The Helix Relation between Two Curves
1Department of Mathematics, University of Gaziantep, Gaziantep, Turkey
2Department of Mathematics, Ankara University, Ankara, Turkey
In this study, we give the relation of being general helix and slant helix of two curves by using the equation between them. Also we find some results and express the characterizations of these curves.
Keywords: general helix, Frenet frame, Bertrand mates
Turkish Journal of Analysis and Number Theory, 2013 1 (1),
Received September 14, 2013; Revised September 21, 2013; Accepted September 23, 2013Copyright © 2013 Science and Education Publishing. All Rights Reserved.
Cite this article:
- GÜVEN, İLKAY ARSLAN, and YUSUF YAYLI. "The Helix Relation between Two Curves." Turkish Journal of Analysis and Number Theory 1.1 (2013): 23-25.
- GÜVEN, İ. A. , & YAYLI, Y. (2013). The Helix Relation between Two Curves. Turkish Journal of Analysis and Number Theory, 1(1), 23-25.
- GÜVEN, İLKAY ARSLAN, and YUSUF YAYLI. "The Helix Relation between Two Curves." Turkish Journal of Analysis and Number Theory 1, no. 1 (2013): 23-25.
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The characterization of general helix was given with Lancret's theorem in 1802. Similarly. The general helix and the associated plane curve in the Euclidean 3-space were studied in . It was clearly given that how to convert the associated plane curve to the general helix and vice versa. The equations of the general helix and associated plane curve were given to make this convertion. Also in , a similar equation was given to show that cylindrical helices can be constructed from plane curves.
In , the general helix and associated plane curve were studied in Minkowski 3-space finding new equations which showed how to obtain the general helix from the plane curve were given in .
In this paper, we take the equation of two curves which is similar to the equation in . Then we calculate the Frenet vectors and axis of symmetry of each curve and obtain the relation between them of how to be a general helix, slant helix and Bertrand mates.
We now recall some basic notions about classical differential geometry of space curves in Euclidean space .
Let be a curve with arc-length parameter s and let denote the Frenet frame of β. is called the unit tangent vector of β at s. The curvature of β is given by . The unit principal normal vector N(s) of β at s is given by . Also the unit vector is called the unit binormal vector of β at s. Then the famous Frenet formula holds
where τ (s) is the torsion of β at s.
Also the Frenet vectors of a curve α, which is not given by arc-length parameter can be calculated as;
A curve is called a general helix if its tangent line forms a constant angle with a fixed straight line. This straight line is the axis of general helix. A classical result stated by Lancret says that "a curve is a general helix if and only if the ratio of the curvature to torsion is constant". If both curvature and torsion are non-zero constant, it is of course a general helix which is called circular helix.
A slant helix in is defined by the property that the principal normal line makes a constant angle with a fixed direction. In , it is shown that α is a slant helix in if and only if the geodesic curvature of the principal normal of the space curve α is a constant function.
Let two curves be α and β in. They are called Bertrand curves if their principal normal vectors are linearly dependent. We say that α and β are Bertrand mates.
3. The Equation of Two Curves
Let be a curve and α be a unit speed general helix in . s denotes arc-length parameter of α. The Frenet frame of and α are indicated by and , respectively. The curvatures of and α are and κ; the torsions are and τ. a is the constant axis of general helix, θ is the angle between a and T. The axis is given by
The equation between and α is denoted in  as
where and are arbitrary constant vector and point.
Now, let find the Frenet vectors of . Since s is not arc-length parameter of , we use the equations in (1).
The tangent vector of is
the principal normal vector of is
Here if we say
then we can denote the principal normal vectors as .
Also the binormal vector of is found as
We will state the following theorems whose proofs will be done by these calculations.
Theorem 1. Let a curve and a general helix be and α, respectively. The equation between them is given by where a is the axis of general helix. If α is general helix then is a general helix.
Proof. Let the curve α be a general helix. The tangent and binormal vectors T and B of α make constant angle with a constant vector which can be a, the axis of α.
Since the tangent vector of depends on T and B, it also make constant angle with that constant vector.
So is a general helix.
Theorem 2. Let a curve and a general helix be and α, respectively. The equation between them is given by where a is the axis of general helix. If the number c between the principal normal vectors of and α is constant, then α is slant helix if and only if is a slant helix.
Proof. The relation of principal normal vectors and N of and α is calculated as;
which we denoted by .
Here if c is constant then and N are linearly dependent.
Firstly let α be a slant helix. Then the principal normal vector of α makes a constant angle with a fixed direction. Since the principal normal vector is linearly dependent with N, also makes a constant angle with that fixed direction. So is a slant helix.
The opposite of the proposition can be proved by following the same procedure.
Remark 1. The number c between the principal normal vectors was taken constant in the theorem. This number c is constant under the condition of κ and τ are constant. Thus α is a circular helix. If α is a circular helix, then α is slant helix if and only if is a slant helix.
Theorem 3. Let a curve and a general helix be and α, respectively. The equation between them is given by where a is the axis of general helix. If the number c between the principal normal vectors of and α is constant, then and α are Bertrand mates.
Proof. In the equation , let c be a constant number, then the principal normal vectors of and α are linearly dependent. So and α are Bertrand curves.
Remark 2. When c is taken as a constant, then α is a circular helix. If α is a circular helix, then and α are Bertrand mates.
Now we will give an example
Example: Let be a general helix with the curvature and torsion;
The axis of α is calculated by . Here the angle θ; between a and T is
If the vectors are Frenet vectors of α, then
So the axis a(s) is found as
The curve is
by taking the arbitrary constant vector and arbitrary point.
Then the Frenet vectors of are calculated by using T, N, B;
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