A Surface Family with a Common Asymptotic Null Curve in Minkowski 3-Space E 1 3

This approach is on constructing a surface family with a common asymptotic null curve. It has provided the necessary and sufficient condition for the curve to be an asymptotic null curve and extended the study to ruled and developable surfaces. Subsequently, the study has examined the Bertrand offsets of a surface family with a common asymptotic null curve. Lastly, we support the results of this approach by some examples.

The pseudo-null geometric phase along optical fiber

In this study, a pseudo-null space curve in Minkowski 3-space is used to describe an optical fiber that is injected into monochromatic linear polarized light. The direction of the electric field vector with respect to the Frenet frame of a pseudo-null curve determines the state polarization of a monochromatic linearly polarized light wave traveling along an optical fiber. For the Frenet frame of a pseudo-null curve in Minkowski 3-space, the polarization vector [Formula: see text] is assumed to be perpendicular to the tangent vector [Formula: see text] with respect to anholonomic coordinates. Anholonomic coordinates for the Frenet frame of a pseudo-null curve are used to describe pseudo-null electromagnetic curves in the normal and binormal directions along an optical fiber. For the Frenet frame of the pseudo-null curve, Lorentz force equations in the normal and binormal directions along the optical fiber are presented. Pseudo-normal and binormal Rytov parallel transport laws for electric fields in the normal and binormal directions along with the optical fiber for the Frenet frame of the pseudo-null curve via anholonomic coordinates are presented. For anholonomic coordinates in Minkowski 3-space, rotations of the polarization planes of a light wave traveling in the normal and binormal directions along with the optical fiber with respect to the Frenet frame of the pseudo-null curve are obtained. Finally, a pseudo-null curve’s Maxwellian evolution is determined.

Null Darboux Curve Pairs in Minkowski 3-Space

Based on the fundamental theories of null curves in Minkowski 3-space, the null Darboux mate curves of a null curve are defined which can be regarded as a kind of extension for Bertrand curves and Mannheim curves in Minkowski 3-space. The relationships of null Darboux curve pairs are explored and their expression forms are presented explicitly.

Involutes of Pseudo-Null Curves in Lorentz–Minkowski 3-Space

In this paper, we analyze involutes of pseudo-null curves in Lorentz–Minkowski 3-space. Pseudo-null curves are spacelike curves with null principal normals, and their involutes can be defined analogously as for the Euclidean curves, but they exhibit properties that cannot occur in Euclidean space. The first result of the paper is that the involutes of pseudo-null curves are null curves, more precisely, null straight lines. Furthermore, a method of reconstruction of a pseudo-null curve from a given null straight line as its involute is provided. Such a reconstruction process in Euclidean plane generates an evolute of a curve, however it cannot be applied to a straight line. In the case presented, the process is additionally affected by a choice of different null frames that every null curve allows (in this case, a null straight line). Nevertheless, we proved that for different null frames, the obtained pseudo-null curves are congruent. Examples that verify presented results are also given.

A natural Frenet frame for null curves on the lightlike cone in Minkowski space $\mathbb{R} ^{4}_{2}$

In this paper, we investigate the representation of curves on the lightlike cone $\mathbb {Q}^{3}_{2}$ Q 2 3 in Minkowski space $\mathbb {R}^{4}_{2}$ R 2 4 by structure functions. In addition, with this representation, we classify all of the null curves on the lightlike cone $\mathbb {Q}^{3}_{2}$ Q 2 3 in four types, and we obtain a natural Frenet frame for these null curves. Furthermore, for this natural Frenet frame, we calculate curvature functions of a null curve, especially the curvature function $\kappa _{2}=0$ κ 2 = 0 , and we show that any null curve on the lightlike cone is a helix. Finally, we find all curves with constant curvature functions.

Dual Associate Null Scrolls with Generalized 1-Type Gauss Maps

In this work, a pair of dual associate null scrolls are defined from the Cartan Frenet frame of a null curve in Minkowski 3-space. The fundamental geometric properties of the dual associate null scrolls are investigated and they are related in terms of their Gauss maps, especially the generalized 1-type Gauss maps. At the same time, some representative examples are given and their graphs are plotted by the aid of a software programme.

Normal Partner Curves of a Pseudo Null Curve on Dual Space Forms

In this work, a kind of normal partner curves of a pseudo null curve on dual space forms is defined and studied. The Frenet frames and curvatures of a pseudo null curve and its associate normal curve on de-Sitter space, its associate normal curve on hyperbolic space, are related by some particular function and the angles between their tangent vector fields, respectively. Meanwhile, the relationships between the normal partner curves of a pseudo null curve are revealed. Last but not least, some examples are given and their graphs are plotted by the aid of a software programme.