Abstract
In a theory of space curves, especially, a helix is the most elementary and interesting topic. A helix, moreover, pays attention to natural scientists as well as mathematicians because of its various applications, for example, DNA, carbon nanotube, screws, springs and so on. Also there are many applications of helix curve or helical structures in Science such as fractal geometry, in the fields of computer aided design and computer graphics. Helices can be used for the tool path description, the simulation of kinematic motion or the design of highways, etc. The problem of the determination of parametric representation of the position vector of an arbitrary space curve according to the intrinsic equations is still open in the Euclidean space E<sup>3</sup> and in the Minkowski space <img src=image/13414896_01.gif>. In this paper, we introduce some characterizations of a non-null slant helix which has a spacelike or timelike axis in <img src=image/13414896_01.gif>. We use vector differential equations established by means of Frenet equations in Minkowski space <img src=image/13414896_01.gif>. Also, we investigate some differential geometric properties of these curves according to these vector differential equations. Besides, we illustrate some examples to confirm our findings.
Highlights
The curves are a fundamental structure of differential geometry
In the differential geometry of a regular curve in the Euclidean 3-space E3, it is well-known that one of the important problem is the characterization of a regular curve [1,2,3,4,5]
A necessary and sufficient condition that a curve to be general helix in Minkowski space E13 is that ratio of curvature to torsion be constant
Summary
The curves are a fundamental structure of differential geometry. In the differential geometry of a regular curve in the Euclidean 3-space E3, it is well-known that one of the important problem is the characterization of a regular curve [1,2,3,4,5]. A helix is a special case of the general helix If both curvature and torsion are non-zero constants, it is called a helix or a W-curve [2,3,4,5,6,7,8]. It is known that a space curve α whose normal lines make a constant angle with a fixed direction is called a slant helix [6,10]. We hope our results will be helpful to mathematicians who are specialized on mathematical modeling
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