Osculating Mate of a Curve in Minkowski 3-Space

Abstract

In this paper, we introduce and develop the concept of osculating curve pairs in the three-dimensional Minkowski space. By defining a vector lying in the intersection of osculating planes of two non-lightlike curves, we characterize osculating mates based on their Frenet frames. We then derive the transformation matrix between these frames and investigate the curvature and torsion relations under varying causal characterizations of the curves—timelike and spacelike. Furthermore, we determine the conditions under which these generalized osculating pairs reduce to well-known curve pairs such as Bertrand, Mannheim, and Bäcklund pairs. Our results extend existing theories by unifying several known curve pair classifications under a single geometric framework in Lorentzian space.

1. Introduction

The study of the differential geometry of curves has long been a cornerstone of geometric analysis, offering profound insights into both the intrinsic and extrinsic properties of space. Classical investigations have focused on special types of associated curve pairs, wherein two regular space curves exhibit systematic relationships dictated by the interplay of their respective Frenet frames. These relationships have significant applications in mechanics, kinematics, and even the theory of relativity, where understanding the relative alignment and motion of trajectories is essential.

While the classical notions of curve pairs—such as Bertrand, Mannheim, and Bäcklund curves—have been extensively explored in Euclidean 3-space ${\mathbb{E}}^3$, their generalization to Minkowski 3-space ${\mathbb{E}}_1^3$ continues to stimulate considerable research interest. Minkowski space, with its pseudo-Euclidean structure and indefinite Lorentzian inner product, introduces unique geometrical phenomena due to its causal structure. In this setting, vectors (and hence curves) are characterized as timelike, spacelike, or lightlike, depending on the sign of their Lorentzian norm. This additional nuance not only affects the definition of the classical Frenet frame but also demands a thorough re-examination of the conditions defining curve pairs.

Given a Lorentzian scalar product defined by

$${〈\mathbf{q},\mathbf{p}〉}_L=-q_1{p}_1+q_2{p}_2+q_3{p}_3$$

for vectors $\mathbf{q}=(q_1,q_2,q_3)$ and $\mathbf{p}=(p_1,p_2,p_3)$ in ${\mathbb{E}}_1^3$, a vector is classified as follows [1]:

• Timelike if ${〈\mathbf{q},\mathbf{q}〉}_L<0$.
• Spacelike if ${〈\mathbf{q},\mathbf{q}〉}_L>0$.
• Lightlike (null) if ${〈\mathbf{q},\mathbf{q}〉}_L=0$ and $\mathbf{q}\ne 0$.

A smooth, regular curve $\zeta :I\to {\mathbb{E}}_1^3$ parametrized by arc-length s is then categorized by the causal character of its velocity vector ${\zeta }^{\prime }\left(s\right)$. For non-lightlike curves, one may construct the Lorentzian Frenet frame $\left\{{\mathbf{f}}_1\left(s\right),{\mathbf{f}}_2\left(s\right),{\mathbf{f}}_3\left(s\right)\right\}$ where

• ${\mathbf{f}}_1\left(s\right)={\zeta }^{\prime }\left(s\right)$ is the unit tangent;
• ${\mathbf{f}}_2\left(s\right)={\displaystyle \frac{{\mathbf{f}}_1^{\prime }\left(s\right)}{{∥{\mathbf{f}}_1^{\prime }\left(s\right)∥}_L}}$ is the principal normal;
• ${\mathbf{f}}_3\left(s\right)={\mathbf{f}}_1\left(s\right)\times {\mathbf{f}}_2\left(s\right)$ is the binormal vector defined through the Lorentzian cross-product [1].

These vectors obey the Lorentzian Frenet–Serret equations of the form

$$\left[\begin{array}{c}{\mathbf{f}}_1^{\prime }\\ {\mathbf{f}}_2^{\prime }\\ {\mathbf{f}}_3^{\prime }\end{array}\right]=\left[\begin{array}{ccc}0& \kappa & 0\\ {ϵ}_1\kappa & 0& \tau \\ 0& {ϵ}_2\tau & 0\end{array}\right]\left[\begin{array}{c}{\mathbf{f}}_1\\ {\mathbf{f}}_2\\ {\mathbf{f}}_3\end{array}\right]$$

(1)

where $\kappa \left(s\right)$ is the curvature, $\tau \left(s\right)$ is the torsion, and ${ϵ}_1={〈{\mathbf{f}}_2,{\mathbf{f}}_2〉}_L$, ${ϵ}_2={〈{\mathbf{f}}_3,{\mathbf{f}}_3〉}_L$ take the values $\pm 1$ in accordance with the causal character of ${\mathbf{f}}_2$ and ${\mathbf{f}}_3$. The Lorentzian cross-product of two vectors $\mathbf{u}=(u_1,u_2,u_3)$ and $\mathbf{v}=(v_1,v_2,v_3)$ in Minkowski 3-space ${\mathbb{E}}_1^3$ is defined as the determinant

$$\mathbf{u}\times \mathbf{v}=\left|\begin{array}{ccc}-{\mathbf{e}}_1& {\mathbf{e}}_2& {\mathbf{e}}_3\\ u_1& u_2& u_3\\ v_1& v_2& v_3\end{array}\right|=(u_3{v}_2-u_2{v}_3,u_3{v}_1-u_1{v}_3,u_1{v}_2-u_2{v}_1).$$

This reflects the orientation and metric of Minkowski 3-space with signature $(-,+,+)$ [1].

Understanding the Frenet apparatus in this setting is crucial when defining relationships between curves, as it allows us to characterize how their respective frames align or differ at corresponding points. This, in turn, enables the classification of associated curve types—such as Bertrand, Mannheim, and osculating mates—based on the relative orientation of their Frenet vectors or osculating planes. Several classical mate curve types are defined via specific alignments among the Frenet vectors:

Bertrand Curves: Two curves $\gamma$ and ${\gamma }^{*}$ form a Bertrand pair if there exists a smooth function $\mu \left(s\right)$ such that

$${\gamma }^{*}\left(s\right)=\gamma \left(s\right)+\mu \left(s\right){\mathbf{f}}_2\left(s\right)\mathrm{and}{\mathbf{f}}_2^{*}\left(s\right)=\pm {\mathbf{f}}_2\left(s\right),$$

meaning the principal normals at corresponding points remain collinear [2,3,4,5,6,7,8,9,10,11,12,13,14].

Mannheim Curves: In a Mannheim pair, the principal normal of one curve coincides with the binormal of its mate, i.e., [5,11,15,16,17,18]

$${\phi }^{*}\left(s\right)=\phi \left(s\right)+\mu \left(s\right){\mathbf{f}}_2\left(s\right)\mathrm{with}{\mathbf{f}}_3^{*}\left(s\right)=\pm {\mathbf{f}}_2\left(s\right).$$

Bäcklund Curves: Curves derived from the Bäcklund transformation are known as Bäcklund curves, and are typically characterized by having constant torsion. Originally formulated for surfaces of constant negative Gaussian curvature, the Bäcklund transformation was later adapted to the theory of space curves, particularly in Euclidean 3-space [19]. In this context, the transformation maps a curve $\delta \left(s\right)$ to a new curve $\tilde{\delta }\left(s\right)$ by

$$\tilde{\delta }=\delta +{\displaystyle \frac{2c}{c^2+{\tau }^2}}\left(cos\eta {\mathbf{f}}_1+sin\eta {\mathbf{f}}_3\right)$$

where $\eta$ is the angle between the displacement vector $\tilde{\delta }-\delta$ and the Frenet frame $\left\{{\mathbf{f}}_1,{\mathbf{f}}_2,{\mathbf{f}}_3\right\}$, with $\kappa$, $\tau$ denoting curvature and torsion, respectively.

This transformation satisfies the differential condition

$${\displaystyle \frac{\partial \eta }{\partial s}}=csin\eta -\kappa .$$

Furthermore, the torsions of Bäcklund pairs at corresponding points are equal and given by

$$\tau ={\tau }^{*}={\displaystyle \frac{sin\eta }{\mu }}$$

where $\mu$ denotes the separation distance, and $\eta$ is the angle between the binormal vectors at corresponding points [20,21].

In Minkowski 3-space, these transformations have been extended to accommodate the Lorentzian setting. Although Minkowski 4-space plays a central role in the formalism of special and general relativity, working in Minkowski 3-space provides a geometrically meaningful and computationally manageable setting to investigate curve behavior under Lorentzian geometry. It allows the extension of classical differential geometry techniques while avoiding the full complexity of spacetime structures. This dimensionally reduced approach has been effectively employed in numerous studies to gain insights into relativistic curvature and frame-based associations. Abdel-Baky established that the curvatures of surfaces associated with Bäcklund congruences are given by the following [22]:

• $-{\displaystyle \frac{{sinh}^2\eta }{{\rho }^2}}$ for spacelike surfaces with spacelike congruence;
• ${\displaystyle \frac{{sin}^2\eta }{{\rho }^2}}$ for timelike surfaces with spacelike congruence;
• ${\displaystyle \frac{{sinh}^2\eta }{{\rho }^2}}$ for timelike surfaces with timelike congruence.

In addition, Akdeniz rigorously studied Bäcklund transformations for both spacelike and timelike curves in Minkowski space and extended the theory to null Cartan curves, showing that Bäcklund transformations map one null Cartan spiral to another compatible one [23,24,25].

Natural Mates: A natural mate is defined by the alignment of the tangent vector of one curve with the normal vector of another, thereby creating a less rigid association than that required for Bertrand pairs [26].

Parallel Mates: In a parallel mate configuration, the tangent vectors of the two curves are parallel (or anti-parallel), and the line joining corresponding points is orthogonal to the common tangent, suggesting a form of parallel transport along the curves.

Each of these classical definitions captures a unique geometric interaction; however, they are typically treated as distinct cases. A unified framework that systematically encompasses all classical curve pair relationships in Minkowski space—while incorporating the causal structure and adapted Frenet frames—remains an active area of research. The present study contributes to this effort by extending the concept of osculating mates and demonstrating how various known curve associations can be interpreted within this generalized setting.

In this direction, Çelik and Özdemir introduced a new class of associated curve pairs, termed osculating curve mates, in the Euclidean setting [27].

Their construction served as a generalization of certain well-known curve pairs, including Mannheim and Bäcklund curves. Motivated by their approach, the present study extends the concept of osculating mates to the three-dimensional Minkowski space. One of our goals is to determine whether specific instances of this generalized configuration correspond to previously established mate types under the Lorentzian metric.

Motivation

Despite the extensive literature on associated curve pairs in both Euclidean and Lorentzian geometries, existing classifications such as Bertrand, Mannheim, and Bäcklund curves are each limited to very specific configurations—typically involving the alignment of a single Frenet vector (e.g., normal or binormal) with another. While valuable, these frameworks do not fully capture the broader spectrum of possible geometric similarities and structural relationships between two curves, especially those involving multiple frame components or combined plane constraints.

Furthermore, most known mate curve definitions assume either a fixed distance or linear dependency between frame vectors, often reducing their applicability to more general or dynamic geometric contexts. In particular, little has been done to systematically explore how the osculating planes of two non-lightlike curves interact in Minkowski 3-space, or how the angle between shared directional vectors and tangent lines can be used to construct more flexible notions of curve association—namely, associations defined not solely by fixed distances or linear dependencies, but also by angular constraints and geometric alignment within shared planes.

Motivated by this gap, the present study aims to introduce and develop a new class of associated curves—osculating curve pairs—defined via a shared directional vector lying in the intersection of the osculating planes of two curves and forming a fixed angle with their respective tangent vectors. This approach provides a framework that contributes to the development of a unified theoretical structure for understanding diverse curve relationships in Minkowski space.

Our goal is to construct this theory rigorously, derive transformation formulas for Frenet frames, establish curvature and torsion relationships, and interpret the results in both matrix and algebraic (split quaternionic) settings. This contributes to the general effort of extending classical geometric concepts to the setting of pseudo-Euclidean spaces and enhances our understanding of how curves relate through their osculating planes, shared frame directions, and the evolution of their respective geometric invariants in relativistic geometries.

In this paper, we introduce and investigate the concept of osculating curve pairs in Minkowski 3-space as a novel generalization of known associated curves. We rigorously define these pairs based on a shared directional vector lying in the intersection of the osculating planes of two non-lightlike curves and forming a fixed angle with their tangent vectors. We classify these osculating pairs into various configurations depending on the causal character of the curves (timelike or spacelike) and derive explicit transformation matrices between their Frenet frames. Moreover, we formulate relationships between the curvature and torsion functions of the original curve and its osculating mate. These transformations are also represented algebraically using split quaternions—a hyperbolic analogue of Hamilton’s quaternions suited to pseudo-Euclidean geometry—to highlight their rotational structure in Minkowski 3-space. Special cases are discussed in which the osculating pair reduces to classical Bertrand, Mannheim, or Bäcklund curves. The results demonstrate how this framework unifies and extends existing theories of associated curves within the context of Lorentzian geometry.

2. Osculating Curve Pairs

Definition 1.

Let ξ and ${\xi }^{*}$ be two regular, non-lightlike space curves in Minkowski 3-space, each defined over an open interval $I\subset \mathbb{R}$ for every $s\in I$. Define the map

$${\xi }^{*}\left(s\right)=\xi \left(s\right)+\mu \left(s\right)\mathbf{x}\left(s\right)$$

where $\mu \left(s\right)$ represents the function that gives the distance, and $\mathbf{x}\left(s\right)$ is a unit vector that is not lightlike for each $s\in I$. When the vector $\mathbf{x}\left(s\right)$ lies along the line where the osculating planes of the two curves ξ and ${\xi }^{*}$ intersect, and the angle between $\mathbf{x}\left(s\right)$ and the tangent vectors at the corresponding points of the curves is $\eta \left(s\right)$, the pair of curves $\left\{\xi ,{\xi }^{*}\right\}$ is referred to as osculating curve pairs or osculating mates. In this context, ξ is referred to as the osculating curve and ${\xi }^{*}$ as the equiosculating curve. Additionally, ${\xi }^{*}\left(s\right)$ is considered the osculating mate of $\xi \left(s\right)$. If the curves $\left\{\xi ,{\xi }^{*}\right\}$ are osculating mates, the following conditions are met:

i.

${\xi }^{*}\left(s\right)-\xi \left(s\right)\in {\mathbb{OP}}_{\xi \left(s\right)}\cap {\mathbb{OP}}_{{\xi }^{*}\left(s\right)}$.

ii.

$\mu \left(s\right)=\left|\xi \left(s\right){\xi }^{*}\left(s\right)\right|$.

iii.

$\langle {\mathbf{f}}_1\left(s\right),\mathbf{x}\left(s\right)\rangle =\langle {\mathbf{f}}_1^{*}\left(s\right),\mathbf{x}\left(s\right)\rangle =cos\eta \left(s\right)$.

where ${\mathbb{OP}}_{\xi \left(s\right)}$, ${\mathbb{OP}}_{{\xi }^{*}\left(s\right)}$ refer to the planes that osculate the curves ξ and ${\xi }^{*}$ at the respective positions $\xi \left(s\right)$ and ${\xi }^{*}\left(s\right)$.

Proposition 1.

If ${\xi }^{*}$ is the osculating mate of the non-lightlike curve ξ in the space ${\mathbb{E}}_1^3,$ then the type (spacelike or timelike) of ${\xi }^{*}$ is the same as the type of the curve ξ.

Proof. 

Suppose $\xi$ and ${\xi }^{*}$ are of different types and $\left\{{\mathbf{f}}_1,{\mathbf{f}}_2,{\mathbf{f}}_3\right\}$ and $\left\{{\mathbf{f}}_1^{*},{\mathbf{f}}_2^{*},{\mathbf{f}}_3^{*}\right\}$ are the Frenet vectors of these curves. Let $\xi$ and ${\xi }^{*}$ be a timelike and spacelike curve with a timelike normal, respectively. Since $\mathbf{x}$ lies along the line formed by the intersection of the osculating planes of the curves $\xi$ and ${\xi }^{*}$, it follows that $\mathbf{x}\in Sp\left\{{\mathbf{f}}_1,{\mathbf{f}}_2\right\}$ and $\mathbf{x}\in Sp\left\{{\mathbf{f}}_1^{*},{\mathbf{f}}_2^{*}\right\}$. If the curve $\xi$ is timelike, then ${\mathbf{f}}_1$ is timelike and ${\mathbf{f}}_2$ is spacelike. Furthermore, since the vector $\mathbf{x}$ forms an angle $\omega$ with the tangent vector ${\mathbf{f}}_1$, the timelike vector $\mathbf{x}$ is given by

$$\mathbf{x}=cosh\omega {\mathbf{f}}_1+sinh\omega {\mathbf{f}}_2.$$

Thus, we can express the following equation as

$${\xi }^{*}=\xi +\mu \left(cosh\omega {\mathbf{f}}_1+sinh\omega {\mathbf{f}}_2\right).$$

Additionally, since $\mathbf{x}$ is orthogonal to the spacelike binormal vectors ${\mathbf{f}}_3^{*}$ and ${\mathbf{f}}_3$, the vectors $\mathbf{y}={\mathbf{f}}_3\times \mathbf{x}$ (spacelike) and ${\mathbf{y}}^{*}={\mathbf{f}}_3^{*}\times \mathbf{x}$ (timelike) are confined to the osculating planes. Moreover, the triplets $\{\mathbf{x},{\mathbf{y}}^{*},{\mathbf{f}}_3^{*}\}$ and $\left\{\mathbf{x},\mathbf{y},{\mathbf{f}}_3\right\}$ create pseudo-orthonormal bases for the curves $\xi$ and ${\xi }^{*}$. In consequence, the following system of matrix equations holds:

$$\left[\begin{array}{c}\mathbf{x}\\ \mathbf{y}\end{array}\right]=\left[\begin{array}{cc}cosh\omega & sinh\omega \\ sinh\omega & cosh\omega \end{array}\right]\left[\begin{array}{c}{\mathbf{f}}_1\\ {\mathbf{f}}_2\end{array}\right]$$

(2)

and

$$\left[\begin{array}{c}\mathbf{x}\\ {\mathbf{y}}^{*}\end{array}\right]=\left[\begin{array}{cc}cosh\omega & sinh\omega \\ -sinh\omega & -cosh\omega \end{array}\right]\left[\begin{array}{c}{\mathbf{f}}_1^{*}\\ {\mathbf{f}}_2^{*}\end{array}\right].$$

(3)

Let $\eta$ denote the angle formed between ${\mathbf{f}}_3^{*}$ and ${\mathbf{f}}_3$. Given that the vectors ${\mathbf{f}}_3$, $\mathbf{y}$, ${\mathbf{f}}_3^{*}$, ${\mathbf{y}}^{*}$ are all pseudo-orthogonal to $\mathbf{x}$, we can derive the following expressions for ${\mathbf{f}}_3^{*}$ and ${\mathbf{y}}^{*}$:

$${\mathbf{f}}_3^{*}=sin\eta \mathbf{y}+cos\eta {\mathbf{f}}_3=sin\eta sinh\omega {\mathbf{f}}_1+sin\eta cosh\omega {\mathbf{f}}_2+cos\eta {\mathbf{f}}_3$$

and

$${\mathbf{y}}^{*}=cosh\eta \mathbf{y}-sinh\eta {\mathbf{f}}_3=cosh\eta sinh\omega {\mathbf{f}}_1+cosh\eta cosh\omega {\mathbf{f}}_2-sinh\eta {\mathbf{f}}_3.$$

These relationships are derived from the equalities in references (2) and (3). Thus, we obtain

$$\begin{array}{ccc}\hfill {\mathbf{f}}_1^{*}& =& cosh\omega \mathbf{x}+sinh\omega {\mathbf{y}}^{*}\hfill \\ & =& cosh\omega \left({\mathbf{f}}_{1}cosh\omega +{\mathbf{f}}_{2}sinh\omega \right)+sinh\omega cosh\eta sinh\omega {\mathbf{f}}_1\hfill \\ & & +cosh\eta cosh\omega {\mathbf{f}}_2-sinh\eta {\mathbf{f}}_3\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill {\mathbf{f}}_2^{*}& =& -sinh\omega \mathbf{x}-cosh\omega {\mathbf{y}}^{*}\hfill \\ & =& -sinh\omega \left({\mathbf{f}}_{1}cosh\omega +{\mathbf{f}}_{2}sinh\omega \right)-cosh\omega cosh\eta sinh\omega {\mathbf{f}}_1\hfill \\ & & -{cosh}^2\omega cosh\eta {\mathbf{f}}_2+cosh\omega sinh\eta {\mathbf{f}}_3.\hfill \end{array}$$

Thus, the vector fields ${\mathbf{f}}_1^{*},$${\mathbf{f}}_2^{*}$ and ${\mathbf{f}}_3^{*}$ are given by the following expressions:

$$\left[\begin{array}{c}{\mathbf{f}}_1^{*}\\ {\mathbf{f}}_2^{*}\\ {\mathbf{f}}_3^{*}\end{array}\right]=\left[\begin{array}{ccc}cosh\eta +(1-cosh\eta ){cosh}^2\omega & (1-cosh\eta )cosh\omega sinh\omega & sinh\omega sinh\eta \\ (cosh\eta -1)sinh\omega cosh\omega & {cosh}^2\omega cosh\eta -{sinh}^2\omega & -cosh\omega sinh\eta \\ sin\eta sinh\omega & sin\eta cosh\omega & cos\eta \end{array}\right]\left[\begin{array}{c}{\mathbf{f}}_1\\ {\mathbf{f}}_2\\ {\mathbf{f}}_3\end{array}\right].$$

Here, we get a contradiction, since the determinant of the matrix transforming the orthonormal frame $\left\{{\mathbf{f}}_1^{*},{\mathbf{f}}_2^{*},{\mathbf{f}}_3^{*}\right\}$ and $\left\{{\mathbf{f}}_1,{\mathbf{f}}_2,{\mathbf{f}}_3\right\}$ is not 1. Therefore, our assumption is wrong. It can be proved similarly that this contradiction is encountered in other different choices of mates of curve. □

Corollary 1.

As a conclusion to this proposition, we will examine osculating curve mates in three cases.

i.

ξ and ${\xi }^{*}$ are timelike.

ii.

ξ and ${\xi }^{*}$ are spacelike curves with spacelike normals.

iii.

ξ and ${\xi }^{*}$ are spacelike curves with timelike normals.

Theorem 1.

Consider $\xi \left(s\right)$ as a timelike curve, with its tangent and normal vector fields denoted by ${\mathbf{f}}_1\left(s\right)$ and ${\mathbf{f}}_2\left(s\right)$, respectively. The osculating mate ${\xi }^{*}$ of $\xi \left(s\right)$ can subsequently be expressed as

$${\xi }^{*}\left(s\right)=\xi \left(s\right)+\mu \left(s\right)\left[cosh\eta \left(s\right){\mathbf{f}}_1\left(s\right)+sinh\eta \left(s\right){\mathbf{f}}_2\left(s\right)\right]$$

(4)

where $\mu \left(s\right)\ne 0$ represents the distance function between the curves ξ and ${\xi }^{*}$. Furthermore, the following relationship holds between the corresponding points of the timelike curves ξ and ${\xi }^{*}$:

$${\mu }^{\prime }=cosh\eta \left(s\right)\left[{\delta }^{*}\left(s\right)-\delta \left(s\right)\right]$$

(5)

where $\delta =∥{\xi }^{\prime }∥$ and ${\delta }^{*}=∥{\xi }^{*\prime }∥$.

Proof. 

Given that ${\xi }^{*}$ is the osculating mate of the curve $\xi$, the expression $\mu \mathbf{x}\left(s\right)={\xi }^{*}\left(s\right)-\xi \left(s\right)$ forms a pseudo-angle $\eta \left(s\right)$ with the tangent timelike vector fields ${\mathbf{f}}_1\left(s\right)$ and ${\mathbf{f}}_1^{*}\left(s\right)$ along the respective timelike curves $\xi$ and ${\xi }^{*}$. Additionally, since the unit vector $\mathbf{x}\left(s\right)$ lies within the osculating plane, we can represent $\mathbf{x}\left(s\right)$ as

$$\mathbf{x}=cosh\eta \left(s\right){\mathbf{f}}_1\left(s\right)+sinh\eta \left(s\right){\mathbf{f}}_2\left(s\right).$$

and

$${\xi }^{*}=\xi +\mu \left[cosh\eta \left(s\right){\mathbf{f}}_1\left(s\right)+sinh\eta \left(s\right){\mathbf{f}}_2\left(s\right)\right].$$

It is evident that $\mathbf{x}$ is timelike. Taking the derivative of this equation with respect to s and applying the Frenet formulas (1), we get

$$\begin{array}{cc}{\delta }^{*}{\mathbf{f}}_1^{*}\hfill & =\left(\delta +{\mu }^{\prime }cosh\eta +{\eta }^{\prime }\mu sinh\eta +\delta \kappa \mu sinh\eta \right){\mathbf{f}}_1\hfill \\ \hfill & +\left({\mu }^{\prime }sinh\eta +\mu \delta \kappa cosh\eta +{\eta }^{\prime }\mu cosh\eta \right){\mathbf{f}}_2+\left(\mu \delta \tau sinh\eta \right){\mathbf{f}}_3.\hfill \end{array}$$

(6)

By applying the Lorentzian scalar product to both sides with the vector $\mathbf{x}$, and assuming that $\eta$ is positively oriented in both planes, we obtain

$$\begin{array}{cc}\hfill {\delta }^{*}〈\mathbf{x},{\mathbf{f}}_1^{*}〉& =\left(\delta +{\mu }^{\prime }cosh\eta +{\eta }^{\prime }\mu sinh\eta +\delta \kappa \mu sinh\eta \right)\left(-cosh\eta \right)\hfill \\ \hfill & +\left({\mu }^{\prime }sinh\eta +\mu \delta \kappa cosh\eta +{\eta }^{\prime }\mu cosh\eta \right)sinh\eta -{\delta }^{*}cosh\eta \hfill \\ \hfill & =-{\mu }^{\prime }-\delta cosh\eta .\hfill \end{array}$$

Consequently, the following equation is obtained:

$${\mu }^{\prime }=\left(cosh\eta \right)\left({\delta }^{*}-\delta \right).$$

□

Theorem 2.

Let $\xi \left(s\right)$ be a spacelike curve with a timelike normal. Given that its tangent vector field is ${\mathbf{f}}_1\left(s\right)$ and its normal vector field is ${\mathbf{f}}_2\left(s\right)$, the osculating mate ${\xi }^{*}$ of ξ can be expressed as follows:

$${\xi }^{*}\left(s\right)=\xi \left(s\right)+\mu \left(s\right)\left[cosh\eta \left(s\right){\mathbf{f}}_1\left(s\right)+sinh\eta \left(s\right){\mathbf{f}}_2\left(s\right)\right]$$

(7)

where $\mu \left(s\right)\ne 0$ represents the distance function between the curves ξ and ${\xi }^{*}$. Additionally, the following equality holds:

$${\mu }^{\prime }=cosh\eta \left(s\right)\left[{\delta }^{*}\left(s\right)-\delta \left(s\right)\right]$$

(8)

at the corresponding points of the spacelike curves ξ and ${\xi }^{*},$ where $\delta =∥{\xi }^{\prime }∥$ and ${\delta }^{*}=∥{\xi }^{*\prime }∥$.

Proof. 

This can be proved similarly to the previous theorem. Also, we have the equality

$$\begin{array}{cc}{\delta }^{*}{\mathbf{f}}_1^{*}\hfill & =\left(\delta +{\mu }^{\prime }cosh\eta +{\eta }^{\prime }\mu sinh\eta +\delta \kappa \mu sinh\eta \right){\mathbf{f}}_1\hfill \\ \hfill & +\left({\mu }^{\prime }sinh\eta +\mu \delta \kappa cosh\eta +{\eta }^{\prime }\mu cosh\eta \right){\mathbf{f}}_2+\left(\mu \delta \tau sinh\eta \right){\mathbf{f}}_3.\hfill \end{array}$$

(9)

for this case. □

Corollary 2.

If the curves ξ and ${\xi }^{*}$ are both timelike curves or spacelike curves with a timelike normal, then the distance between the reciprocal points of the curves ξ and ${\xi }^{*}$ is constant if and only if $\delta ={\delta }^{*},$ since $cosh\eta >0$.

Theorem 3.

Let $\xi \left(s\right)$ be a spacelike curve with a spacelike normal. Given that its tangent vector field is ${\mathbf{f}}_1\left(s\right)$ and its normal vector field is ${\mathbf{f}}_2\left(s\right)$, the osculating mate ${\xi }^{*}$ of ξ can be expressed as follows:

$${\xi }^{*}\left(s\right)=\xi \left(s\right)+\mu \left(s\right)\left[cos\eta \left(s\right){\mathbf{f}}_1\left(s\right)+sin\eta \left(s\right){\mathbf{f}}_2\left(s\right)\right]$$

(10)

where $\eta \left(s\right)\in [0,2\pi )$ and $\mu \left(s\right)\ne 0$ represents the distance function between the curves ξ and ${\xi }^{*}$. Furthermore, the following equality holds:

$${\mu }^{\prime }=cos\eta \left(s\right)\left[{\delta }^{*}\left(s\right)-\delta \left(s\right)\right]$$

(11)

which relates the corresponding points of the spacelike curves ξ and ${\xi }^{*},$ where $\delta =∥{\xi }^{\prime }∥$ and ${\delta }^{*}=∥{\xi }^{*\prime }∥$.

Proof. 

Consider ${\xi }^{*}$ as the osculating mate of the curve $\xi$. In this case, the vector difference $\mu \mathbf{x}\left(s\right)={\xi }^{*}\left(s\right)-\xi \left(s\right)$ forms a pseudo-angle $\eta \left(s\right)$ with the tangent spacelike vector fields ${\mathbf{f}}_1\left(s\right)$ and ${\mathbf{f}}_1^{*}\left(s\right)$ along the spacelike curves $\xi$ and ${\xi }^{*}$. Conversely, given that the unit vector $\mathbf{x}\left(s\right)$ lies within the osculating plane, we can represent it as

$$\mathbf{x}=cos\eta {\mathbf{f}}_1+sin\eta {\mathbf{f}}_2$$

and consequently, the osculating mate ${\xi }^{*}$ can be written as

$${\xi }^{*}=\xi +\mu \left[cos\eta {\mathbf{f}}_1+sin\eta {\mathbf{f}}_2\right].$$

Now, differentiating both sides of this equation with respect to s, and applying the Frenet formulas, we obtain the following result:

$$\begin{array}{cc}{\delta }^{*}{\mathbf{f}}_1^{*}\hfill & =\left(\delta +{\mu }^{\prime }cos\eta -{\eta }^{\prime }\mu sin\eta -\delta \kappa \mu sin\eta \right){\mathbf{f}}_1\hfill \\ \hfill & +\left({\mu }^{\prime }sin\eta +\mu \delta \kappa cos\eta +{\eta }^{\prime }\mu cos\eta \right){\mathbf{f}}_2+\left(\mu \delta \tau sin\eta \right){\mathbf{f}}_3\hfill \end{array}.$$

(12)

By applying the Lorentzian scalar product to both sides with the vector $\mathbf{x}$, we obtain

$${\delta }^{*}cos\eta ={\mu }^{\prime }+\delta cos\eta .$$

From this, we obtain the relation ${\mu }^{\prime }=\left(cos\eta \right)\left({\delta }^{*}-\delta \right).$ □

Corollary 3.

i.

The distance between the matching points of the curves ξ and ${\xi }^{*}$ remains constant, provided that ${\delta }^{*}=\delta$, given that $\eta \ne \pi /2$.

ii.

Given that η equals $\pi /2$ along the curves and ${\mu }^{\prime }=0$, the distance between the matching points of the curves ξ and ${\xi }^{*}$ is

$$l=\left|{\xi }^{*}\left(s\right)-\xi \left(s\right)\right|=\left|\mu \right|$$

which remains constant along the curves. It is important to emphasize that this does not imply ${\delta }^{*}=\delta$.

3. Frenet Apparatus and Its Relation in Osculating Curve Pairs

Theorem 4.

Consider the osculating timelike curve pair $\left\{\xi ,{\xi }^{*}\right\}$, defined by the equation

$${\xi }^{*}=\xi +\mu \left[cosh\omega {\mathbf{f}}_1+sinh\omega {\mathbf{f}}_2\right],$$

where the Frenet vector fields associated with the curves ξ and ${\xi }^{*}$ are $\left\{{\mathbf{f}}_1,{\mathbf{f}}_2,{\mathbf{f}}_3\right\}$ and $\left\{{\mathbf{f}}_1^{*},{\mathbf{f}}_2^{*},{\mathbf{f}}_3^{*}\right\}$, respectively. Let $\eta \in [0,2\pi )$ represent the angle between the reciprocal binormal vectors of the timelike curves. In this case, the following matrix equality holds:

$$\left[\begin{array}{c}{\mathbf{f}}_1^{*}\\ {\mathbf{f}}_2^{*}\\ {\mathbf{f}}_3^{*}\end{array}\right]=\left[\begin{array}{ccc}-{sinh}^2\omega cos\eta +{cosh}^2\omega & \left(sinh\omega cosh\omega \right)\left(1-cos\eta \right)& sinh\omega sin\eta \\ \left(sinh\omega cosh\omega \right)\left(cos\eta -1\right)& {cosh}^2\omega cos\eta -{sinh}^2\omega & -cosh\omega sin\eta \\ sin\eta sinh\omega & sin\eta cosh\omega & cos\eta \end{array}\right]\left[\begin{array}{c}{\mathbf{f}}_1\\ {\mathbf{f}}_2\\ {\mathbf{f}}_3\end{array}\right].$$

(13)

Proof. 

As the vector field $\mathbf{x}=cosh\omega {\mathbf{f}}_1+sinh\omega {\mathbf{f}}_2$ lies within the osculating plane, it is orthogonally related to both binormal vectors ${\mathbf{f}}_3$ and ${\mathbf{f}}_3^{*}$. Consequently, the vectors $\mathbf{y}={\mathbf{f}}_3\times \mathbf{x}$ and ${\mathbf{y}}^{*}={\mathbf{f}}_3^{*}\times \mathbf{x}$ are contained within the respective osculating planes of the timelike curves. Moreover, the pairs $\{\mathbf{x},{\mathbf{y}}^{*},{\mathbf{f}}_3^{*}\}$ and $\left\{\mathbf{x},\mathbf{y},{\mathbf{f}}_3\right\}$ form pseudo-orthonormal frames for the curves $\xi$ and ${\xi }^{*}$, respectively. Thus, the matrix equations can be formulated as follows:

$$\left[\begin{array}{c}\mathbf{x}\\ \mathbf{y}\end{array}\right]=\left[\begin{array}{cc}cosh\omega & sinh\omega \\ sinh\omega & cosh\omega \end{array}\right]\left[\begin{array}{c}{\mathbf{f}}_1\\ {\mathbf{f}}_2\end{array}\right]$$

(14)

and

$$\left[\begin{array}{c}\mathbf{x}\\ {\mathbf{y}}^{*}\end{array}\right]=\left[\begin{array}{cc}cosh\omega & sinh\omega \\ sinh\omega & cosh\omega \end{array}\right]\left[\begin{array}{c}{\mathbf{f}}_1^{*}\\ {\mathbf{f}}_2^{*}\end{array}\right].$$

(15)

Let $\eta$ represent the angle formed between ${\mathbf{f}}_3$ and ${\mathbf{f}}_3^{*}$. Given that the vectors ${\mathbf{f}}_3,$$\mathbf{y},$${\mathbf{f}}_3^{*},$${\mathbf{y}}^{*}$ are all pseudo-orthogonal to the vector $\mathbf{x}$, we can apply the equality (14) to obtain

$${\mathbf{f}}_3^{*}=sin\eta \mathbf{y}+cos\eta {\mathbf{f}}_3=sin\eta sinh\omega {\mathbf{f}}_1+sin\eta cosh\omega {\mathbf{f}}_2+cos\eta {\mathbf{f}}_3$$

and

$${\mathbf{y}}^{*}=cos\eta \mathbf{y}-sin\eta {\mathbf{f}}_3=cos\eta sinh\omega {\mathbf{f}}_1+cos\eta cosh\omega {\mathbf{f}}_2-sin\eta {\mathbf{f}}_3.$$

Thus, by utilizing the equalities (14) and (15), we derive

$$\begin{array}{cc}\hfill \left[\begin{array}{c}{\mathbf{f}}_1^{*}\\ {\mathbf{f}}_2^{*}\end{array}\right]& =\left[\begin{array}{cc}cosh\omega & -sinh\omega \\ -sinh\omega & cosh\omega \end{array}\right]\left[\begin{array}{c}\mathbf{x}\\ {\mathbf{y}}^{*}\end{array}\right]\hfill \\ \hfill & =\left[\begin{array}{c}\left\{\begin{array}{c}\left({cosh}^2\omega -{sinh}^2\omega cos\eta \right){\mathbf{f}}_1+\left(cosh\omega sinh\omega -cosh\omega sinh\omega cos\eta \right){\mathbf{f}}_2\\ +sinh\omega sin\eta {\mathbf{f}}_3\end{array}\right\}\\ \left\{\begin{array}{c}\left(cosh\omega sinh\omega cos\eta -cosh\omega sinh\omega \right){\mathbf{f}}_1+\left({cosh}^2\omega cos\eta -{sinh}^2\omega \right){\mathbf{f}}_2\\ -cosh\omega sin\eta {\mathbf{f}}_3\end{array}\right\}\end{array}\right].\hfill \end{array}$$

As a result, we obtain the matrix equality (13). □

Theorem 5.

Consider the osculating spacelike curve pair $\left\{\xi ,{\xi }^{*}\right\}$, with timelike normals, defined by the equation

$${\xi }^{*}=\xi +\mu \left[cosh\gamma {\mathbf{f}}_1+sinh\gamma {\mathbf{f}}_2\right],$$

where $\left\{{\mathbf{f}}_1^{*},{\mathbf{f}}_2^{*},{\mathbf{f}}_3^{*}\right\}$ and $\left\{{\mathbf{f}}_1,{\mathbf{f}}_2,{\mathbf{f}}_3\right\}$ represent the Frenet vector fields of the curves ${\xi }^{*}$ and ξ, respectively. If η denotes the pseudo-angle between the corresponding binormals of the curves, then the matrix equality is given by

$$\left[\begin{array}{c}{\mathbf{f}}_1^{*}\\ {\mathbf{f}}_2^{*}\\ {\mathbf{f}}_3^{*}\end{array}\right]=\left[\begin{array}{ccc}{cosh}^2\omega -{sinh}^2\omega cosh\eta & -sinh\omega cosh\omega \left(cosh\eta -1\right)& -sinh\omega sinh\eta \\ sinh\omega cosh\omega \left(cosh\eta -1\right)& -{sinh}^2\omega +cosh\eta {cosh}^2\omega & cosh\omega sinh\eta \\ sinh\eta sinh\omega & sinh\eta cosh\omega & cosh\eta \end{array}\right]\left[\begin{array}{c}{\mathbf{f}}_1\\ {\mathbf{f}}_2\\ {\mathbf{f}}_3\end{array}\right].$$

(16)

Proof. 

Since the spacelike vector field $\mathbf{x}=cosh\omega {\mathbf{f}}_1+sinh\omega {\mathbf{f}}_2$ lies within the osculating plane, it remains pseudo-orthogonal to the binormal vectors ${\mathbf{f}}_3$ and ${\mathbf{f}}_3^{*}$. As a result, the vector fields $-\mathbf{y}={\mathbf{f}}_3\times \mathbf{x}$ and $-{\mathbf{y}}^{*}={\mathbf{f}}_3^{*}\times \mathbf{x}$ are contained within the osculating planes of the spacelike curves. Furthermore, the sets $\{\mathbf{x},{\mathbf{y}}^{*},{\mathbf{f}}_3^{*}\}$ and $\left\{\mathbf{x},\mathbf{y},{\mathbf{f}}_3\right\}$ constitute pseudo-orthonormal frames corresponding to the spacelike curves $\xi$ and ${\xi }^{*}$, respectively. Thus, the following matrix equations can be expressed as

$$\left[\begin{array}{c}\mathbf{x}\\ \mathbf{y}\end{array}\right]=\left[\begin{array}{cc}cosh\omega & sinh\omega \\ sinh\omega & cosh\omega \end{array}\right]\left[\begin{array}{c}{\mathbf{f}}_1\\ {\mathbf{f}}_2\end{array}\right]$$

(17)

and

$$\left[\begin{array}{c}\mathbf{x}\\ {\mathbf{y}}^{*}\end{array}\right]=\left[\begin{array}{cc}cosh\omega & sinh\omega \\ sinh\omega & cosh\omega \end{array}\right]\left[\begin{array}{c}{\mathbf{f}}_1^{*}\\ {\mathbf{f}}_2^{*}\end{array}\right].$$

(18)

Let $\eta$ represent the pseudo-angle between ${\mathbf{f}}_3^{*}$ and ${\mathbf{f}}_3$. Given that the vectors ${\mathbf{f}}_3^{*}$, ${\mathbf{y}}^{*}$$\mathbf{x}$, it follows from (17) that we obtain

$$\begin{array}{ccc}\hfill {\mathbf{f}}_3^{*}& =& sinh\eta sinh\omega {\mathbf{f}}_1+sinh\eta cosh\omega {\mathbf{f}}_2+cosh\eta {\mathbf{f}}_3,\hfill \\ \hfill {\mathbf{y}}^{*}& =& cosh\eta sinh\omega {\mathbf{f}}_1+cosh\omega cosh\eta {\mathbf{f}}_2+sinh\eta {\mathbf{f}}_3.\hfill \end{array}$$

Hence, by applying the equalities (17) and (18), we obtain

$$\left[\begin{array}{c}{\mathbf{f}}_1^{*}\\ {\mathbf{f}}_2^{*}\end{array}\right]=\left[\begin{array}{c}\left\{\begin{array}{c}\left({cosh}^2\omega -cosh\eta {sinh}^2\omega \right){\mathbf{f}}_1-\left(sinh\omega cosh\omega \right)\left(cosh\eta -1\right){\mathbf{f}}_2\\ -\left(sinh\omega sinh\eta \right){\mathbf{f}}_3\end{array}\right\}\\ \left\{\begin{array}{c}\left(sinh\omega cosh\omega \right)\left(cosh\eta -1\right){\mathbf{f}}_1+\left(-{sinh}^2\omega +cosh\eta {cosh}^2\omega \right){\mathbf{f}}_2\\ +cosh\omega sinh\eta {\mathbf{f}}_3\end{array}\right\}\end{array}\right].$$

Thus, we obtain the matrix equality (16). □

Theorem 6.

Consider the pair of osculating spacelike curves $\left\{\xi ,{\xi }^{*}\right\}$ with spacelike normals, which are defined by the equation

$${\xi }^{*}=\xi +\mu \left[cos\omega {\mathbf{f}}_1+sin\omega {\mathbf{f}}_2\right]$$

where $\left\{{\mathbf{f}}_1,{\mathbf{f}}_2,{\mathbf{f}}_3\right\}$ and $\left\{{\mathbf{f}}_1^{*},{\mathbf{f}}_2^{*},{\mathbf{f}}_3^{*}\right\}$ represent the Frenet vector fields of the curves ξ and ${\xi }^{*}$, respectively. Let η denote the pseudo-angle between the reciprocal binormals of these curves. From this, we derive the matrix equality

$$\left[\begin{array}{c}{\mathbf{f}}_1^{*}\\ {\mathbf{f}}_2^{*}\\ {\mathbf{f}}_3^{*}\end{array}\right]=\left[\begin{array}{ccc}{cos}^2\omega +cosh\eta {sin}^2\omega & -sin\omega cos\omega \left(cosh\eta -1\right)& -sinh\eta sin\omega \\ -sin\omega cos\omega \left(cosh\eta -1\right)& {sin}^2\omega +cosh\eta {cos}^2\omega & sinh\eta cos\omega \\ -sinh\eta sin\omega & sinh\eta cos\omega & cosh\eta \end{array}\right]\left[\begin{array}{c}{\mathbf{f}}_1\\ {\mathbf{f}}_2\\ {\mathbf{f}}_3\end{array}\right].$$

(19)

Proof. 

Since the vector field $\mathbf{x}=cos\omega {\mathbf{f}}_1+sin\omega {\mathbf{f}}_2$ resides within the osculating plane, it is orthogonally related to the binormal vectors ${\mathbf{f}}_3$ and ${\mathbf{f}}_3^{*}$. Consequently, the vectors $\mathbf{y}={\mathbf{f}}_3\times \mathbf{x}$ and ${\mathbf{y}}^{*}={\mathbf{f}}_3^{*}\times \mathbf{x}$ lie within the respective osculating planes of the spacelike curves. Additionally, the sets $\{\mathbf{x},{\mathbf{y}}^{*},{\mathbf{f}}_3^{*}\}$ and $\left\{\mathbf{x},\mathbf{y},{\mathbf{f}}_3\right\}$ define pseudo-orthonormal frames for the curves $\xi$ and ${\xi }^{*}$, respectively. As a result, we can express the matrix equations as follows:

$$\left[\begin{array}{c}\mathbf{x}\\ \mathbf{y}\end{array}\right]=\left[\begin{array}{cc}cos\omega & sin\omega \\ -sin\omega & cos\omega \end{array}\right]\left[\begin{array}{c}{\mathbf{f}}_1\\ {\mathbf{f}}_2\end{array}\right]$$

(20)

and

$$\left[\begin{array}{c}\mathbf{x}\\ {\mathbf{y}}^{*}\end{array}\right]=\left[\begin{array}{cc}cos\omega & sin\omega \\ -sin\omega & cos\omega \end{array}\right]\left[\begin{array}{c}{\mathbf{f}}_1^{*}\\ {\mathbf{f}}_2^{*}\end{array}\right].$$

(21)

Let $\eta$ denote the pseudo-angle between ${\mathbf{f}}_3^{*}$ and ${\mathbf{f}}_3$. Since all vectors ${\mathbf{f}}_3^{*}$, ${\mathbf{y}}^{*}$, ${\mathbf{f}}_3$, and $\mathbf{y}$ are pseudo-orthogonal to the vector $\mathbf{x}$, by utilizing the relation in (20), we derive the following:

$${\mathbf{f}}_3^{*}=-sinh\eta sin\omega {\mathbf{f}}_1+sinh\eta cos\omega {\mathbf{f}}_2+cosh\eta {\mathbf{f}}_3$$

and

$${\mathbf{y}}^{*}=-\left(cosh\eta sin\omega \right){\mathbf{f}}_1+\left(cos\omega cosh\eta \right){\mathbf{f}}_2+\left(sinh\eta \right){\mathbf{f}}_3.$$

Thus, by applying the equalities in (20) and (21), we obtain

$$\left[\begin{array}{c}{\mathbf{f}}_1^{*}\\ {\mathbf{f}}_2^{*}\end{array}\right]=\left[\begin{array}{c}\left({cos}^2\omega +cosh\eta {sin}^2\omega \right){\mathbf{f}}_1-\left(sin\omega cos\omega \right)\left(cosh\eta -1\right){\mathbf{f}}_2-\left(sinh\eta sin\omega \right){\mathbf{f}}_3\\ -\left(sin\omega cos\omega \right)\left(cosh\eta -1\right){\mathbf{f}}_1+\left({sin}^2\omega +cosh\eta {cos}^2\omega \right){\mathbf{f}}_2+\left(sinh\eta cos\omega \right){\mathbf{f}}_3\end{array}\right]$$

Consequently, the matrix equality (19) follows. □

4. Relations for the Curvature and Torsion of the Osculating Curve $\xi$

Theorem 7.

Consider an osculating timelike curve pair $\left\{\xi ,{\xi }^{*}\right\}$ defined by the equation

$${\xi }^{*}=\xi +\mu \left(\left(cosh\omega \right){\mathbf{f}}_1+\left(sinh\omega \right){\mathbf{f}}_2\right).$$

When the pseudo-angle between the binormals at corresponding points of ξ and ${\xi }^{*}$ satisfies $\eta \ne 0$ and the condition $\delta \ne {\delta }^{*}$ holds, the torsion and curvature of the curve $\xi \left(s\right)$ are expressed as

$$\begin{array}{c}\tau ={\displaystyle \frac{{\delta }^{*}sin\eta }{\delta \mu }},\hfill \\ \\ \hfill \kappa =-{\displaystyle \frac{\mu {\omega }^{\prime }-\delta sinh\omega +\left(sinh\omega cos\eta \right){\delta }^{*}}{\delta \mu }},\end{array}$$

(22)

respectively.

Proof. 

By comparing the equality (6) with

$${\mathbf{f}}_1^{*}=\left(-{sinh}^2\omega cos\eta +{cosh}^2\omega \right){\mathbf{f}}_1+\left(sinh\omega cosh\omega \right)\left(1-cos\eta \right){\mathbf{f}}_2+\left(sinh\omega sin\eta \right){\mathbf{f}}_3,$$

we derive the following system of equations:

$$\left\{\begin{array}{c}{\delta }^{*}\left(-{sinh}^2\omega cos\eta +{cosh}^2\omega \right)=\delta +{\mu }^{\prime }cosh\omega +{\omega }^{\prime }\mu sinh\omega +\delta \kappa \mu sinh\omega ,\hfill \\ {\delta }^{*}\left(sinh\omega cosh\omega \right)\left(1-cos\eta \right)=\left({\mu }^{\prime }sinh\omega +\mu \delta \kappa cosh\omega +{\omega }^{\prime }\mu cosh\omega \right),\hfill \\ {\delta }^{*}\left(sinh\omega sin\eta \right)=\mu \delta \tau sinh\omega .\hfill \end{array}\right.$$

For the case where $\delta \ne {\delta }^{*}$, by considering ${\mu }^{\prime }=\left(cosh\omega \right)\left({\delta }^{*}-\delta \right)$, we obtain the following expressions for the torsion and curvature:

$$\tau ={\displaystyle \frac{{\delta }^{*}sin\eta }{\delta \mu }}\mathrm{and}\kappa =-{\displaystyle \frac{\mu {\omega }^{\prime }-\delta sinh\omega +\left(sinh\omega cos\eta \right){\delta }^{*}}{\delta \mu }}.$$

□

Theorem 8.

Consider the osculating spacelike curve pair $\left\{\xi ,{\xi }^{*}\right\}$ with timelike normals, defined by the equation

$${\xi }^{*}=\xi +\mu \left(\left(cosh\omega \right){\mathbf{f}}_1+\left(sinh\omega \right){\mathbf{f}}_2\right).$$

When the pseudo-angle between the binormals at corresponding points of ξ and ${\xi }^{*}$ satisfies $\eta \ne 0$ and the condition $\delta \ne {\delta }^{*}$ holds, the torsion and curvature of the curve $\xi \left(s\right)$ are expressed as

$$\begin{array}{c}\tau =-{\displaystyle \frac{{\delta }^{*}sinh\eta }{\delta \mu }},\hfill \\ \\ \hfill \kappa ={\displaystyle \frac{\left(sinh\omega \right)\left(\delta -{\delta }^{*}\left(cosh\eta \right)\right)-\mu {\omega }^{\prime }}{\delta \mu }},\end{array}$$

(23)

respectively.

Proof. 

Similar to the proof of the preceding theorem, by comparing the equality (9) with

$${\mathbf{f}}_1^{*}=\left({cosh}^2\omega -\left({sinh}^2\omega \right)cosh\eta \right){\mathbf{f}}_1-\left(sinh\omega cosh\omega \right)\left(cosh\eta -1\right){\mathbf{f}}_2-\left(sinh\omega sinh\eta \right){\mathbf{f}}_3,$$

we derive formulas (23) under the condition $\delta \ne {\delta }^{*}$. □

Theorem 9.

Consider the osculating spacelike curve pair $\left\{\xi ,{\xi }^{*}\right\}$ with spacelike normals, defined by the equation

$${\xi }^{*}=\xi +\mu \left(\left(cos\omega \right){\mathbf{f}}_1+\left(sin\omega \right){\mathbf{f}}_2\right)$$

where $\omega \left(s\right)\in [0,2\pi )-\pi /2$. When the pseudo-angle between the binormals at corresponding points of ξ and ${\xi }^{*}$ satisfies $\eta \ne 0$ and the condition $\delta \ne {\delta }^{*}$ holds, the torsion and curvature of the curve $\xi \left(s\right)$ are expressed as

$$\begin{array}{c}\tau =-{\displaystyle \frac{{\delta }^{*}sinh\eta }{\delta \mu }},\hfill \\ \\ \hfill \kappa ={\displaystyle \frac{-\mu {\omega }^{\prime }+\left(sin\omega \right)\left(\delta -\left(cosh\eta \right){\delta }^{*}\right)}{\delta \mu }},\end{array}$$

(24)

respectively.

Proof. 

By comparing the equality (12) with

$${\mathbf{f}}_1^{*}=\left({cos}^2\omega +cosh\eta {sin}^2\omega \right){\mathbf{f}}_1-\left(sin\omega cos\omega \right)\left(cosh\eta -1\right){\mathbf{f}}_2-\left(sinh\eta sin\omega \right){\mathbf{f}}_3,$$

we derive the following system of equations:

$$\left\{\begin{array}{c}{\delta }^{*}\left({cos}^2\omega +cosh\eta {sin}^2\omega \right)=\delta +\left(\left(cos\omega \right)\left({\delta }^{*}-\delta \right)\right)\left(cos\omega \right)-{\omega }^{\prime }\mu \left(sin\omega \right)-\delta \kappa \mu \left(sin\omega \right),\hfill \\ -{\delta }^{*}\left(sin\omega cos\omega \right)\left(cosh\eta -1\right)=\left(\left(cos\omega \right)\left({\delta }^{*}-\delta \right)\right)\left(sin\omega \right)+\mu \delta \kappa \left(cos\omega \right)+{\omega }^{\prime }\mu \left(cos\omega \right),\hfill \\ -{\delta }^{*}\left(sinh\eta sin\omega \right)=\mu \delta \tau \left(sin\omega \right).\hfill \end{array}\right.$$

For the case where $\delta \ne {\delta }^{*}$, by considering ${\mu }^{\prime }=\left(cosh\omega \right)\left({\delta }^{*}-\delta \right)$, we obtain the following expressions for the torsion and curvature:

$$\tau =-{\displaystyle \frac{{\delta }^{*}sinh\eta }{\delta \mu }}\mathrm{and}\kappa ={\displaystyle \frac{-\mu {\omega }^{\prime }+\left(sin\omega \right)\left(\delta -\left(cosh\eta \right){\delta }^{*}\right)}{\delta \mu }}.$$

□

5. Analysis of Curvature and Torsion of the Osculating Curve ${\xi }^{*}$

Theorem 10.

Consider the osculating timelike curve pair $\left\{\xi ,{\xi }^{*}\right\}$ defined by the equation

$${\xi }^{*}=\xi +\mu \left(\left(cosh\omega \right){\mathbf{f}}_1+\left(sinh\omega \right){\mathbf{f}}_2\right).$$

where the pseudo-angle between the binormal vectors of the corresponding points of ξ and ${\xi }^{*}$ is denoted by $\eta \ne k\pi ,$$k\in \mathbb{Z}$. The torsion and curvature of the curve ${\xi }^{*}\left(s\right)$ are then given by the following expressions:

$$\begin{array}{c}{\tau }^{*}={\displaystyle \frac{sin\eta }{\mu }}-{\displaystyle \frac{{\eta }^{\prime }}{{\delta }^{*}cosh\omega }}\hfill \\ \\ \hfill {\kappa }^{*}=-{\displaystyle \frac{{\omega }^{\prime }}{{\delta }^{*}}}-{\displaystyle \frac{{\eta }^{\prime }tanh\omega cot\eta }{{\delta }^{*}}}-{\displaystyle \frac{\delta \tau sinh\omega }{{\delta }^{*}}}tan\left({\displaystyle \frac{\eta }{2}}\right).\end{array}$$

(25)

Additionally, the relation

$$\mu =ctan\left({\displaystyle \frac{\eta }{2}}\right)$$

holds between the angle $\eta \in [0,2\pi )$ and the distance function μ, where $c\in \mathbb{R}$ is a constant.

Proof. 

Let us now compute the derivative of the timelike tangent vector field ${\mathbf{f}}_1^{*}$, as presented in Equation (13). By utilizing the Frenet formulas for timelike curves, we derive the following expression:

$$\begin{array}{cc}{\delta }^{*}{\kappa }^{*}{\mathbf{f}}_2^{*}\hfill & =\left(\begin{array}{c}-{\omega }^{\prime }sinh2\omega cos\eta +{\eta }^{\prime }{sinh}^2\omega sin\eta +{\omega }^{\prime }sinh2\omega \\ +\delta \kappa sinh\omega cosh\omega \left(1-cos\eta \right)\end{array}\right){\mathbf{f}}_1\hfill \\ \hfill & +\left(\begin{array}{c}\left(-{sinh}^2\omega cos\eta +{cosh}^2\omega \right)\delta \kappa +{\omega }^{\prime }cosh2\omega \left(1-cos\eta \right)\\ +{\eta }^{\prime }sinh\omega cosh\omega sin\eta -\delta \tau sinh\omega sin\eta \end{array}\right){\mathbf{f}}_2\hfill \\ \hfill & +\left(\delta \tau sinh\omega cosh\omega \left(1-cos\eta \right)+{\omega }^{\prime }cosh\omega sin\eta +{\eta }^{\prime }sinh\omega cos\eta \right){\mathbf{f}}_3.\hfill \end{array}$$

(26)

By taking the Lorentzian scalar product of both sides with ${\mathbf{f}}_1$, we derive the relation

$${\delta }^{*}{\kappa }^{*}={\displaystyle \frac{-2{\omega }^{\prime }cos\eta }{cos\eta -1}}+{\displaystyle \frac{2{\omega }^{\prime }}{cos\eta -1}}+{\displaystyle \frac{{\eta }^{\prime }sinh\omega sin\eta }{cosh\omega \left(cos\eta -1\right)}}-\delta \kappa$$

given that $〈{\mathbf{f}}_2^{*},{\mathbf{f}}_1〉=-\left(sinh\omega cosh\omega \right)\left(cos\eta -1\right)$, as obtained from Equation (13). Hence, we obtain

$${\delta }^{*}{\kappa }^{*}={\displaystyle \frac{{\eta }^{\prime }tanh\omega sin\eta }{cos\eta -1}}-2{\omega }^{\prime }-\delta \kappa$$

(27)

for $\eta \ne 2\pi k$, where $k\in \mathbb{Z}$. Furthermore, taking the Lorentzian scalar product of both sides of Equation (26) with the spacelike vector field ${\mathbf{f}}_3$ yields

$${\delta }^{*}{\kappa }^{*}=-{\omega }^{\prime }-{\eta }^{\prime }tanh\omega cot\eta -\delta \tau sinh\omega \left({\displaystyle \frac{1-cos\eta }{sin\eta }}\right)$$

since $〈{\mathbf{f}}_2^{*},{\mathbf{f}}_3〉=-cosh\omega sin\eta$. Thus, we obtain

$${\kappa }^{*}=-{\displaystyle \frac{{\omega }^{\prime }}{{\delta }^{*}}}-{\displaystyle \frac{{\eta }^{\prime }tanh\omega cot\eta }{{\delta }^{*}}}-{\displaystyle \frac{\delta \tau sinh\omega }{{\delta }^{*}}}tan{\displaystyle \frac{\eta }{2}}.$$

Now, substituting the equality $\tau ={\displaystyle \frac{{\delta }^{*}sin\eta }{\delta \mu }}$ into this equation, we derive

$${\delta }^{*}{\kappa }^{*}=-{\omega }^{\prime }-{\eta }^{\prime }tanh\omega cot\eta -\left({\displaystyle \frac{{\delta }^{*}}{\mu }}\right)sinh\omega \left(1-cos\eta \right).$$

(28)

Additionally, by utilizing the equalities (27) and (28), we establish

$${\eta }^{\prime }tanh\omega \left({\displaystyle \frac{sin\eta }{cos\eta -1}}\right)-\delta \kappa ={\omega }^{\prime }-{\eta }^{\prime }tanh\omega cot\eta -\left({\displaystyle \frac{{\delta }^{*}}{\mu }}\right)sinh\omega \left(1-cos\eta \right).$$

By incorporating the expressions for $\kappa$ and ${\mu }^{\prime }$ from (22) and (5), we obtain the separable differential equation

$${\displaystyle \frac{{\eta }^{\prime }}{sin\eta }}={\displaystyle \frac{{\mu }^{\prime }}{\mu }},$$

which has the solution

$$\mu =ctan{\displaystyle \frac{\eta }{2}}.$$

(29)

The relation (29) suggests that if $\mu$ is constant, then $\eta$ must also remain constant. Next, we differentiate ${\mathbf{f}}_3^{*}$ as given in (13), leading to

$$\begin{array}{ccc}\hfill -{\delta }^{*}{\tau }^{*}{\mathbf{f}}_2^{*}& =& \left({\eta }^{\prime }cos\eta sinh\omega +{\omega }^{\prime }sin\eta cosh\omega +\delta \kappa sin\eta cosh\omega \right){\mathbf{f}}_1\hfill \\ & & +\left(\delta \kappa sin\eta sinh\omega +{\eta }^{\prime }cos\eta cosh\omega +{\omega }^{\prime }sin\eta sinh\omega -\delta \tau cos\eta \right){\mathbf{f}}_2\hfill \\ & & +\left(\delta \tau sin\eta cosh\omega -{\eta }^{\prime }sin\eta \right){\mathbf{f}}_3.\hfill \end{array}$$

Using the relation $〈{\mathbf{f}}_2^{*},{\mathbf{f}}_3〉=-cosh\omega sin\eta$, we finally obtain

$${\tau }^{*}={\displaystyle \frac{sin\eta }{\mu }}-{\displaystyle \frac{{\eta }^{\prime }}{{\delta }^{*}cosh\omega }}.$$

□

Theorem 11.

Consider an osculating mate of spacelike curves $\left\{\xi ,{\xi }^{*}\right\}$ with a timelike normal, which satisfies the following relation:

$${\xi }^{*}=\xi +\mu \left(\left(cosh\omega \right){\mathbf{f}}_1+\left(sinh\omega \right){\mathbf{f}}_2\right)$$

where $\omega \ne 0$. If the angle η between the binormals at the corresponding points of ξ and ${\xi }^{*}$ is non-zero, the curvature and torsion of the curve ${\xi }^{*}\left(s\right)$ are expressed as

$$\begin{array}{c}{\tau }^{*}=-{\displaystyle \frac{sinh\eta }{\mu }}+{\displaystyle \frac{{\eta }^{\prime }}{{\delta }^{*}cosh\omega }}\hfill \\ \\ \hfill {\kappa }^{*}=-{\displaystyle \frac{{\omega }^{\prime }}{{\delta }^{*}}}-{\displaystyle \frac{{\eta }^{\prime }tanh\omega coth\eta }{{\delta }^{*}}}-{\displaystyle \frac{\delta \tau sinh\omega }{{\delta }^{*}}}tanh{\displaystyle \frac{\eta }{2}}.\end{array}$$

(30)

Furthermore, the relationship

$$\mu =ctanh{\displaystyle \frac{\eta }{2}}$$

establishes a connection between the distance function μ and the angle η, where c is a real constant.

Proof. 

Now, let us compute the derivative of

$${\mathbf{f}}_1^{*}=\left({cosh}^2\omega -cosh\eta {sinh}^2\omega \right){\mathbf{f}}_1-\left(sinh\omega cosh\omega \right)\left(cosh\eta -1\right){\mathbf{f}}_2-\left(sinh\omega sinh\eta \right){\mathbf{f}}_3$$

as presented in (16). By applying the Frenet formulas, we arrive at the following expression:

$$\begin{array}{ccc}\hfill {\delta }^{*}{\kappa }^{*}{\mathbf{f}}_2^{*}& =& \left(\begin{array}{c}2{\omega }^{\prime }cosh\omega sinh\omega -{\eta }^{\prime }sinh\eta {sinh}^2\omega -2{\omega }^{\prime }cosh\eta sinh\omega cosh\omega \\ -\delta \kappa \left(sinh\omega cosh\omega \right)\left(cosh\eta -1\right)\end{array}\right){\mathbf{f}}_1\hfill \\ & & +\left(\begin{array}{c}-{\omega }^{\prime }{cosh}^2\omega \left(cosh\eta -1\right)-{\omega }^{\prime }{sinh}^2\omega \left(cosh\eta -1\right)\\ -{\eta }^{\prime }sinh\omega cosh\omega sinh\eta \\ -\delta \tau \left(sinh\omega sinh\eta \right)+\delta \kappa \left({cosh}^2\omega -cosh\eta {sinh}^2\omega \right)\end{array}\right){\mathbf{f}}_2\hfill \\ & & +\left(-\delta \tau \left(sinh\omega cosh\omega \right)\left(cosh\eta -1\right)-{\eta }^{\prime }sinh\omega cosh\eta -{\omega }^{\prime }cosh\omega sinh\eta \right){\mathbf{f}}_3.\hfill \end{array}$$

(31)

By taking the inner product of both sides with ${\mathbf{f}}_1$, we derive

$$\begin{array}{ccc}\hfill {\delta }^{*}{\kappa }^{*}\left(sinh\omega cosh\omega \right)\left(cosh\eta -1\right)& =& 2{\omega }^{\prime }cosh\omega sinh\omega -{\eta }^{\prime }sinh\eta {sinh}^2\omega \hfill \\ & & -2{\omega }^{\prime }cosh\eta sinh\omega cosh\omega \hfill \\ & & -\delta \kappa \left(sinh\omega cosh\omega \right)\left(cosh\eta -1\right)\hfill \end{array}$$

which can be rewritten as

$${\delta }^{*}{\kappa }^{*}={\displaystyle \frac{2{\omega }^{\prime }}{cosh\eta -1}}-{\displaystyle \frac{{\eta }^{\prime }sinh\eta tanh\omega }{cosh\eta -1}}-{\displaystyle \frac{2{\omega }^{\prime }cosh\eta }{cosh\eta -1}}-\delta \kappa .$$

Since the relation $〈{\mathbf{f}}_2^{*},{\mathbf{f}}_1〉=\left(sinh\omega cosh\omega \right)\left(cosh\eta -1\right)$ holds, we conclude that

$${\delta }^{*}{\kappa }^{*}=-2{\omega }^{\prime }-{\displaystyle \frac{{\eta }^{\prime }sinh\eta tanh\omega }{\left(cosh\eta -1\right)}}-\delta \kappa .$$

(32)

Similarly, taking the inner product of both sides of (31) with the binormal vector field ${\mathbf{f}}_3$, we obtain

$${\delta }^{*}{\kappa }^{*}=-{\omega }^{\prime }-{\eta }^{\prime }tanh\omega coth\eta -\delta \tau sinh\omega \left({\displaystyle \frac{cosh\eta -1}{sinh\eta }}\right).$$

(33)

Since $〈{\mathbf{f}}_2^{*},{\mathbf{f}}_3〉=cosh\omega sinh\eta$, it follows that

$${\kappa }^{*}=-{\displaystyle \frac{{\omega }^{\prime }}{{\delta }^{*}}}-{\displaystyle \frac{{\eta }^{\prime }tanh\omega coth\eta }{{\delta }^{*}}}-{\displaystyle \frac{\delta \tau sinh\omega }{{\delta }^{*}}}tanh{\displaystyle \frac{\eta }{2}}$$

By substituting $\tau =-{\displaystyle \frac{{\delta }^{*}sinh\eta }{\delta \mu }}$ into the equation above, we obtain

$${\delta }^{*}{\kappa }^{*}=-{\omega }^{\prime }-{\eta }^{\prime }tanh\omega coth\eta +\left({\displaystyle \frac{{\delta }^{*}}{\mu }}\right)sinh\omega \left(cosh\eta -1\right).$$

By utilizing the previously established equalities (32) and (33), we derive the following relation:

$$-{\displaystyle \frac{{\eta }^{\prime }sinh\eta tanh\omega }{cosh\eta -1}}-\delta \kappa ={\omega }^{\prime }-{\eta }^{\prime }tanh\omega coth\eta +\left({\displaystyle \frac{{\delta }^{*}}{\mu }}\right)sinh\omega \left(cosh\eta -1\right).$$

Substituting the expression for the curvature $\kappa$ from Equation (23), we obtain

$$-{\displaystyle \frac{{\eta }^{\prime }}{sinh\eta }}=-{\displaystyle \frac{\left({\delta }^{*}-\delta \right)cosh\omega }{\mu }}.$$

Since it is known that ${\mu }^{\prime }=cosh\omega \left({\delta }^{*}-\delta \right)$, we deduce the separable differential equation

$${\displaystyle \frac{{\eta }^{\prime }}{sinh\eta }}={\displaystyle \frac{{\mu }^{\prime }}{\mu }}.$$

Solving this equation yields

$$\mu =ctanh{\displaystyle \frac{\eta }{2}}.$$

(34)

This result, denoted by (34), implies that if $\mu$ remains constant, then $\eta$ must also be constant.

Next, we differentiate ${\mathbf{f}}_3^{*}$, as given in (16), which leads to

$${\mathbf{f}}_3^{*}=\left(sinh\eta sinh\omega \right){\mathbf{f}}_1+\left(sinh\eta cosh\omega \right){\mathbf{f}}_2+cosh\eta {\mathbf{f}}_3.$$

Taking the derivative, we obtain

$$\begin{array}{ccc}\hfill {\delta }^{*}{\tau }^{*}{\mathbf{f}}_2^{*}& =& \left({\eta }^{\prime }cosh\eta sinh\omega +{\omega }^{\prime }sinh\eta cosh\omega +\delta \kappa sinh\eta cosh\omega \right){\mathbf{f}}_1\hfill \\ & & +\left(\delta \kappa sinh\eta sinh\omega +{\eta }^{\prime }cosh\eta cosh\omega +{\omega }^{\prime }sinh\eta sinh\omega +\delta \tau cosh\eta \right){\mathbf{f}}_2\hfill \\ & & +\left(\delta \tau sinh\eta cosh\omega +{\eta }^{\prime }sinh\eta \right){\mathbf{f}}_3.\hfill \end{array}$$

Utilizing the identity $〈{\mathbf{f}}_2^{*},{\mathbf{f}}_3〉=cosh\omega sinh\eta$, we arrive at the expression

$${\tau }^{*}=-{\displaystyle \frac{sinh\eta }{\mu }}+{\displaystyle \frac{{\eta }^{\prime }}{{\delta }^{*}cosh\omega }}.$$

□

Theorem 12.

Consider the pair of osculating mate curves $\left\{\xi ,{\xi }^{*}\right\}$ consisting of spacelike curves with a spacelike normal, related by the equation

$${\xi }^{*}=\xi +\mu \left(\left(cos\omega \right){\mathbf{f}}_1+\left(sin\omega \right){\mathbf{f}}_2\right)$$

for $\omega \ne {\displaystyle \frac{\pi }{2}},$$\omega \ne {\displaystyle \frac{3\pi }{2}}$ and $\omega \in [0,2\pi )$. If the angle η between the binormals of the corresponding points on ξ and ${\xi }^{*}$ satisfies $\eta \ne 0$, then the curvature and torsion of the curve ${\xi }^{*}\left(s\right)$ are given by

$$\begin{array}{c}{\tau }^{*}=-{\displaystyle \frac{sinh\eta }{\mu }}+{\displaystyle \frac{{\eta }^{\prime }}{{\delta }^{*}cos\omega }}\hfill \\ \\ \hfill {\kappa }^{*}=-{\displaystyle \frac{{\omega }^{\prime }}{{\delta }^{*}}}-{\displaystyle \frac{{\eta }^{\prime }tan\omega coth\eta }{{\delta }^{*}}}-{\displaystyle \frac{\delta \tau sin\omega }{{\delta }^{*}}}tanh{\displaystyle \frac{\eta }{2}}.\end{array}$$

Furthermore, there exists the following relation between the angle η and the distance function μ:

$$\mu =ctanh{\displaystyle \frac{\eta }{2}},$$

where c is a real constant.

Proof. 

Let us compute the derivative of

$${\mathbf{f}}_1^{*}=\left({cos}^2\omega +cosh\eta {sin}^2\omega \right){\mathbf{f}}_1-\left(sin\omega cos\omega \right)\left(cosh\eta -1\right){\mathbf{f}}_2-sinh\eta sin\omega {\mathbf{f}}_3$$

from (19), and applying the Frenet formulas, we obtain

$$\begin{array}{cc}\hfill {\delta }^{*}{\kappa }^{*}{\mathbf{f}}_2^{*}& =\left(\begin{array}{c}-2{\omega }^{\prime }cos\omega sin\omega +{\eta }^{\prime }sinh\eta {sin}^2\omega +{\omega }^{\prime }cosh\eta sin2\omega \\ +\delta \kappa \left(sin\omega cos\omega \right)\left(cosh\eta -1\right)\end{array}\right){\mathbf{f}}_1\hfill \\ \hfill & -\left(\begin{array}{c}{\omega }^{\prime }(cosh\eta -1)cos2\omega +{\displaystyle \frac{{\eta }^{\prime }}{2}}sin2\omega sinh\eta +\delta \tau (sin\omega sinh\eta )\\ -\delta \kappa ({cos}^2\omega +cosh\eta {sin}^2\omega )\end{array}\right){\mathbf{f}}_2\hfill \\ \hfill & +\left(-\delta \tau \left(sin\omega cos\omega \right)\left(cosh\eta -1\right)-{\eta }^{\prime }cosh\eta sin\omega -{\omega }^{\prime }cos\omega sinh\eta \right){\mathbf{f}}_3.\hfill \end{array}$$

(35)

By taking the scalar product with ${\mathbf{f}}_1$, we obtain

$${\delta }^{*}{\kappa }^{*}={\displaystyle \frac{2{\omega }^{\prime }}{cosh\eta -1}}-{\displaystyle \frac{2{\omega }^{\prime }cosh\eta }{cosh\eta -1}}-{\displaystyle \frac{{\eta }^{\prime }sinh\eta tan\omega }{cosh\eta -1}}-\delta \kappa ,$$

since $〈{\mathbf{f}}_2^{*},{\mathbf{f}}_1〉=-\left(sin\omega cos\omega \right)\left(cosh\eta -1\right)$. This simplifies to

$${\delta }^{*}{\kappa }^{*}=-2{\omega }^{\prime }-{\displaystyle \frac{{\eta }^{\prime }sinh\eta tan\omega }{cosh\eta -1}}-\delta \kappa .$$

(36)

If we now take the inner product of both sides of (35) with the vector field ${\mathbf{f}}_3$, we obtain

$${\delta }^{*}{\kappa }^{*}=-{\omega }^{\prime }-{\eta }^{\prime }tan\omega coth\eta -\delta \tau sin\omega \left({\displaystyle \frac{cosh\eta -1}{sinh\eta }}\right)$$

using the fact that $〈{\mathbf{f}}_2^{*},{\mathbf{f}}_3〉=-sinh\eta cos$. Consequently, we derive

$${\kappa }^{*}=-{\displaystyle \frac{{\omega }^{\prime }}{{\delta }^{*}}}-{\displaystyle \frac{{\eta }^{\prime }tan\omega coth\eta }{{\delta }^{*}}}-{\displaystyle \frac{\delta \tau sin\omega }{{\delta }^{*}}}tanh{\displaystyle \frac{\eta }{2}}.$$

Substituting $\tau =-{\displaystyle \frac{{\delta }^{*}sinh\eta }{\delta \mu }}$ into this equation yields

$${\delta }^{*}{\kappa }^{*}=-{\omega }^{\prime }-{\eta }^{\prime }tan\omega coth\eta +\left({\displaystyle \frac{{\delta }^{*}}{\mu }}\right)sin\omega \left(cosh\eta -1\right).$$

(37)

By utilizing the equalities (36) and (37), we can derive the expression

$$-{\displaystyle \frac{{\eta }^{\prime }sinh\eta tan\omega }{cosh\eta -1}}-\delta \kappa ={\omega }^{\prime }-{\eta }^{\prime }tan\omega coth\eta +\left({\displaystyle \frac{{\delta }^{*}}{\mu }}\right)sin\omega \left(cosh\eta -1\right).$$

Upon substituting the curvature $\kappa$ obtained in (24), we establish the equation

$${\displaystyle \frac{-{\eta }^{\prime }}{sinh\eta }}={\displaystyle \frac{\left(\delta -{\delta }^{*}\right)cos\omega }{\mu }}.$$

Since we recognize that ${\mu }^{\prime }=\left(cos\omega \right)\left({\delta }^{*}-\delta \right)$, it follows that

$${\displaystyle \frac{{\eta }^{\prime }}{sinh\eta }}={\displaystyle \frac{{\mu }^{\prime }}{\mu }}.$$

Solving this differential equation yields

$$\mu =ctanh\left({\displaystyle \frac{\eta }{2}}\right),$$

(38)

which is labeled as (38).

The expression (38) implies that if $\mu$ remains constant, then $\eta$ must also be constant. To further analyze this, we now compute the derivative of ${\mathbf{f}}_3^{*}$ as given in (19). The resulting expression is

$${\mathbf{f}}_3^{*}=-sinh\eta sin\omega {\mathbf{f}}_1+sinh\eta cos\omega {\mathbf{f}}_2+cosh\eta {\mathbf{f}}_3.$$

Additionally, differentiating further provides

$$\begin{array}{ccc}\hfill {\delta }^{*}{\tau }^{*}{\mathbf{f}}_2^{*}& =& -\left({\eta }^{\prime }cosh\eta sin\omega +{\omega }^{\prime }sinh\eta cos\omega +\delta \kappa sinh\eta cos\omega \right){\mathbf{f}}_1\hfill \\ & & +\left(-\delta \kappa sinh\eta sin\omega +{\eta }^{\prime }cosh\eta cos\omega -{\omega }^{\prime }sinh\eta sin\omega +\delta \tau cosh\eta \right){\mathbf{f}}_2\hfill \\ & & +\left(\delta \tau sinh\eta cos\omega +{\eta }^{\prime }sinh\eta \right){\mathbf{f}}_3.\hfill \end{array}$$

Given the relation $〈{\mathbf{f}}_2^{*},{\mathbf{f}}_3〉=-sinh\eta cos\omega$, we conclude that

$${\tau }^{*}=-{\displaystyle \frac{sinh\eta }{\mu }}+{\displaystyle \frac{{\eta }^{\prime }}{{\delta }^{*}cos\omega }}.$$

□

6. Rotation Matrix Between Frenet Frame of Osculating Curve

The rotation matrix between the $\left\{{\mathbf{f}}_1,{\mathbf{f}}_2,{\mathbf{f}}_3\right\}$ and $\left\{{\mathbf{f}}_1^{*},{\mathbf{f}}_2^{*},{\mathbf{f}}_3^{*}\right\}$ Frenet frame of the osculating mate of curves $\left\{\xi ,{\xi }^{*}\right\}$ will be obtained by using the rotation transformation (39) in the generalized unit quaternion 3-space ${\mathbb{E}}_{\omega \eta }^3$ [28] and using the rotation matrix (40) corresponding to a unit timelike quaternion [29].

$$\left[\begin{array}{ccc}{q}_1^2+\omega q_2^2-\eta q_3^2-\omega \eta q_4^2& 2\eta \left(q_2{q}_3-q_1{q}_4\right)& 2\eta \left(q_1{q}_3+\omega q_2{q}_4\right)\\ 2\omega \left(q_1{q}_4+q_2{q}_3\right)& q_1^2-\omega q_2^2+\eta q_3^2-\omega \eta q_4^2& 2\omega \left(\eta q_3{q}_4-q_1{q}_2\right)\\ 2\omega q_2{q}_4-2q_1{q}_3& 2q_1{q}_2+2\eta q_3{q}_4& q_1^2-\omega q_2^2-\eta q_3^2+\omega \eta q_4^2\end{array}\right]$$

(39)

$$\left[\begin{array}{ccc}{q}_1^2+q_2^2+q_3^2+q_4^2& 2q_1{q}_4-2q_2{q}_3& -2q_1{q}_3-2q_2{q}_4\\ 2q_1{q}_4+2q_2{q}_3& q_1^2-q_2^2-q_3^2+q_4^2& -2q_1{q}_2-2q_3{q}_4\\ -2q_1{q}_3+2q_2{q}_4& 2q_1{q}_2-2q_3{q}_4& q_1^2-q_2^2+q_3^2-q_4^2\end{array}\right].$$

(40)

Proposition 2.

Consider the Frenet frames $\left\{{\mathbf{f}}_1,{\mathbf{f}}_2,{\mathbf{f}}_3\right\}$ and $\left\{{\mathbf{f}}_1^{*},{\mathbf{f}}_2^{*},{\mathbf{f}}_3^{*}\right\}$, which are associated with the osculating mates of the timelike curves $\left\{\xi ,{\xi }^{*}\right\}$ at their respective points. The following matrix relations hold:

$$\left[\begin{array}{c}{\mathbf{f}}_1^{*}\\ {\mathbf{f}}_2^{*}\\ {\mathbf{f}}_3^{*}\end{array}\right]=\left[\begin{array}{ccc}-{sinh}^2\omega cos\eta +{cosh}^2\omega & sinh\omega cosh\omega \left(1-cos\eta \right)& sinh\omega sin\eta \\ sinh\omega cosh\omega \left(cos\eta -1\right)& {cosh}^2\omega cos\eta -{sinh}^2\omega & -cosh\omega sin\eta \\ sin\eta sinh\omega & sin\eta cosh\omega & cos\eta \end{array}\right]\left[\begin{array}{c}{\mathbf{f}}_1\\ {\mathbf{f}}_2\\ {\mathbf{f}}_3\end{array}\right].$$

Similarly, we also have the inverse relation

$$\left[\begin{array}{c}{\mathbf{f}}_1\\ {\mathbf{f}}_2\\ {\mathbf{f}}_3\end{array}\right]=\left[\begin{array}{ccc}\begin{array}{c}{cosh}^2\omega \left(1-cos\eta \right)\\ +cos\eta \end{array}& cosh\omega sinh\omega \left(1-cos\eta \right)& -sinh\omega sin\eta \\ cosh\omega sinh\omega \left(cos\eta -1\right)& {cosh}^2\omega \left(cos\eta -1\right)+1& cosh\omega sin\eta \\ -sinh\omega sin\eta & -cosh\omega sin\eta & cos\eta \end{array}\right]\left[\begin{array}{c}{\mathbf{f}}_1^{*}\\ {\mathbf{f}}_2^{*}\\ {\mathbf{f}}_3^{*}\end{array}\right],$$

where $\eta \in [0,2\pi )$. The matrix $R_{\eta }$ represents a rotation transformation of a vector through the pseudo-angle η around the axis $\mathbf{r}=\left(cosh\omega ,-sinh\omega ,0\right)$ and is given by

$$R_{\eta }=\left[\begin{array}{ccc}-{sinh}^2\omega cos\eta +{cosh}^2\omega & sinh\omega cosh\omega \left(1-cos\eta \right)& sinh\omega sin\eta \\ sinh\omega cosh\omega \left(cos\eta -1\right)& {cosh}^2\omega cos\eta -{sinh}^2\omega & -cosh\omega sin\eta \\ sin\eta sinh\omega & sin\eta cosh\omega & cos\eta \end{array}\right].$$

Finally, the corresponding unit timelike split quaternion $\mathbf{q}$ that relates to $R_{\eta }$ is expressed as

$$\mathbf{q}=\pm cos\left({\displaystyle \frac{\eta }{2}}\right)\pm \mathbf{i}sin\left({\displaystyle \frac{\eta }{2}}\right)cosh\omega \mp \mathbf{j}sin\left({\displaystyle \frac{\eta }{2}}\right)sinh\omega .$$

Proof. 

It is a well-established fact that a unit timelike split quaternion $q=q_1+q_2\mathbf{i}+q_3\mathbf{j}+q_4\mathbf{k}$ corresponds to the rotation matrix R given by (40):

$$R=\left[\begin{array}{ccc}{q}_1^2+q_2^2+q_3^2+q_4^2& 2q_1{q}_4-2q_2{q}_3& -2q_1{q}_3-2q_2{q}_4\\ 2q_1{q}_4+2q_2{q}_3& q_1^2-q_2^2-q_3^2+q_4^2& -2q_1{q}_2-2q_3{q}_4\\ -2q_1{q}_3+2q_2{q}_4& 2q_1{q}_2-2q_3{q}_4& q_1^2-q_2^2+q_3^2-q_4^2\end{array}\right].$$

In the case where $q_1\ne 0$, the following relations hold:

$$\begin{array}{ll}{q}_1^2={\displaystyle \frac{1}{4}}\left(1+{\widehat{R}}_{11}+{\widehat{R}}_{22}+{\widehat{R}}_{33}\right),\hfill & q_2={\displaystyle \frac{1}{4q_1}}\left({\widehat{R}}_{32}-{\widehat{R}}_{23}\right),\\ q_3=-{\displaystyle \frac{1}{4q_1}}\left({\widehat{R}}_{13}+{\widehat{R}}_{31}\right),\hfill & q_4={\displaystyle \frac{1}{4q_1}}\left({\widehat{R}}_{21}+{\widehat{R}}_{12}\right).\end{array}$$

From these relations, the quaternion components are derived as

$$q_1=\pm cos{\displaystyle \frac{\eta }{2}},q_2=\pm sin{\displaystyle \frac{\eta }{2}}cosh\omega ,q_3=\mp sin{\displaystyle \frac{\eta }{2}}sinh\omega q_4=0.$$

□

Corollary 4.

In the context of a matrix R representing a rotation about a timelike axis in 3-dimensional Lorentzian space, the rotation angle η is related to the matrix by the following equation:

$$cos\eta ={\displaystyle \frac{limtr\left(R\right)-1}{2}}.$$

From this, we can derive the expression

$$cos\eta ={\displaystyle \frac{{\widehat{R}}_{11}+{\widehat{R}}_{22}+{\widehat{R}}_{33}-1}{2}}={\displaystyle \frac{2cos\eta +1-1}{2}}=cos\eta .$$

Proposition 3.

The Frenet frames $\left\{{\mathbf{f}}_1,{\mathbf{f}}_2,{\mathbf{f}}_3\right\}$ and $\left\{{\mathbf{f}}_1^{*},{\mathbf{f}}_2^{*},{\mathbf{f}}_3^{*}\right\}$ correspond to the osculating partners of the spacelike curves $\left\{\xi ,{\xi }^{*}\right\}$, where the normal vector is timelike at the relevant points. In this context, we have the matrix relations

$$\left[\begin{array}{c}{\mathbf{f}}_1^{*}\\ {\mathbf{f}}_2^{*}\\ {\mathbf{f}}_3^{*}\end{array}\right]=\left[\begin{array}{ccc}{cosh}^2\omega {sinh}^2\omega cosh\eta & sinh\omega cosh\omega \left(1-cosh\eta \right)& -sinh\omega sinh\eta \\ sinh\omega cosh\omega \left(cosh\eta -1\right)& -{sinh}^2\omega +cosh\eta {cosh}^2\omega & cosh\omega sinh\eta \\ sinh\eta sinh\omega & sinh\eta cosh\omega & cosh\eta \end{array}\right]\left[\begin{array}{c}{\mathbf{f}}_1\\ {\mathbf{f}}_2\\ {\mathbf{f}}_3\end{array}\right]$$

and

$$\left[\begin{array}{c}{\mathbf{f}}_1\\ {\mathbf{f}}_2\\ {\mathbf{f}}_3\end{array}\right]=\left[\begin{array}{ccc}{cosh}^2\omega -{sinh}^2\omega cosh\eta & sinh\omega cosh\omega \left(1-cosh\eta \right)& sinh\eta sinh\omega \\ sinh\omega cosh\omega \left(cosh\eta -1\right)& cosh\eta {cosh}^2\omega -{sinh}^2\omega & -cosh\omega sinh\eta \\ -sinh\eta sinh\omega & -cosh\omega sinh\eta & cosh\eta \end{array}\right]\left[\begin{array}{c}{\mathbf{f}}_1^{*}\\ {\mathbf{f}}_2^{*}\\ {\mathbf{f}}_3^{*}\end{array}\right].$$

Here, the matrix

$$R_{\eta }=\left[\begin{array}{ccc}{cosh}^2\omega -\left({sinh}^2\omega \right)cosh\eta & sinh\omega cosh\omega \left(1-cosh\eta \right)& -sinh\omega sinh\eta \\ sinh\omega cosh\omega \left(cosh\eta -1\right)& -{sinh}^2\omega +cosh\eta {cosh}^2\omega & cosh\omega sinh\eta \\ sinh\eta sinh\omega & sinh\eta cosh\omega & cosh\eta \end{array}\right]$$

represents a rotation matrix that rotates a vector through the pseudo-angle η around the axis $\mathbf{r}=\left(cosh\omega ,-sinh\omega ,0\right)$. The corresponding unit timelike quaternion $\mathbf{q}$ is given by

$$q=\pm cosh\left({\displaystyle \frac{\eta }{2}}\right)\pm \left(sinh{\displaystyle \frac{\eta }{2}}cosh\omega \right)\mathbf{i}\pm \left(sinh{\displaystyle \frac{\eta }{2}}sinh\omega \right)\mathbf{j}.$$

Proof. 

A unit spacelike split quaternion $\mathbf{q}=q_1+q_2\mathbf{i}+q_3\mathbf{j}+q_4\mathbf{k}$ is known to correspond to the rotation matrix derived from (39):

$$R=\left[\begin{array}{ccc}{q}_1^2-q_2^2-q_3^2+q_4^2& -2q_1{q}_4+2q_2{q}_3& 2q_1{q}_3-2q_2{q}_4\\ -\left(2q_1{q}_4+2q_2{q}_3\right)& q_1^2+q_2^2+q_3^2+q_4^2& 2q_1{q}_2-2q_3{q}_4\\ -2q_1{q}_3-2q_2{q}_4& 2q_1{q}_2+2q_3{q}_4& q_1^2+q_2^2-q_3^2-q_4^2\end{array}\right].$$

If $q_1\ne 0$, the following relations hold:

$$\begin{array}{ll}{q}_1^2={\displaystyle \frac{1}{4}}\left(1+{\widehat{R}}_{11}+{\widehat{R}}_{22}+{\widehat{R}}_{33}\right),\hfill & q_2={\displaystyle \frac{1}{4q_1}}\left({\widehat{R}}_{32}+{\widehat{R}}_{23}\right),\\ q_3={\displaystyle \frac{1}{4q_1}}\left({\widehat{R}}_{13}-{\widehat{R}}_{31}\right),\hfill & q_4=-{\displaystyle \frac{1}{4q_1}}\left({\widehat{R}}_{21}+{\widehat{R}}_{12}\right).\end{array}$$

Thus, using these expressions, we arrive at:

$$q_1=\pm cosh\left({\displaystyle \frac{\eta }{2}}\right),q_2=\pm sinh\left({\displaystyle \frac{\eta }{2}}\right)cosh\omega ,q_3=\pm sinh\left({\displaystyle \frac{\eta }{2}}\right)sinh\omega q_4=0.$$

□

Corollary 5.

In a matrix R that represents a rotation about a spacelike axis in 3-dimensional Lorentzian space, the rotation angle t is related to the trace of R through the following formula:

$$cosht={\displaystyle \frac{limtr\left(R\right)-1}{2}}.$$

From this, we can derive

$$cosht={\displaystyle \frac{{\widehat{R}}_{11}+{\widehat{R}}_{22}+{\widehat{R}}_{33}-1}{2}}={\displaystyle \frac{2cosh\eta +1-1}{2}}=cosh\eta .$$

Proposition 4.

Consider the Frenet frames $\left\{{\mathbf{f}}_1,{\mathbf{f}}_2,{\mathbf{f}}_3\right\}$ and $\left\{{\mathbf{f}}_1^{*},{\mathbf{f}}_2^{*},{\mathbf{f}}_3^{*}\right\}$ corresponding to the osculating mates of the spacelike curves $\left\{\xi ,{\xi }^{*}\right\}$, where the normal vector is spacelike at each respective point. The following matrix relationships hold between the frames:

$$\left[\begin{array}{c}{\mathbf{f}}_1^{*}\\ {\mathbf{f}}_2^{*}\\ {\mathbf{f}}_3^{*}\end{array}\right]=\left[\begin{array}{ccc}{cos}^2\omega +cosh\eta {sin}^2\omega & -sin\omega cos\omega \left(cosh\eta -1\right)& -sinh\eta sin\omega \\ -sin\omega cos\omega \left(cosh\eta -1\right)& {sin}^2\omega +cosh\eta {cos}^2\omega & sinh\eta cos\omega \\ -sinh\eta sin\omega & sinh\eta cos\omega & cosh\eta \end{array}\right]\left[\begin{array}{c}{\mathbf{f}}_1\\ {\mathbf{f}}_2\\ {\mathbf{f}}_3\end{array}\right]$$

and conversely,

$$\left[\begin{array}{c}{\mathbf{f}}_1\\ {\mathbf{f}}_2\\ {\mathbf{f}}_3\end{array}\right]=\left[\begin{array}{ccc}cosh\eta {sin}^2\omega -{sin}^2\omega +1& -sin\omega cos\omega \left(cosh\eta -1\right)& sinh\eta sin\omega \\ -sin\omega cos\omega \left(cosh\eta -1\right)& cosh\eta -{sin}^2\omega \left(cosh\eta -1\right)& -sinh\eta cos\omega \\ sinh\eta sin\omega & -sinh\eta cos\omega & cosh\eta \end{array}\right]\left[\begin{array}{c}{\mathbf{f}}_1^{*}\\ {\mathbf{f}}_2^{*}\\ {\mathbf{f}}_3^{*}\end{array}\right]$$

where $\omega \in [0,2\pi )$. The rotation matrix $R_{\eta }$ represents a rotation by the pseudo-angle η around the axis $\mathbf{r}=\left(cos\omega ,-sin\omega ,0\right)$, and is given by

$$R_{\eta }=\left[\begin{array}{ccc}{cos}^2\omega +cosh\eta {sin}^2\omega & -sin\omega cos\omega \left(cosh\eta -1\right)& -sinh\eta sin\omega \\ -sin\omega cos\omega \left(cosh\eta -1\right)& {sin}^2\omega +cosh\eta {cos}^2\omega & sinh\eta cos\omega \\ -sinh\eta sin\omega & sinh\eta cos\omega & cosh\eta \end{array}\right]$$

Finally, the corresponding unit timelike quaternion $\mathbf{q}$ is given by

$$\mathbf{q}=\pm cosh\left({\displaystyle \frac{\eta }{2}}\right)\pm \mathbf{i}sinh\left({\displaystyle \frac{\eta }{2}}\right)cos\omega \mp \mathbf{j}sinh\left({\displaystyle \frac{\eta }{2}}\right)sin\omega .$$

Proof. 

A unit quaternion $\mathbf{q}=q_1+q_2\mathbf{i}+q_3\mathbf{j}+q_4\mathbf{k}$ is associated with the corresponding rotation matrix R as outlined in Equation (39):

$$R=\left[\begin{array}{ccc}{q}_1^2-q_2^2+q_3^2-q_4^2& 2q_1{q}_4-2q_2{q}_3& -2q_1{q}_3+2q_2{q}_4\\ -2q_1{q}_4-2q_2{q}_3& q_1^2+q_2^2-q_3^2-q_4^2& 2q_1{q}_2+2q_3{q}_4\\ -2q_1{q}_3-2q_2{q}_4& 2q_1{q}_2-2q_3{q}_4& q_1^2+q_2^2+q_3^2+q_4^2\end{array}\right].$$

When $q_1\ne 0$ is non-zero, the following relations hold:

$$\begin{array}{ll}{q}_1^2={\displaystyle \frac{1}{4}}\left(1+{\widehat{R}}_{11}+{\widehat{R}}_{22}+{\widehat{R}}_{33}\right),\hfill & q_2={\displaystyle \frac{1}{4q_1}}\left({\widehat{R}}_{32}+{\widehat{R}}_{23}\right),\hfill \\ q_3=-{\displaystyle \frac{1}{4q_1}}\left({\widehat{R}}_{13}+{\widehat{R}}_{31}\right),\hfill & q_4={\displaystyle \frac{1}{4q_1}}\left({\widehat{R}}_{12}-{\widehat{R}}_{21}\right).\hfill \end{array}$$

By applying these relations, we determine

$$q_1=\pm cosh{\displaystyle \frac{\eta }{2}},q_2=\pm sinh{\displaystyle \frac{\eta }{2}}cos\omega ,q_3=\mp sinh{\displaystyle \frac{\eta }{2}}sin\omega q_4=0.$$

□

Corollary 6.

The rotation angle is $\eta ,$ since

$$cosht={\displaystyle \frac{{\widehat{R}}_{11}+{\widehat{R}}_{22}+{\widehat{R}}_{33}-1}{2}}={\displaystyle \frac{2cosh\eta +1-1}{2}}=cosh\eta .$$

Now, let us examine spacial cases.

7. Special Cases for Osculating Curve Pairs

Case 1.

Given that $\left\{\xi ,{\xi }^{*}\right\}$ represents the osculating mate of timelike curves, when we set $\omega =0$, the relation

$${\xi }^{*}=\xi +\mu {\mathbf{f}}_1$$

holds true. Under these conditions, the Frenet frame of ${\xi }^{*}$ is obtained by rotating the Frenet frame of ξ around the tangent vector through an angle η, and the corresponding transformation matrix is

$$\left[\begin{array}{c}{\mathbf{f}}_1^{*}\\ {\mathbf{f}}_2^{*}\\ {\mathbf{f}}_3^{*}\end{array}\right]=\left[\begin{array}{ccc}1& 0& 0\\ 0& cos\eta & -sin\eta \\ 0& sin\eta & cos\eta \end{array}\right]\left[\begin{array}{c}{\mathbf{f}}_1\\ {\mathbf{f}}_2\\ {\mathbf{f}}_3\end{array}\right].$$

When $\eta =0$ and $\eta \in [0,2\pi )$$\left(\mathrm{with}\omega \ne 0\right)$, we have

$${\xi }^{*}=\xi +\mu \left[cosh\omega {\mathbf{f}}_1+sinh\omega {\mathbf{f}}_2\right]$$

and in this case, the Frenet frames of ξ and ${\xi }^{*}$ are identical, as the angle $\eta =0$. The transformation matrix for this scenario is

$$\left[\begin{array}{c}{\mathbf{f}}_1^{*}\\ {\mathbf{f}}_2^{*}\\ {\mathbf{f}}_3^{*}\end{array}\right]=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]\left[\begin{array}{c}{\mathbf{f}}_1\\ {\mathbf{f}}_2\\ {\mathbf{f}}_3\end{array}\right].$$

Case 2.

(Timelike Bertrand Mate) In the context of timelike curves, the curves $\left\{\xi ,{\xi }^{*}\right\}$ are considered osculating mates when the condition $\omega =0$ and $\eta =\pi$ holds, or when $\eta =0$ with $\eta \in [0,2\pi )$. Under these circumstances, $\left\{\xi ,{\xi }^{*}\right\}$ form a pair of Bertrand mates. When $\omega =0$, the relationship between ξ and ${\xi }^{*}$ can be expressed as

$${\xi }^{*}=\xi +\mu {\mathbf{f}}_1.$$

Furthermore, applying a rotation of angle $\eta =\left(2k+1\right)\pi$ about the tangent vector to the Frenet frame of ξ yields the Frenet frame of ${\xi }^{*}$, where k is an integer. This transformation can be represented by the following matrix:

$$\left[\begin{array}{c}{\mathbf{f}}_1^{*}\\ {\mathbf{f}}_2^{*}\\ {\mathbf{f}}_3^{*}\end{array}\right]=\left[\begin{array}{ccc}1& 0& 0\\ 0& -1& 0\\ 0& 0& -1\end{array}\right]\left[\begin{array}{c}{\mathbf{f}}_1\\ {\mathbf{f}}_2\\ {\mathbf{f}}_3\end{array}\right].$$

On the other hand, when $\eta =0$, the relation between ξ and ${\xi }^{*}$ is modified to

$${\xi }^{*}=\xi +\mu \left[cosh\omega {\mathbf{f}}_1+sinh\omega {\mathbf{f}}_2\right].$$

In this case, the Frenet frames of ξ and ${\xi }^{*}$ remain identical with the angle $\eta =0$, and they are represented by the following matrix:

$$\left[\begin{array}{c}{\mathbf{f}}_1^{*}\\ {\mathbf{f}}_2^{*}\\ {\mathbf{f}}_3^{*}\end{array}\right]=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]\left[\begin{array}{c}{\mathbf{f}}_1\\ {\mathbf{f}}_2\\ {\mathbf{f}}_3\end{array}\right].$$

Case 3.

When $\left\{\xi ,{\xi }^{*}\right\}$ form an osculating mate of timelike curves and $\omega =0$, with $\eta ={\displaystyle \frac{\pi }{2}}$ or ${\displaystyle \frac{3\pi }{2}}$, they are considered Mannheim mates. This is because, under the condition $\omega =0$, we have

$${\xi }^{*}=\xi +\mu {\mathbf{f}}_1.$$

In addition, by rotating the Frenet frame of ξ around its tangent vector by an angle of ${\displaystyle \frac{\pi }{2}}$ or ${\displaystyle \frac{3\pi }{2}}$, one obtains the Frenet frame of ${\xi }^{*}$. This transformation is represented by the following matrix:

$$\left[\begin{array}{c}{\mathbf{f}}_1^{*}\\ {\mathbf{f}}_2^{*}\\ {\mathbf{f}}_3^{*}\end{array}\right]=\left[\begin{array}{ccc}1& 0& 0\\ 0& 0& \mp 1\\ 0& \pm 1& 0\end{array}\right]\left[\begin{array}{c}{\mathbf{f}}_1\\ {\mathbf{f}}_2\\ {\mathbf{f}}_3\end{array}\right].$$

Case 4.

Consider $\left\{\xi ,{\xi }^{*}\right\}$ as an osculating mate of timelike curves. If the angle η is constant and $\mu =ctan{\displaystyle \frac{\eta }{2}}$, then μ remains constant, where $\eta \in [0,2\pi )-\left\{\pi \right\}$. Additionally, the condition of μ being constant implies that $\delta ={\delta }^{*}$, because we know that ${\mu }^{\prime }=\left(\delta -{\delta }^{*}\right)cosh\omega$, and since $cosh\omega >0$ for all $\omega \in \mathbb{R}$, it follows that $\delta ={\delta }^{*}$. If η is fixed, the torsions and curvatures of ξ and ${\xi }^{*}$ take the following form:

$$\tau ={\tau }^{*}={\displaystyle \frac{sin\eta }{\mu }},\kappa =-{\displaystyle \frac{{\omega }^{\prime }}{\delta }}+\tilde{c}sinh\omega sin\eta ,{\kappa }^{*}=-{\displaystyle \frac{{\omega }^{\prime }}{\delta }}-\tilde{c}sinh\omega sin\eta$$

where $\tilde{c}={\displaystyle \frac{1}{c}}$. Therefore, the relationship between the curvatures is

$$\kappa ={\kappa }^{*}+2\tilde{c}sinh\omega sin\eta .$$

Thus, ξ and ${\xi }^{*}$ are Backlund mates, and the transformation that relates them is the Backlund transform.

Case 5.

Let $\left\{\xi ,{\xi }^{*}\right\}$ be an osculating mate of spacelike curves with a timelike normal. If we set $\omega =0$, then the relationship between the curves is given by

$${\xi }^{*}=\xi +\mu {\mathbf{f}}_1.$$

Applying a rotation of pseudo-angle η around the tangent vector, the Frenet frame of ${\xi }^{*}$ is determined as follows:

$$\left[\begin{array}{c}{\mathbf{f}}_1^{*}\\ {\mathbf{f}}_2^{*}\\ {\mathbf{f}}_3^{*}\end{array}\right]=\left[\begin{array}{ccc}1& 0& 0\\ 0& cosh\eta & sinh\eta \\ 0& sinh\eta & cosh\eta \end{array}\right]\left[\begin{array}{c}{\mathbf{f}}_1\\ {\mathbf{f}}_2\\ {\mathbf{f}}_3\end{array}\right].$$

If we further assume that $\eta =0$ (while $\omega \ne 0$), then

$${\xi }^{*}=\xi +\mu \left[cosh\omega {\mathbf{f}}_1+sinh\omega {\mathbf{f}}_2\right].$$

In this case, the Frenet frames of both ξ and ${\xi }^{*}$ remain identical when the angle $\eta =0$ (with $\omega \ne 0$). This can be written as

$$\left[\begin{array}{c}{\mathbf{f}}_1^{*}\\ {\mathbf{f}}_2^{*}\\ {\mathbf{f}}_3^{*}\end{array}\right]=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]\left[\begin{array}{c}{\mathbf{f}}_1\\ {\mathbf{f}}_2\\ {\mathbf{f}}_3\end{array}\right].$$

Case 6.

If $\left\{\xi ,{\xi }^{*}\right\}$ are osculating mates of spacelike curves with a timelike normal, and both $\omega =0$ and $\eta =0$, then ξ and ${\xi }^{*}$ are considered to be Bertrand mates. For $\omega =0$, we have

$${\xi }^{*}=\xi +\mu {\mathbf{f}}_1.$$

Additionally, the Frenet frames of ξ and ${\xi }^{*}$ remain the same when the pseudo-angle $\eta =0$. This relationship can be expressed as

$$\left[\begin{array}{c}{\mathbf{f}}_1^{*}\\ {\mathbf{f}}_2^{*}\\ {\mathbf{f}}_3^{*}\end{array}\right]=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]\left[\begin{array}{c}{\mathbf{f}}_1\\ {\mathbf{f}}_2\\ {\mathbf{f}}_3\end{array}\right].$$

Case 7.

Let $\left\{\xi ,{\xi }^{*}\right\}$ be osculating mates of spacelike curves with a timelike normal. If η is constant and $\mu =ctanh{\displaystyle \frac{\eta }{2}}$, then μ remains constant. Furthermore, the constancy of μ implies that $\delta ={\delta }^{*}$, since ${\mu }^{\prime }=cosh\omega \left(\delta -{\delta }^{*}\right)$ and for all $\omega \in \mathbb{R}$, $cosh\omega >0$. As a result, when η remains unchanged, the torsion and curvatures of ξ and ${\xi }^{*}$ satisfy the following relations:

$$\tau ={\tau }^{*}=-{\displaystyle \frac{sinh\eta }{\mu }},\kappa ={\kappa }^{*}-{\displaystyle \frac{{\omega }^{\prime }}{\delta }}-2\tilde{c}sinh\omega sinh\eta$$

where $\tilde{c}={\displaystyle \frac{1}{c}}$. Hence, $\left\{\xi ,{\xi }^{*}\right\}$ forms a Backlund pair, and the transformation that relates ξ and ${\xi }^{*}$ is referred to as the Backlund transformation.

Case 8.

Let $\left\{\xi ,{\xi }^{*}\right\}$ be osculating mates of spacelike curves with spacelike normals. If $\omega =\pi$ and $\omega \in [0,2\pi )$, then the relationship between ξ and ${\xi }^{*}$ is given by

$${\xi }^{*}=\xi \pm \mu {\mathbf{f}}_1.$$

The Frenet frame of ${\xi }^{*}$ is obtained by applying a rotation to the Frenet frame of ξ about its tangent vector by the pseudo-angle η, and it can be expressed as follows:

$$\left[\begin{array}{c}{\mathbf{f}}_1^{*}\\ {\mathbf{f}}_2^{*}\\ {\mathbf{f}}_3^{*}\end{array}\right]=\left[\begin{array}{ccc}1& 0& 0\\ 0& cosh\eta & \pm sinh\eta \\ 0& \pm sinh\eta & cosh\eta \end{array}\right]\left[\begin{array}{c}{\mathbf{f}}_1\\ {\mathbf{f}}_2\\ {\mathbf{f}}_3\end{array}\right].$$

If we take $\eta =0$ with $\omega \ne \pi$, then

$${\xi }^{*}=\xi +\mu \left[cos\omega {\mathbf{f}}_1+sin\omega {\mathbf{f}}_2\right]$$

In this case, the Frenet frames of both ξ and ${\xi }^{*}$ are identical when $\eta =0$ (with $\omega \ne \pi$), and the transformation is represented as

$$\left[\begin{array}{c}{\mathbf{f}}_1^{*}\\ {\mathbf{f}}_2^{*}\\ {\mathbf{f}}_3^{*}\end{array}\right]=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]\left[\begin{array}{c}{\mathbf{f}}_1\\ {\mathbf{f}}_2\\ {\mathbf{f}}_3\end{array}\right].$$

Case 9.

Let $\left\{\xi ,{\xi }^{*}\right\}$ be an osculating mate of spacelike curves with spacelike normals. If $\omega =\left({\displaystyle \frac{\pi }{2}}\right),\left({\displaystyle \frac{3\pi }{2}}\right)$ and $\omega \in [0,2\pi )$, then the relationship between ξ and ${\xi }^{*}$ is given by

$${\xi }^{*}=\xi \pm \mu {\mathbf{f}}_2.$$

Applying a rotation of the Frenet frame of ξ about the normal vector ${\mathbf{f}}_2$ by an angle η, we derive the Frenet frame of ${\xi }^{*}$, which is given by

$$\left[\begin{array}{c}{\mathbf{f}}_1^{*}\\ {\mathbf{f}}_2^{*}\\ {\mathbf{f}}_3^{*}\end{array}\right]=\left[\begin{array}{ccc}cosh\eta & 0& \pm sinh\eta \\ 0& 1& 0\\ \pm sinh\eta & 0& cosh\eta \end{array}\right]\left[\begin{array}{c}{\mathbf{f}}_1\\ {\mathbf{f}}_2\\ {\mathbf{f}}_3\end{array}\right].$$

If $\eta =0$ (with $\omega \ne \left({\displaystyle \frac{\pi }{2}}\right),\left({\displaystyle \frac{3\pi }{2}}\right)$), then

$${\xi }^{*}=\xi +\mu \left[cos\omega {\mathbf{f}}_1+sin\omega {\mathbf{f}}_2\right].$$

In this case, the Frenet frames of both ξ and ${\xi }^{*}$ are identical when $\eta =0$$\left(with\omega \ne \left({\displaystyle \frac{\pi }{2}}\right),\left({\displaystyle \frac{3\pi }{2}}\right)\right)$, and the transformation is represented as

$$\left[\begin{array}{c}{\mathbf{f}}_1^{*}\\ {\mathbf{f}}_2^{*}\\ {\mathbf{f}}_3^{*}\end{array}\right]=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]\left[\begin{array}{c}{\mathbf{f}}_1\\ {\mathbf{f}}_2\\ {\mathbf{f}}_3\end{array}\right].$$

Case 10.

The pair $\left\{\xi ,{\xi }^{*}\right\}$ forms an osculating mate of spacelike curves with spacelike normals. If η remains constant and $\mu =ctanh\left(\eta /2\right)$, then μ also remains constant. Furthermore, for μ to be constant, it is necessary that either $\delta ={\delta }^{*}$ or $\omega =\left({\displaystyle \frac{\pi }{2}}\right),\left({\displaystyle \frac{3\pi }{2}}\right)$, given that ${\mu }^{\prime }=cos\omega \left(\delta -{\delta }^{*}\right)$.

Consequently, under the assumption that η is constant and $\delta ={\delta }^{*}$, the torsion and curvatures of ξ and ${\xi }^{*}$ are given by

$$\tau ={\tau }^{*}=-{\displaystyle \frac{sinh\eta }{\mu }},\kappa =-{\displaystyle \frac{{\omega }^{\prime }}{\delta }}-\tilde{c}sin\omega sinh\eta ,{\kappa }^{*}=-{\displaystyle \frac{{\omega }^{\prime }}{\delta }}+\tilde{c}sin\omega sinh\eta$$

where $\tilde{c}={\displaystyle \frac{1}{c}}$. This establishes the relation

$$\kappa ={\kappa }^{*}-2\tilde{c}sin\omega sinh\eta .$$

If $\omega =\left({\displaystyle \frac{\pi }{2}}\right),\left({\displaystyle \frac{3\pi }{2}}\right)$, then the torsion and curvature expressions simplify to

$$\tau ={\tau }^{*}=-{\displaystyle \frac{sinh\eta }{\mu }},\kappa ={\kappa }^{*}-2\tilde{c}sin\omega sinh\eta$$

Thus, $\left\{\xi ,{\xi }^{*}\right\}$ constitutes a Bäcklund mate, and the transformation mapping ξ to ${\xi }^{*}$ corresponds to a Bäcklund transformation.

8. Conclusions

In this study, we have extended the concept of osculating curve mates to the Minkowski 3-space and examined their geometric behavior under Lorentzian inner product structures. We defined osculating mates via the intersection of osculating planes and a common directional vector, and we explored how the causal character of the curves (timelike or spacelike) influences their curvature and torsion relations. Furthermore, we derived the rotational transformation matrices between the Frenet frames of these osculating mates and identified conditions under which the pairs coincide with known curve configurations such as Bertrand, Mannheim, and Bäcklund pairs.

Our findings show that the osculating curve concept not only unifies several previously defined mate curve types but also offers a broader geometric framework for analyzing the interaction of curves in Lorentzian space. The quaternionic representation of the transformation between Frenet frames highlights the deep algebraic structure of these relationships. Future studies may consider extending this framework to higher-dimensional Lorentzian spaces or to null curves and their associated invariants.