One-Parameter Hyperbolic Dual Spherical Movements and Timelike Ruled Surfaces

Abstract

In this paper, explicit expressions were improved for timelike ruled surfaces with the similarity of hyperbolic dual spherical movements. From this, the well known Hamilton and Mannhiem formulae of surfaces theory are attained at the hyperbolic line space. Then, by employing the E. Study map, a new timelike Plücker conoid is immediately founded and its geometrical properties are examined. In addition, via the height dual function, we specified the connection among the timelike ruled surface and the order of contact with its timelike Disteli-axis. Lastly, a classification for a timelike line to be a stationary timelike Disteli-axis is attained and explained in detail. Our findings contribute to a deeper realization of the cooperation between hyperbolic spatial movements and timelike ruled surfaces, with potential implementations in fields such as robotics and mechanical engineering.

1. Introduction

In the context of spatial kinematics, the trajectories of oriented lines embedded in a moving rigid body are generally ruled surfaces. The importance of the ruled surface lies in the reality that it is utilized in numerous areas of manufacturing and engineering, including modeling of apparel, automobile parts and ship hulls (see, e.g., [1,2,3,4]). One of the most convenient processes for considering the movement of the line space seems to find a relationship through this space and dual numbers. Via the E. Study map in screw and dual number algebra, the set of all oriented lines in Euclidean 3-space ${\mathbb{E}}^3$ is immediately linked to the set of points on the dual unit sphere in the dual 3-space ${\mathbb{D}}^3$. The E. Study map allows a perfect generalization of the mathematical statement for a spherical point geometry to a spatial line geometry by manner of dual number extension, that is, replacing all ordinary quantities by the corresponding dual number quantities. There exists a vast literature on the E. Study map, including several monographs, for example, [1,2,3,4,5,6,7,8].

In the Minkowski 3-space ${\mathbb{E}}_1^3$ the study of the ruled surfaces is more interesting than the Euclidean case; Lorentzian distance can be negative, positive or zero, whereas the Euclidean distance can only be positive. Then, if we occupy the Minkowski 3-space ${\mathbb{E}}_1^3$ as a substitutional of the Euclidean 3-space ${\mathbb{E}}^3$ the E. Study map can be presented as follows: The timelike (spacelike) oriented lines are represented with the timelike (spacelike) dual points on hyperbolic (Lorentzian) dual unit sphere in the Lorentzian Dual 3-space ${\mathbb{D}}_1^3$. A spacelike regular curve on ${\mathbb{H}}_{+}^2$ matches a timelike ruled surface at ${\mathbb{E}}_1^3$. Similarly the spacelike (timelike) curve on ${\mathbb{S}}_1^2$ matches the timelike (spacelike) ruled surface at ${\mathbb{E}}_1^3$. In view of its connections with engineering and physical sciences in Minkowski space, a considerable number of geometers and engineers have studied and purchased many of the ruled surfaces (see [9,10,11,12,13,14]).

The paper studies the kinematic geometry of a timelike line in a one-parameter hyperbolic spatial movement by means of the E. Study map. Then, some unprecedented and well known formulae of surface theory concerning hyperbolic line space and their geometrical explanations are presented. Afterward, the corresponding timelike Plücker conoid associated with the movement is acquired. In addition, a sufficient and necessary condition for a timelike line to be a stationary Disteli-axis is introduced and examined in detail.

2. Preliminaries

In this section, we give a brief outline of the dual numbers and dual vectors (see [1,2,3,4,5,6,7,8,15,16]): A directed (non-null) line L in Minkowski 3-space ${\mathbb{E}}_1^3$ can be identified by a point $\mathbf{a}\in L$ and a normalized vector $\mathbf{e}$ of L; that is, ${∥\mathbf{e}∥}^2=\pm 1$. To have coordinates for L, one arranges the moment vector ${\mathbf{e}}^{*}=\mathbf{a}\times \mathbf{e}$ in connection with the origin point in ${\mathbb{E}}_1^3$. If $\mathbf{a}$ is changed by any point $\mathbf{b}=\mathbf{a}+t\mathbf{e}$,$t\in \mathbb{R}$ on L, this reveals that x${}^{*}$ is independent of $\mathbf{a}$ on L. The two non-null vectors $\mathbf{e}$ and ${\mathbf{e}}^{*}$ are dependent; they satisfy the following two equations:

$$<\mathbf{e},\mathbf{e}>=\pm 1,<{\mathbf{e}}^{*},\mathbf{e}>=0.$$

The six components $e_i,e_i^{*}(i=1,2,3)$ of $\mathbf{e}$ and ${\mathbf{e}}^{*}$ define the normalized Plucker coordinates of L. Hence, the two non-null vectors $\mathbf{e}$ and ${\mathbf{e}}^{*}$ locate the directed line $L.$

A dual number $\widehat{e}$ is a number $e+\epsilon e^{*}$, where $e,$ $e^{*}$ in $\mathbb{R}$ and $\epsilon$ is a dual unit with $\epsilon \ne 0$ and${\epsilon }^2=0$. Then, the set

$${\mathbb{D}}^3=\{\widehat{\mathbf{e}}:=\mathbf{e}+\epsilon {\mathbf{e}}^{*}=({\widehat{e}}_1,{\widehat{e}}_2,{\widehat{e}}_3)\},$$

with the Lorentzian scalar product

$$<\widehat{\mathbf{e}},\widehat{\mathbf{e}}>=-{\widehat{e}}_1^2+{\widehat{e}}_2^2+{\widehat{e}}_3^2,$$

forms the dual Lorentzian 3-space ${\mathbb{D}}_1^3$. Then,

$$\begin{array}{c}-<{\widehat{\mathbf{f}}}_1,{\widehat{\mathbf{f}}}_1>=<{\widehat{\mathbf{f}}}_2,{\widehat{\mathbf{f}}}_2>=<{\widehat{\mathbf{f}}}_3,{\widehat{\mathbf{f}}}_3>=1,\\ {\widehat{\mathbf{f}}}_1\times {\widehat{\mathbf{f}}}_2={\widehat{\mathbf{f}}}_3,{\widehat{\mathbf{f}}}_2\times {\widehat{\mathbf{f}}}_3=-{\widehat{\mathbf{f}}}_1,{\widehat{\mathbf{f}}}_3\times {\widehat{\mathbf{f}}}_1={\widehat{\mathbf{f}}}_2.\end{array}$$

where ${\widehat{\mathbf{f}}}_1$, ${\widehat{\mathbf{f}}}_2$ and ${\widehat{\mathbf{f}}}_3$ are the dual base at the origin point $\widehat{\mathbf{0}}\left(0,0,0\right)$ of the dual Lorentzian 3-space ${\mathbb{D}}_1^3$. Then, a point $\widehat{\mathbf{e}}={({\widehat{e}}_1,{\widehat{e}}_2,{\widehat{e}}_3)}^t$ has dual coordinates ${\widehat{e}}_i=(e_i+\epsilon e_i^{*})\in \mathbb{D}$. If $\mathbf{e}\ne \mathbf{0}$ the norm $∥\widehat{\mathbf{e}}∥$ of $\widehat{\mathbf{e}}=\mathbf{e}+\epsilon {\mathbf{e}}^{*}$ is

$$∥\widehat{\mathbf{e}}∥=\sqrt{\left|<\widehat{\mathbf{e}},\widehat{\mathbf{e}}>\right|}=∥\mathbf{e}∥(1+\epsilon \frac{<\mathbf{e},{\mathbf{e}}^{*}>}{{∥\mathbf{e}∥}^2}).$$

So, the vector $\widehat{\mathbf{e}}$ defines a timelike (spacelike) dual unit vector if ${∥\widehat{\mathbf{e}}∥}^2=-1$(${∥\widehat{\mathbf{e}}∥}^2=1$). Then,

$${∥\widehat{\mathbf{e}}∥}^2=\pm 1⟺{∥\mathbf{e}∥}^2=\pm 1,<\mathbf{e},{\mathbf{e}}^{*}>=0.$$

The hyperbolic and Lorentzian (de Sitter space) dual unit spheres with the center $\widehat{\mathbf{0}}$, respectively, are:

$${\mathbb{H}}_{+}^2=\left\{\widehat{\mathbf{e}}\in {\mathbb{D}}_1^3\mid -{\widehat{e}}_1^2+{\widehat{e}}_2^2+{\widehat{e}}_3^2=-1\right\},$$

and

$${\mathbb{S}}_1^2=\left\{\widehat{\mathbf{e}}\in {\mathbb{D}}_1^3\mid -{\widehat{e}}_1^2+{\widehat{e}}_2^2+{\widehat{e}}_3^2=1\right\}.$$

Hence, we have the following map (E. Study’s map): The ring shaped hyperboloid compatibility with the set of spacelike lines, the common asymptotic cone matching the set of null lines and the oval shaped hyperboloid matching with the set of timelike lines (see Figure 1). Therefore, a regular curve on ${\mathbb{H}}_{+}^2$ matches a timelike ruled surface in ${\mathbb{E}}_1^3$. Moreover, a regular curve on ${\mathbb{S}}_1^2$ matches a spacelike or timelike ruled surface in ${\mathbb{E}}_1^3$.

Definition 1. 

For any two (non-null) dual vectors $\widehat{\mathbf{a}}$ and $\widehat{\mathbf{b}}$ in ${\mathbb{D}}_1^3$ [9,10,11,12,13,14], we have:

(i) 

If $\widehat{\mathbf{a}}$ and $\widehat{\mathbf{b}}$ are two dual spacelike vectors, then:

• If they define a dual spacelike plane, there is a single dual number $\widehat{\theta }=\theta +\epsilon {\theta }^{*}$; $0\le \theta \le \pi$ and ${\theta }^{*}\in \mathbb{R}$ such that $<\widehat{\mathbf{a}},\widehat{\mathbf{b}}>=∥\widehat{\mathbf{a}}∥∥\widehat{\mathbf{b}}∥cos\widehat{\theta }$. This number is named the spacelike dual angle from $\widehat{\mathbf{a}}$ to $\widehat{\mathbf{b}}$.
• If they define a dual timelike plane; there is a single dual number $\widehat{\theta }=\theta +\epsilon {\theta }^{*}\ge 0$ such that $<\widehat{\mathbf{a}},\widehat{\mathbf{b}}>=ϵ∥\widehat{\mathbf{a}}∥∥\widehat{\mathbf{b}}∥cosh\widehat{\theta }$, where $ϵ=+1$ or $ϵ=-1$ via $sign\left({\widehat{a}}_2\right)=sign\left({\widehat{b}}_2\right)$ or $sign\left({\widehat{a}}_2\right)\ne sign\left({\widehat{b}}_2\right)$, respectively. This number is named the central dual angle from $\widehat{\mathbf{a}}$ to $\widehat{\mathbf{b}}$.

(ii) 

If $\widehat{\mathbf{a}}$ and $\widehat{\mathbf{b}}$ are dual timelike vectors, then there is a single dual number $\widehat{\theta }=\theta +\epsilon {\theta }^{*}\ge 0$ such that $<\widehat{\mathbf{a}},\widehat{\mathbf{b}}>=ϵ∥\widehat{\mathbf{a}}∥∥\widehat{\mathbf{b}}∥cosh\widehat{\theta }$, where $ϵ=+1$ or $ϵ=-1$ via $\widehat{\mathbf{a}}$ and $\widehat{\mathbf{b}}$ have different time-direction or the same time-direction, respectively. This dual number is named the Lorentzian timelike dual angle from $\widehat{\mathbf{a}}$ to $\widehat{\mathbf{b}}$.

(iii) 

If $\widehat{\mathbf{a}}$ is dual spacelike and $\widehat{\mathbf{b}}$ is dual timelike, then there is a single dual number $\widehat{\theta }=\theta +\epsilon {\theta }^{*}\ge$ 0 such that $<\widehat{\mathbf{a}},\widehat{\mathbf{b}}>=ϵ∥\widehat{\mathbf{a}}∥∥\widehat{\mathbf{b}}∥sinh\widehat{\theta }$, where $ϵ=+1$ or $ϵ=-1$ via $sign\left({\widehat{a}}_2\right)=sign\left({\widehat{b}}_1\right)$ or $sign\left({\widehat{a}}_2\right)\ne sign\left({\widehat{b}}_1\right)$. This number is named the Lorentzian timelike dual angle from $\widehat{\mathbf{a}}$ to $\widehat{\mathbf{b}}$.

Definition 2. 

A set of non-null oriented lines $\widehat{\mathbf{e}}=(\mathbf{e},{\mathbf{e}}^{*})\in {\mathbb{E}}_1^3$ satisfies:

$$<{\mathbf{a}}^{*},\mathbf{e}>+<{\mathbf{e}}^{*},\mathbf{a}>=0,$$

where ${∥\widehat{\mathbf{e}}∥}^2=1({∥\mathbf{e}∥}^2=-1$) is named a spacelike (timelike) line complex when $<\mathbf{e},{\mathbf{e}}^{*}>\ne 0$ and is a spacelike (timelike) singular line complex when $<{\mathbf{e}}^{*},\mathbf{e}>=0$,and ${∥\widehat{\mathbf{x}}∥}^{\mathbf{2}}=\pm 1$.

Geometrically, a non-null singular line complex is a set of all non-null lines $\widehat{\mathbf{a}}=(\mathbf{a},{\mathbf{a}}^{*})$ intersecting the non-null line $\widehat{\mathbf{e}}=(\mathbf{e},{\mathbf{e}}^{*})$. Then, we can realize a non-null line congruence by mutual lines of any two non-null line complexes. The mutual lines of two non-null line congruences forms a differentiable set of non-null lines in ${\mathbb{E}}_1^3$ realized as a non-null ruled surface. Non-null ruled surfaces (such as cones and cylinders ) include non-null lines in which the tangent plane touches the surface over the non-null generator (ruling). Such non-null lines are named non-null torsal lines.

3. One-Parameter Hyperbolic Dual Spherical Movements

Let us have two hyperbolic dual unit spheres ${\mathbb{H}}_{+m}^2$ and ${\mathbb{H}}_{+f}^2$. Let $\widehat{\mathbf{0}}$ be the mutual center and two orthonormal dual frames $\left\{\widehat{\mathbf{e}}\right\}=\{\widehat{\mathbf{0}}$;${\widehat{\mathbf{e}}}_1($timelike$),{\widehat{\mathbf{e}}}_2,{\widehat{\mathbf{e}}}_3$} and $\{\widehat{\mathbf{f}}\}=\{\widehat{\mathbf{0}}$;${\widehat{\mathbf{f}}}_1$(timelike),${\widehat{\mathbf{f}}}_2$,${\widehat{\mathbf{f}}}_3$} be rigidly tied to ${\mathbb{H}}_{+m}^2$ and ${\mathbb{H}}_{+f}^2,$ respectively. We keep $\{\widehat{\mathbf{f}}\}$ as it is, whereas the set $\left\{\widehat{\mathbf{e}}\right\}$ are functions of a real parameter $t\in \mathbb{R}$ (say the time). Then, we say that the ${\mathbb{S}}_{1m}^2$ elements of the set moves with respect to ${\mathbb{S}}_{1f}^2$. Such movement is a one-parameter hyperbolic dual spherical movements and is indicated by ${\mathbb{H}}_{+m}^2/{\mathbb{H}}_{+f}^2$. If ${\mathbb{H}}_{+m}^2$ and ${\mathbb{H}}_{+f}^2$ act as the hyperbolic line spaces ${\mathbb{L}}_m$ and ${\mathbb{L}}_f$, respectively, then ${\mathbb{H}}_{+m}^2/{\mathbb{H}}_{+f}^2$ act as the one-parameter hyperbolic spatial movements ${\mathbb{L}}_m/{\mathbb{L}}_f$. Then, ${\mathbb{L}}_m$ is the movable hyperbolic space with respect to the stationary hyperbolic space ${\mathbb{L}}_f$. Suppose $<{\widehat{\mathbf{f}}}_i,{\widehat{\mathbf{e}}}_j>={\widehat{l}}_{ij}$ and the dual matrix $\widehat{l}\left(t\right)=\left(l_{ij}\right)+\epsilon \left(l_{ij}^{*}\right)$; we can perform ${\mathbb{H}}_{+m}^2/{\mathbb{H}}_{+f}^2$ as follows:

$${\mathbb{H}}_{+m}^2/{\mathbb{H}}_{+f}^2:\left(\begin{array}{c}{\widehat{\mathbf{f}}}_1\\ {\widehat{\mathbf{f}}}_2\\ {\widehat{\mathbf{f}}}_3\end{array}\right)=\left(\begin{array}{ccc}{\widehat{l}}_{11}& {\widehat{l}}_{12}& {\widehat{l}}_{13}\\ {\widehat{l}}_{21}& {\widehat{l}}_{22}& {\widehat{l}}_{23}\\ {\widehat{l}}_{31}& {\widehat{l}}_{32}& {\widehat{l}}_{33}\end{array}\right)\left(\begin{array}{c}{\widehat{\mathbf{e}}}_1\\ {\widehat{\mathbf{e}}}_2\\ {\widehat{\mathbf{e}}}_3\end{array}\right).$$

Then, the signature matrix $ϵ$ describing the inner product in ${\mathbb{D}}_1^3$ is given by:

$$ϵ=\left(\begin{array}{ccc}-1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right).$$

The dual matrix $\widehat{l}$ has the property that ${\widehat{l}}^T=ϵ{\widehat{l}}^{-1}ϵ,$${\widehat{l}}^{-1}=ϵ{\widehat{l}}^Tϵ\widehat{l}$. So, we have

$$\widehat{l}{\widehat{l}}^{-1}=\widehat{l}ϵ{\widehat{l}}^Tϵ={\widehat{l}}^{-1}\widehat{l}=ϵ{\widehat{l}}^Tϵ\widehat{l}=I,$$

where I is the $3\times 3$ unit matrix. This outcome indicates that if a one-parameter hyperbolic spatial movement is given in ${\mathbb{E}}_1^3$, we can find a dual orthogonal $3\times 3$ matrix $\widehat{l}\left(t\right)=\left({\widehat{l}}_{ij}\right)$, where $\left({\widehat{l}}_{ij}\right)$ are dual functions of one parameter $t\in \mathbb{R}$. As the set of real orthogonal matrices, the set of dual orthogonal $3\times 3$ matrices, specified by $\mathbb{O}\left({\mathbb{D}}_1^{3\times 3}\right)$, create a group with matrix multiplication as the group operation (real hyperbolic orthogonal matrices are a subgroup of hyperbolic dual orthogonal matrices). The identity element of $\mathbb{O}\left({\mathbb{D}}_1^{3\times 3}\right)$ is the $3\times 3$ unit matrix. Since the center of the hyperbolic dual unit sphere in ${\mathbb{D}}_1^3$ has to stay stationary, the transformation group in ${\mathbb{D}}_1^3$ (the resemblance of hyperbolic movements in the Minkowski 3-space ${\mathbb{E}}_1^3$) does not depend on any translations.

The Lie algebra $\mathbb{L}\left({\mathbb{O}}_{{\mathbb{D}}_1^{3\times 3}}\right)$ of the group $\mathbb{GL}$ of $3\times 3$ positive orthogonal dual matrices $\widehat{l}$ is the algebra of skew-adjoint $3\times 3$ dual matrices

$$\widehat{\omega }\left(t\right):={\widehat{l}}^{{}^{\prime }}ϵ{\widehat{l}}^Tϵ=\left(\begin{array}{ccc}0\hfill & {\widehat{\omega }}_3\hfill & {\widehat{\omega }}_2\hfill \\ {\widehat{\omega }}_3\hfill & 0\hfill & {\widehat{\omega }}_1\hfill \\ {\widehat{\omega }}_2\hfill & -{\widehat{\omega }}_1\hfill & 0\hfill \end{array}\right);\left({}^{\prime }=\frac{d}{dt}\right).$$

Then, ${\mathbb{H}}_{+m}^2/{\mathbb{H}}_{+f}^2$ is realized by

$$\left(\begin{array}{c}{\widehat{\mathbf{e}}}_1^{{}^{\prime }}\hfill \\ {\widehat{\mathbf{e}}}_2^{{}^{\prime }}\hfill \\ {\widehat{\mathbf{e}}}_3^{{}^{\prime }}\hfill \end{array}\right)=\left(\begin{array}{ccc}0\hfill & {\widehat{\omega }}_3\hfill & {\widehat{\omega }}_2\hfill \\ {\widehat{\omega }}_3\hfill & 0\hfill & {\widehat{\omega }}_1\hfill \\ {\widehat{\omega }}_2\hfill & -{\widehat{\omega }}_1\hfill & 0\hfill \end{array}\right)\left(\begin{array}{c}{\widehat{\mathbf{e}}}_1\\ {\widehat{\mathbf{e}}}_2\\ {\widehat{\mathbf{e}}}_3\end{array}\right)=\widehat{\omega }\times \left(\begin{array}{c}{\widehat{\mathbf{e}}}_1\\ {\widehat{\mathbf{e}}}_2\\ {\widehat{\mathbf{e}}}_3\end{array}\right).$$

$\widehat{\omega }\left(t\right)=\omega \left(t\right)+\epsilon {\omega }^{*}\left(t\right)=({\widehat{\omega }}_1,{\widehat{\omega }}_2,{\widehat{\omega }}_3)$ is the instantaneous dual rotation vector of ${\mathbb{H}}_{+m}^2/{\mathbb{H}}_{+f}^2$. $\omega$ and ${\omega }^{*}$ are the instantaneous rotational differential velocity vector and the instantaneous translational differential velocity vector of the movement ${\mathbb{L}}_m/{\mathbb{L}}_f$, respectively.

3.1. The Blaschke Approach for Timelike Ruled Surface

Through the movement ${\mathbb{H}}_{+m}^2/{\mathbb{H}}_{+f}^2$, any fixed point $\widehat{\mathbf{x}}\in {\mathbb{H}}_{+m}^2$, normally, drawing a spacelike dual curve $\widehat{\mathbf{x}}\left(t\right)$ on ${\mathbb{H}}_{+f}^2$ acts as a timelike ruled surface in ${\mathbb{L}}_f$. Let us mark this surface by ($\widehat{x}$). Therefore, ($\widehat{x}$) is parametrized by a spacelike dual curve $\widehat{\mathbf{x}}\left(t\right)\in {\mathbb{H}}_{+f}^2$. As usual, the Blaschke frame {$\widehat{\mathbf{0}};\widehat{\mathbf{x}}\left(t\right)$, $\widehat{\mathbf{t}}\left(t\right)$, $\widehat{\mathbf{g}}\left(t\right)\}$ is:

$$\begin{array}{c}\widehat{\mathbf{x}}=\widehat{\mathbf{x}}\left(t\right),\widehat{\mathbf{t}}\left(t\right)={\widehat{\mathbf{x}}}^{{}^{\prime }}{∥{\widehat{\mathbf{x}}}^{{}^{\prime }}∥}^{-1},\widehat{\mathbf{x}}\times \widehat{\mathbf{t}}=-\widehat{\mathbf{g}},\\ -<\widehat{\mathbf{x}},\widehat{\mathbf{x}}>=<\widehat{\mathbf{t}},\widehat{\mathbf{t}}>=<\widehat{\mathbf{g}},\widehat{\mathbf{g}}>=1,\\ \widehat{\mathbf{x}}\times \widehat{\mathbf{t}}=\widehat{\mathbf{g}},\widehat{\mathbf{x}}\times \widehat{\mathbf{g}}=-\widehat{\mathbf{t}},\widehat{\mathbf{t}}\times \widehat{\mathbf{g}}=-\widehat{\mathbf{x}}.\end{array}$$

(1)

The dual unit vectors $\widehat{\mathbf{x}}$, $\widehat{\mathbf{t}}$ and $\widehat{\mathbf{g}}$ act out three concurrent mutually orthogonal oriented lines in ${\mathbb{L}}_f$ and they meet at a point $\mathbf{c}$ on ($\widehat{x}$) named the central point. The trajectory of the central point traces the striction curve $\mathbf{c}\left(t\right)$ on ($\widehat{x}$). Then, the Blaschke formula is found as:

$$\left(\begin{array}{c}{\widehat{\mathbf{x}}}^{{}^{\prime }}\hfill \\ {\widehat{\mathbf{t}}}^{{}^{\prime }}\hfill \\ {\widehat{\mathbf{g}}}^{{}^{\prime }}\hfill \end{array}\right)=\left(\begin{array}{ccc}0\hfill & \widehat{p}\hfill & 0\hfill \\ \widehat{p}\hfill & 0\hfill & \widehat{q}\hfill \\ 0\hfill & -\widehat{q}\hfill & 0\hfill \end{array}\right)\left(\begin{array}{c}\widehat{\mathbf{x}}\hfill \\ \widehat{\mathbf{t}}\hfill \\ \widehat{\mathbf{g}}\hfill \end{array}\right)=\widehat{\omega }\times \left(\begin{array}{c}\widehat{\mathbf{x}}\hfill \\ \widehat{\mathbf{t}}\hfill \\ \widehat{\mathbf{g}}\hfill \end{array}\right),$$

(2)

where $\widehat{\omega }\left(t\right)=(\widehat{q},0,\widehat{p})$, and

$$\widehat{p}\left(t\right)=p\left(t\right)+\epsilon p^{*}\left(t\right)=∥{\widehat{\mathbf{x}}}^{{}^{\prime }}∥,\widehat{q}=q+\epsilon q^{*}=det(\widehat{\mathbf{x}},{\widehat{\mathbf{x}}}^{{}^{\prime }},{\widehat{\mathbf{x}}}^{″}),$$

are the Blaschke invariants of the timelike dual curve $\widehat{\mathbf{x}}\left(t\right)\in {\mathbb{H}}_{+f}^2$. The tangent of the striction curve is:

$${\mathbf{c}}^{{}^{\prime }}\left(t\right)=-q^{*}\left(t\right)\mathbf{x}\left(t\right)+p^{*}\left(t\right)\mathbf{g}\left(t\right).$$

(3)

The curvature functions of ($\widehat{x}$) are:

$$\gamma \left(t\right)=\frac{q\left(t\right)}{p\left(t\right)},\mathrm{\Gamma }\left(t\right)=\frac{q^{*}\left(t\right)}{q\left(t\right)}\mathrm{and}\mu \left(t\right)=\frac{p^{*}\left(t\right)}{p\left(t\right)}\mathrm{with}p\left(t\right)\ne 0.$$

(4)

The geometric clarifications of $\gamma \left(t\right)$, $\mathrm{\Gamma }\left(t\right)$ and $\mu \left(t\right)$ are as follows:$\gamma$ is the spherical curvature of the image curve $\mathbf{x}=\mathbf{x}\left(t\right)$; $\mathrm{\Gamma }$ is the angle through the tangent to the striction curve and the ruling of ($\widehat{x}$); and $\mu$ is its distribution parameter at the ruling. Thus, a timelike ruled surface can be performed by

$$\left(\widehat{x}\right):\mathbf{y}(t,v)=\stackrel{t}{\underset{0}{\int }}\left(-q^{*}\left(t\right)\mathbf{x}\left(t\right)+p^{*}\left(t\right)\mathbf{g}\left(t\right)\right)dt+v\mathbf{x}\left(t\right).$$

(5)

The timelike Disteli-axis or evolute of $\widehat{\mathbf{x}}\left(t\right)\in {\mathbb{H}}_{1f}^2$ is defined as:

$$\widehat{\mathbf{b}}\left(t\right)=\mathbf{b}\left(t\right)+\epsilon {\mathbf{b}}^{*}\left(t\right)=\frac{\widehat{\omega }\left(t\right)}{∥\widehat{\omega }\left(t\right)∥}=\frac{\widehat{q}\widehat{\mathbf{x}}-\widehat{p}\widehat{\mathbf{g}}}{\sqrt{{\widehat{q}}^2-{\widehat{p}}^2}},\mathrm{with}\left|\widehat{q}\right|>\left|\widehat{p}\right|.$$

(6)

Then, we obtain:

$$\left(\begin{array}{c}{\widehat{\mathbf{x}}}^{{}^{\prime }}\hfill \\ {\widehat{\mathbf{t}}}^{{}^{\prime }}\hfill \\ {\widehat{\mathbf{g}}}^{{}^{\prime }}\hfill \end{array}\right)=∥\widehat{\omega }∥\widehat{\mathbf{b}}\times \left(\begin{array}{c}\widehat{\mathbf{x}}\hfill \\ \widehat{\mathbf{t}}\hfill \\ \widehat{\mathbf{g}}\hfill \end{array}\right).$$

Thus, at any instant $t\in \mathbb{R}$, we gain

$${\omega }^{*}\left(t\right)=\frac{q{q}^{*}-p{p}^{*}}{\sqrt{q^2-p^2}},\omega \left(t\right)=\sqrt{q^2-p^2}.$$

(7)

Then, the timelike Disteli-axis is the instantaneous screw axis ($\mathbb{ISA}$) of the movement ${\mathbb{L}}_m/{\mathbb{L}}_f$. Consequently, we have:

Proposition 1. 

At any instant $t\in \mathbb{R}$, the pitch of the Blaschke frame along the $\mathbb{ISA}$ is

$$h\left(t\right):=\frac{<\omega ,{\omega }^{*}>}{{∥\omega ∥}^2}=\frac{q{q}^{*}-p{p}^{*}}{q^2-p^2}.$$

(8)

Furthermore, we have:

(1) 

The timelike Disteli-axis $\widehat{\mathbf{b}}\left(t\right)$ can be located via Equation (6).

(2) 

The dual angular speed can be designed as $∥\widehat{\omega }\left(t\right)∥=\omega \left(t\right)(1+\epsilon h\left(t\right))$.

(3) 

If $\mathbf{y}(x,y,z)$ is a point on the timelike Disteli-axis $\widehat{\mathbf{b}}\left(t\right)$, then

$$\mathbf{y}(t,v)=\mathbf{b}\left(t\right)\times {\mathbf{b}}^{*}\left(t\right)+v\mathbf{b}\left(t\right),v\in \mathbb{R}.$$

(9)

is a non-developable timelike ruled surface $\left(\widehat{b}\right)$.

In the case of $h\left(t\right)=0$, that is, the movement ${\mathbb{L}}_m/{\mathbb{L}}_f$is pure rotation, then

$$\widehat{\mathbf{b}}\left(t\right)=\mathbf{b}\left(t\right)+\epsilon {\mathbf{b}}^{*}\left(t\right)=\frac{1}{∥\omega ∥}(\omega +\epsilon {\omega }^{*}),$$

(10)

whereas if $h\left(t\right)=0$ and ${∥\omega \left(t\right)∥}^2=1$, then $\widehat{\omega }\left(t\right)$ is a timelike line. However, if $\widehat{\omega }\left(t\right)=$0$+\epsilon {\omega }^{*}\left(t\right)$, that is, the movement ${\mathbb{L}}_m/{\mathbb{L}}_f$is pure translational, we let ${\omega }^{*}\left(t\right)=∥{\omega }^{*}\left(t\right)∥;$ ${\omega }^{*}\mathbf{b}\left(t\right)={\omega }^{*}$ for arbitrary ${\mathbf{b}}^{*}\left(t\right)$ such that ${\omega }^{*}\left(t\right)\ne 0$; in other cases, $\mathbf{b}\left(t\right)$ can be an arbitrary selection, too.

Let $\widehat{\psi }\left(t\right)=\psi \left(t\right)+\epsilon {\psi }^{*}\left(t\right)$ be the dual radius of curvature from $\widehat{\mathbf{b}}$ to $\widehat{\mathbf{x}}$ (see Figure 2). Then,

$$\widehat{\mathbf{b}}\left(t\right)=cosh\widehat{\psi }\widehat{\mathbf{x}}-sinh\widehat{\psi }\widehat{\mathbf{g}},$$

(11)

where

$$coth\widehat{\psi }=coth\psi +\epsilon {\psi }^{*}(1-{coth}^2\psi )=\frac{\widehat{q}}{\widehat{p}}.$$

(12)

From Equations (4), (8) and (12), we obtain:

$$\left.\begin{array}{c}h\left(t\right)=\mathrm{\Gamma }{cosh}^2\psi -\mu {sinh}^2\psi ,\hfill \\ {\psi }^{*}\left(t\right)=\frac{1}{2}\left(\mathrm{\Gamma }-\mu \right)sinh2\psi .\hfill \end{array}\right\}$$

(13)

The first expression is due to Hamilton and the second one is due to Mannhiem, (see [1,2,3,4]).

3.2. Timelike Plücker’s Conoid

We now are dealing with the kinematics-geometrical of the Hamilton and Mannhiem formulae. To this aim, the surface ${\psi }^{*}$ is a Minkowski version of the well known Plücker conoid or cylindroid as follows: let $\widehat{\mathbf{t}}$ use the y-axis of a stationary hyperbolic frame $\left(oxyz\right)$ and let the location of $\widehat{\mathbf{b}}$ be designated by angle $\psi$ and distance ${\psi }^{*}$ on the positive orientation of the y-axis. The dual unit vectors $\widehat{\mathbf{x}}($timelike) and $\widehat{\mathbf{g}}($spacelike) can be taken with the x- and z-axes, respectively. Then, {$\widehat{\mathbf{x}}$, $\widehat{\mathbf{g}},\widehat{\mathbf{t}}$} identifies the coordinate system of the timelike Plücker conoid (Figure 2). Here $\widehat{\mathbf{t}}$ represents a directed mutual perpendicular of the given lines $\widehat{\mathbf{x}}$ and $\widehat{\mathbf{g}},$ as well as the sign of $\psi$ and ${\psi }^{*}$ is related to the orientation of $\widehat{\mathbf{t}}$. In view of Equations (9) and (11), we have

$$\left(\widehat{b}\right):\mathbf{y}(t,v)=(0,-{\psi }^{*},0)+v(cosh\psi ,0,sinh\psi ),v\in \mathbb{R},$$

(14)

from which we have

$${\psi }^{*}:=y=\frac{1}{2}\left(\mathrm{\Gamma }-\mu \right)sinh2\psi ,x=vcosh\psi \mathrm{and}z=vsinh\psi .$$

By an easy calculation, we gain

$$\left(\widehat{b}\right):\left(x^2-z^2\right)y+\left(\mu -\mathrm{\Gamma }\right)xz=0,$$

(15)

which is the algebraic equation for $\left(\widehat{b}\right)$. It is clear that Equation (15) is based only on the two integral invariants of the first order; $\mu -\mathrm{\Gamma }=1,0\le \psi \in \mathbb{R}$, $-2\le \upsilon \le 2$ (Figure 3). Further, one can obtain a second-order algebraic equation in $x/z$ as

$$\frac{x}{z}=\frac{1}{2y}\left[-\left(\mu -\mathrm{\Gamma }\right)\pm \sqrt{{\left(\mu -\mathrm{\Gamma }\right)}^2+4y^2}\right].$$

(16)

For the limits of $\left(\widehat{b}\right),$ we put ${\left(\mu -\mathrm{\Gamma }\right)}^2+4y^2=0$. Thus, the two boundaries of $\left(\widehat{b}\right)$ are:

$$y=\pm i\left(\mu -\mathrm{\Gamma }\right)/2,\mathrm{with}i=\sqrt{-1}.$$

(17)

Equation (17) gives two isotropic torsal timelike planes, each of which contains one isotropic torsal timelike line L. Hence, the geometric aspects of $\left(\widehat{b}\right)$ are as follows:

(i)

If $h\left(t\right)\ne 0$, then we have two rulings through the point $(0,y,0)$; and for the two limit isotropic torsal timelike planes $y=\pm i\left(\mu -\mathrm{\Gamma }\right)/2$, the rulings and the principal axes are identical.

(ii)

If $h\left(t\right)=0,$ then we have two torsal isotropic timelike lines $L_1$, $L_2$ specified as

$$L_1,L_2:\frac{x}{z}=tanh\psi =\pm \sqrt{\frac{\mu }{\mathrm{\Gamma }}},y=\pm i\left(\mu -\mathrm{\Gamma }\right)/2.$$

(18)

Equation (18) shows that the two-isotropic torsal timelike lines $L_1$ and$L_2$ are orthogonal to each other. So, if $\mu$ and $\mathrm{\Gamma }$ are equal, then the timelike Plücker conoid degenerates to a family of timelike lines in the origin “$\mathbf{o}$” in the timelike torsal plane $y=0$. In this case, $L_1$ and $L_2$ are the principal axes of an elliptic timelike line congruence. However, if $\mu$ and $\mathrm{\Gamma }$ have opposite signs, then $L_1$ and $L_2$ are coincident with the principal axes of a timelike hyperbolic line congruence.

However, to transform from polar coordinates to Cartesian, we use the transformation

$$x=\frac{1}{\sqrt{h}}cosh\psi \mathrm{and}z=\frac{1}{\sqrt{h}}sinh\psi ,$$

into Hamilton’s formula; we obtain the equation

$$D:\left|\mu \right|x^2-\left|\mathrm{\Gamma }\right|z^2=1,$$

of a hyperbolic conic section. This conic section is a Minkowski form of Dupin’s indicatrix of the surface theory in Euclidean 3-space ${\mathbb{E}}^3$.

Serret–Frenet Frame

In Equation (3), if $p^{*}=0$, then $\left(\widehat{x}\right)$ is a timelike tangential developable ruled surface. In this case, Dupin’s indicatrix is a set of parallel isotropic timelike lines located via $z=\pm i/\left|\mathrm{\Gamma }\right|$. Let u be an arc length parameter of $\mathbf{c}\left(t\right)$ and $\left\{\kappa \right(u)$, $\tau \left(u\right)$, ${\mathbf{\upsilon }}_1\left(u\right),$ ${\mathbf{\upsilon }}_2\left(u\right)$, ${\mathbf{\upsilon }}_3\left(u\right)\}$ be the usual movable Serret–Frenet frame. Then,

$$\frac{d}{du}\left(\begin{array}{c}{\mathbf{\upsilon }}_1\hfill \\ {\mathbf{\upsilon }}_2\hfill \\ {\mathbf{\upsilon }}_3\hfill \end{array}\right)=\left(\begin{array}{ccc}0\hfill & \kappa \left(u\right)\hfill & 0\hfill \\ \kappa \left(u\right)\hfill & 0\hfill & -\tau \left(u\right)\hfill \\ 0\hfill & \tau \left(u\right)\hfill & 0\hfill \end{array}\right)\left(\begin{array}{c}{\mathbf{\upsilon }}_1\hfill \\ {\mathbf{\upsilon }}_2\hfill \\ {\mathbf{\upsilon }}_3\hfill \end{array}\right),$$

where

$$\kappa \left(u\right)=\frac{1}{\mathrm{\Gamma }\left(u\right)},\tau \left(u\right)=\frac{\gamma \left(u\right)}{\mathrm{\Gamma }\left(u\right)},\mathrm{with}\mathrm{\Gamma }\left(u\right)\ne 0.$$

Therefore, the curvature function $\mathrm{\Gamma }\left(u\right)$ is the radius of curvature of the timelike striction curve $\mathbf{c}\left(u\right)$. Then,

$$\left.\begin{array}{c}h\left(u\right)=\frac{1}{\kappa }{cosh}^2\psi ,{\psi }^{*}\left(u\right)=\frac{1}{2\kappa }sinh2\psi ,\hfill \\ \left(\widehat{b}\right):\left(x^2-z^2\right)y-\frac{1}{\kappa }xz=0.\hfill \end{array}\right\}$$

(b) If $q^{*}=0$, then the striction curve is tangent to $\mathbf{g}$; it is normal to the ruling through $\mathbf{c}\left(u\right)$, where ${\mathbf{\upsilon }}_1=\mathbf{g}\left(u\right)$, ${\mathbf{\upsilon }}_2=-\mathbf{t}\left(u\right)$, ${\mathbf{\upsilon }}_3=\mathbf{x}\left(u\right)$. In this case, ($\widehat{x}$) a timelike binormal ruled surface. Moreover, the Dupin’s indicatrix is a set of parallel spacelike lines located via $x=\pm 1/\left|\mu \right|$. Similarly, we have

$$\frac{d}{du}\left(\begin{array}{c}{\mathbf{\upsilon }}_1\hfill \\ {\mathbf{\upsilon }}_2\hfill \\ {\mathbf{\upsilon }}_3\hfill \end{array}\right)=\left(\begin{array}{ccc}0\hfill & \kappa \left(u\right)\hfill & 0\hfill \\ -\kappa \left(u\right)\hfill & 0\hfill & -\tau \left(u\right)\hfill \\ 0\hfill & -\tau \left(u\right)\hfill & 0\hfill \end{array}\right)\left(\begin{array}{c}{\mathbf{\upsilon }}_1\hfill \\ {\mathbf{\upsilon }}_2\hfill \\ {\mathbf{\upsilon }}_3\hfill \end{array}\right),$$

where

$$\kappa \left(u\right)=\frac{\gamma \left(u\right)}{\mu \left(u\right)},\tau \left(u\right)=\frac{1}{\mu \left(u\right)},\mathrm{with}\mu \left(u\right)\ne 0.$$

Then, the curvature function $\mu \left(u\right)$ is the radius of torsion of the spacelike striction curve $\mathbf{c}\left(s\right)$. Similarly, we have:

$$\left.\begin{array}{c}h\left(u\right)=-\frac{1}{\tau }{sinh}^2\psi ,{\psi }^{*}\left(u\right)=-\frac{1}{2\kappa }sinh2\psi ,\hfill \\ \left(\widehat{b}\right):\left(x^2-z^2\right)y+\frac{1}{\tau }xz=0.\hfill \end{array}\right\}$$

3.3. Timelike Ruled Surface with Stationary Disteli-Axis

In the following, when we say ($\widehat{x}$) is a timelike ruled surface with stationary timelike Disteli-axis, we mean that all the rulings of ($\widehat{x}$) have a stationary dual angle from its Disteli-axis.

Let $d\widehat{s}=ds+\epsilon d{s}^{*}$ define the dual arc length of $\widehat{\mathbf{x}}\left(t\right)$. Then, we have

$$\widehat{s}\left(t\right)=\stackrel{t}{\underset{0}{\int }}\widehat{p}dt=\stackrel{t}{\underset{0}{\int }}p(1+\epsilon \mu )dt.$$

(19)

By employing $\widehat{s}$ instead of t, from Equations (2) and (19), we have:

$$\left(\begin{array}{c}{\widehat{\mathbf{x}}}^{{}^{\prime }}\hfill \\ {\widehat{\mathbf{t}}}^{{}^{\prime }}\hfill \\ {\widehat{\mathbf{g}}}^{{}^{\prime }}\hfill \end{array}\right)=\left(\begin{array}{ccc}0\hfill & 1\hfill & 0\hfill \\ 1\hfill & 0\hfill & \widehat{\gamma }\hfill \\ 0\hfill & -\widehat{\gamma }\hfill & 0\hfill \end{array}\right)\left(\begin{array}{c}\widehat{\mathbf{x}}\hfill \\ \widehat{\mathbf{t}}\hfill \\ \widehat{\mathbf{g}}\hfill \end{array}\right)=\widehat{\omega }\times \left(\begin{array}{c}\widehat{\mathbf{x}}\hfill \\ \widehat{\mathbf{t}}\hfill \\ \widehat{\mathbf{g}}\hfill \end{array}\right);({}^{\prime }=\frac{d}{d\widehat{s}}),$$

(20)

where $\widehat{\omega }\left(\widehat{s}\right)=(\widehat{\gamma },0,1)$, $\widehat{\gamma }\left(\widehat{s}\right):=\gamma +\epsilon {\gamma }^{*}=\frac{\widehat{q}}{\widehat{p}}$ is the dual spherical curvature of $\widehat{\mathbf{x}}\left(\widehat{s}\right)\in {\mathbb{H}}_{+f}^2$. Then,

$$\left.\begin{array}{c}\widehat{\gamma }\left(\widehat{s}\right)=\gamma +\epsilon \left(\mathrm{\Gamma }-\gamma \mu \right)=coth\widehat{\psi },\hfill \\ \widehat{\kappa }\left(\widehat{s}\right):=\kappa +\epsilon {\kappa }^{*}=\sqrt{{\widehat{\gamma }}^2-1}=\frac{1}{sinh\widehat{\psi }}=\frac{1}{\widehat{\rho }\left(\widehat{s}\right)},\hfill \\ \widehat{\tau }\left(\widehat{s}\right):=\tau +\epsilon {\tau }^{*}=\pm {\widehat{\psi }}^{{}^{\prime }}=\pm \frac{{\widehat{\gamma }}^{{}^{\prime }}}{{\widehat{\gamma }}^2-1},\hfill \end{array}\right\}$$

(21)

where $\widehat{\kappa }\left(\widehat{s}\right)$ is the dual curvature and $\widehat{\tau }\left(\widehat{s}\right)$ is the dual torsion of the dual curve $\widehat{\mathbf{x}}\left(\widehat{s}\right)\in {\mathbb{H}}_{+f}^2$.

Height Dual Functions

In analogy with [17,18], a dual point ${\widehat{\mathbf{b}}}_0$ of ${\mathbb{H}}_{+f}^2$ will be said to be a ${\widehat{\mathbf{b}}}_k$ evolute of the dual curve$\widehat{\mathbf{x}}\left(\widehat{s}\right)$on ${\mathbb{H}}_{+f}^2$; for all i such that $1\le i\le k$, $<{\widehat{\mathbf{b}}}_0,{\widehat{\mathbf{x}}}^i\left(\widehat{s}\right)>=0$, but $<{\widehat{\mathbf{b}}}_0,{\widehat{\mathbf{x}}}^{k+1}\left(\widehat{s}\right)>\ne 0$. Here ${\widehat{\mathbf{x}}}^i$ signalizes the i-th derivatives of $\widehat{\mathbf{x}}$ with respect to the dual arc length of$\widehat{\mathbf{x}}\left(\widehat{s}\right)$on ${\mathbb{H}}_{+f}^2$. For the first evolute $\widehat{\mathbf{b}}$ of$\widehat{\mathbf{x}}\left(\widehat{s}\right)$, we have $<\widehat{\mathbf{b}},{\widehat{\mathbf{x}}}^{{}^{\prime }}>=\pm <\widehat{\mathbf{b}},\widehat{\mathbf{t}}>=0$ and $<\widehat{\mathbf{b}},{\widehat{\mathbf{x}}}^{{}^{″}}>=\pm <\widehat{\mathbf{b}},\widehat{\mathbf{x}}+\widehat{\gamma }\widehat{\mathbf{g}}>\ne 0$. So, $\widehat{\mathbf{b}}$ is at least a ${\widehat{\mathbf{b}}}_2$ evolute of $\widehat{\mathbf{x}}\left(\widehat{s}\right)\in {\mathbb{H}}_{+f}^2$.

We now address a regular dual function $\widehat{h}:\widehat{I}\times {\mathbb{H}}_{+f}^2\to \mathbb{D}$, by $\widehat{h}(\widehat{s},{\widehat{\mathbf{b}}}_0)=<{\widehat{\mathbf{b}}}_0,\widehat{\mathbf{x}}>$. We call $\widehat{h}$ a height dual function on $\widehat{\mathbf{x}}\left(\widehat{s}\right)\in {\mathbb{H}}_{+f}^2$. We use $\widehat{h}\left(\widehat{s}\right)=<{\widehat{\mathbf{b}}}_0,\widehat{\mathbf{x}}>$ for any stationary point ${\widehat{\mathbf{b}}}_0\in {\mathbb{H}}_{+f}^2$.

Proposition 2. 

Let $\widehat{\mathbf{x}}:\widehat{I}\subseteq \mathbb{D}\to {\mathbb{H}}_{+f}^2$ be a dual curve $\widehat{\mathbf{x}}\left(\widehat{s}\right)$ in ${\mathbb{H}}_{+f}^2$ with $\left|\widehat{\gamma }\right|>1$. Then, the following holds:

1-h will be stationary in the first approximation iff ${\widehat{\mathbf{b}}}_0\in Sp\{\widehat{\mathbf{x}}$,$\widehat{\mathbf{g}}\}$; that is,

$$h^{{}^{\prime }}=0\iff <{\widehat{\mathbf{x}}}^{{}^{\prime }},{\widehat{\mathbf{b}}}_0>=0\iff <\widehat{\mathbf{t}},{\widehat{\mathbf{b}}}_0>=0\iff {\widehat{\mathbf{b}}}_0={\widehat{a}}_1\widehat{\mathbf{x}}+{\widehat{a}}_2\widehat{\mathbf{g}};$$

for some dual numbers ${\widehat{a}}_1,{\widehat{a}}_2\in \mathbb{D},$ and $-{\widehat{a}}_1^2+{\widehat{a}}_2^2=-1$.

2-h will be stationary in the second approximation iff ${\widehat{\mathbf{b}}}_0$ is ${\widehat{\mathbf{b}}}_2$ evolute of $\widehat{\mathbf{x}}\left(\widehat{s}\right)\in {\mathbb{H}}_{+f}^2$; that is,

$$h^{{}^{\prime }}=h^{″}=0\iff {\widehat{\mathbf{b}}}_0=\pm \widehat{\mathbf{b}},\mathit{and}\left|\widehat{\gamma }\right|>1.$$

3-h will be stationary in the third approximation iff ${\widehat{\mathbf{b}}}_0$ is ${\widehat{\mathbf{b}}}_3$ evolute of $\widehat{\mathbf{x}}\left(\widehat{s}\right)\in {\mathbb{H}}_{+f}^2$; that is,

$$h^{{}^{\prime }}=h^{″}=h^{‴}=0\iff {\widehat{\mathbf{b}}}_0=\pm \widehat{\mathbf{b}},\left|\widehat{\gamma }\right|>1\mathit{and}{\widehat{\gamma }}^{{}^{\prime }}\ne 0.$$

4-h will be stationary in the fourth approximation iff ${\widehat{\mathbf{b}}}_0$ is ${\widehat{\mathbf{b}}}_4$ evolute of $\widehat{\mathbf{x}}\left(\widehat{s}\right)\in {\mathbb{H}}_{+f}^2$; that is,

$$h^{{}^{\prime }}=h^{″}=h^{‴}=h^{{}^{{}^{\left(iv\right)}}}=0\iff {\widehat{\mathbf{b}}}_0=\pm \widehat{\mathbf{b}},\left|\widehat{\gamma }\right|>1,{\widehat{\gamma }}^{{}^{\prime }}=0\mathit{and}{\widehat{\gamma }}^{″}\ne 0.$$

Proof. 

For the first derivative of h, we obtain:

$$h^{{}^{\prime }}=<{\widehat{\mathbf{x}}}^{{}^{\prime }},{\widehat{\mathbf{b}}}_0>.$$

(22)

So, we obtain:

$$h^{{}^{\prime }}=0\iff <\widehat{\mathbf{t}},{\widehat{\mathbf{b}}}_0>=0\iff {\widehat{\mathbf{b}}}_0={\widehat{a}}_1\widehat{\mathbf{x}}+{\widehat{a}}_2\widehat{\mathbf{g}};$$

for some dual numbers ${\widehat{a}}_1,{\widehat{a}}_2\in \mathbb{D}$ and $-{\widehat{a}}_1^2+{\widehat{a}}_2^2=-1$, the result is clear.

2-Derivation of Equation (22) leads to:

$$h^{{}^{″}}=<{\widehat{\mathbf{x}}}^{{}^{{}^{″}}},{\widehat{\mathbf{b}}}_0>=<\widehat{\mathbf{x}}+\widehat{\gamma }\widehat{\mathbf{g}},{\widehat{\mathbf{b}}}_0>.$$

(23)

Then,

$$h^{{}^{\prime }}=h^{{}^{″}}=0\iff <{\widehat{\mathbf{x}}}^{{}^{\prime }},{\widehat{\mathbf{b}}}_0>=<{\widehat{\mathbf{x}}}^{{}^{{}^{″}}},{\widehat{\mathbf{b}}}_0>=0\iff {\widehat{\mathbf{b}}}_0=\pm \frac{{\widehat{\mathbf{x}}}^{{}^{\prime }}\times {\widehat{\mathbf{x}}}^{{}^{{}^{″}}}}{∥{\widehat{\mathbf{x}}}^{{}^{\prime }}\times {\widehat{\mathbf{x}}}^{{}^{{}^{″}}}∥}=\pm \widehat{\mathbf{b}}.$$

(24)

3-Derivation of Equation (23) leads to:

$$h^{{}^{{}^{‴}}}=<{\widehat{\mathbf{x}}}^{{}^{‴}},{\widehat{\mathbf{b}}}_{\mathbf{0}}>=\left(1-{\widehat{\gamma }}^2\right)<\widehat{\mathbf{t}},{\widehat{\mathbf{b}}}_0>+{\widehat{\gamma }}^{{}^{\prime }}<\widehat{\mathbf{g}},{\widehat{\mathbf{b}}}_0>$$

Hence, we have:

$$h^{{}^{\prime }}=h^{{}^{″}}=h^{{}^{{}^{‴}}}=0\iff {\widehat{\mathbf{b}}}_0=\pm \widehat{\mathbf{b}},\left|\widehat{\gamma }\right|>1\mathrm{and}{\widehat{\gamma }}^{{}^{\prime }}\ne 0.$$

4-By similar arguments, we can also obtain:

$$h^{{}^{\prime }}=h^{{}^{″}}=h^{{}^{{}^{‴}}}=h^{{}^{{}^{\left(iv\right)}}}=0\iff {\widehat{\mathbf{b}}}_0=\pm \widehat{\mathbf{b}},\left|\widehat{\gamma }\right|>1,{\widehat{\gamma }}^{{}^{\prime }}=0\mathrm{and}{\widehat{\gamma }}^{{}^{″}}\ne 0.$$

The proof is completed. □

According to the above proposition, we obtain the following:

(a)

The osculating circle $\mathbb{S}(\widehat{\rho },{\widehat{\mathbf{b}}}_0)$ of $\widehat{\mathbf{x}}\left(\widehat{s}\right)$ in ${\mathbb{H}}_{+f}^2$ is specified by the equations

$$<{\widehat{\mathbf{b}}}_0,\mathbf{X}>=\widehat{\rho }\left(\widehat{s}\right),<{\widehat{\mathbf{x}}}^{\prime },{\widehat{\mathbf{b}}}_0>=0,<{\widehat{\mathbf{x}}}^{{}^{{}^{″}}},{\widehat{\mathbf{b}}}_0>=0,$$

which are gained from the condition that the osculating circle must be at least 3rd order at $\widehat{\mathbf{x}}\left({\widehat{s}}_0\right)$ iff ${\widehat{\gamma }}^{{}^{\prime }}\ne 0$.

(b)

The osculating circle $\mathbb{S}(\widehat{\rho },{\widehat{\mathbf{b}}}_0)$ and the curve $\widehat{\mathbf{x}}\left(\widehat{s}\right)$ in ${\mathbb{H}}_{+f}^2$ are at least 4th order at $\widehat{\mathbf{x}}\left(s_0\right)$ iff ${\widehat{\gamma }}^{{}^{\prime }}=0$ and ${\widehat{\gamma }}^{{}^{″}}\ne 0$.

In this way, by taking into consideration the evolutes of $\widehat{\mathbf{x}}\left(\widehat{s}\right)$ in ${\mathbb{H}}_{+f}^2$, we can gain a sequence of evolutes ${\widehat{\mathbf{b}}}_2$, ${\widehat{\mathbf{b}}}_3$, …,${\widehat{\mathbf{b}}}_n$. The ownerships and the connection through these evolutes and their involutes are very interesting problems. For example, it is simple to see that when ${\widehat{\mathbf{b}}}_0=\pm \widehat{\mathbf{b}}$ and ${\widehat{\gamma }}^{{}^{\prime }}=0$, $\widehat{\mathbf{x}}\left(\widehat{s}\right)$ exist, $\widehat{\psi }$ is dual constant relative to ${\widehat{\mathbf{b}}}_0$. In this case, the timelike Disteli-axis is stationary up to second order and the line $\widehat{\mathbf{x}}$ moves over it with stationary pitch. Thus, the timelike ruled surface ($\widehat{x}$) with stationary timelike Disteli-axis is created by timelike line $\widehat{\mathbf{x}}$ located at a Lorentzian stationary distance ${\psi }^{*}$ and Lorentzian stationary angle $\psi$ with respect to the timelike Disteli-axis $\widehat{\mathbf{b}}$.

Theorem 1. 

A non-developable timelike ruled surface $\left(\widehat{x}\right)$ is a stationary timelike Disteli-axis if and only if $\gamma \left(s\right)$ = constant, and $\mathrm{\Gamma }\left(s\right)-\gamma \left(s\right)\mu \left(s\right)$ = constant.

We will now construct a timelike ruled surface with stationary timelike Disteli-axis. From Equation (20), we have ODE ${\widehat{\mathbf{t}}}^{{}^{{}^{{}^{″}}}}+{\widehat{\kappa }}^2\widehat{\mathbf{t}}=\mathbf{0}$. The general solution of this equation is:

$$\widehat{\mathbf{x}}\left(\widehat{\phi }\right)=\left(cosh\widehat{\psi },sinh\widehat{\psi }cos\widehat{\phi },sinh\widehat{\psi }sin\widehat{\phi }\right),$$

(25)

where $\widehat{\phi }:=\phi +\epsilon {\phi }^{*}=\widehat{\kappa }\widehat{s}$. Therefore, we have:

$$\left(\begin{array}{c}\widehat{\mathbf{x}}\hfill \\ \widehat{\mathbf{t}}\hfill \\ \widehat{\mathbf{g}}\hfill \end{array}\right)=\left(\begin{array}{ccc}cosh\widehat{\psi }\hfill & sinh\widehat{\psi }cos\widehat{\phi }\hfill & sinh\widehat{\psi }sin\widehat{\phi }\hfill \\ 0\hfill & -sin\widehat{\phi }\hfill & cos\widehat{\phi }\hfill \\ sinh\widehat{\psi }\hfill & cosh\widehat{\psi }cos\widehat{\phi }\hfill & cosh\widehat{\psi }sin\widehat{\phi }\hfill \end{array}\right)\left(\begin{array}{c}{\widehat{\mathbf{f}}}_1\hfill \\ {\widehat{\mathbf{f}}}_2\hfill \\ {\widehat{\mathbf{f}}}_3\hfill \end{array}\right).$$

(26)

From Equations (11) and (26), we have:

$$\widehat{\mathbf{b}}=cosh\widehat{\psi }\widehat{\mathbf{x}}-sinh\widehat{\psi }\widehat{\mathbf{g}}={\widehat{\mathbf{f}}}_1.$$

(27)

This indicates that the timelike Disteli-axis $\widehat{\mathbf{b}}$ is coincident with ${\widehat{\mathbf{f}}}_1$. From the real and the dual parts of $\widehat{\mathbf{x}}$ in Equation (25), we obtain

$$\left.\begin{array}{c}{x}_1=cosh\psi ,x_1^{*}={\psi }^{*}sinh\psi ,\hfill \\ x_2=sinh\psi cos\phi ,x_2^{*}={\psi }^{*}cosh\psi cos\phi -{\phi }^{*}sinh\psi sin\phi ,\hfill \\ x_3=sinh\psi sin\phi ,x_3^{*}={\psi }^{*}cosh\psi sin\phi +{\phi }^{*}sinh\psi cos\phi .\hfill \end{array}\right\}$$

(28)

Let y$(y_1,y_2,y_3)$ be a point on $\widehat{\mathbf{x}}$. Since ${\mathbf{x}}^{*}=$y$\times \mathbf{x}$, we have that the system of linear equations in $y_i$(i = 1, 2, 3 and $y_{is}$ are the coordinates ofy):

$$\left.\begin{array}{c}\hfill c-y_{2}sinh\psi sin\phi +y_{3}sinh\psi cos\phi =x_1^{*},\hfill \\ \hfill -y_{1}sinh\psi sin\phi +y_{3}cosh\psi =x_2^{*},\hfill \\ \hfill y_{1}sinh\psi cos\phi -y_{2}cosh\psi =x_3^{*}.\hfill \end{array}\right\}$$

(29)

The matrix of coefficients of unknowns $y_i$ is the skew symmetric matrix

$$\left(\begin{array}{ccc}0\hfill & -sinh\psi sin\phi \hfill & sinh\psi cos\phi \hfill \\ -sinh\psi sin\phi \hfill & 0\hfill & cosh\psi \hfill \\ sinh\psi cos\phi \hfill & -cosh\psi \hfill & 0\hfill \end{array}\right),$$

and thus its rank is 2 with $\psi \ne 2\pi k$ (k is an integer). The rank of the augmented matrix

$$\left(\begin{array}{cccc}0\hfill & -sinh\psi sin\phi \hfill & sinh\psi cos\phi \hfill & x_1^{*}\\ -sinh\psi sin\phi \hfill & 0\hfill & cosh\psi \hfill & x_2^{*}\\ sinh\psi cos\phi \hfill & -cosh\psi \hfill & 0\hfill & x_2^{*}\end{array}\right),$$

is also 2. Then, this system has infinite solutions given by

$$\begin{array}{c}\hfill c-y_{2}sin\phi +y_{3}cos\phi ={\psi }^{*}\hfill \\ \hfill y_2=(y_1-{\phi }^{*})tanh\psi cos\phi -{\psi }^{*}sin\phi ,\hfill \\ \hfill y_3=(y_1-{\phi }^{*})tanh\psi sin\phi +{\psi }^{*}cos\phi .\hfill \end{array}$$

(30)

Since $y_1$ can be arbitrary, we may take $y_1-{\phi }^{*}=0$. In this case, Equation (30) becomes

$$y_1={\phi }^{*},y_2=-{\psi }^{*}sin\phi ,y_3={\psi }^{*}cos\phi .$$

(31)

If we set ${\phi }^{*}=h\phi$ and φ as the movement parameter, then $\left(\widehat{x}\right)$ is timelike ruled in ${\mathbb{L}}_f$-space. We now simply find the base curve as

$$\mathbf{y}\left(\phi \right)=\left(h\phi ,-{\psi }^{*}sin\phi ,{\psi }^{*}cos\phi \right).$$

(32)

It can be shown that $<{\mathbf{y}}^{{}^{\prime }},{\mathbf{x}}^{{}^{\prime }}>=0$; (${}^{\prime }=\frac{d}{d\phi }$) so the base curve $\mathbf{y}\left(\phi \right)$ of ($\widehat{x}$) is its striction curve. The curvature $\kappa \left(\phi \right)$ and torsion $\tau \left(\phi \right)$ of $\mathbf{y}\left(\phi \right)$ can be given by

$$\kappa \left(\phi \right)=\frac{{\psi }^{*}}{{\psi }^{*2}-h^2},\tau \left(\phi \right)=\frac{h}{{\psi }^{*2}-h^2},$$

which means that $\mathbf{y}\left(\phi \right)$ is a spacelike ($\left|{\psi }^{*}\right|>\left|h\right|$) or timelike ($\left|{\psi }^{*}\right|<\left|h\right|$) circular helix. From Equations (26) and (32), it can be found that

$$\left(\begin{array}{c}-<{\mathbf{y}}^{{}^{\prime }},\mathbf{x}>\\ -<{\mathbf{y}}^{{}^{\prime }},\mathbf{g}>\end{array}\right)=\left(\begin{array}{cc}cosh\psi & sinh\psi \\ sinh\psi & cosh\psi \end{array}\right)\left(\begin{array}{c}h\\ {\psi }^{*}\end{array}\right).$$

(33)

Hence, the major geometrical characteristics of $\left(\widehat{x}\right)$ can be described as follows: $\left(\widehat{x}\right)$ is a stationary timelike Disteli-axis ruled surface, $<{\mathbf{y}}^{{}^{\prime }},\mathbf{x}>$ is constant, $<{\mathbf{y}}^{{}^{\prime }},\mathbf{g}>$ is constant and $\mathbf{y}\left(\phi \right)$ is a spacelike ($\left|{\psi }^{*}\right|>\left|h\right|$) or timelike ($\left|{\psi }^{*}\right|<\left|h\right|$) circular helix.

Theorem 2. 

Let $\left(\widehat{x}\right)$ be any non-developable ruled surface in Minkowski 3-space ${\mathbb{E}}_1^3$. Then, $\left(\widehat{x}\right)$ is a timelike ruled Weingarten surface if and only if $\left(\widehat{x}\right)$ is a stationary timelike Disteli-axis ruled.

On the the other hand, let $\mathbf{p}(x,y,z)$ be a point on the timelike oriented line $\widehat{\mathbf{x}}$. Then, from Equations (28) and (32), we gain

$$\left(\widehat{x}\right):\mathbf{p}(\phi ,v)=\left(\begin{array}{c}h\phi +vcosh\psi \\ -{\psi }^{*}sin\phi +vsinh\psi cos\phi \\ {\psi }^{*}cos\phi +vsinh\psi sin\phi \end{array}\right),v\in \mathbb{R}.$$

(34)

The constants h, ψ and ${\psi }^{*}$ can control the shape of the surface $\left(\widehat{x}\right)$. The timelike ruled surface $\left(\widehat{x}\right)$ can be classified into four types according to their striction curves:

(1) 

Timelike helicoidal surface with its striction curve is a spacelike cylindrical helix: for $h=2$, ${\psi }^{*}=0.5$, $\psi =2.1$, $0\le v\le 5$ and $0\le \phi \le 2\pi$ (Figure 4).

(2) 

Lorentzian sphere with its striction curve is a spacelike circle: for $h=0,$${\psi }^{*}=2$, $\psi =2.1$, $-3\le v\le 3$ and $0\le \phi \le 2\pi$ (Figure 5).

(3) 

Timelike Archimedes with its striction curve is a timelike line: for $h=2$, ${\psi }^{*}=0$, $\psi =2.1$, $-3\le v\le 3$ and $0\le \phi \le 2\pi$ (Figure 6).

(4) 

Timelike circular cone with its striction curve is a fixed point: for $h={\psi }^{*}=0$, $\psi =0.5$, $-3\le v\le 3$ and $0\le \phi \le 2\pi$ (Figure 7).

4. Conclusions

In this work, we utilized E. Study’s map as a direct procedure for inspecting the kinematic geometry of a timelike ruled surface with stationary timelike Disteli-axis by the similarity with hyperbolic dual spherical kinematics. This provides the ability to have a set of curvature functions that locate the local shape of timelike ruled surface. Hence, the hyperbolic version of the well known equation of the Plücker conoid has been presented and its properties are explained in details. In addition, a characterization for a timelike line to be a stationary timelike Disteli-axis is inspected and outlined. Our results in this paper can contribute to the field of spatial kinematics and have practical applications in mechanical mathematics and engineering. In future work, we plan to proceed to study some applications of kinematic geometry of one parameter hyperbolic spatial movement combine with skew timelike axes such that at any instant the contact points are located on a timelike line and so forth, offered in [19,20,21,22]. Moreover, we believe this work can be used to study some applications of kinematic geometry of one parameter hyperbolic spatial movement combine with singularity theory and submanifold theory; visualizing data can help especially in the field of relativity, mathematical physics, etc. More new results and properties can be found at [23,24,25].