Position Vector of Spacelike Slant Helices in Minkowski 3-Space

Abstract

Abstract. In this paper, position vector of a spacelike slant helix with respect to standard frame are deduced in Minkowski space E 3 1 . Some new characterizations of a spacelike slant helices are presented. Also, a vector differential equation of third order is constructed to determine position vector of an arbitrary spacelike curve. In terms of solution, we determine the parametric representation of the spacelike slant helices from the intrinsic equations. Thereafter, we apply this method to find the parametric representation of some special spacelike slant helices such as: Salkowski and anti-Salkowski curves.

On k‐type null Cartan slant helices in Minkowski 3‐space

“…18 Isophotic curves play an important role in visual psychophysics. 19 In semi-Euclidean spaces E 4 1 and E 5 2 , null Cartan slant helices are characterized in Nešović et al 20 and Uçum et al 21 In particular, null Cartan slant helices in E 3 1 are studied in Ali and López 22 as null Cartan curves that satisfy the condition that the scalar product of their principal normal vector field N and a constant vector is constant. Null Cartan general helices in Lorentz-Minkowski spaces L n , n ≥ 3, are introduced in Ferrández et al 23 It is proved in Ferrández et al 23 that null Cartan curve in E 3 1 is the general helix if and only if it is null Cartan helix.…”

Pseudo‐spherical Darboux images and lightcone images of principal‐directional curves of nonlightlike curves in Minkowski 3‐space

“…Geometric properties of these associated curves can reflect more characteristics of an original curve and explain the behavior of it. Following the notion of slant helix in R 3 1 defined by Izumiya, 8 Ali and Lopez 22 present the notion of a slant helix in 3-dimensional Minkowski space R 3 1 by saying that the principal normal lines make a constant angle with a fixed direction. They characterize a slant helix by an equivalent definition (cf.…”

“…Inspired by the study of, 6,22,23 in this paper, we consider the nonlightlike curves (ie, timelike curves or spacelike curves) in Lorentz-Minkowski 3-space and analyze the geometric properties of the nonlightlike curves by means of some associated curves, there exist three kinds of Frenet-type frames associated to nonlightlike curves and some special vector fields along the nonlightlike curve. Choosing the principal normal vector, Darboux vector of Lorentzian version and the pseudo-vector product of these two vectors, we obtain an alternative frame along the original nonlightlike curve.…”

On spacelike rectifying slant helices in Minkowski 3-space

“…If n is a non-lightlike curve, we know that ε 3 = sgn[g(n ′ , n ′ )] = ±1 . Note that when n is a timelike curve, ε 3 = −1 in equation (1). …”

“…Later, Ali published two papers about characterizations of slant helices in Minkowski 3-space [1,2].…”