Dear Label_Name,

Greetings! I hope you are well.

I am writing on behalf of an international publishing house, Science and Education Publishing, USA (http://www.sciepub.com).

Science and Education Publishing (SciEP) is a multidisciplinary open access publisher of Journals covering the fields of Science, Technology and Medicine. All SciEP publications are subject to high-quality peer review, editorial and production processes and are freely available online.

Welcome to the April, 2013 edition of SciEP newsletter, providing monthly summaries of articles on science & education related topics of interest to scientist and professionals. The mission of our journals is to provide an open access forum for publishing new findings on issues pertinent to scientific and academic field. We also invite you to submit original research papers, review articles, short reports, or case studies are considered for publication in our Multidisciplinary Journals.

American Journal of Public Health Research
DOI: 10.12691/ajphr-1-1-3

The Implementation of a Tele-Homecare System with Service Oriented Architecture and HL7 Message Transmission Standard
Ching-Sung Wang

American Journal of Medical and Biological Research
DOI: 10.12691/ajmbr-1-1-2

The Influence of Ethanol on Pyruvate Kinases Activity in Vivo, in Vitro, in Silico
Lelevich Sergey Vladimirovich, Khrustalev Vladislav Victorovich, Barkovsky Eugene Victorovich, Shedogubova Tatyana Aleksandrovna

American Journal of Medical Sciences and Medicine
DOI: 10.12691/ajmsm-1-1-1

Clinical and Histological Features in Wegener Granulomatosis
Li Cavoli Gioacchino, Passantino Rita, Bono Luisa, Tortorici Calogera, Ferrantelli Angelo, Giammarresi Carlo, Rotolo Ugo

American Journal of Infectious Diseases and Microbiology
DOI: 10.12691/ajidm-1-1-2

The Curse of Lymphatic Filariasis: Would the Continual Use of Diethylcarbamazine Eliminate this Scourge in Papua New Guinea?
Anne Lanham, Lillian Mwanri

We have released issues of some journals on February 2013.
American Journal of Public Health Research
American Journal of Medical Sciences and Medicine
American Journal of Medical and Biological Research
American Journal of Pharmacological Sciences
American Journal of Clinical Medicine Research
American Journal of Nursing Research
American Journal of Cancer Prevention
American Journal of Medicine Studies
American Journal of Infectious Diseases and Microbiology
American Journal of Cardiovascular Disease Research
American Journal of Biomedical Research
American Journal of Mechanical Engineering
American Journal of Microbiological Research
Journal of Preventive Medicine
American Journal of Food and Nutrition
American Journal of Hypertension Research
American Journal of Sports Science and Medicine
Journal of Cancer Research and Treatment
American Journal of Zoological Research
Journal of Biomedical Engineering and Technology
American Journal of Microbiological Research
Biomedical Science and Engineering

We are pleased to announce that submissions are now being accepted for peer review for all the journals across all our subject areas in SciEP.

Please submit your manuscript via our online submission system:
http://www.sciepub.com/author/signin

To unsubscribe from our mailing list, please click on the following link: unsubscribe

I am looking forward to hearing from you.

Kind regards,

David Rain
Email:join@sciepub.com
Science and Education Publishing (http://www.sciepub.com)

The Helix Relation between Two Curves

İLKAY ARSLAN GÜVEN*, YUSUF YAYLI

Department of Mathematics, University of Gaziantep, Gaziantep, Turkey

Abstract: 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

Received September 14, 2013; Revised September 21, 2013; Accepted September 23, 2013

Citation: İLKAY ARSLAN GÜVEN, YUSUF YAYLI, The Helix Relation between Two Curves. Turkish Journal of Analysis and Number Theory, 1 (1). 23-25. January 2013. doi:10.12691/tjant-1-1-6

1. Introduction

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 [7]. 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 [3], a similar equation was given to show that cylindrical helices can be constructed from plane curves.

In [2], 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 [7]. 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.

2. Preliminaries

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;

[picture] | (1)

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 [4], 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 [7] 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

where

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;

References

[1]   | M. Barros, General helices and a theorem of Lancret, Proceedings of the American Mathematical Society, 125-5 (1997),1503-1509.
[2]   | I.A. Güven, S. Kaya, Y. Yayli, General helix and associated plane curve in Minkowski 3-space, Far East Journal of Math. Sciences, 47-2 (2010), 225-233.
[3]   | S. Izumiya and N. Takeuchi, Generic properties of helices and Bertrand curves, Journal of Geometry, 74 (2002), 97-109.
[4]   | L. Kula and Y. Yayli, On slant helix and its spherical indicatrix, Applied Mathematics and Computation, 169 (2005), 600-607.
[5]   | W. Kühnel, Differential Geometry, second ed., Am. Math. Soc., 2006.
[6]   | D.J. Struik, Lectures on Classical Differential Geometry, Dover, New York, 1988.
[7]   | S. Sy, General helices and other topics in the differential geometry of curves, Master of Sci. in Mathematics, Michigan Technological Uni., 2001.