On Helices and Bertrand Curves in Euclidean 3-Space

In this article, we investigate Bertrand curves corresponding to the spherical images of the tangent, binormal, principal normal and Darboux indicatrices of a space curve in Euclidean 3-space. As a result, in case of a space curve is a general helix, we show that the curves corresponding to the spherical images of its the tangent indicatrix and binormal indicatrix are both Bertrand curves and circular helices. Similarly, in case of a space curve is a slant helix, we demonstrate that the curve corresponding to the spherical image of its the principal normal indicatrix is both a Bertrand curve and a circular helix.


INTRODUCTION
In the differential geometry of a regular curve in Euclidean 3-space, it is well known that, one of the important problems is characterization of a regular curve. The curvature  and the torsion  of a regular curve play an important role to determine the shape and size of the curve [1,2]. For example: if 0,   then the curve is a geodesic. If 0   (constant) and 0,   then the curve is a circle with radius 1/ .  A curve of constant slope or general helix in Euclidean 3-space is characterized by the property that the tangent lines make a constant angle with a fixed direction (the axis of the general helix). A classical result about helix stated by Lancret in 1802 and first proved by de Saint Venant in 1845 (see [3] for details) says that: A necessary and sufficient condition that a curve be a general helix is that the ratio /  is constant along the curve, where 0.
  If both  and  are non-zero constants, it is called a circular helix.
There are a lot of interesting applications of helices (e.g., DNA double and collagen triple helix, helical staircases, helical structures in fractal geometry and so on). All these make authors say that the helix is one of the most fascinated curves in science and nature.
In the study of fundamental theory and the characterizations of space curves, the corresponding relations between the curves are very interesting problem. The wellknown Bertrand curve is characterized as a kind of such corresponding relation between the two curves. Bertrand curves discovered by J. Bertrand in 1850 are one of the important and interesting topics of classical special curve theory. A Bertrand curve is defined as a special curve which shares its the principal normals with another special 2 curve, called Bertrand mate or Bertrand partner curve. The curve  is a Bertrand curve if and only if there exist non-zero real numbers , [1,6]. So a circular helix is a Bertrand curve. Bertrand mates represent particular examples of offset curves [4] which are used in computer-aided design (CAD) and computer-aided manufacture (CAM).
Izumiya and Takeuchi [5] have shown that cylindrical helices can be constructed from plane curves and Bertrand curves can be constructed from spherical curves. After that, they [6] have studied cylindrical helices and Bertrand curves from the view point as curves on ruled surfaces. Schief [7] has given a study of the integrability of Bertrand curves. Kula and Yayli [8] have studied the spherical images of the tangent indicatrix and binormal indicatrix of a slant helix and they have shown that the spherical images are spherical helices. Camci et al. [9] have given some characterizations for a nondegenerate curve  to be a generalized helix by using harmonic curvatures of the curve in n -dimensional Euclidean space .
n The aim of this paper is to investigate Bertrand curves corresponding to the spherical images of a space curve in Euclidean 3-space. In case of a space curve is general helix, we show that the curves corresponding to the spherical images of its the tangent indicatrix and binormal indicatrix are both Bertrand curves and circular helices. Furthermore, in case of a space curve is slant helix, we demonstrate that the curve corresponding to the spherical image of its the principal normal indicatrix is both a Bertrand curve and a circular helix.

PRELIMINARIES
To meet the requirements in the next section, here, the basic elements of the theory of curves in the Euclidean space are briefly presented. After that, we describe the method to construct Bertrand curves from spherical curves. A curve 3 : is called the unit binormal vector of  at . s For the derivations of the Frenet frame, the Frenet-Serret formulae hold: is the torsion of the curve  at . s For any unit speed curve 3 : as the Darboux vector field of .  Spherical images (indicatrices) are well-known in classical differential geometry of curves. Let us define the curve C on 2 by the help of vector field ( ) This curve is 3 called the spherical Darboux image or the Darboux indicatrix of .  Similarly, the unit tangent vectors along the curve  generate a curve () T on . 2 The curve () T is called the spherical indicatrix of T or more commonly, will be a representation of ( ). [3]. For a general parameter t of a space curve ,  we can calculate the curvature and the torsion as follows: is the geodesic curvature of the curve  on 2 which is given by We now define a space curve where , a  are constant numbers and c is a constant vector [5]. The following theorem and its result are the key in this article. Theorem 1. Under the above notation,  is a Bertrand curve. Moreover, all Bertrand curves can be constructed by the above method [5].

HELICES AND BERTRAND CURVES IN EUCLIDEAN 3-SPACE
In this section, we investigate Bertrand curves corresponding to the spherical images of the tangent indicatrix, binormal indicatrix, principal normal indicatrix and Darboux indicatrix of a space curve in Euclidean 3-space and give some results of helices and Bertrand curves. Moreover an example of the spherical image of the tangent indicatrix of a space curve and the corresponding Bertrand curve is presented.
The proof is completed. Thus we can give the following corollary of Theorem 3.

Corollary 3.
Let  be a general helix parametrized by arc-length parameter .
s Then is both a Bertrand curve and a circular helix. The picture of the space curve  is given by Figure 1.