Let us address that ${\mathbb{H}}_{+m}^2$ and ${\mathbb{H}}_{+f}^2$ are two hyperbolic $\mathcal{DUS}$ centered at the origin $\widehat{\mathbf{0}}$ in ${\mathbb{D}}_1^3$. Let $\left\{\widehat{\mathbf{u}}\right\}$ = {$\widehat{\mathbf{0}}$;${\widehat{\mathbf{u}}}_1$($\mathbb{T}-like$)$,{\widehat{\mathbf{u}}}_2,{\widehat{\mathbf{u}}}_3$}, and $\left\{\widehat{\mathsf{\zeta }}\right\}=\{\widehat{\mathbf{0}}$;${\widehat{\mathsf{\zeta }}}_1$($\mathbb{T}-like$),${\widehat{\mathsf{\zeta }}}_2$,${\widehat{\mathsf{\zeta }}}_3$} be two orthonormal dual frames of ${\mathbb{H}}_{+m}^2$ and ${\mathbb{H}}_{+f}^2$, respectively. If we say $\left\{\widehat{\mathsf{\zeta }}\right\}$ is stationary, whereas the elements of $\left\{\widehat{\mathbf{u}}\right\}$ are functions of a real parameter $t\in \mathbb{R}$ (say the time). Then, we say that ${\mathbb{H}}_{+m}^2$ moves on ${\mathbb{H}}_{+f}^2.$ This is a one-parameter Lorentzian dual spherical ($\mathcal{DS}$) locomotion and will signalize by ${\mathbb{H}}_{+m}^2/{\mathbb{H}}_{+f}^2$. Via the E. Study map, the hyperbolic $\mathcal{DUS}$ ${\mathbb{H}}_{+m}^2$ and ${\mathbb{H}}_{+f}^2$ matches the hyperbolic line spaces $\mathbb{L}$${}_m$(mobile) and $\mathbb{L}$${}_f$ (immobile), respectively. Therefore, ${\mathbb{L}}_m/{\mathbb{L}}_f$ is the mobile hyperbolic line space against the hyperbolic immobile space ${\mathbb{L}}_f$, because at any instant the instantaneous screw axis ($\mathbb{ISA}$) of ${\mathbb{L}}_m/{\mathbb{L}}_f$ creates a mobile $\mathbb{T}-like$ axode ${\pi }_m$ in ${\mathbb{L}}_m$, and immobile $\mathbb{T}-like$ axode ${\pi }_f$ in ${\mathbb{L}}_f$. Therefore, we insert an orthonormal dual frame $\left\{\widehat{\mathbf{r}}\right\}=\{\widehat{\mathbf{0}}$;${\widehat{\mathbf{r}}}_1$($\mathbb{T}-like$),${\widehat{\mathbf{r}}}_2$,${\widehat{\mathbf{r}}}_3$}, which is specified as follows: we set ${\widehat{\mathbf{r}}}_1\left(t\right)={\mathbf{r}}_1\left(t\right)+\epsilon {\mathbf{r}}_1^{*}\left(t\right)$, as the $\mathbb{ISA}$ and ${\widehat{\mathbf{r}}}_2\left(t\right):={\mathbf{r}}_2\left(t\right)+\epsilon {\mathbf{r}}_2^{*}\left(t\right)=\frac{d{\widehat{\mathbf{r}}}_1}{dt}{∥\frac{d{\widehat{\mathbf{r}}}_1}{dt}∥}^{-1}$ as the joint central normal of two disjoint screw axes. A third dual unit vector is designated as ${\widehat{\mathbf{r}}}_3\left(t\right)={\widehat{\mathbf{r}}}_1\times {\widehat{\mathbf{r}}}_2$. Then, The set $\left\{\widehat{\mathbf{r}}\right\}$ is the relative Blaschke frame, and ${\widehat{\mathbf{r}}}_1,$${\widehat{\mathbf{r}}}_2,$ and ${\widehat{\mathbf{r}}}_3$ are intersected at the joint striction (central) point $\mathbf{s}\left(t\right)$ of the $\mathbb{T}-like$ axodes ${\pi }_i$$(i=m,$$f)$. The dual arc length $d{\widehat{s}}_i=d{s}_i+\epsilon d{s}_i^{*}$ of ${\widehat{\mathbf{r}}}_1\left(t\right)$ is $\widehat{\sigma }\left(t\right)=\sigma \left(t\right)+\epsilon {\sigma }^{*}\left(t\right)$ is the first order asset of the locomotions ${\mathbb{H}}_{+m}^2/{\mathbb{H}}_{+f}^2$. We set $d\widehat{s}=ds+\epsilon d{s}^{*}$ to represent $d{\widehat{s}}_i$, since they are equal to each other. Then, is the distribution parameter ($\mathcal{D}-par$) of the $\mathbb{T}-like$ axodes. Via the E. Study map: for the locomotion the $\mathbb{T}-like$ axodes have the $\mathbb{ISA}$ in mutual; that is the mobile axode osculating with the immobile axode along the $\mathbb{ISA}$ in the first order (compared with [1,2,3]). For the Blaschke formulae with respect to ${\mathbb{H}}_{+i}^2(i=m,$$f)$, we find where ${\widehat{\mathsf{\varrho }}}_i:={\mathsf{\varrho }}_i+\epsilon {\mathsf{\varrho }}_i^{*}={\widehat{\beta }}_i{\widehat{\mathbf{r}}}_1-{\widehat{\mathbf{r}}}_3$ is the Darboux vector, and is the radii of curvature of the $\mathbb{T}-like$ axodes ${\pi }_i$. ${\beta }_i\left(s\right)$, ${\mathsf{\Omega }}_i\left(s\right)$ and $\lambda \left(s\right)$ are the curvature (construction) functions of the $\mathbb{T}-like$ axodes. By setting $\left|{\widehat{\beta }}_i\right|<1$, the $\mathbb{S}-like$ Disteli-axes ($\mathbb{DA}$) of the $\mathbb{T}-like$ axodes is where ${\widehat{\phi }}_i\left(\widehat{s}\right)={\phi }_i+\epsilon {\phi }_i^{*}$ is a Lorentzian $\mathbb{T}-like$ dual angle (radius of curvature) among ${\widehat{\mathbf{r}}}_1$ and ${\widehat{\mathbf{b}}}_i$. Then, Equation (14) is a novel dual hyperbolic version of the $\mathbb{ES}$ formula (Compared with [1,2,3]). Via the real and the dual parts, respectively, we attain and Equation (15) in conjuction with (16) are new Disteli formulae ($\mathbb{DF}$) for the $\mathbb{T}-like$ axodes of the locomotion ${\mathbb{L}}_m/{\mathbb{L}}_f$.

Now let us assume that $\left\{\widehat{\mathbf{r}}\right\}$ is stabilized in ${\mathbb{H}}_{+m}^2$. Then, where is the relative Darboux vector. $∥\widehat{\mathsf{\varrho }}∥=\widehat{\varrho }=\widehat{\varrho }+\epsilon {\widehat{\varrho }}^{*}={\beta }_r+\epsilon \left({\mathsf{\Omega }}_r+\lambda {\beta }_r\right)$ is the relative radii of curvature; $\varrho ={\beta }_f-{\beta }_m$, and ${\varrho }^{*}={\mathsf{\Omega }}_f-{\mathsf{\Omega }}_m-\lambda \left({\beta }_f-{\beta }_m\right)$ are the rotational angular speed and translational angular speed of the locomotion ${\mathbb{L}}_m/{\mathbb{L}}_f$, as well they are both invariants in kinematics, respectively. As a result, the following corollary can be stated:

In this study, we deviate from the exclusive use of translational locomotion, namely when ${\varrho }^{*}\ne 0$. Moreover, we impose the condition of excluding zero divisors, denoted by $\varrho =0$. Consequently, our investigation will solely focus on non-torsional locomotions, ensuring that the axodes associated with these motions are non-developable $\mathbb{T}-like$ ruled surfaces, characterized by $\lambda \ne 0$.

In the context of planar locomotions, the $\mathbb{ES}$ formula associates the locus of a point to its curvature center and is the main ingredient for a graphical structure producing one assigned the other [1,2,3]. In 1914, Disteli [20] assigned a curvature axis for the ruling of a ruled surface and extended the planar $\mathbb{ES}$ formula to spatial locomotions. However, the $\mathbb{DF}$ of a line trajectory had been acquired in [4,5,6,7,8,21], around inscription should be refind as follows: we shall define a new manner to have $\mathbb{DF}$ by dual function approximations. Thus, we request the $\mathbb{T}-like$ line $\widehat{\mathbf{x}}\in {\mathbb{L}}_m$, which at a steady dual angle from a steady $\mathbb{S}-like$ line $\widehat{\mathbf{y}}\in {\mathbb{L}}_f$. So, if $\widehat{\psi }=\psi +\epsilon {\psi }^{*}$ is the dual angle of $\widehat{\mathbf{x}}$( $\mathbb{T}-like$), and $\widehat{\mathbf{y}}$ ($\mathbb{S}-like$), then For $\widehat{\psi }$ is steady up to the 2nd order at $\widehat{w}={\widehat{w}}_0$, we have and Hence, for the 1st order $\langle {\widehat{\mathbf{x}}}^{{}^{\prime }},\widehat{\mathbf{y}}\rangle =0$, and for the 2nd order $\langle {\widehat{\mathbf{x}}}^{{}^{{}^{″}}},\widehat{\mathbf{y}}\rangle =0$. Therefore, $\widehat{\psi }$ will be steady in the 2nd approximation if and only if $\widehat{\mathbf{y}}$ is the $\mathbb{S}-like$ $\mathbb{DA}$$\widehat{\mathbf{b}}$ of ($\widehat{x}$), that is,

Hence, from Equations (32) and (39), the following corollary can be given.

Via this corollary, and based on Equation (34), it can be concluded that the rulings of ($\widehat{x}$) are the constant dual angle $\widehat{\vartheta }$ with respect to the $\mathbb{S}-like$$\mathbb{DA}$ if and only if ${\widehat{\beta }}^{{}^{\prime }}=0$. Therefore, the $\mathbb{T}-like$ ruled surface ($\widehat{x}$) is generated locally by a one-parameter hyperbolic spatial locomotion with pitch $h\left(s\right)$ along the steady $\mathbb{DA}$$\widehat{\mathbf{b}}$, This locomotion is performed by the $\mathbb{T}-like$ line $\widehat{\mathbf{x}}$, which is positioned at a constant hyperbolic distance ${\vartheta }^{*}$ and a constant angle $\vartheta$ relative to $\widehat{\mathbf{b}}$. This indicates that the striction curve of ($\widehat{x}$) can be classified as either a $\mathbb{S}-like$ or $\mathbb{T}-like$ cylindrical helix. The corollary shown below can be used to identify the circumstances of steady $\mathbb{DA}$.

Furthermore, from Equations (32) and (39), we find that So, $\widehat{\mathbf{b}}$ is the osculating circle of $\widehat{\mathbf{x}}\left(\widehat{u}\right)\in {\mathbb{H}}_{+f}^2$. Further, it can be seen from Equations (22), (27) and (32) that Then, all ${\widehat{\mathbf{r}}}_1$, $\widehat{\mathbf{x}}$ and $\widehat{\mathbf{b}}$ belong to a $\mathbb{T}-like$ line congruence whose focus line is the $\mathbb{S}-like$ line $\widehat{\mathbf{t}}$. This can be realized as follows: we set $\widehat{\mathbf{t}}$ with respect to the set $\left\{\widehat{\mathbf{r}}\right\}$ by its intercept distance ${\phi }^{*}$, control on the $\mathbb{ISA}$ and the angle $\phi$, control with respect to ${\widehat{\mathbf{r}}}_2$. We set the dual angle $\widehat{\alpha }=\alpha +\epsilon {\alpha }^{*}$, which realizes the attitude of $\widehat{\mathbf{b}}$ over $\widehat{\mathbf{t}}$. These dual angles are all estimated relative to the $\mathbb{ISA}$ (see Figure 2). The following governs the signals: $(\vartheta ,$${\vartheta }^{*})$ and $(\alpha ,$${\alpha }^{*})$ are via the right-hand screw rule with the thumb pointing on $\widehat{\mathbf{t}}$; the sense of $\widehat{\mathbf{t}}$ is such that $\widehat{\vartheta }=\vartheta +\epsilon {\vartheta }^{*}\ge 0$, and $0\le \phi \le 2\pi$, ${\phi }^{*}\in \mathbb{R}$ are explained with the thumb in the direction of the $\mathbb{ISA}$. Since $\widehat{\mathbf{x}}$ is a $\mathbb{T}-like$ dual unit vector, we can write out the components of $\widehat{\mathbf{x}}$ in the following form:Therefore, the Blaschke frame of $\widehat{\mathbf{x}}=\widehat{\mathbf{x}}\left(\widehat{w}\right)$ can be written as Comparably, the $\mathbb{S}-like$$\mathbb{DA}$ is Substituting from Equations (23) and (45) into the third term of Equation (38) yields Into Equation (46) we substitute from Equation (43) to obtain Equation (47) is a new hyperbolic $\mathbb{ES}$ formula that fastens a $\mathbb{T}-like$ ruled surface and its osculating circle in terms of the dual angle $\widehat{\phi }$ as well as the second order invariant $\widehat{\varrho }$. Via the real and the dual parts, respectively, we obtain and Equation (48) with (49) are novel $\mathbb{DF}$ in the context of one-parameter hyperbolic spatial locomotions. The former equation establishes a relationship between the positions of the $\mathbb{T}-like$ line in the space ${\mathbb{L}}_m$ and the $\mathbb{S}-like$$\mathbb{DA}$ denoted as $\widehat{\mathbf{b}}$. Based on the information provided in Figure 2, the presence of the signal ${\alpha }^{*}$ (+ or −) in Equation (49) indicates whether the positions of the $\mathbb{DA}$$\widehat{\mathbf{b}}$ are located on the positive or negative direction of the mutual central normal $\widehat{\mathbf{t}}$.

However, we can derive the Equation (47) as follows: the hyperbolic radii of curvature $\widehat{\psi }$ can be written as (see Figure 2):Then, we have Substituting Equation (51), into Equation (33), with awareness of (43), we obtain After some algebraic manipulations, we find as asserted. Moreover, in the case of axodes, it is possible to derive a second dual formulation of the $\mathbb{ES}$ formulae in the following manner: from Equations (22) and (43), one finds facilely Moreover, from Equation (44) we have A simple computation offers that and The amalgamation of Equations (44) and (57) leads to Then, by equating the coefficients of $\widehat{\mathbf{x}},$ $\widehat{\mathbf{t}}$, and $\widehat{\mathbf{g}}$ in Equations (56) and (58), we have and Substituting this into the left hand side of Equation (53), one finds Finally, by substituting $\widehat{\varrho }:={\widehat{\beta }}_f-{\widehat{\beta }}_m=tanh{\widehat{\phi }}_f-tanh{\widehat{\phi }}_m$ into Equation (59), one obtains Equation (62) presents a novel hyperbolic dual variant of the widely recognized $\mathbb{ES}$ formula in the context of conventional spherical kinematics, as discussed in References [1,2,3,4,5,6,7,8,9,21]. This narrative provides a link between the two $\mathbb{T}-like$ axodes in the locomotion of ${\mathbb{L}}_m/{\mathbb{L}}_f$. It should be noted that the striction point is the origin of the relative Blaschke frame, denoted as $\mathbf{s}=\mathbf{0}$, see Figure 2.

We present a method for locating a $\mathbb{T}-like$ line congruence. Therefore, from the real and the dual parts of $\widehat{\mathbf{x}}$ in Equation (43), respectively, we obtain and Since ${\mathbf{x}}^{*}=$$ϵ$$\times \mathbf{x}$, we possess the system of linear equations in ${ϵ}_i$ (i = 1, 2, 3):The coefficient matrix of unknowns ${ϵ}_i$(i = 1, 2, 3) is the skew-adjoint matrix and thus its rank is 2 with $\vartheta \ne 0$, and $\phi \ne 2\pi k$ (k is an integer). The rank of the augmented matrix is also 2. Hence, this system possesses an infinite number of solutions that are specified by Since ${ϵ}_1$ can be arbitrary, we may then put ${ϵ}_1={\phi }^{*}$. In this affair, we have which is the base (director) surface of the $\mathbb{T}-like$ line congruence. Let $\mathsf{\xi }({\xi }_1,{\xi }_2,{\xi }_3)$ be a point on the directed $\mathbb{T}-like$ line $\widehat{\mathbf{x}}$. We can write that where $\rho \in \mathbb{R}$. Given that $\phi$ and ${\phi }^{*}$ are two independent variables, it may be said that $\widehat{x}$ is a $\mathbb{T}-like$ line congruence in ${\mathbb{L}}_f$-space in general. If we define ${\phi }^{*}=h\phi$ and $\phi$ as the parameter for locomotion, then $\left(\widehat{x}\right)$ can be considered as a $\mathbb{T}-like$ ruled in ${\mathbb{L}}_f$-space. As a result, the director surface represented by Equation (69) is constrained by the striction curve on ($\widehat{x}$), which implies that

The curvature ${\kappa }_c\left(\phi \right)$ and torsion ${\tau }_c\left(\phi \right)$ can be given by Then, $\mathbf{c}\left(\phi \right)$ is a $\mathbb{S}-like$ ($\left|{\vartheta }^{*}\right|>\left|h\right|$) or $\mathbb{T}-like$ ($\left|{\vartheta }^{*}\right|<\left|h\right|$) cylindrical helix with the $\mathbb{ISA}$ as its axis. Further, the $\mathbb{T}-like$ ruled surface is The constants h, $\vartheta$ and ${\vartheta }^{*}$ can control the shape of $\left(\widehat{x}\right)$. In the case of $0\le \phi \le 2\pi$, and ${\vartheta }^{*}\ne 0$, we attain where $\varpi ={\vartheta }^{*}tanh\vartheta$, and ${\mathsf{\Psi }}_1={\xi }_1-h\phi$. So, $\left(\widehat{x}\right)$ is a two-parameter family of one-sheeted hyperboloids. The intersection of each hyperboloid and the $\mathbb{S}-like$ plane ${\xi }_1=h\phi$ is a one-parameter family of Lorentzian cylinder $\left(c\right)$: ${\xi }_2^2+{\xi }_3^2={\vartheta }^{*2}$ which is the envelope of $\left(\widehat{x}\right)$. The $\mathbb{T}-like$ ruled surface $\left(\widehat{x}\right)$ can be classified into 4-kinds via their striction curves: