On Integral Invariants and Dralls for a Dual Lorentzian Closed Strip

Abstract

In this paper, a dual Lorentzian closed strip $\{\tilde{X}\left(s\right),N\left(s\right)\}$ is considered within the context of Lorentzian 3-space, with a real parameter s, a timelike moving frame $\left\{A\right(s),G(s),N(s\left)\right\}$ is defined, which moves along the curve of this dual Lorentzian closed strip. A timelike vector D, which is fixed in this frame, is investigated alongside the dual integral invariants (dual angles of pitch, dual pitch) and dralls of the dual Lorentzian closed strip corresponding to the dual Lorentzian closed curve traced by vectors $A,G,N$ and D. Some fundamental relations among these integral invariants of the dual Lorentzian closed strip are studied. Furthermore, these dual results are mapped to the line-space $R_1^3$ and several theorems are presented regarding the dual Lorentzian geodesic curvature $K_g$, dual Lorentzian normal curvature $K_n$ and dual Lorentzian geodesic torsion $T_g$ of the strip, accounting for the causal characters (timelike) inherent in Lorentzian geometry.

1. Introduction

The theory of dual numbers, introduced by Clifford [1] in 1873, has played a fundamental role in the development of kinematics and line geometry. Subsequently, E. Study [2] established a one-to-one correspondence between oriented lines in Euclidean three-dimensional space and points on the dual unit sphere, which is now known as the E. Study mapping. This correspondence provides an effective geometric framework in which differentiable dual spherical curves can be associated with ruled surfaces in line space. The extension of the E. Study mapping to space-like and time-like lines in Minkowski 3-space has also been investigated, establishing a correspondence between directed lines and ordered pairs in Lorentzian geometry [3].

Ruled surfaces and their dual representations have attracted considerable attention 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 obtained fundamental geometric relations for ruled surfaces. Later, Yapar and Sağıroğlu [5] introduced curvature motions on the dual hyperbolic unit sphere and investigated the associated ruled surfaces and line congruences by means of E. Study’s correspondence.

Integral invariants are among the most important tools for the global analysis of ruled surfaces. Özyılmaz and Yaylı [6] studied the integral invariants of time-like ruled surfaces and derived relations involving dual spherical motions and dual angles of pitch. Bektaş and Şenyurt [7] investigated ruled surfaces generated by closed time-like curves in dual Lorentzian space and obtained 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.

Recent studies have further expanded the theory of dual curves, dual frames, and their associated ruled surfaces. Gilani et al. [9] investigated dual curves and dual focal curves in dual Lorentzian space and established relationships between dual frames and curvature properties. In a related study, Çalışkan [10] examined dual Darboux ruled surfaces and discussed geometric relations between dual curves and ruled surface invariants. Moreover, investigations based on dual spherical indicatrices and E. Study correspondence have continued to provide new perspectives for the characterization of ruled surfaces [11,12]. These studies demonstrate that dual geometric methods remain an effective approach for analyzing the differential properties and global invariants of ruled surfaces.

Although significant progress has been achieved in the study of ruled surfaces, dual spherical motions, and integral invariants, several aspects of dual Lorentzian closed strips remain unexplored. In particular, while dual integral invariants such as pitch, angle of pitch, and drall have been extensively studied in Euclidean and certain Lorentzian settings, a unified framework for these invariants associated with dual Lorentzian closed strips has not yet been established. Furthermore, explicit relations between the dual Darboux–Ribaucour frame, dual integral invariants, and the corresponding Lorentzian line-space quantities require further investigation.

Motivated by these considerations, this paper develops a systematic study of dual Lorentzian closed strips in Lorentzian line space $R_1^3$. The main contributions of this work can be summarized as follows:

• A dual Lorentzian closed strip together with its associated dual Darboux–Ribaucour moving frame is introduced, and the corresponding differential characterizations are obtained.
• Explicit formulas for the dual integral invariants, including dual angles of pitch, dual pitches, and dralls of the ruled surfaces generated by the frame vectors, are derived. Moreover, fundamental relations among these invariants are established.
• By means of E. Study’s correspondence, the obtained dual results are transferred to Lorentzian line space $R_1^3$. New expressions for the dual Lorentzian geodesic curvature, normal curvature, and geodesic torsion are derived, extending the theory of dual closed strips to the Lorentzian setting.

2. Preliminaries

Basic definitions and fundamental properties of dual numbers can be found in the work of Guggenheimer [13]. The essential concepts of dual Lorentzian vectors, including the dual Lorentzian inner product and the Lorentzian cross product, are presented in detail by Şentürk and Yüce [14].

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 denotes a real parameter, which is generally chosen as the arc-length parameter of the indicatrix $x\left(s\right)$. The dual arc-length of $\tilde{X}\left(s\right)$ is denoted by $\tilde{s}=s+ϵs*$ where the real part s represents the arc length of the indicatrix, while the dual part describes the translational displacement associated with the corresponding ruled surface in the Lorentzian line space $R_1^3$. The derivative of the dual position vector with respect to the dual arc-length parameter defines the unit dual tangent vector of the ruled surface and plays a fundamental role in its differential geometry.

Through E. Study’s correspondence, every differentiable curve on the dual Lorentzian unit sphere corresponds uniquely to a Lorentzian ruled surface, that is, to a differentiable one-parameter family of oriented lines in the Lorentzian line space.

Suppose that $\tilde{X}:I\to {\tilde{D}}_1^3$, $\tilde{X}\left(s\right)=x\left(s\right)+ϵx*\left(s\right)$ is a dual Lorentzian closed timelike curve parametrized by the arc length s of its indicatrix. Since $x=a$ is a unit timelike vector tangent to the indicatrix, we have $<x^{\prime },x^{\prime }>=1$ where the prime denotes differentiation with respect to s.

The relation $p\left(s\right)\wedge x\left(s\right)=x*\left(s\right)$ admits infinitely many solutions for the vector function $p\left(s\right)$. If $p_0\left(s\right)$ is one such solution, then the general solution can be written as $p\left(s\right)=p_0\left(s\right)+u\left(s\right)x\left(s\right)$ where $u\left(s\right)$ is an arbitrary real-valued function. Taking the Lorentzian inner product yields

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

(1)

Choosing $u=u_0=-\int <p_0^{\prime },x>$ ensures that $v\left(s\right)=p_0\left(s\right)+u_0\left(s\right)x\left(s\right)$ satisfies $<v^{\prime },x> = 0$. Consequently, the dual Lorentzian closed curve admits the unique representation

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

(2)

It should be noted that the vector function $v\left(s\right)$ is uniquely determined by the curve $\tilde{X}\left(s\right)$. Accordingly, the dual arc length of $\tilde{X}\left(s\right)$ is given by

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

(3)

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

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: The strip is called a dual Lorentzian closed curvature, geodesic, or asymptotic strip if $\tilde{X}$ is a Lorentzian curvature line $(T_g=0)$, geodesic curve $(K_g=0)$, or asymptotic curve $(K_n=0)$, respectively.

Throughout this paper, all strips and ruled surfaces are assumed to be positively oriented. From Equation (3), we have ${\displaystyle \frac{ds*}{ds}}=1+ϵ\sigma$, or equivalently, ${\displaystyle \frac{ds}{ds*}}=1-ϵ\sigma$. Moreover, since $<a,{(x*)}^{\prime }>$ is equivalent to $<v^{\prime }\wedge x,a> = \sigma$, it follows that $v^{\prime }\wedge x=-\sigma a$. Consequently,

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

(4)

Now consider the orientable dual Lorentzian closed strip $\tilde{X}\left(s\right)=x\left(s\right)+ϵv\left(s\right)\wedge x\left(s\right),N\left(s\right)=n\left(s\right)+ϵv\left(s\right)\wedge n\left(s\right)$, where $<A,N> = 0, \Vert \tilde{X}\Vert =\Vert N\Vert =1$ and N is spacelike. Here, $A={\displaystyle \frac{d\tilde{X}}{ds*}}$. Defining the dual spacelike unit vector $N\wedge A=G=g+ϵg*$, it follows immediately from Equation (4) that $N=n+ϵv\wedge n$.

The trihedron $A,G,N$ is called the dual timelike Darboux–Ribaucour frame associated with the dual Lorentzian closed strip. Here, A is timelike, whereas G and N are spacelike. 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:

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

(5)

In Equation (5), the dual functions $K_g=k_g+ϵk_g^{*},K_n=k_n+ϵk_n^{*}$ and $T_g=t_g+ϵt_g^{*}$ denote the dual geodesic curvature, dual normal curvature, and dual geodesic torsion of the dual Lorentzian closed strip, respectively.

Since $v^{\prime }\wedge x=-\sigma a$, it follows that $n\wedge (v^{\prime }\wedge x)=-\sigma g$. Hence,

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

(6)

where $\delta = <n,v^{\prime }>$ and $\lambda = <n,x>$. Using the above relation, we obtain

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

(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 $t_g$ denote the geodesic curvature, normal curvature, and geodesic torsion of the indicatrix x, respectively. Accordingly, Equations (7)–(9) provide explicit representations of the dual geodesic curvature, dual normal curvature, and dual geodesic torsion of the dual Lorentzian closed strip in terms of the corresponding real geometric invariants.

By means of E. Study’s mapping, the motion of the dual Lorentzian unit sphere K with respect to a fixed dual Lorentzian unit sphere $K^{\prime }$ represents the spatial kinematic relationship between the moving Lorentzian space H and the fixed Lorentzian space $H^{\prime }$. Consequently, the one-parameter motion $H/H^{\prime }$ generates a closed Lorentzian ruled surface in the fixed space. The present study is based on the theory of dual Lorentzian spherical motions and E. Study’s correspondence established in the literature, with particular emphasis on the characterization of dual pitch, dual angle of pitch, and the associated integral invariants in the Lorentzian line space $R_1^3$.

Definition 2. 

The dual Lorentzian Steiner vector of the closed motion $K/K^{\prime }$ is 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)

which is the dual analogue of the classical Steiner vector introduced in [15].

For a closed Lorentzian ruled surface x in the Lorentzian line space $R_1^3$, the real angle of pitch is defined by

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

(12)

where d denotes the real part of the Steiner vector associated with the motion. Likewise, the real pitch of the ruled surface is given by

$$L_x = <d*,x>+<d,x*>$$

(13)

where d and $d*$ denote the real and dual parts of the dual Lorentzian Steiner vector, respectively [15].

Definition 3. 

Let $\tilde{X}\left(s\right)=x\left(s\right)+ϵx*\left(s\right)$ be an orientable closed Lorentzian ruled surface satisfying $\parallel \tilde{X}\parallel = 1$. Using Equations (12) and (13), the dual angle of pitch of the ruled surface is defined as

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

(14)

3. Dual Pitch Angles of Closed Lorentzian Ruled Surfaces Generated by Vectors A, G, N and D

Definition 4. 

Using Equations (11) and (14), the dual angles of pitch of the closed Lorentzian ruled surfaces generated in $K^{\prime }$ by the vectors A, G, and N are defined as follows:

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

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

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

(15)

For G,

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

Equivalently, the real and dual components of ${\Lambda }_g$ are given by

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

(16)

Similarly for N,

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

the real and dual components of ${\Lambda }_n$ are given by

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

(17)

Consequently, substituting Equations (15)–(17) into the definition of the dual Lorentzian Steiner vector yields

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

(18)

Theorem 1. 

During the one-parameter motion of the dual Lorentzian closed strip $\{\tilde{X}\left(s\right),N\left(s\right)\}$ associated with the moving frame $\{A,G,N\}$, the components of the dual Lorentzian Steiner vector are precisely the dual angles of pitch of the closed Lorentzian ruled surfaces generated in $K^{\prime }$ by the vectors A, G, and N.

Next, consider a fixed dual timelike unit vector D attached to the moving frame $\{A,G,N\}$, defined by

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

(19)

where $\varphi ={\varphi }_1+ϵ{\varphi }_1^{*}$ and $\theta ={\theta }_1+ϵ{\theta }_1^{*}$ are fixed dual hyperbolic angles. According to Definition 4, the dual angle of pitch of the closed Lorentzian ruled surface generated in $K^{\prime }$ by the vector D is given by

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

(20)

Substituting Equation (18) into Equation (20), we obtain

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

(21)

Separating the real and dual parts of Equation (21) gives

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

(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. 

For the one-parameter motion associated with a dual Lorentzian closed strip, Equations (22) and (23) establish the relationships between the integral invariants of the Lorentzian line space and the real angles of pitch and real pitches of the closed Lorentzian ruled surfaces generated by the lines a, g, and n.

Furthermore, the distribution parameter of the closed Lorentzian ruled surface generated during the closed motion by a dual unit vector $M=m+ϵm*$ is defined by

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

(24)

According to Equation (24), the distribution parameters of the closed Lorentzian ruled surfaces generated by the vectors A, G, N, and D are given by

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

(25)

where

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

(26)

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

Assuming that the vector D lies in the plane spanned by $\{A,G\}$, Equation (21) reduces to

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

(27)

Separating the real and dual parts of Equation (27) gives

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

(28)

and the corresponding 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 (26).

Special Cases

Assume that the strip is a dual Lorentzian closed curvature strip. Then $T_g=0$ and consequently ${\Lambda }_a=0$. Furthermore, suppose that the vector D lies in the plane $\{A,G\}$ of the moving frame. Under these assumptions, Equations (27) and (29) reduce to

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

(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. 

Let the strip be a dual Lorentzian closed curvature strip. During the one-parameter dual spherical motion $K/K^{\prime }$, the dual angle of pitch of the closed Lorentzian ruled surface generated in $K^{\prime }$ by the fixed timelike vector D lying in the plane $\{A,G\}$ is completely determined by the dual angle of pitch of the ruled surface generated by the vector G together with the fixed dual angle θ.

Consequently,

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

can be reformulated in the following manner:

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

(32)

Hence, the subsequent theorem can be presented:

Theorem 4. 

For a dual Lorentzian closed curvature strip, the ratio of the dual angle of pitch of the ruled surface generated by the vector D to that generated by the vector G is constant throughout the one-parameter dual spherical motion $K/K^{\prime }$. In particular, this ratio is independent of the motion.

The real and dual parts of Equation (32) are

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

(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 the closed spatial motion $H/H^{\prime }$ corresponding to the one-parameter dual spherical motion $K/K^{\prime }$ associated with a dual Lorentzian closed curvature strip. Let d be a fixed line in the moving space H that is parallel to the plane $\{a,g\}$, and let this line generate a closed Lorentzian ruled surface in the fixed space $H^{\prime }$. Then the ratio of the real angle of pitch of this ruled surface to that of the ruled surface generated by the line g is constant. Moreover, this ratio is independent of the motion and is equal to $sinh{\theta }_1$.

Theorem 6. 

Consider the closed spatial motion $H/H^{\prime }$ corresponding to the one-parameter dual spherical motion $K/K^{\prime }$ associated with a dual Lorentzian closed curvature strip. Let d be a fixed line in the moving space H, parallel to the plane $\{a,g\}$, and let it generate a closed Lorentzian ruled surface in $H^{\prime }$. The real pitch of this ruled surface is given by Equation (34), where $L_g$ and ${\lambda }_g$ denote the real integral invariants of the ruled surface generated by the line g, while ${\theta }_1$ and ${\theta }_1^{*}$ represent the angle and the shortest distance between the lines d and g, respectively. Furthermore, the following relation holds in the Lorentzian line space:

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

(35)

Proof. 

Since $cos{h}^2{\theta }_1-sin{h}^2{\theta }_1=1$, from Equation (33), we have

$$cosh{\theta }_1=\pm {\displaystyle \frac{\sqrt{{\lambda }_g^2+{\lambda }_d^2}}{{\lambda }_g}}$$

(36)

The sign is determined by the sign of ${\lambda }_g$. Substituting the expressions for $sinh{\theta }_1$ and $cosh{\theta }_1$ into Equation (34), we obtain

$$L_d=L_g{\displaystyle \frac{{\lambda }_d}{{\lambda }_g}}\mp {\theta }_1^{*}\sqrt{{\lambda }_g^2+{\lambda }_d^2}$$

(37)

Multiplying both sides by ${\lambda }_g$ and rearranging the resulting expression completes the proof. □

Assume that the strip is a dual Lorentzian closed geodesic strip. Then $K_g=0$ and consequently ${\Lambda }_n=0$. Suppose further that the vector D lies in the plane $\{A,G\}$ of the moving frame. Under these assumptions, Equations (27) and (28) remain valid, while the distribution parameter reduces to

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

(38)

Now assume that the strip is a dual Lorentzian closed asymptotic strip. Then $K_n=0$ and hence ${\Lambda }_g=0$. If the vector D lies in the plane $\{A,G\}$, Equations (27) and (29) reduce to

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

(39)

and

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

(40)

where

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

(41)

This allows us to state the following theorems:

Theorem 7. 

Let the strip be a dual Lorentzian closed asymptotic strip. During the one-parameter dual spherical motion $K/K^{\prime }$, the dual angle of pitch of the ruled surface generated by the fixed timelike vector D lying in the plane $\{A,G\}$ is determined by the dual angle of pitch of the ruled surface generated by the vector A and the fixed dual angle θ.

Consequently, Equation (37) yields the following relationship:

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

(42)

Theorem 8. 

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

Separating the real and dual parts of Equation (42), we obtain

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

(43)

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

(44)

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

Theorem 9. 

During the closed spatial motion $H/H^{\prime }$ corresponding to the dual closed spherical motion $K/K^{\prime }$ associated with a dual Lorentzian closed asymptotic strip, the ratio between the real pitch angle of the closed Lorentzian ruled surface generated in $H^{\prime }$ by the fixed line d and that of the ruled surface generated by the line a is constant. Here, the line d is assumed to be fixed in the moving space H and parallel to the plane $\{a,g\}$. This ratio is independent of the motion and is equal to $cosh{\theta }_1$.

Theorem 10. 

Let $H/H^{\prime }$ be a closed spatial motion corresponding to the dual closed spherical motion $K/K^{\prime }$ associated with a dual Lorentzian closed asymptotic strip. Suppose that the line d is fixed in the moving space H and is parallel to the plane spanned by $\{a,g\}$. Then, the real pitch of the closed Lorentzian ruled surface generated by d in $H^{\prime }$ is determined by Equation (44). In this formulation, ${\lambda }_a$ and $L_a$ denote the real integral invariants of the closed Lorentzian ruled surface generated by the line a in $H^{\prime }$. Furthermore, ${\theta }_1$ and ${\theta }_1^{*}$ represent the angle and the shortest distance between the lines d and a, respectively. The following relation holds in Lorentzian line space:

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

(45)

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

Assume that the timelike vector D is expressed in terms of the vectors G and N of the moving frame $\{A,G,N\}$. Let the strip be a dual Lorentzian closed curvature strip, satisfying $T_g=0$ and consequently, ${\Lambda }_a=0$. Moreover, suppose that the timelike vector D is fixed with respect to the moving frame $\{A,G,N\}$. Under these assumptions, the dual instantaneous Pfaffian vector W reduces to

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

(46)

Therefore, the dual angle of pitch ${\Lambda }_d$ corresponding to the ruled surface generated by the vector D can be obtained from the inner product between the dual instantaneous Pfaffian vector and D as

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

(47)

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

Consider the case in which the timelike vector D lies entirely in the plane spanned by $\{A,N\}$ of the moving frame $\{A,G,N\}$. Under this restriction, we have $\theta =0$. Substituting this condition into Equation (21), we obtain

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

(48)

Separating the real and dual parts of the above expression gives

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

(49)

The corresponding distribution parameter is expressed as

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

(50)

where

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

Special Cases

First, consider the case in which the strip is a dual Lorentzian closed curvature strip. Hence, $T_g=0$, and therefore ${\Lambda }_a=0$. Assume additionally that the timelike vector D lies in the plane $\{A,N\}$ of the moving frame $\{A,G,N\}$. Under these assumptions, we get

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

(51)

Consequently, Equation (50) reduces to

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

(52)

Therefore, we can state the following theorem:

Theorem 11. 

Let the strip be a dual Lorentzian closed curvature strip. During the one-parameter dual spherical motion $K/K^{\prime }$, the dual angle of pitch of the ruled surface generated on $K^{\prime }$ by the fixed timelike vector D lying in the plane $\{A,N\}$ is determined by the dual angle of pitch corresponding to the vector N and the dual angle ϕ.

Thus the Equation (51) holds and it can be written by the following form:

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

(53)

Hence, the subsequent theorem can be presented:

Theorem 12. 

For a dual Lorentzian closed curvature strip, the ratio between the dual angle of pitch associated with the fixed timelike vector D in the plane $\{A,N\}$ and the dual angle of pitch associated with the vector N remains constant during the one-parameter dual spherical motion $K/K^{\prime }$. This ratio is independent of the motion.

The real and dual components of Equation (53) are given by:

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

(54)

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

(55)

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

Theorem 13. 

Consider a closed spatial motion $H/H^{\prime }$ corresponding to the one-parameter dual spherical motion $K/K^{\prime }$ associated with a dual Lorentzian closed curvature strip. Let d be a fixed line in the moving space H parallel to the plane $\{a,n\}$. Then, the ratio between the real pitch angle of the ruled surface generated by d in $H^{\prime }$ and that of the ruled surface generated by the line n is constant and equal to $sinh{\varphi }_1$. Therefore, this ratio is invariant with respect to the motion.

Theorem 14. 

Under the closed spatial motion $H/H^{\prime }$ corresponding to the one-parameter dual spherical motion $K/K^{\prime }$ of the dual Lorentzian closed curvature strip, the real pitch of the ruled surface generated in $H^{\prime }$ by the fixed line d can be determined by Equation (55). Here, d is assumed to be parallel to the plane $\{a,n\}$ in the moving space H. The quantities ${\lambda }_n$ and $L_n$ denote the real integral invariants of the ruled surface generated by the line n, while ${\varphi }_1$ and ${\varphi }_1^{*}$ represent the angle and the shortest distance between the lines d and n, respectively. Therefore, the following relation holds in Lorentzian line space:

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

(56)

Suppose that the strip is a dual Lorentzian closed geodesic strip. Then, $K_g=0$ and consequently ${\Lambda }_n=0$. Furthermore, assume that the timelike vector D lies in the plane $\{A,N\}$ of the moving frame $\{A,G,N\}$. Under these assumptions, Equation (48) reduces to

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

(57)

The corresponding distribution parameter is given by

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

(58)

The following theorems are obtained as a consequence.

Theorem 15. 

Let $K/K^{\prime }$ be a one-parameter dual spherical motion in Lorentzian space associated with a dual Lorentzian closed geodesic strip. Then, the dual angle of pitch of the closed Lorentzian ruled surface generated on $K^{\prime }$ by the fixed timelike vector D lying in the plane $\{A,N\}$ of the moving frame $\{A,G,N\}$ is determined by the dual angle of pitch of the ruled surface generated by the vector A as

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

(59)

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 (59) can be explicitly structured as follows:

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

(60)

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

(61)

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

Theorem 17. 

Consider a closed spatial motion $H/H^{\prime }$ corresponding to the dual closed spherical motion $K/K^{\prime }$ associated with a dual closed geodesic strip in the Lorentzian space. Let the line d be fixed in the space H and parallel to the plane spanned by $\{a,n\}$. Then, the ratio between the real pitch angle of the closed Lorentzian ruled surface generated by d in $H^{\prime }$ and the real pitch angle of the closed Lorentzian ruled surface generated by a in $H^{\prime }$ remains constant throughout the motion. This invariant ratio is given by $cosh{\varphi }_1$.

Theorem 18. 

Let $H/H^{\prime }$ represent a closed spatial motion corresponding to the dual closed spherical motion $K/K^{\prime }$ in the Lorentzian space associated with a dual closed geodesic strip. In the moving space H, let d be a fixed line parallel to the plane spanned by $\{a,n\}$. The real pitch of the closed Lorentzian ruled surface generated by the line d in $H^{\prime }$ is determined by Equation (61). Here, ${\lambda }_a$ and $L_a$ denote the real integral invariants of the closed Lorentzian ruled surface generated by the line a in $H^{\prime }$. Moreover, ${\varphi }_1$ and ${\varphi }_1$ represent the angle and the distance between the lines d and a, respectively. Then, in the Lorentzian line space, the following relation holds:

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

(62)

Next, consider the case where the strip is characterized as a dual Lorentzian closed asymptotic strip, satisfying $K_n=0$ and ${\Lambda }_g=0$. Assume that the fixed timelike vector D lies in the plane spanned by $\{A,N\}$ of the moving frame $\{A,G,N\}$. By combining Equations (48)–(50), we obtain the following relations:

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

(63)

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

(64)

and

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

(65)

Consequently, the relation in (25) can be reformulated as:

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

(66)

For the case of a dual curvature strip, the above relation takes the simplified form:

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

(67)

For a dual geodesic strip, the fundamental relation reduces to:

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

(68)

Alternatively, for a dual asymptotic strip, the following geometric relation is obtained:

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

(69)

7. Conclusions

In this paper, a theoretical framework for dual Lorentzian closed strips was developed within the Lorentzian line space $R_1^3$. By introducing the dual Darboux–Ribaucour moving frame associated with a dual Lorentzian closed strip, the differential properties of the strip were characterized through the dual geodesic curvature, dual normal curvature, and dual geodesic torsion. The obtained results provide explicit relations between these geometric quantities and the corresponding dual integral invariants.

The dual angles of pitch, dual pitches, and drall invariants of the ruled surfaces generated by the frame vectors $A,G,N$, and an arbitrary fixed timelike vector D were derived. Moreover, several relations among these invariants were established for different special classes of dual Lorentzian closed strips, including curvature, geodesic, and asymptotic strips. Through E. Study’s correspondence, these results were interpreted in the Lorentzian line space, providing a geometric connection between dual spherical motions and ruled surface invariants.

The developed theoretical framework may provide a useful geometric foundation for applications involving spatial motions, kinematic modeling, and robotic mechanisms where Lorentzian line geometry and dual representations are relevant. In particular, the obtained invariant relations may contribute to the geometric description of constrained motions and motion planning problems. Furthermore, the Lorentzian formulation may offer potential connections with geometric models arising in relativistic settings, where the causal character of curves and surfaces plays an essential role.

Future research may focus on the computational realization of the obtained formulas, including numerical examples and application-oriented models. Extending the present results to other classes of dual Lorentzian motions, investigating the behavior of invariants under different causal conditions, and applying the developed theory to mechanical and robotic systems constitute promising directions for further studies.