Position vector of a time-like slant helix in Minkowski 3-space

Abstract

In this paper, position vector of a time-like slant helix with respect to standard frame of Minkowski space E 3 1 is studied in terms of Frenet equations. First, a vector differential equation of third order is constructed to determine position vector of an arbitrary timelike slant helix. In terms of solution, we determine the parametric representation of the slant helices from the intrinsic equations. Thereafter, we apply this method to find the representation of a time-like Salkowski and time-like anti-Salkowski curves as examples of a slant helices, by means of intrinsic equations. Moreover, we present some new characterizations of slant helices and illustrate some examples of our main results.

Classical Differential Geometry of Curves According to Type-2 Bishop Trihedra

“…The curve is called a slant helix if normal lines makes a constant angle with a fixed direction. In further researches, spherical images, the tangent and the binormal indicatrix and some characterizations of such curves in Euclidean space and Lorentz-Minkowski spaces are presented (see [1], [2], [5], [15] and [16]). This fact has also been extended to the isotropic curves in the complex space by [23].…”

“…Let m and r > 0 be given and f(s) = (α -c) 2 . We say that α has jth order spherical contact with the sphere of radius r and center c at s =s 0 if Note that the larger j is, the closer f(s) is to being a constant function and so the closer a is to lying on a sphere of radius r and center c. …”

“…Chen to to classify curves with the solution of differential equations with constant coefficients with respect to standard frame of the space. This method generally uses ordinary vector differential equations as well as Frenet equations [2]. By this way, curves of a finite Chen type and some of classifications are given by the researchers in Euclidean space or another spaces, see [8], [9], [18], [23].…”

Characterizations of Spacelike Slant Helices In Minkowski 3-Space

“…They showed that the spherical indicatrix of a slant helix in E 3 1 are helices. Also in [3], Ali and Turgut, studied the position vector of a timelike slant helix in E 3 1 .…”

NOTE ON NULL HELICES IN <TEX>$\mathbb{E}_1^3$</TEX>

“…A certain property of a helix that its tangent lines make a constant angle with a fixed direction leads to the definitions of another types of the helices, for example, the general helices, slant helices and binormal slant helices. Naturally, many mathematicians have been studying another type of helices and they introduced many interesting results different from the ordinary helices (see [1,2,3,4,5,7,8,20,21]). …”