, a helix, or a curve with a constant curvature. The most natural geometric object in differential geometry of curves in Euclidian 3-space is a position vector. The position vector is very important, owing to its applications in mathematics, engineering, physics, and other natural sciences.
In this paper, we investigate the position vectors of the natural mate and the conjugate of a given space curve using the Frenet frame of the given curve as a basis for . The position vectors of the natural mate and conjugate curves will be useful for studying the surfaces generated by these curves such as ruled and translation surfaces. In Section 2 of this paper, we review some basic concepts of space curves which will be used in the rest of this study. In Section 3, we study the position vectors of natural mate and conjugate of a unit speed curve with nonvanishing curvature and torsion. Using the position vector of the natural mate, we give the necessary and sufficient condition for a given curve with nonvanishing curvature and torsion to be a Bertrand curve. Also, a new characterization for a general helix is proven.
2. Preliminaries
In this section, we review some basic concepts of the differential geometry of curves in Euclidean 3-space, and for more detail, we refer the reader to [2, 3, 11–16]. First, we start with the definition of a smooth space curve. A parametrized curve in is a map , where is a real interval, given by such that and are smooth functions on . is a regular curve if for all . The unit tangent vector, the unit principal normal vector, and the unit binormal vector of a regular curve are defined by , , and , respectively. At any point of , there are three planes spanned by the vectors , , and , these planes are the normal plane, the rectifying plane, and the osculating plane, respectively. The curvature and the torsion of a regular curve are given by and , respectively, and is called a Frenet curve if and . The Serret Frenet apparatus associated to is given by .
If for all , then is called a unit speed curve and the Frenet- Serret equations are given by
In the rest of this section, we state definitions of some special curves.
Definition 1. Let be a smooth space curve. Then, is called a helix if its tangent makes a fixed angle with a fixed direction.
Definition 2. Let be a smooth space curve. Then, is called a slant helix if its principal normal makes a fixed angle with a fixed direction.
Definition 3. Let be a smooth space curve. Then, is called a rectifying curve if it lies in the rectifying plane at each point.
It has been obtained by Chen in [3] that the distance squared function of a rectifying curve is a quadratic polynomial in its arc length.
Definition 4. Let be a smooth space curve. Then, is called a spherical curve if it lies in a sphere.
For a spherical curve, it is obvious to obtain that its distance from the center of the sphere, which the curve lies on, is equal to the radius of the sphere. This will play a role in the proof of Theorem 10.
Definition 5. Let be a smooth space curve. Then, is called a Salkowski curve if it has constant curvature and nonconstant torsion.
Definition 6. Given a unit speed curve with nonvanishing curvature and torsion. The natural mate of is defined by . If has negative torsion, then its conjugate is given by .
3. Natural and Conjugate Mates Associated with a Smooth Space Curve
In this section, we give the position vectors of natural and conjugate curves. Using position vectors of the mentioned curves, we give more brand results which carry interesting relationship between a given smooth curve and its associated natural and conjugate curves.
Theorem 7. Let be a unit speed Frenet curve with Serret-Frenet apparatus . The natural mate of is given by where , is the distance function of , and is the derivative of with respect to .
Proof. Since the unit tangent of is given by , we have Now, differentiating equation (3), we get Thus, from equation (4), we have the following equations: Therefore, we have Our task now is to find . The distance squared function, , of is given by Now, differentiating equation (7), we get Hence, using equations (5) and (6) in equation (8), we obtain which completes the proof.
Now, as an application of Theorem 7, we give a criterion for a Bertrand curve with a neat proof. First, we state the following definition and a well-known result regarding the Bertrand curve.
Definition 8. A curve is called a Bertrand curve if there is another curve , different from , and a bijection between and such that and have the same principal normal at each pair of corresponding points under .
The following theorem is a well-know result, and it can be found in many books of elementary differential geometry of curves and surfaces.
Theorem 9. A curve with and is called a Bertrand curve if and only if it satisfies the condition where and are constants.
Now, we give a criterion for a Bertrand curve in term of natural mate.
Theorem 10. Let be a unit speed curve with nonzero curvature and nonzero torsion and Serret-Frenet apparatus and be its natural mate. Then, the following assertions are equivalent: (1) is a Bertrand curve(2) is a spherical curve(3) lies in the rectifying plane of
Proof. Let be a unit speed curve with and . Then, from Theorem 7, the naturel mate of is given by where and is the distance function of . Now, is a spherical curve if and only if is a positive nonzero constant if and only if where and are constants (i.e., lies in the rectifying plane of ), if and only if if and only if if and only if is a Bertrand curve.
Remark 11. Theorem 10 gives a method to create a spherical curve using a Bertrand curve. In [9], a method to create a Bertrand curve using a spherical curve with Sabban frame was introduced. Also, in [17], some methods to create special curves such as helix, slant helix, Bertrand curves, and Mannheim curves were introduced.
Corollary 12. If is a Salkowski curve, then its natural mate is given by
In what follows, we give an example of the natural mate of a Salkowski curve, and for more detail in Salkowski curve, we refer the reader to [4].
Example 13. Let , be a Salkowski curve given by where,
where and . This curve was investigated by Salkowski in 1909.
This curve has and torsion . The unit tangent of is given by where
Now, using Theorem 10, the natural mate of is given by . Now, we draw pictures for and its natural mate when as shown in Figure 1. In this, Figure 1(a) is the curve and (b) is the natural mate of . It can be observed from (b) that the natural mate of lies on a unit sphere.
(a) The curve
(b) The blue curve is lying on the unit sphere centered at the origin
It can be easily observed from Theorem 10 that if is a circular helix or a Salkowski curve, then its natural mate always lies on the rectifying planes of .
In the coming theorem, we give a new criterion for a general helix.
Theorem 14. Let be a unit speed curve with nonvanishing curvature and torsion. Then, is a general helix if and only if there exists a fixed direction orthogonal to its natural mate.
Proof. Let be a unit speed curve with nonvanishing curvature and torsion, and be its natural mate. First, assume that is a general helix, then there exists a fixed direction makes a constant angle with its tangent. Let be a unit constant vector lies on that direction, then , and . Now, using Theorem 7, we have
Since is a helix, then . Therefore, , which means that is orthogonal to .
Conversely, assume that there exists a fixed direction orthogonal to . Let be a unit constant vector lies on that direction, then ; therefore, which implies that , which means that is a general helix.
Remark 15. In [7, 11], it has been proved that a smooth curve with nonvanishing curvature and torsion is a general helix if and only if its natural mate is a plane curve. Using this fact and the result in Theorem 14, it can be concluded that the axis of a helix is always normal to the plane containing its natural mate.
Now, we give the position vector for conjugate.
Theorem 16. Let be a Frenet curve with Serret-Frenet apparatus with negative torsion. The conjugate mate of is given by where and is the distance function of .
Proof. Since the unit tangent of is given by , we have Now, differentiating equation (18) w.r.t the arc length, we get Thus, from equation (19), we have the following equations: Therefore, we have Our task now is to find . The distance squared function, , of is given by Now, differentiating equation (22), we get Hence, using equations (20) and (21) in equation (23), we obtain which implies that which completes the proof.
From Theorem 16, we have the following corollary.
Corollary 17. Let be a unit speed curve with Serret-Frenet apparatus with negative torsion. Then, (1)If is a spherical curve, then (2)If is a rectifying curve, then
Proof. If is a spherical curve, then its distance function is a constant. Thus, . If is a rectifying curve, then it has been proved in [3] that its distance function satisfies for some constants and . Therefore, .
4. Conclusion
The position vector of a curve in the Euclidian 3-space is the most natural geometric object. It is important in many applications in several areas such as mathematics, engineering, and natural sciences. Throughout this paper, we study the position vectors of the natural and conjugate mates associated with a given smooth space curve in Euclidian 3-space. The position vectors of the natural and the conjugate mates associated with a given smooth curve are very useful for studying these curves. Also, the position vector of the natural mate associated to a given curve is used to prove a new criterion for a helix and Bertrand curves. Moreover, it can be easy to study the surfaces generated by the natural and conjugate mates associated with a smooth curve using their position vectors.
Data Availability
No external data has been used in this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.