On Timelike Rectifying Slant Helices in Minkowski 3-Space

Abstract

In this work, we study timelike rectifying slant helices in E31. First, we find general equations ofthe curvature and the torsion of timelike rectifying slant helices. After that, by solving second order linear differential equations, we obtain families of timelike rectifying slant helices that lieon cones.Helices arise in nanosprings, carbon nanotubes, DNA double and collagen triple helices. The double helix shape is commonly associated with DNA [1]. In differential geometry, a general helix in Euclidean 3-space is characterized by the property that the tangent lines make a constant angle with a fixed direction [12, 13]. Similarly, the notion of slant helix was introduced by Izuyama and Takeuchi by the property that the principal normal lines make a constant angle with a fixed direction [8, 9]. They showed that a space curve is a slant helix if and only if the geodesic curvature of the principal normal of the curve is a constant function. In [10, 11], Kula et al. studied the spherical images of slant helices. Later, Ahmet T. Ali studied slant helices in Minkowski 3-space [1, 2]. The notion of rectifying curve has been introduced by Chen [5, 6]. Chen proposed the conditions under which the position vector of a unit speed curve lies in its rectifying plane. Besides, he stated the importance of rectifying curves in Physics.