On the Integral Invariants and Dralls for a Dual Lorentzian Closed Strip

1. Introduction

The study of kinematics and line geometry has been profoundly influenced by the introduction of dual numbers by Clifford in 1873 [1]. Subsequently, E. Study [2] established a one-to-one correspondence between oriented lines in Euclidean three-space and points on the dual unit sphere, now known as the E. Study mapping. This correspondence provides a powerful geometric framework in which differentiable dual spherical curves correspond to ruled surfaces in line space. The E. Study mapping for space-like and time-like lines in Minkowski 3-space has been investigated previously, where a one-to-one correspondence between directed lines and ordered pairs was established [3].

Ruled surfaces and their dual representations have attracted considerable interest due to their applications in differential geometry, kinematics, and motion theory. In this context, Önder and Uğurlu [4] characterized normal and spherical curves in dual space and established important geometric relations for the study of ruled surface invariants. Later, Yapar and Sağıroğlu [5] introduced curvature motion on the dual hyperbolic unit sphere and investigated the associated ruled surfaces and line congruences through E. Study’s correspondence.

Integral invariants constitute one of the most important tools in the global analysis of ruled surfaces. Özyılmaz and Yaylı [6] investigated the integral invariants of timelike ruled surfaces and obtained relations involving dual spherical motions and dual angles of pitch. Bektaş and Şenyurt [7] further studied ruled surfaces generated by closed timelike curves in dual Lorentzian space and derived several characterizations in terms of pitch, angle of pitch, and drall invariants. More recently, Gür Mazlum, Şenyurt, and Grilli [8] examined dual parallel equidistant ruled surfaces and established new relations between Gaussian curvatures and dual integral invariants.

A significant contribution to the theory of closed strips was made by Yapar [9], who investigated dual closed strips in Euclidean line space and obtained important relationships among the integral invariants of the corresponding ruled surfaces. However, analogous results for dual Lorentzian closed strips have not been studied in sufficient detail.

Motivated by these developments, the present paper investigates the integral invariants and dralls of dual Lorentzian closed strips in the line space $\left(R_1^3\right)$. Using E. Study’s correspondence and the differential geometry of dual Lorentzian curves, we derive new relations among the Lorentzian pitch, dual angle of pitch, and associated drall invariants. Furthermore, explicit expressions for the dual Lorentz geodesic curvature, dual Lorentz normal curvature, and dual Lorentz geodesic torsion are obtained. These results extend Yapar’s theory of dual closed strips from Euclidean line space to the Lorentzian setting and contribute to the geometric analysis of closed strip motions in the Lorentzian space.

2. Preliminaries

Basic definitions and theorems concerning dual numbers are given by Guggenheimer [10]. Also, fundamental definitions and theorems regarding dual Lorentzian vectors which include dual Lorentzian inner product and Lorentzian cross product can be found in the work of Şentürk and Yüce [11].

A differentiable curve $\tilde{X}\left(s\right)=x\left(s\right)+ϵx^{*}\left(s\right)$ on the dual Lorentzian unit sphere ${\tilde{S}}_1^2$ is called a dual Lorentzian curve. Here, s is a real parameter that typically represents the arc-lenght of the curve $x\left(s\right)$. The dual arc-lenght of the curve $\tilde{X}\left(s\right)$, denoted by $\tilde{s}=s+ϵs^{*}$, is a dual scalar that characterizes the total displacement of the corresponding ruled surface in the Lorentzian line space $R_1^3$. The real part s represents the translational shift $pitch$ along the line of motion. The derivative of the dual position vector with respect to the dual arc-lenght gives the unit dual tangent vector of the Lorentzian ruled surface, providing a complete description of its differential geometry. By means of E. Study mapping, a differentiable Lorentzian curve $\tilde{X}\left(s\right)$ on the dual unit Lorentzian sphere represents a Lorentzian ruled surface, which is differentiable family of lines in Lorentzian line space.

Suppose we have a dual Lorentzian closed curve, denoted as $\tilde{X}:I\to {\tilde{D}}_1^3$, $\tilde{X}\left(s\right)=x\left(s\right)+ϵx^{*}\left(s\right)$, which is parametrized by the arc-lenght s of its indicatrix. Given that $x=a$ represents a unit vector aligned with the tangent of this indicatrix, we establish that $<x^{\prime },x^{\prime }>=1$ for the Lorentzian inner product, where the prime symbol indicates differentiation relative to s.

The relationship $p\left(s\right)\wedge x\left(s\right)=x^{*}\left(s\right)$ typically yields an infinitive number of solutions for the vector function $p\left(s\right)$. If $p_0\left(s\right)$ represents one such valid solution, the entire set of solutions can be expressed as $p\left(s\right)=p_0\left(s\right)+u\left(s\right)x\left(s\right)$. In this context, u is a real scalar function of s. By analyzing the Lorentzian inner product,

$$<p^{\prime },x>=<p_0^{\prime }+u^{\prime }.x+u.x^{\prime },x>=<p_0^{\prime },x>+u^{\prime }$$

(1)

we can define $u=u_0=-\int <p_0^{\prime },x>$. This specific choice ensures that $p_0\left(s\right)+u_0\left(s\right)x\left(s\right)=v\left(s\right)$ becomes the unique solution where the condition $<v^{\prime },x>=0$ is satisfied. Consequently, the dual Lorentzian closed curve can be uniquely represented as:

$$\tilde{X}\left(s\right)=x\left(s\right)+ϵ(v\left(s\right)\wedge x\left(s\right))$$

(2)

It is important to note that v is determined solely by the curve $\tilde{X}\left(s\right)$. Following this logic, the dual arc-lenght of the curve $\tilde{X}\left(s\right)$ is formulated as:

$$s^{*}={\int }_{t_1}^t\left|\right|x\left|\right|dt+ϵ{\int }_{t_1}^t<t,x^{*}>dt=s+ϵ{\int }_0^s<a,{\left(x^{*}\right)}^{\prime }>ds=s+ϵ{\int }_0^s\sigma ds$$

(3)

where we define $\sigma =<a,{\left(x^{*}\right)}^{\prime }>$.

Definition 1. 

Let us examine a Lorentz strip positioned along the dual Lorentzian closed curve $\tilde{X}\left(s\right)$ within the dual Lorentzian space ${\tilde{D}}_1^3$. This structure is identified as a dual Lorentzian closed strip. Depending on the geometric nature of the curve $\tilde{X}$ within this space, the strip is classified as follows:

1.

It is a dual Lorentzian closed curvature strip if $\tilde{X}$ represents the Lorentzian curvature line of thee strip.

2.

It is a dual Lorentzian closed geodesic strip if $\tilde{X}$ serves as the Lorentzian geodesic curve.

3.

It is dual Lorentzian closed asymptotic strip if $\tilde{X}$ corresponds to the Lorentzian asymptotic curve of the strip.

The strips and ruled surfaces discussed throughout this study are assumed to be positively orientable. Based on the previously established Equation (1.3), the relationship between the parameters can be expressed as ${\displaystyle \frac{d{s}^{*}}{ds}}=1+ϵ\sigma$ or, conversely, ${\displaystyle \frac{ds}{d{s}^{*}}}=1-ϵ\sigma$. Furthermore, since the inner product $<a,{\left(x^{*}\right)}^{\prime }>$ is equivalent to $<v^{\prime }\wedge x,a>=\sigma$, it follows that $v^{\prime }\wedge x=\sigma a$. This leads to the derivation

$${\displaystyle \frac{d\tilde{X}}{d{s}^{*}}}={\displaystyle \frac{d\tilde{X}}{ds}}.{\displaystyle \frac{ds}{d{s}^{*}}}=A=a+ϵa^{*}=a+ϵv\wedge a$$

(4)

Now, let $\tilde{X}\left(s\right)=x\left(s\right)+ϵv\left(s\right)\wedge x\left(s\right),N\left(s\right)=n\left(s\right)+ϵv\left(s\right)\wedge n\left(s\right)$ be an orientable Lorentzian closed strip where $<A,N>=0$ and $\left|\right|\tilde{X}\left|\right|=\left|\right|N\left|\right|=1$ (N is spacelike). Here the derivative ${\displaystyle \frac{dX}{d{s}^{*}}}$ is denoted by A. By defining the dual spacelike unit vector as $N\wedge A=G=g+ϵg^{*}$, we can deduce from Equation (2.4) that $N=n+ϵv\wedge n$.

We define $A,G,N$ as the dual timelike Darboux Riboucour frame associated with the curve $\tilde{X}\left(s\right)$ on the dual Lorentzian closed strip. Given that $A,G,N$ constitutes a timelike frame, A is defined as a timelike vector, while G and N are designed as spacelike vectors. By applying the principles of inner products and their respective derivatives to these vectors, the system of differential equation governing the frame can be organized into the following matrix representation:

$${\displaystyle \frac{d}{d{s}^{*}}}\left[\begin{array}{c}A\\ G\\ N\end{array}\right]=\left[\begin{array}{ccc}0& K_g& K_n\\ K_g& 0& T_g\\ K_n& -T_g& 0\end{array}\right].\left[\begin{array}{c}A\\ G\\ N\end{array}\right]$$

(5)

In Equation (2.5), the dual functions $K_g=k_g+ϵk_g^{*},K_n=k_n+ϵk_n^{*}$ and $T_g=t_g+ϵt_g^{*}$ are defined as geodesic curvature, normal curvature and geodesic torsion of the dual Lorentzian closed strip.

Since $v^{\prime }\wedge x=-\sigma a$, we get $n\wedge (v^{\prime }\wedge x)=-\sigma g$. From this, we obtain the following result:

$$v^{\prime }={\displaystyle \frac{1}{\lambda }}.(\delta x-\sigma g)$$

(6)

where $\delta =<n,v^{\prime }>$ and $\lambda =<n,x>$. From here, we derive that:

$$\begin{array}{cc}\hfill K_g& =<G,{\displaystyle \frac{dA}{d{s}^{*}}}>\hfill \\ \hfill & =<G,{\displaystyle \frac{d}{ds}}[a+ϵ(v\wedge a)].{\displaystyle \frac{ds}{d{s}^{*}}}>\hfill \\ \hfill & =<g+ϵ(v\wedge g),{\displaystyle \frac{d}{ds}}[a+ϵ(v\wedge a)].{\displaystyle \frac{ds}{d{s}^{*}}}>\hfill \\ \hfill & =k_g+{\displaystyle \frac{ϵ}{\lambda }}.(\delta c_1-\sigma \lambda k_g)\hfill \end{array}$$

(7)

$$K_n=k_n+{\displaystyle \frac{ϵ}{\lambda }}.(\delta c_2+\sigma -\sigma \lambda k_n)$$

(8)

and

$$T_g=t_g+{\displaystyle \frac{ϵ}{\lambda }}.(\delta c_3-\sigma \lambda t_g)$$

(9)

where $\lambda =<n,x>,\delta =<n,v^{\prime }>,c_1=<g,x\wedge a>,c_2=<n,x\wedge a>$ and $c_3=<n,x\wedge g>$. Here $k_g,k_n$ and $t_g$ correspond to the geodesic curvature, normal curvature and geodesic torsion of the curve x, respectively. The resulting structure provide new descriptions of $K_g,K_n$ and $T_g$ of the dual Lorentzian closed strip.

By employing E. Study’s mapping, the motion of a dual Lorentzian unit sphere K relative to a fixed Lorentzian sphere $K^{\prime }$ is utilized to represent the spatial kinematic relationship between the moving Lorentzian space H and fixed Lorentzian space $H^{\prime }$. This one-parameter spatial motion $H/H^{\prime }$ inherently construct a closed Lorentzian ruled surface within the fixed frame. This study builds upon the foundational dual Lorentzian element correspondences and Lorentzian spherical motion theories established in previous literature, specificially focusing on the characterization of pitch and real angle properties in $R_1^3$.

Definition 2. 

The dual Lorentzian Steiner vector of the closed motion $K/K^{\prime }$ is denoted by W and defined by;

$$\begin{array}{cc}\hfill W& =\oint (T_g.A-K_n.G+K_g.N)\hfill \end{array}$$

(10)

$$\begin{array}{cc}\hfill & =\oint (t_{g}a-k_{n}g+k_{g}n)+ϵ\oint (t_g{a}^{*}+t_g^{*}a-k_n{g}^{*}-k_n^{*}g+k_g{n}^{*}+k_g^{*}n)\hfill \end{array}$$

(11)

establishing an analogy to the definition in [9].

For a dual closed Lorentzian ruled surface x embedded in $R_1^3$, its real pitch angle is characterized by

$${\lambda }_x=-<d,x>$$

(12)

given that d is the Steiner vector representing the motion. Concurrently, the real pitch of the surface x is computed by

$$L_x=<d^{*},x>+<d,x^{*}>$$

(13)

where the real numbers d and $d^{*}$ constitute the dual Lorentzian Steiner vector components [9].

Definition 3. 

Let $\tilde{X}\left(s\right)=x\left(s\right)+ϵx^{*}\left(s\right)$ signify an orientable closed Lorentzian ruled surface under the condition $\left|\right|\tilde{X}\left|\right|=1$. Based on relations (2.11) and (2.13), the dual pitch angle for the closed Lorentzian ruled surface x is derived and formulated by;

$$\begin{array}{cc}\hfill {\mathrm{\Lambda }}_x& =-<W,X>=-<w+ϵw^{*},x+ϵx^{*}>\hfill \\ \hfill & =-<w,x>-ϵ(<w^{*},x>+<w,x^{*}>)\hfill \\ \hfill & ={\lambda }_x-ϵL_x\hfill \end{array}$$

(14)

3. Dual Pitch Angles of Closed Lorentzian Ruled Surfaces Generated by Vectors $\mathbf{A},\mathbf{G},\mathbf{N}$ and $\mathbf{D}$

Definition 4. 

By employing Equations (2.11) and (2.14), the dual angles of pitch for the closed Lorentzian ruled surfaces generated on $K^{\prime }$ via the vectors $A,G,N$ and D can be expressed respectively as follows:

$${\mathrm{\Lambda }}_a=-<W,A>$$

alternatively, the real and dual components of ${\mathrm{\Lambda }}_a$ are given by;

$${\lambda }_a=\oint t_g,L_a=-\oint t_{g^{*}}$$

(15)

For G,

$${\mathrm{\Lambda }}_g=-<W,G>$$

or, equivalently, the real and dual parts of ${\mathrm{\Lambda }}_g$ can be written as;

$${\lambda }_g=\oint k_n,L_g=-\oint k_{n^{*}}$$

(16)

Similarly for N,

$${\mathrm{\Lambda }}_n=-<W,N>$$

or corresponding the real and dual parts of ${\mathrm{\Lambda }}_n$ are;

$${\lambda }_n=-\oint k_g,L_n=\oint k_{g^{*}}$$

(17)

Consequently, by substituting from relations (3.1), (3.2) and (3.3), the vector W is formulated as,

$$W={\mathrm{\Lambda }}_{a}A-{\mathrm{\Lambda }}_{g}G-{\mathrm{\Lambda }}_{n}N$$

(18)

Theorem 1. 

Throughout the displacement along the dual Lorentzian closed strip $\{\tilde{X}\left(s\right),N\left(s\right)\}$ belonging to the moving frame $\{A,G,N\}$, the constituents of the Steiner vector characterizing the motion $K/K^{\prime }$ are composed of the dual pitch angles of the closed Lorentzian ruled surfaces generated on $K^{\prime }$ via the vectors $A,G$ and N.

At this stage, let us introduce a constant unit dual timelike vector D that is rigidly linked to the moving frame $\{A,G,N\}$, formulated as follows:

$$D=cosh\theta cosh\varphi A+sinh\theta cosh\varphi G+sinh\varphi N$$

(19)

where $\varphi ={\varphi }_1+ϵ{\varphi }_1^{*}$ and $\theta ={\theta }_1+ϵ{\theta }_1^{*}$ denote fixed dual hiporbolic angles. Hence, by virtue of Definition (3.1), the dual pitch angle corresponding to the closed Lorentzian ruled surface generated on $K^{\prime }$ via the vector D can be formulated as:

$${\mathrm{\Lambda }}_d={\lambda }_d-ϵL_d=-<W,D>$$

(20)

or, equivalently,

$${\mathrm{\Lambda }}_d={\mathrm{\Lambda }}_{a}cosh\theta cosh\varphi +{\mathrm{\Lambda }}_{g}sinh\theta cosh\varphi +{\mathrm{\Lambda }}_{n}sinh\varphi$$

(21)

written in terms of its real and dual components, ${\mathrm{\Lambda }}_d$ yields:

$${\lambda }_d={\lambda }_{a}cosh{\theta }_{1}cosh{\varphi }_1+{\lambda }_{g}sinh{\theta }_{1}cosh{\varphi }_1+{\lambda }_{n}sinh{\varphi }_1$$

(22)

$$\begin{array}{c}\hfill L_d=L_{a}cosh{\theta }_{1}cosh{\varphi }_1+L_{g}sinh{\theta }_{1}cosh{\varphi }_1+L_{n}sinh{\varphi }_1\\ \hfill -{\lambda }_a({\theta }_1^{*}sinh{\theta }_{1}cosh{\varphi }_1+{\varphi }_1^{*}sinh{\varphi }_{1}cosh{\theta }_1)\\ \hfill -{\lambda }_g({\theta }_1^{*}cosh{\theta }_{1}cosh{\varphi }_1+{\varphi }_1^{*}sinh{\varphi }_{1}sinh{\theta }_1)\\ \hfill -{\lambda }_n{\varphi }_1^{*}cosh{\varphi }_1\end{array}$$

(23)

Consequently, we can state the following theorem.

Theorem 2. 

By considering the one-parameter dual constrained motion along a given dual Lorentzian closed strip, it is established that relation (3.8) and (3.9) connect the Lorentzian line-space properties with the actual angles of pitch and the real pitch of the Lorentzian closed ruled surfaces generated by lines $a,g$ and n, respectively.

Furthermore, the distribution parameter ${\displaystyle \frac{1}{d_m}}$ belonging to a closed Lorentzian ruled surface traced during a closed motion via a dual unit vector $M=m+ϵm^{*}$ can be formulated as:

$${\displaystyle \frac{1}{d_m}}={\displaystyle \frac{<m^{\prime },{\left(m^{*}\right)}^{\prime }>}{<m^{\prime },m^{\prime }>}}$$

(24)

In accordance with expression (3.10), the specific distribution parameters corresponding to the closed Lorentzian ruled surfaces generated by the lines $A,G,N$ and D can be stated in the following manner:

$${\displaystyle \frac{1}{d_a}}={\displaystyle \frac{k_g{k}_g^{*}+k_n{k}_n^{*}}{k_g^2+k_n^2}},{\displaystyle \frac{1}{d_g}}={\displaystyle \frac{-k_g{k}_g^{*}+t_g{t}_g^{*}}{-k_g^2+t_g^2}},{\displaystyle \frac{1}{d_n}}={\displaystyle \frac{-k_n{k}_n^{*}+t_g{t}_g^{*}}{-k_n^2+t_g^2}},{\displaystyle \frac{1}{d_d}}={\displaystyle \frac{-Ad+Be+Cf}{-A^2+B^2+C^2}}$$

(25)

where the constituent variables are defined as:

$$\begin{array}{cc}\hfill A& =sinh{\theta }_{1}cosh{\varphi }_1{k}_g+sinh{\varphi }_1{k}_n\hfill \\ \hfill B& =cosh{\theta }_{1}cosh{\varphi }_1{k}_g-sinh{\varphi }_1{t}_g\hfill \\ \hfill C& =cosh{\theta }_{1}cosh{\varphi }_1{k}_n+sinh{\theta }_{1}cosh{\varphi }_1{t}_g\hfill \\ \hfill d& =sinh{\theta }_{1}cosh{\varphi }_1{k}_g^{*}+sinh{\varphi }_1{k}_n^{*}+{\varphi }_1^{*}cosh{\varphi }_1{k}_n+{\theta }_1^{*}cosh{\theta }_{1}cosh{\varphi }_1{k}_g+{\varphi }_1^{*}sinh{\varphi }_{1}sinh{\theta }_1{k}_g\hfill \\ \hfill e& =cosh{\theta }_{1}cosh{\varphi }_1{k}_g^{*}+{\theta }_1^{*}snh{\theta }_{1}cosh{\varphi }_1{k}_g+{\varphi }_1^{*}sinh{\varphi }_{1}cosh{\theta }_1{k}_g-sinh{\varphi }_1{t}_g^{*}-{\varphi }_1^{*}cosh{\varphi }_1{t}_g\hfill \\ \hfill f& =cosh{\theta }_{1}cosh{\varphi }_1{k}_n^{*}+{\theta }_1^{*}sinh{\theta }_{1}cosh{\varphi }_1{k}_n+{\varphi }_1^{*}sinh{\varphi }_{1}cosh{\theta }_1{k}_n\hfill \\ \hfill & +sinh{\theta }_{1}cosh{\varphi }_1{t}_g^{*}+{\theta }_1^{*}cosh{\theta }_{1}cosh{\varphi }_1{t}_g+{\varphi }_1^{*}sinh{\varphi }_{1}sinh{\theta }_1{t}_g\hfill \end{array}$$

(26)

4. Analysis in the Plane $\{\mathbf{A},\mathbf{G}\}$ of the Moving Frame $\{\mathbf{A},\mathbf{G},\mathbf{N}\}$

Under these conditions, utilizing Equation (3.7) subject to the constraints, we obtain the following formulated relationship:

$${\mathrm{\Lambda }}_d={\mathrm{\Lambda }}_{a}cosh\theta +{\mathrm{\Lambda }}_{g}sinh\theta$$

(27)

Alternatively, the real along with the the dual components of ${\mathrm{\Lambda }}_d$ can be expressed as:

$$\begin{array}{cc}\hfill {\lambda }_d& ={\lambda }_{a}cosh{\theta }_1+{\lambda }_{g}sinh{\theta }_1\hfill \\ \hfill L_d& =L_{a}cosh{\theta }_1+L_{g}sinh{\theta }_1-{\lambda }_g{\theta }_1^{*}cosh{\theta }_1\hfill \end{array}$$

(28)

and the distribution parameter is given by;

$${\displaystyle \frac{1}{d_d}}={\displaystyle \frac{-dsinh{\theta }_1{k}_g+ecosh{\theta }_1{k}_g+f(k_{n}cosh{\theta }_1+t_{g}sinh{\theta }_1)}{k_g^2+{(k_{n}cosh{\theta }_1+t_{g}sinh{\theta }_1)}^2}}$$

(29)

where $d,e$ and f are formulated as Equation (3.12).

4.1. Special Cases

Let the strip be dual closed Lorentzian curvature strip ($T_g=0$, and so ${\mathrm{\Lambda }}_a=0$) and D in the plane $\{A,G\}$ of the frame $\{A,G,N\}$. Under these assumptions, Equations (4.1) and (4.3) yield the following result:

$$\begin{array}{cc}\hfill {\mathrm{\Lambda }}_d& ={\mathrm{\Lambda }}_{g}sinh\theta \hfill \\ \hfill {\displaystyle \frac{1}{d_d}}& ={\displaystyle \frac{-usinh{\theta }_1{k}_g+vcosh{\theta }_1{k}_g+wcosh{\theta }_1{k}_n}{k_g^2+cos{h}^2{\theta }_1{k}_n^2}}\hfill \end{array}$$

(30)

where,

$$\begin{array}{cc}\hfill u& =sinh{\theta }_1{k}_g^{*}+{\varphi }_1^{*}{k}_n+{\theta }_1^{*}cosh{\theta }_1{k}_g\hfill \\ \hfill v& =cosh{\theta }_1{k}_g^{*}+{\theta }_1^{*}sinh{\theta }_1{k}_g\hfill \\ \hfill w& =cosh{\theta }_1{k}_n^{*}+{\theta }_1^{*}sinh{\theta }_1{k}_n.\hfill \end{array}$$

(31)

Consequently, we can state the following theorem:

Theorem 3. 

Supposing that the strip represents a curvature strip, throughout the one-parameter dual closed spherical motion $K/K^{\prime }$ in Lorentzian space, the dual pitch angle corresponding to the closed Lorentzian ruled surface generated on $K^{\prime }$ via the fixed timelike vector D located in the plane $\{A,G\}$ of the moving frame $\{A,G,N\}$ can be evaluated through the dual pitch angle of the closed Lorentzian ruled surface generated on $K^{\prime }$ via the vector G alongside the dual angle θ.

Thus the relation,

$${\mathrm{\Lambda }}_d={\mathrm{\Lambda }}_{g}sinh\theta$$

can be reformulated in the following manner:

$${\displaystyle \frac{{\mathrm{\Lambda }}_d}{{\mathrm{\Lambda }}_g}}=sinh\theta$$

(32)

Hence, the subsequent theorem can be presented:

Theorem 4. 

In the case where the strip is a curvature strip, during one-parameter dual closed spherical motion $K/K^{\prime }$ the ratio between the dual pitch angle of the closed Lorentzian ruled surface traced on $K^{\prime }$ by the fixed timelike vector D within the plane $\{A,G\}$ of the moving frame $\{A,G,N\}$ and the dual pitch angle of the closed Lorentzian ruled surface generated on $K^{\prime }$ by the vector G remains constant and varies independently of the underlying motion.

The real along with the dual component of expression Equation (4.6) are specified respectively as follows:

$${\displaystyle \frac{{\lambda }_d}{{\lambda }_g}}=sinh{\theta }_1$$

(33)

$$L_d=L_{g}sinh{\theta }_1-{\lambda }_g{\theta }_1^{*}cosh{\theta }_1$$

(34)

In the Lorentzian line space, we can establish the following theoretical principles:

Theorem 5. 

Consider a closed spatial motion $H/H^{\prime }$ associated with a one-parameter closed dual spherical motion $K/K^{\prime }$ that is linked to a closed Lorentzian curvature strip. Let d be a fixed line in space H that lies parallel to the $\{a,g\}$ plane, and let it generate a closed Lorentzian ruled surface in $H^{\prime }$. The ratio of the real pitch angle of this surface to the real pitch real pich angle of the closed Lorentzian ruled surface generated in $H^{\prime }$ by the line g remains constant. This ratio is invariant with respect to the motion itself and is precisely given by $sinh{\theta }_1$.

Theorem 6. 

Under the closed spatial motion $H/H^{\prime }$ corresponding to the one-parameter closed dual spherical motion $K/K^{\prime }$ of the closed Lorentzian dual curvature strip, the real pitch of the closed Lorentzian ruled surface generated in $H^{\prime }$ by the fixed line d (which resides in the moving space H and is parallel to the plane $\{a,g\}$) can be computed via Equation (4.8). Within this formulation, $L_g$ and ${\lambda }_g$ represent the real integral invariants belonging to the closed Lorentzian ruled surface generated in $H^{\prime }$ by the line g. Furthermore, ${\theta }_1$ and ${\theta }_1^{*}$ denote the spatial angle and the shortest distance between lines d and g, respectively. Consequently, in the Lorentzian line space, the following structural relation holds:

$${\lambda }_g{L}_d-L_g{\lambda }_d\mp {\theta }_1^{*}{\lambda }_g{({\lambda }_g^2+{\lambda }_d^2)}^{{\displaystyle \frac{1}{2}}}=0$$

(35)

Suppose that the strip constitutes a dual Lorentzian closed geodesic strip, which implies $K_g=0$ and consequently ${\mathrm{\Lambda }}_n=0$. Additionally, let D lie within the plane defined by $\{A,G\}$ with respect to the moving frame $\{A,G,N\}$. Under these conditions, the Equations (4.1) and (4.2) hold true, which distribution parameter is given by:

$${\displaystyle \frac{1}{d_d}}={\displaystyle \frac{cosh{\theta }_1{k}_n^{*}+{\theta }_1^{*}sinh{\theta }_1{k}_n+sinh{\theta }_1{t}_g^{*}+{\theta }_1^{*}cosh{\theta }_1{t}_g}{cosh{\theta }_1{k}_n+sinh{\theta }_1{t}_g}}$$

(36)

Next, let the strip be characterized as a dual Lorentzian closed asymptotic strip, such that $K_n=0$ and ${\mathrm{\Lambda }}_g=0$. Assuming D is embedded in the $\{A,G\}$ plane of the moving frame $\{A,G,N\}$, combining Equations (4.1) and (4.3) gives:

$${\mathrm{\Lambda }}_d={\mathrm{\Lambda }}_{a}cosh\theta$$

(37)

and

$${\displaystyle \frac{1}{d_d}}={\displaystyle \frac{-psinh{\theta }_1{k}_g+vcosh{\theta }_1{k}_g+rsinh{\theta }_1{t}_g}{k_g^2+sin{h}^2{\theta }_1{t}_g^2}}$$

(38)

where

$$\begin{array}{cc}\hfill p& =sinh{\theta }_1{k}_g^{*}+{\theta }_1^{*}cosh{\theta }_1{k}_g\hfill \\ \hfill v& =cosh{\theta }_1{k}_g^{*}+{\theta }_1^{*}sinh{\theta }_1{k}_g\hfill \\ \hfill r& =sinh{\theta }_1{t}_g^{*}+{\theta }_1^{*}cosh{\theta }_1{t}_g.\hfill \end{array}$$

(39)

This allows us to state the following theorems:

Theorem 7. 

Consider a one-parameter dual spherical motion denoted by $K/K\prime$ in the Lorentzian space. If the strip represents the dual asymptotic strip, the dual angle of pitch for the closed Lorentzian ruled surface generated on $K^{\prime }$ by the fixed timelike vector D located in the plane $\{A,G\}$ of the moving frame $\{A,G,N\}$ is identifiable via the dual angle of pitch of the closed Lorentzian ruled surface traced on $K^{\prime }$ by the vector A, given that the dual angle is constant.

Consequently, Equation (4.11) yields the following relationship:

$${\displaystyle \frac{{\mathrm{\Lambda }}_d}{{\mathrm{\Lambda }}_a}}=cosh\theta$$

(40)

Theorem 8. 

Under the condition that the strip serves as the dual asymptotic strip during the closed spherical motion $K/K^{\prime }$ in the Lorentzian space, the proportion between the dual angle of pitch of the closed Lorentzian ruled surface generated on $K^{\prime }$ by the fixed timelike vector D in the $\{A,G\}$ plane of the moving frame $\{A,G,N\}$ and the dual angle of pitch of the closed Lorentzian ruled surface generated on $K^{\prime }$ by the vector A remains constant and invariant under the motion.

The real and dual components of Equation (4.14) can be explicitly structured as follows:

$${\displaystyle \frac{{\lambda }_d}{{\lambda }_a}}=cosh{\theta }_1$$

(41)

$$L_d=L_{a}cosh{\theta }_1-{\lambda }_a{\theta }_1^{*}sinh{\theta }_1$$

(42)

These formulations lead to the subsequent theorems within the Lorentzian line space:

Theorem 9. 

During a closed spatial motion $H/H^{\prime }$ that maps to the dual closed spherical motion $K/K^{\prime }$ linked with the dual closed asymptotic strip in the Lorentzian space, the ratio of the real pitch angle of the closed Lorentzian ruled surface traced in $H^{\prime }$ by the line d (which is fixed in the space H and parallel to the $\{a,g\}$ plane) to the real pitch angle of the closed Lorentzian ruled surface generated in $H^{\prime }$ by the line a is constant. This ratio is independent of the underlying motion and equals $cosh{\theta }_1$.

Theorem 10. 

Let $H/H^{\prime }$ represents a closed spatial motion that corresponds to the dual closed spatial motion $K/K^{\prime }$ in the Lorentzian space, which is associated with the dual closed asymptotic strip. In the moving space H, a line d is fixed such that it remains parallel to the plane spanned by $\{a,g\}$. The real pitch of the closed Lorentzian ruled surface generated by this line d within $H^{\prime }$ can be evaluated using Equation (4.16). Within this framework, ${\lambda }_a$ and $L_a$ denote the real integral invariants characterizing the closed Lorentzian ruled surface generated by the line a in $H^{\prime }$. The parameters ${\theta }_1$ and ${\theta }_1^{*}$ represent angle and distance between the lines d and a, respectively. Furthermore, within the Lorentzian line space, following analytical relationship is satisfied:

$$L_d{\lambda }_a\mp {\lambda }_a{\theta }_1^{*}{({\lambda }_a^2+{\lambda }_d^2)}^{{\displaystyle \frac{1}{2}}}-L_a{\lambda }_d=0$$

(43)

5. Analysis in the Plane $\{\mathbf{G},\mathbf{N}\}$ of the Moving Frame $\{\mathbf{A},\mathbf{G},\mathbf{N}\}$

Assume that the timelike vector D depends on the vectors G and N of this moving frame. Let the strip be a dual closed spherical curvature strip such that $T_g=0$ and ${\mathrm{\Lambda }}_a=0$. Furthermore, let the timelike vector D be rigidly linked to the moving frame $\{A,G,N\}$. Under these conditions, the dual instantaneous Pfaffian vector W reduces to:

$$W=-{\mathrm{\Lambda }}_{g}G-{\mathrm{\Lambda }}_{n}N$$

(44)

Consequently, the dual angle of pitch ${\mathrm{\Lambda }}_d$ can be determined via the inner product of the dual instantaneous velocity vector and the vector D as follows:

$${\mathrm{\Lambda }}_d=-<W,D>={\mathrm{\Lambda }}_{g}sinh\theta cosh\varphi +{\mathrm{\Lambda }}_{n}sinh\varphi$$

(45)

6. Analysis in the Plane $\{\mathbf{A},\mathbf{N}\}$ of the Moving Frame $\{\mathbf{A},\mathbf{G},\mathbf{N}\}$

Let us evaluate the configuration where the timelike vector D is situated entirely in the plane spanned by $\{A,N\}$ within the moving frame $\{A,G,N\}$. Under this constraint, the condition $\theta =0$ is imposed, which simplifies the geometric relations. By substituting these parameters into Equation (3.7), we arrive at:

$${\mathrm{\Lambda }}_d={\mathrm{\Lambda }}_{a}cosh\varphi +{\mathrm{\Lambda }}_{n}sinh\varphi$$

(46)

Alternatively, this equation can be decoupled into its respective real and dual components as follows:

$$\begin{array}{cc}\hfill {\lambda }_d& ={\lambda }_{a}cosh{\varphi }_1+{\lambda }_{n}sinh{\varphi }_1\hfill \\ \hfill L_d& =L_{a}cosh{\varphi }_1+L_{n}sinh{\varphi }_1-{\lambda }_a{\varphi }_1^{*}sinh{\varphi }_1-{\lambda }_n{\varphi }_1^{*}cosh{\varphi }_1\hfill \end{array}$$

(47)

and the distribution parameter is given by;

$${\displaystyle \frac{1}{d_d}}={\displaystyle \frac{k_n{k}_n^{*}+s(cosh{\varphi }_1{k}_g^{*}-sinh{\varphi }_1{t}_g^{*}-{\varphi }_1^{*}cosh{\varphi }_1{t}_g)}{k_n^2+s^2}}$$

(48)

where

$$s=cosh{\varphi }_1{k}_g-sinh{\varphi }_1{t}_g$$

6.1. Special Cases

Now, consider case where the strip is a dual closed spherical curvature strip (which implies $T_g=0$, and consequently ${\mathrm{\Lambda }}_a=0$). Under this geometric condition, suppose that the timelike vector D lies entirely within the $\{A,N\}$ plane of the moving frame $\{A,G,N\}$. In this case, the dual pitch expression simplifies to:

$${\mathrm{\Lambda }}_d={\mathrm{\Lambda }}_{n}sinh\varphi$$

(49)

Consequently, this yields the following relation from the Equation (6.3):

$${\displaystyle \frac{1}{d_d}}={\displaystyle \frac{k_n{k}_n^{*}+cos{h}^2{\varphi }_1{k}_g{k}_g^{*}}{k_n^2+cos{h}^2{\varphi }_1{k}_g^2}}$$

(50)

Consequently, we can state the following theorem:

Theorem 11. 

Supposing that the strip represents a curvature strip, throughout the one-parameter dual closed spherical motion $K/K^{\prime }$ in Lorentzian space, the dual pitch angle corresponding to the closed Lorentzian ruled surface generated on $K^{\prime }$ via the fixed timelike vector D located in the plane $\{A,N\}$ of the moving frame $\{A,G,N\}$ can be evaluated through the dual pitch angle of the closed Lorentzian ruled surface generated on $K^{\prime }$ via the vector N alongside the dual angle ϕ.

Thus the Equation (6.4) holds and it can be written by the following form:

$${\displaystyle \frac{{\mathrm{\Lambda }}_d}{{\mathrm{\Lambda }}_n}}=sinh\varphi$$

(51)

Hence, the subsequent theorem can be presented:

Theorem 12. 

In the case where the strip is a curvature strip, during one-parameter dual closed spherical motion $K/K^{\prime }$ the ratio between the dual pitch angle of the closed Lorentzian ruled surface traced on $K^{\prime }$ by the fixed timelike vector D within the plane $\{A,N\}$ of the moving frame $\{A,G,N\}$ and the dual pitch angle of the closed Lorentzian ruled surface generated on $K^{\prime }$ by the vector N remains constant and varies independently of the underlying motion.

The real along with the dual component of expression Equation (6.6) are specified respectively as follows:

$${\displaystyle \frac{{\lambda }_d}{{\lambda }_n}}=sinh{\varphi }_1$$

(52)

$$L_d=L_{n}sinh{\varphi }_1-{\lambda }_n{\varphi }_1^{*}cosh{\varphi }_1$$

(53)

In the Lorentzian line space, we can establish the following theoretical principles:

Theorem 13. 

Consider a closed spatial motion $H/H^{\prime }$ associated with a one-parameter closed dual spherical motion $K/K^{\prime }$ that is linked to a closed Lorentzian curvature strip. Let d be a fixed line in space H that lies parallel to the $\{a,n\}$ plane, and let it generate a closed Lorentzian ruled surface in $H^{\prime }$. The ratio of the real pitch angle of this surface to the real pitch real pitch angle of the closed Lorentzian ruled surface generated in $H^{\prime }$ by the line n remains constant. This ratio is invariant with respect to the motion itself and is precisely given by $sinh{\varphi }_1$.

Theorem 14. 

Under the closed spatial motion $H/H^{\prime }$ corresponding to the one-parameter closed dual spherical motion $K/K^{\prime }$ of the closed Lorentzian dual curvature strip, the real pitch of the closed Lorentzian ruled surface generated in $H^{\prime }$ by the fixed line d (which resides in the moving space H and is parallel to the plane $\{a,n\}$) can be computed via Equation (6.8). Within this formulation, $L_n$ and ${\lambda }_n$ represent the real integral invariants belonging to the closed Lorentzian ruled surface generated in $H^{\prime }$ by the line n. Furthermore, ${\varphi }_1$ and ${\varphi }_1^{*}$ denote the spatial angle and the shortest distance between lines d and n, respectively. Consequently, in the Lorentzian line space, the following structural relation holds:

$${\lambda }_n{L}_d-L_n{\lambda }_d\mp {\varphi }_1^{*}{\lambda }_n{({\lambda }_n^2+{\lambda }_d^2)}^{{\displaystyle \frac{1}{2}}}=0$$

(54)

Suppose that the strip constitutes a dual Lorentzian closed geodesic strip, which implies $K_g=0$ and consequently ${\mathrm{\Lambda }}_n=0$. Additionally, let D lie within the plane defined by $\{A,N\}$ with respect to the moving frame $\{A,G,N\}$. Under these conditions, the Equation (6.1) lead to the relation;

$${\mathrm{\Lambda }}_d={\mathrm{\Lambda }}_{a}cosh\varphi$$

(55)

which distribution parameter is given by:

$${\displaystyle \frac{1}{d_d}}={\displaystyle \frac{k_n{k}_n^{*}+sinh{\varphi }_1{t}_g(sinh{\varphi }_1{t}_g^{*}+{\varphi }_1^{*}cosh{\varphi }_1{t}_g)}{k_n^2+sin{h}^2{\varphi }_1{t}_g^2}}$$

(56)

This allows us to state the following theorems:

Theorem 15. 

Consider a one-parameter dual spherical motion denoted by $K/K\prime$ in the Lorentzian space. If the strip represents the dual closed geodesic strip, the dual angle of pitch for the closed Lorentzian ruled surface generated on $K^{\prime }$ by the fixed timelike vector D located in the plane $\{A,N\}$ of the moving frame $\{A,G,N\}$ is identifiable via the dual angle of pitch of the closed Lorentzian ruled surface traced on $K^{\prime }$ by the vector A, given that the dual angle is constant.

Consequently, Equation (6.10) yields the following relationship:

$${\displaystyle \frac{{\mathrm{\Lambda }}_d}{{\mathrm{\Lambda }}_a}}=cosh\varphi$$

(57)

Theorem 16. 

Under the condition that the strip serves as the dual geodesic strip during the closed spherical motion $K/K^{\prime }$ in the Lorentzian space, the proportion between the dual angle of pitch of the closed Lorentzian ruled surface generated on $K^{\prime }$ by the fixed timelike vector D in the $\{A,N\}$ plane of the moving frame $\{A,G,N\}$ and the dual angle of pitch of the closed Lorentzian ruled surface generated on $K^{\prime }$ by the vector A remains constant and invariant under the motion.

The real and dual components of Equation (6.12) can be explicitly structured as follows:

$${\displaystyle \frac{{\lambda }_d}{{\lambda }_a}}=cosh{\varphi }_1$$

(58)

$$L_d=L_{a}cosh{\varphi }_1-{\lambda }_a{\varphi }_1^{*}sinh{\varphi }_1$$

(59)

These formulations lead to the subsequent theorems within the Lorentzian line space:

Theorem 17. 

During a closed spatial motion $H/H^{\prime }$ that maps to the dual closed spherical motion $K/K^{\prime }$ linked with the dual closed geodesic strip in the Lorentzian space, the ratio of the real pitch angle of the closed Lorentzian ruled surface traced in $H^{\prime }$ by the line d (which is fixed in the space H and parallel to the $\{a,n\}$ plane) to the real pitch angle of the closed Lorentzian ruled surface generated in $H^{\prime }$ by the line a is constant. This ratio is independent of the underlying motion and equals $cosh{\varphi }_1$.

Theorem 18. 

Let $H/H^{\prime }$ represents a closed spatial motion that corresponds to the dual closed spatial motion $K/K^{\prime }$ in the Lorentzian space, which is associated with the dual closed geodesic strip. In the moving space H, a line d is fixed such that it remains parallel to the plane spanned by $\{a,n\}$. The real pitch of the closed Lorentzian ruled surface generated by this line d within $H^{\prime }$ can be evaluated using Equation (6.14). Within this framework, ${\lambda }_a$ and $L_a$ denote the real integral invariants characterizing the closed Lorentzian ruled surface generated by the line a in $H^{\prime }$. The parameters ${\varphi }_1$ and ${\varphi }_1^{*}$ represent angle and distance between the lines d and a, respectively. Furthermore, within the Lorentzian line space, following analytical relationship is satisfied:

$$L_d{\lambda }_a-L_a{\lambda }_d\mp {\lambda }_a{\varphi }_1^{*}{({\lambda }_d^2-{\lambda }_a^2)}^{{\displaystyle \frac{1}{2}}}=0$$

(60)

Next, let the strip be characterized as a dual Lorentzian closed asymptotic strip, such that $K_n=0$ and ${\mathrm{\Lambda }}_g=0$. Assuming D is embedded in the $\{A,N\}$ plane of the moving frame $\{A,G,N\}$, combining Equations (6.1), (6.2) and (6.3) gives:

$${\mathrm{\Lambda }}_d={\lambda }_{a}cosh{\varphi }_1+{\lambda }_{n}sinh{\varphi }_1$$

(61)

$$L_d=L_{a}cosh{\varphi }_1+L_{n}sinh{\varphi }_1-{\lambda }_a{\varphi }_1^{*}sinh{\varphi }_1-{\lambda }_n{\varphi }_1^{*}cosh{\varphi }_1$$

(62)

and

$${\displaystyle \frac{1}{d_d}}={\displaystyle \frac{cosh{\varphi }_1{k}_g^{*}-sinh{\varphi }_1{t}_g^{*}-{\varphi }_1^{*}cosh{\varphi }_1{t}_g}{cosh{\varphi }_1{k}_g-sinh{\varphi }_1{t}_g}}$$

(63)

Consequently, the relation in (3.11) can be reformulated as:

$$2k_n{k}_n^{*}=k_n^2.({\displaystyle \frac{1}{d_a}}+{\displaystyle \frac{1}{d_n}})+t_g^2.({\displaystyle \frac{1}{d_g}}-{\displaystyle \frac{1}{d_n}})+k_g^2.({\displaystyle \frac{1}{d_a}}-{\displaystyle \frac{1}{d_g}})$$

(64)

Assuming the dual Lorentzian closed strip constitutes a dual curvature strip, the expression simplifies to:

$$2k_n{k}_n^{*}=k_n^2.({\displaystyle \frac{1}{d_a}}+{\displaystyle \frac{1}{d_n}})+k_g^2.({\displaystyle \frac{1}{d_a}}-{\displaystyle \frac{1}{d_g}})$$

(65)

Under the assumption that the dual Lorentzian closed strip characterizes a dual geodesic strip, the main relation reduces to:

$$2k_n{k}_n^{*}=k_n^2.({\displaystyle \frac{1}{d_a}}+{\displaystyle \frac{1}{d_n}})+t_g^2.({\displaystyle \frac{1}{d_g}}-{\displaystyle \frac{1}{d_n}})$$

(66)

Alternatively, if the dual Lorentzian closed strip is treated as a dual asymptotic strip, one obtains the following geometric ratio:

$${\displaystyle \frac{t_g^2}{k_g^2}}=\left({\displaystyle \frac{d_g-d_a}{d_g-d_n}}\right).\left({\displaystyle \frac{d_n}{d_a}}\right)$$

(67)

7. Conclusion

In this paper, we have modeled a dual Lorentzian closed strip characterized by the system $\{\tilde{X}\left(s\right)=x\left(s\right)+ϵv\left(s\right)\wedge x\left(s\right),N\left(s\right)=n\left(s\right)+ϵv\left(s\right)\wedge n\left(s\right)\}$. By employing a moving frame $\left\{A\right(s),G(s),N(s\left)\right\}$ and a fixed line D, the distribution parameters and dual integral invariants of respective dual closed Lorentzian strip were systematically evaluated. The core findings reveal explicit dependencies between the integral invariants of the closed strips that trace out dual closed curves via the vectors $A,G,N$ and D in the Lorentzian space. Using Study’s mapping principle, these relations are effectively transferred to the line space $R_1^3$, providing innovative representations for the dual Lorentzian strip’s geodesic curvature $\left(K_g\right)$, normal curvature $\left(K_n\right)$ and geodesic torsion $\left(T_g\right)$. Looking forward, the proposed methodology establishes a foundation for identifying further distinct relations among the integral invariants of the dual Lorentzian closed strips.