The Geometrical Characterizations of the Bertrand Curves of the Null Curves in Semi-Euclidean 4-Space

According to the Frenet equations of the null curves in semi-Euclidean 4-space, the existence conditions and the geometrical characterizations of the Bertrand curves of the null curves are given in this paper. The examples and the graphs of the Bertrand pairs with two different conditions are also given in order to supplement the conclusion of this paper more intuitively.

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.

Mannheim B-curve couples in Minkowski 3-space

In this paper, we define null Cartan and pseudo null Mannheim curves in Minkowski 3-space according to their Bishop frames. We obtain the necessary and the sufficient conditions for pseudo null curves to be Mannheim B-curves in terms of their Bishop curvatures. We prove that there are no null Cartan curves in Minkowski 3-space which are Mannheim B-curves, by considering the cases when their Mannheim B-mate curves are spacelike, timelike, null Cartan and pseudo null curves. Finally, we give some examples of pseudo null Mannheim B-curve pairs.

(k,m)-type slant helices for partially null and pseudo null curves in Minkowski space 𝔼14{\rm{\mathbb E}}_1^4

In this study we define the notion of (k,m)-type slant helices in Minkowski 4-space and express some characterizations for partially and pseudo null curves in {\rm{\mathbb E}}_1^4 .

ON BÄCKLUND TRANSFORMATIONS WITH SPLIT QUATERNIONS

The present paper deals with the introduction of Bäcklund Transformations with split quaternions in Minkowski space. Firstly, we tersely summarized the basic concepts of split quaternion theory and Bishop Frames of non-null curves in Minkowski space. Then, for Bäcklund transformations defined with each case of non-null curves, we give relationships between Bäcklund transformations and split quaternions. It is also presented some special propositions for transformations constructed with split quaternions. At the end, results obtained with the mathematical model have been evaluated.