Timelike Constant Axis Ruled Surface Family in Minkowski 3-Space

Abstract

A timelike ($\mathcal{TL}$) constant axis ruled surface in ${\mathcal{E}}_1^3$ (Minkowski 3-space), as determined by its ruling, forms a constant dual angle with its Disteli-axis (striction axis or curvature axis). In this article, we employ the symmetry through point geometry of Lorentzian dual curves and the line geometry of $\mathcal{TL}$ ruled surfaces. This produces the capability to expound a set of curvature functions that specify the local configurations of $\mathcal{TL}$ ruled surfaces. Then, we gain some new constant axis ruled surfaces in Lorentzian line space and their geometrical illustrations. Further, we also earn several organizations among a $\mathcal{TL}$ constant axis ruled surface and its striction curve.

1. Introduction

In the context of kinematics, the tracks of a directed line linked with a mobile rigid body is normally a ruled surface ($\mathcal{RS}$). The geometry of $\mathcal{RS}$ has been exceedingly utilized in computer-aided manufacturing ($\mathcal{CAM}$), computer-aided geometric design ($\mathcal{CAGD}$), and geometric modeling [1,2,3,4,5]. Nowadays, the assets of ruled surfaces and their applications have been researched in Euclidean and non-Euclidean spaces (see, e.g., [6,7,8,9,10,11,12,13]). One of the popularization comfortable manners to addressing the movement of line space appears to discover an attachment with this space and dual numbers. From the E. Study map in dual number ($\mathcal{DN}$) algebra, the family of all directed lines in Euclidean 3-space ${\mathcal{E}}^3$ is immediately related to the family of points on the dual unit sphere ($\mathcal{DUS}$) in the dual 3-space ${\mathcal{D}}^3$ [1,2,3]. Supplementary allocations on the pivotal crucial annotations of the E. Study map and one-parameter dual spherical locomotion were initiated in [4,5,6,7,8].

In Minkowski 3-space, ${\mathcal{E}}_1^3$, the addressing of $\mathcal{RS}$ is more distant and engaging than the Euclidean case as the Lorentzian metric function can be non-positive, positive, or zero, whereas the Euclidean metric function can only be positive-definite. Then, if we have ${\mathcal{E}}_1^3$ as an alternate of ${\mathcal{E}}^3$, the E. Study map can be given as follows: The family of all timelike ($\mathcal{TL}$) (spacelike $(\mathcal{SL}$)) directed lines in ${\mathcal{E}}_1^3$ is immediately related to the family of points on the hyperbolic ${\mathcal{H}}_{+}^2$ (Lorentzian ${\mathcal{S}}_1^2$) $\mathcal{DUS}$ in the Lorentzian dual 3-space ${\mathcal{D}}_1^3$. It proceeds that an $\mathcal{SL}$ curve on ${\mathcal{H}}_{+}^2$ matches a $\mathcal{TLRS}$ at ${\mathcal{E}}_1^3$. Similarly, an $\mathcal{SL}$ ($\mathcal{TL}$) curve on ${\mathcal{S}}_1^2$ matches $\mathcal{TL}$ ($\mathcal{SL}$) $\mathcal{RS}$ at ${\mathcal{E}}_1^3$. Due to their dealings with engineering and physical sciences in Minkowski space, senior geometers and engineers have researched and acquired extensive ownership of ruled surfaces (see [6,7,8,9,10,11,12,13]).

However, to our knowledge, there is no work on the construction of a $\mathcal{TL}$ constant axis $\mathcal{RS}$ family in Minkowski 3-space ${\mathcal{E}}_1^3$. This work intends to pinpoint a set of Lorentzian invariants that elucidate the local shape of a $\mathcal{TL}$ constant axis ruled surface family. As a result, the Hamilton and Mannheim formulae of surfaces theories are accomplished in Lorentzian line-space and their geometrical clarifications are evaluated. Further, we also elucidate the kinematic geometry of a $\mathcal{TL}$ cylindroid created by the $\mathcal{TL}$ Disteli-axis. Subsequently, we explore some situations that command private $\mathcal{TL}$ constant axis ruled surface families, such as the Archimedes helicoid and developable surfaces.

2. Basic Concepts

In this section, we allocate a concise synopsis of the $\mathcal{DN}$ theory, and Lorentzian vectors [11,12,13]. An oriented (non-null) line in ${\mathcal{E}}_1^3$ may be offered by a point $\mathbf{p}\in \mathcal{L}$ and a normalized vector $\mathbf{v}$ of $\mathcal{L}$, that is, ${∥\mathbf{v}∥}^2=\pm 1$. To earn ingredients for $\mathcal{L}$, one makes the moment vector ${\mathbf{v}}^{*}=\mathbf{p}\times \mathbf{v}$ in regard to the origin point in ${\mathcal{E}}_1^3$. If $\mathbf{p}$ is exchanged by a point $\mathbf{q}=\mathbf{p}+t\mathbf{v}$,$t\in \mathbb{R}$ on $\mathcal{L}$, this leads to ${\mathit{v}}^{*}$ being independent of $\mathbf{p}$ on $\mathcal{L}$. The two vectors $\mathbf{v}$ and ${\mathbf{v}}^{*}$ are not independent of one another; they fulfill the following mutual relations:

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

(1)

The six components $v_i,v_i^{*}(i=1,2,3)$ of $\mathbf{v}$ and ${\mathbf{v}}^{*}$ are named the normalized Plücker coordinates of the line $\mathcal{L}$. Hence, the two non-null vectors $\mathbf{v}$ and ${\mathbf{v}}^{*}$ regulate the oriented (non-null) $\mathcal{L}.$

If $v,$and $v^{*}$ are real numbers, the couple $\widehat{v}=v+\epsilon v^{*}$ is named a $\mathcal{DN}$, such that $\epsilon \ne 0,$and ${\epsilon }^2=0$. This is in actuality very close to the notion of a complex number. Then, the set

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

associated with the Lorentzian metric

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

forms the so-named dual Lorentzian 3-space ${\mathcal{D}}_1^3$. Thus, a point $\widehat{\mathbf{v}}={({\widehat{v}}_1,{\widehat{v}}_2,{\widehat{v}}_3)}^t$ has dual coordinates ${\widehat{v}}_i=(v_i+\epsilon v_i^{*})\in \mathcal{D}$. If $\widehat{\mathbf{v}}=\mathbf{v}+\epsilon {\mathbf{v}}^{*}$ is a non-null dual vector, the norm $∥\widehat{\mathbf{v}}∥$ of $\widehat{\mathbf{v}}$ is elucidated by

$$\begin{array}{ccc}\hfill ∥\widehat{\mathbf{v}}∥& =& \sqrt{\left|<\widehat{\mathbf{v}},\widehat{\mathbf{v}}>\right|}=\sqrt{\left|<\mathbf{v},\mathbf{v}>\right|}+\epsilon \frac{1}{2\sqrt{\left|<\mathbf{v},\mathbf{v}>\right|}}\frac{<\mathbf{v},\mathbf{v}>}{\left|<\mathbf{v},\mathbf{v}>\right|}2<\mathbf{v},{\mathbf{v}}^{*}>\hfill \\ & =& ∥\mathbf{v}∥+\epsilon \frac{1}{∥\mathbf{v}∥}\frac{<\mathbf{v},\mathbf{v}>}{\left|<\mathbf{v},\mathbf{v}>\right|}<\mathbf{v},{\mathbf{v}}^{*}>.\hfill \end{array}$$

If $\mathbf{v}$ is an $\mathcal{SL}$ vector, we possess

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

If $\mathbf{v}$ is a $\mathcal{TL}$ vector, we possess

$$∥\widehat{\mathbf{v}}∥=∥\mathbf{v}∥-\epsilon \frac{1}{∥\mathbf{v}∥}<\mathbf{v},{\mathbf{v}}^{*}>=∥\mathbf{v}∥\left(1-\epsilon \frac{1}{{∥\mathbf{v}∥}^2}<\mathbf{v},{\mathbf{v}}^{*}>\right).$$

Therefore, $\widehat{\mathbf{v}}$ is an $\mathcal{SL}$ ($\mathcal{TL}$) dual unit vector ($\mathcal{DUV}$) if ${∥\widehat{\mathbf{v}}∥}^2=1$ (${∥\mathbf{v}∥}^2=-1$). For any two vectors, $\widehat{\mathbf{v}}=\left({\widehat{v}}_1,{\widehat{v}}_2,{\widehat{v}}_3\right)$ and $\widehat{\mathbf{w}}=({\widehat{w}}_1,{\widehat{w}}_2,{\widehat{w}}_3)$ of ${\mathcal{D}}_1^3$, the vector product is

$$\widehat{\mathbf{v}}\times \widehat{\mathbf{w}}=\left|\begin{array}{ccc}-{\widehat{\mathbf{r}}}_1& {\widehat{\mathbf{r}}}_2& {\widehat{\mathbf{r}}}_3\\ {\widehat{v}}_1& {\widehat{v}}_2& {\widehat{v}}_3\\ w_1& w_2& w_3\end{array}\right|,$$

where ${\widehat{\mathbf{r}}}_1$, ${\widehat{\mathbf{r}}}_2$, ${\widehat{\mathbf{r}}}_3$ is the canonical basis of ${\mathcal{D}}_1^3$. The hyperbolic and Lorentzian $\mathcal{DUS}$, respectively, are

$${\mathcal{H}}_{+}^2=\{\widehat{\mathbf{v}}\in {\mathcal{D}}_1^3\mid {∥\mathbf{v}∥}^2:=-{\widehat{v}}_1^2+{\widehat{v}}_2^2+{\widehat{v}}_3^2=-1\},$$

and

$${\mathcal{S}}_1^2=\{\widehat{\mathbf{v}}\in {\mathcal{D}}_1^3\mid {∥\mathbf{v}∥}^2:=-{\widehat{v}}_1^2+{\widehat{v}}_2^2+{\widehat{v}}_3^2=1\}.$$

It is evident that

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

(2)

It follows that Equations (1) and (2) are congruous. Due to this we include the following map (E. Study map). The ring-shaped hyperboloid is in bijection with the family of $\mathcal{SL}$ lines, the common asymptotic cone is in bijection with the family of null lines, and the oval-shaped hyperboloid is in bijection with the family of $\mathcal{TL}$ lines (see Figure 1).

Figure 1. The hyperbolic and Lorentzian $\mathcal{DU}$ spheres.

In the E. Study map, a differentiable curve $v\in \mathbb{R}\mapsto \widehat{\mathbf{z}}\left(v\right)\in {\mathcal{H}}_{+}^2$ defines a $\mathcal{TLRS}$ in ${\mathcal{E}}_1^3$. Comparably, a regular curve $v\in \mathbb{R}\mapsto \widehat{\mathbf{z}}\left(v\right)\in {\mathcal{S}}_1^2$ defines a $\mathcal{SL}$ or $\mathcal{TLRS}$ in ${\mathcal{E}}_1^3$. $\widehat{\mathbf{z}}\left(v\right)$ is identified with the rulings of the surface.

3. Main Results

In this section, we address the $\mathcal{TLRS}$ by the E. Study map. Therefore, a $\mathcal{SL}$ differentiable curve

$$v\in \mathbb{R}\mapsto \widehat{\mathbf{z}}\left(v\right)\in {\mathcal{H}}_{+}^2,v\in \mathbb{R},$$

demonstrate a $\mathcal{TLRS}$ (($\widehat{z}$) for short) in Minkowski 3-space ${\mathcal{E}}_1^3$. From spherical kinematics, the Blaschke frame ($\mathcal{BF}$) is

$$\widehat{\mathbf{z}}={\widehat{\mathbf{z}}}_1\left(v\right),{\widehat{\mathbf{z}}}_2\left(v\right)=\frac{d\widehat{\mathbf{z}}}{dv}{∥\frac{d\widehat{\mathbf{z}}}{dv}∥}^{-1},{\widehat{\mathbf{z}}}_2\left(v\right)={\widehat{\mathbf{z}}}_1\times {\widehat{\mathbf{z}}}_2,$$

(3)

where

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

(4)

The lines of the $\mathcal{BF}$ intersect at the striction curve ($\mathcal{SC}$) $\mathbf{c}\left(v\right)$, which is the adjacent point through two rulings $\widehat{\mathbf{z}}\left(v\right)$ and $\widehat{\mathbf{z}}(v+dv)$. ${\widehat{\mathbf{z}}}_2$ is the central normal to ($\widehat{z}$) at the striction point. In the context of spherical kinematics, the locomotion of the $\mathcal{BF}$ is a rotation around the Darboux vector $\widehat{\varpi }$, that is,

$$\frac{d}{dv}\left[\begin{array}{c}{\widehat{\mathbf{z}}}_1\hfill \\ {\widehat{\mathbf{z}}}_2\hfill \\ {\widehat{\mathbf{z}}}_3\hfill \end{array}\right]=\left[\begin{array}{ccc}0\hfill & \widehat{a}\hfill & 0\hfill \\ \widehat{a}\hfill & 0\hfill & \widehat{b}\hfill \\ 0\hfill & -\widehat{b}\hfill & 0\hfill \end{array}\right]\left[\begin{array}{c}{\widehat{\mathbf{z}}}_1\hfill \\ {\widehat{\mathbf{z}}}_2\hfill \\ {\widehat{\mathbf{z}}}_3\hfill \end{array}\right]=\widehat{\varpi }\times \left[\begin{array}{c}{\widehat{\mathbf{z}}}_1\hfill \\ {\widehat{\mathbf{z}}}_2\hfill \\ {\widehat{\mathbf{z}}}_3\hfill \end{array}\right],$$

(5)

where $\widehat{\varpi }\left(v\right)=\widehat{b}\left(v\right){\widehat{\mathbf{z}}}_1\left(v\right)+\widehat{a}\left(v\right){\widehat{\mathbf{z}}}_3\left(v\right)$, and

$$\left.\begin{array}{c}\widehat{a}\left(v\right)=a\left(v\right)+\epsilon a^{*}\left(v\right)=∥\frac{d\widehat{\mathbf{z}}}{dv}∥,\hfill \\ \widehat{b}\left(v\right)=b\left(v\right)+\epsilon b^{*}\left(v\right)=det(\widehat{\mathbf{z}},\frac{d\widehat{\mathbf{z}}}{dv},\frac{d^2\widehat{\mathbf{z}}}{d{v}^2}),\hfill \end{array}\right\}$$

(6)

are the Blaschke invariants of $\widehat{\mathbf{z}}\left(v\right)\in {\mathcal{H}}_{+}^2$. The tangent of the $\mathcal{SC}$ is

$$\frac{d\mathbf{c}\left(v\right)}{dv}=-b^{*}\left(v\right){\mathbf{z}}_1\left(v\right)+a^{*}\left(v\right){\mathbf{z}}_3\left(v\right).$$

(7)

The Lorentzian functions of ($\widehat{z}$) are

$$ϵ\left(v\right)=\frac{b\left(v\right)}{a\left(v\right)},\mathrm{\Gamma }\left(v\right)=\frac{b^{*}\left(v\right)}{b\left(v\right)},\mathrm{and}\mu \left(v\right)=\frac{a^{*}\left(v\right)}{a\left(v\right)};b\left(v\right)\ne 0,a\left(v\right)\ne 0.$$

(8)

The expositions of $ϵ\left(v\right)$, $\mathrm{\Gamma }\left(v\right)$, and $\mu \left(v\right)$ are as follows:$ϵ$ is the geodesic curvature of the spherical curve ${\mathbf{z}}_1\left(v\right)$; $\mathrm{\Gamma }\left(v\right)$ is the angle through the tangent of the $\mathcal{SC}$ and the ruling of ($\widehat{z}$); and $\mu$ is its distribution parameter at the ruling. Thus, a $\mathcal{TLRS}$ can be attained as

$$\left(\widehat{z}\right):\mathbf{y}(v,r)=\stackrel{v}{\underset{0}{\int }}\left(-b^{*}\left(v\right){\mathbf{z}}_1\left(v\right)+a^{*}\left(v\right){\mathbf{z}}_3\left(v\right)\right)dv+r{\mathbf{z}}_1\left(v\right),r\in \mathbb{R}.$$

(9)

3.1. Hamilton and Mannheim Formulae

Through the locomotion of the $\mathcal{BF}$, all kinematic–geometric features can be deduced with the Darboux vector $\widehat{\varpi }$. Therefore, from the presumption that $\left|\widehat{b}\right|>\left|\widehat{a}\right|$, we designate the $\mathcal{TL}$ Disteli-axis ($\mathcal{DA}$) as

$$\widehat{\mathbf{e}}\left(v\right):=\mathbf{e}+\epsilon {\mathbf{e}}^{*}=\frac{\widehat{\varpi }}{∥\widehat{\varpi }∥}=\frac{\widehat{b}}{\sqrt{{\widehat{b}}^2-{\widehat{a}}^2}}{\widehat{\mathbf{z}}}_1+\frac{\widehat{a}}{\sqrt{{\widehat{b}}^2-{\widehat{a}}^2}}{\widehat{\mathbf{z}}}_3,$$

(10)

Then, Equation (5) can be formed as

$$\frac{d}{dv}\left[\begin{array}{c}{\widehat{\mathbf{z}}}_1\hfill \\ {\widehat{\mathbf{z}}}_2\hfill \\ {\widehat{\mathbf{z}}}_3\hfill \end{array}\right]=∥\widehat{\varpi }∥\widehat{\mathbf{e}}\times \left[\begin{array}{c}{\widehat{\mathbf{z}}}_1\hfill \\ {\widehat{\mathbf{z}}}_2\hfill \\ {\widehat{\mathbf{z}}}_3\hfill \end{array}\right].$$

(11)

Here, $∥\widehat{\varpi }∥:=\varpi \left(v\right)+\epsilon {\varpi }^{*}\left(v\right)$ is the angular speed of the $\mathcal{BF}$ movement;

$${\omega }^{*}\left(v\right)=\frac{b{b}^{*}-a{a}^{*}}{\sqrt{b^2-a^2}},\omega \left(v\right)=\sqrt{b^2-a^2}.$$

(12)

Therefore, the $\mathcal{TLDA}$ is the instantaneous screw-axis of the $\mathcal{BF}$.

Proposition 1. 

The pitch $p\left(v\right)$ of the $\mathcal{BF}$ movement around the $\mathcal{TLDA}$ is

$$p\left(v\right):=\frac{<\varpi ,{\varpi }^{*}>}{{∥\omega ∥}^2}=\frac{b{b}^{*}-a{a}^{*}}{b^2-a^2}.$$

(13)

In fact, we find the following:

(1)

$∥\widehat{\varpi }∥$ can be distinguished as $∥\widehat{\varpi }∥=\varpi \left(v\right)(1+\epsilon p\left(v\right))$.

(2)

If $\mathbf{y}(x,y,z)$ is a point on the $\mathcal{TLDA}$, then

$$\left(\widehat{e}\right):\mathbf{y}(v,r)=\mathbf{e}\left(v\right)\times {\mathbf{e}}^{*}\left(v\right)+r\mathbf{e}\left(v\right),r\in \mathbb{R}$$

(14)

is a non-developable $\mathcal{TLRS}$.

If the $\mathcal{BF}$- movement is a pure turnover, that is, $p\left(v\right)=0$, then

$$\mathbf{e}\left(v\right)=\mathbf{e}\left(v\right)+\epsilon {\mathbf{e}}^{*}\left(v\right)=\frac{1}{∥\varpi ∥}(\varpi +\epsilon {\varpi }^{*}),$$

(15)

whereas if $p\left(v\right)=0$ and ${∥\varpi \left(v\right)∥}^2=-1$, then $\widehat{\varpi }\left(v\right)$ is a $\mathcal{TL}$ line. However, if $\widehat{\varpi }\left(v\right)=\epsilon {\varpi }^{*}\left(v\right)$, that is, the $\mathcal{BF}$- movement is a pure translational, we let ${\varpi }^{*}\left(v\right)=∥{\varpi }^{*}\left(v\right)∥;$${\varpi }^{*}$e$\left(v\right)={\varpi }^{*}$ for a random ${\mathbf{e}}^{*}\left(v\right)$ with ${\varpi }^{*}\left(v\right)\ne 0$. In another form, $\mathbf{e}\left(v\right)$ can have random volition, too.

Let $\widehat{\omega }\left(v\right)=\omega \left(v\right)+\epsilon {\omega }^{*}\left(v\right)$ be the radii of the curvature between $\widehat{\mathbf{e}}$ and ${\widehat{\mathbf{z}}}_1$ (see Figure 2). Then,

$$\widehat{\mathbf{e}}\left(v\right)=cosh\widehat{\omega }{\widehat{\mathbf{z}}}_1+sinh\widehat{\omega }{\widehat{\mathbf{z}}}_3,$$

(16)

where

$$coth\widehat{\omega }=coth\omega +\epsilon {\omega }^{*}(1-{coth}^2\omega )=\frac{\widehat{b}}{\widehat{a}}.$$

(17)

From the Equations (7), (12), and (16), we attain

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

(18)

Equation (17) is a Lorentzian version of the Hamilton and Mannheim formulae (compared with [1,2,3]).

Figure 2. $\widehat{\mathbf{e}}\left(v\right)=cosh\widehat{\omega }{\widehat{\mathbf{z}}}_1+sinh\widehat{\omega }{\widehat{\mathbf{z}}}_3$.

3.2. Timelike Cylindroid

We now clarify and explore the Hamilton and Mannheim formulae. The surface clarified by ${\omega }^{*}$ represents a $\mathcal{TL}$ cylindroid; let ${\widehat{\mathbf{z}}}_2$ be selected along the $y—$axis of a fixed Lorentzian frame $\left(oxyz\right)$ and the station of $\widehat{\mathbf{e}}$ be distinguished by angle $\omega$ and distance ${\omega }^{*}$ on the non-negative direction of the $y—$axis. The $\mathcal{DU}$ vectors ${\widehat{\mathbf{z}}}_1$ and ${\widehat{\mathbf{z}}}_3$ can be marked with the x- and z-axes, respectively (Figure 2). Therefore, in view of Equations (14), (16), and (18), we locate

$$\left(\widehat{e}\right):{\omega }^{*}:=-y=\frac{1}{2}\left(\mu -\mathrm{\Gamma }\right)sinh2\omega ,x=rcosh\omega ,\mathrm{and}z=rsinh\omega .$$

(19)

For $\mu -\mathrm{\Gamma }=1,-1.5\le \omega \le 1.5$, and $-2\le r\le 2$, the $\mathcal{TL}$ cylindroid is as shown in Figure 3. For the limits of $\left(\widehat{e}\right)$, the algebraic equation is

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

(20)

Note that it is a third-order polynomial at the coordinates $x,y$, and z. Also, we have

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

(21)

So, the two limits of $\left(\widehat{e}\right)$ are demonstrated by

$${\left(\mu -\mathrm{\Gamma }\right)}^2+4y^2=0\Rightarrow y=\pm i\left(\mu -\mathrm{\Gamma }\right)/2,\mathrm{with}i=\sqrt{-1},$$

(22)

which represents two isotropic torsal $\mathcal{SL}$ planes, each of them containing one isotropic $\mathcal{SL}$ torsal line $\mathcal{L}$.

Figure 3. $\mathcal{TL}$ cylindroid.

In Equation (18), $p\left(v\right)$ is not-periodic. This means that $\left(\widehat{e}\right)$ is not a closed $\mathcal{RS}$. Further, it mostly has two extreme values: the Lorentzian functions $\mu$ and $\mathrm{\Gamma }$. This shows that the $\mathcal{DU}$ vectors ${\widehat{\mathbf{z}}}_1$ and ${\widehat{\mathbf{z}}}_3$ are the principal axes of the $\mathcal{SL}$ cylindroid. Moreover, the geometric assets of the $\mathcal{TL}$ cylindroid are as follows:

A:

If $p\left(v\right)\ne 0$, that is, the $\mathcal{BF}$- movement is not a pure turnover, then there are two isotropic rulings crossing through the point $(0,y,0)$.

B:

If $p\left(v\right)=0$, that is, the $\mathcal{BF}$- movement is a pure turnover, then there are two isotropic lines ${\mathcal{L}}_1$, ${\mathcal{L}}_2$ detected by

$${\mathcal{L}}_1,{\mathcal{L}}_2:\frac{x}{z}=coth\beta =\pm \sqrt{\frac{\mu }{\mathrm{\Gamma }}},y=\pm i\left(\mu -\mathrm{\Gamma }\right)/2.$$

(23)

So, if $\mu$ and $\mathrm{\Gamma }$ are commensurate, then the $\mathcal{TL}$ cylindroid deteriorates to a family of $\mathcal{TL}$ lines through the origin “$\mathbf{o}"$ in the torsal plane $y=0$. For this reason, ${\mathcal{L}}_1$, ${\mathcal{L}}_2$ are the principal axes of an elliptic $\mathcal{TL}$ line congruence. However, if $\mu$ and $\mathrm{\Gamma }$ have reverse signs, then ${\mathcal{L}}_1$, ${\mathcal{L}}_2$ are isotropic and are coincident with the principal axes of a $\mathcal{TL}$ hyperbolic line congruence.

3.3. Timelike Constant Axis Ruled Surface Family

From now on, when we say ($\widehat{z}$) is a $\mathcal{TL}$ constant axis $\mathcal{RS}$, we signify that all the rulings of ($\widehat{z}$) have a constant $\mathcal{DA}$ with respect to the $\mathcal{TLDA}$.

Let $d\widehat{\phi }=d\phi +\epsilon d{\phi }^{*}$ be the dual arc length of $\widehat{\mathbf{z}}\left(v\right)\in {\mathcal{H}}_{+}^2$. Then,

$$\widehat{\phi }\left(v\right)=\stackrel{v}{\underset{0}{\int }}\widehat{a}dv=\stackrel{v}{\underset{0}{\int }}a(1+\epsilon \mu )dv.$$

(24)

From the Equations (4) and (23), we have

$$\left[\begin{array}{c}{\widehat{\mathbf{z}}}_1^{\prime }\hfill \\ {\widehat{\mathbf{z}}}_2^{\prime }\hfill \\ {\widehat{\mathbf{z}}}_3^{\prime }\hfill \end{array}\right]=\left[\begin{array}{ccc}0\hfill & 1\hfill & 0\hfill \\ 1\hfill & 0\hfill & \widehat{ϵ}\hfill \\ 0\hfill & -\widehat{ϵ}\hfill & 0\hfill \end{array}\right]\left[\begin{array}{c}{\widehat{\mathbf{z}}}_1\hfill \\ {\widehat{\mathbf{z}}}_2\hfill \\ {\widehat{\mathbf{z}}}_3\hfill \end{array}\right];{(}^{\prime }=\frac{d}{d\widehat{\phi }}),$$

(25)

where $\widehat{ϵ}\left(\phi \right)=ϵ+\epsilon \left(\mathrm{\Gamma }-ϵ\mu \right)$ is the geodesic curvature; $ϵ$, $\mathrm{\Gamma }$, and $\mu$ are the Lorentzian functions of $\left(\widehat{z}\right)$. The $\mathcal{SC}$ is

$$\mathbf{c}\left(\phi \right)=\stackrel{\phi }{\underset{0}{\int }}\left(-\mathrm{\Gamma }\left(\phi \right){\mathbf{z}}_1\left(\phi \right)+\mu \left(\phi \right){\mathbf{z}}_3\left(\phi \right)\right)d\phi .$$

(26)

Then,

$$\left(\widehat{z}\right):\mathbf{y}(\phi ,r)=\mathbf{c}\left(\phi \right)+r{\mathbf{z}}_1\left(\phi \right),r\in \mathbb{R},$$

(27)

is a $\mathcal{TLRS}$ in the Minkowski 3-space ${\mathcal{E}}_1^3$.

Actually, it is essential to possess the curvature $\widehat{\kappa }\left(\widehat{\phi }\right)$ and the torsion $\widehat{\tau }\left(\widehat{\phi }\right)$. Therefore, the Serret–Frenet frame ($\mathcal{SFF}$) of $\widehat{\mathbf{z}}\left(v\right)\in {\mathcal{H}}_{+}^2$ is specified by $\{\widehat{\mathbf{t}}\left(\widehat{\phi }\right),\widehat{\mathbf{p}}\left(\widehat{\phi }\right),\widehat{\mathbf{e}}\left(\widehat{\phi }\right)\}$. Then, the $\mathcal{SFF}$ is shown by a turnover of (${\widehat{\mathbf{z}}}_1$,${\widehat{\mathbf{z}}}_3$) as

$$\left[\begin{array}{c}\widehat{\mathbf{t}}\hfill \\ \widehat{\mathbf{p}}\hfill \\ \widehat{\mathbf{e}}\hfill \end{array}\right]=\left[\begin{array}{ccc}0\hfill & 1\hfill & 0\hfill \\ sinh\widehat{\omega }\hfill & 0\hfill & cosh\widehat{\omega }\hfill \\ cosh\widehat{\omega }\hfill & 0\hfill & sinh\widehat{\omega }\hfill \end{array}\right]\left[\begin{array}{c}{\widehat{\mathbf{z}}}_1\hfill \\ {\widehat{\mathbf{z}}}_2\hfill \\ {\widehat{\mathbf{z}}}_3\hfill \end{array}\right],$$

(28)

where

$$\left.\begin{array}{c}\widehat{\mathbf{t}}\times \widehat{\mathbf{p}}=-\widehat{\mathbf{e}},\widehat{\mathbf{e}}\times \widehat{\mathbf{t}}=\widehat{\mathbf{p}},\widehat{\mathbf{p}}\times \widehat{\mathbf{e}}=\widehat{\mathbf{t}},\hfill \\ <\widehat{\mathbf{t}},\widehat{\mathbf{t}}>=<\widehat{\mathbf{p}},\widehat{\mathbf{p}}>=-<\widehat{\mathbf{e}},\widehat{\mathbf{e}}>=1.\hfill \end{array}\right\}$$

(29)

Comparably, we possess

$$\left[\begin{array}{c}{\widehat{\mathbf{t}}}^{\prime }\hfill \\ {\widehat{\mathbf{p}}}^{\prime }\hfill \\ {\widehat{\mathbf{e}}}^{\prime }\hfill \end{array}\right]=\left[\begin{array}{ccc}0\hfill & \widehat{\kappa }\hfill & 0\hfill \\ -\widehat{\kappa }\hfill & 0\hfill & \widehat{\tau }\hfill \\ 0\hfill & \widehat{\tau }\hfill & 0\hfill \end{array}\right]\left[\begin{array}{c}\widehat{\mathbf{t}}\hfill \\ \widehat{\mathbf{p}}\hfill \\ \widehat{\mathbf{e}}\hfill \end{array}\right],$$

(30)

where

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

(31)

Here, $\widehat{\rho }\left(\widehat{\phi }\right)=\rho +\epsilon {\rho }^{*}$ and $\widehat{\sigma }\left(\widehat{\phi }\right)=\sigma +\epsilon {\sigma }^{*}$ are the radii of curvature and the radii of torsion of $\widehat{\mathbf{z}}\left(\widehat{\phi }\right)\in {\mathcal{H}}_{+}^2$, respectively.

Height Functions

In harmonization with [14], a point ${\widehat{\mathbf{e}}}_0\in {\mathcal{H}}_{+}^2$ will be entitled to a ${\widehat{\mathbf{e}}}_k$ evolute of the $\mathcal{DC}\widehat{\mathbf{z}}\left(\widehat{\phi }\right)\in {\mathcal{H}}_{+}^2$, for all $\widehat{\phi }$, such that $<{\widehat{\mathbf{e}}}_0,\widehat{\mathbf{z}}\left(\widehat{\phi }\right)>=0$ but $<{\widehat{\mathbf{e}}}_0,{\widehat{\mathbf{z}}}_1^{k+1}\left(\widehat{\phi }\right)>\ne 0$. Here, ${\widehat{\mathbf{z}}}^{k+1}$ signals the k-th derivatives of $\widehat{\mathbf{z}}\left(\widehat{\phi }\right)$ with respect to $\widehat{\phi }$. For the first evolute $\widehat{\mathbf{e}}$ of $\widehat{\mathbf{z}}\left(\widehat{\phi }\right)$, we have $<\widehat{\mathbf{e}},{\widehat{\mathbf{z}}}^{\prime }>=\pm <\widehat{\mathbf{e}},{\widehat{\mathbf{z}}}_2>=0$ and $<\widehat{\mathbf{e}},{\widehat{\mathbf{z}}}^{\prime \prime }>=\pm <\widehat{\mathbf{e}},-{\widehat{\mathbf{z}}}_1+\widehat{ϵ}{\widehat{\mathbf{z}}}_3>\ne 0$. So, $\widehat{\mathbf{e}}$ is at least an ${\widehat{\mathbf{e}}}_2$ evolute of $\widehat{\mathbf{z}}\left(\widehat{\phi }\right)\in {\mathcal{H}}_{+}^2$.

We are now heading a dual function $\widehat{\mathrm{\Omega }}:I\times {\mathcal{H}}_{+}^2\to \mathcal{D}$, by $\widehat{\mathrm{\Omega }}(\widehat{\phi },{\widehat{\mathbf{e}}}_0)=<{\widehat{\mathbf{e}}}_0,\widehat{\mathbf{z}}>$. We say that $\widehat{\mathrm{\Omega }}$ is a height function of $\widehat{\mathbf{z}}\left(\widehat{\phi }\right)\in {\mathcal{H}}_{+}^2$. We employ $\widehat{\mathrm{\Omega }}\left(\widehat{\phi }\right)=\widehat{\mathrm{\Omega }}(\widehat{\phi },{\widehat{\mathbf{e}}}_0)$ for any fixed point ${\widehat{\mathbf{e}}}_0\in {\mathcal{H}}_{+}^2$. Hence, we offer the following.

Proposition 2. 

Under the last suppositions, the below holds:

i. 

$\widehat{\mathrm{\Omega }}$ will be invariant in the first evaluation if ${\widehat{\mathbf{e}}}_0\in Sp\{{\widehat{\mathbf{z}}}_1$,${\widehat{\mathbf{z}}}_3\}$, that is,

$${\widehat{\mathrm{\Omega }}}^{\prime }=0\iff <\widehat{\mathbf{z}},{\widehat{\mathbf{e}}}_0>=0\iff <{\widehat{\mathbf{z}}}_2,{\widehat{\mathbf{e}}}_0>=0\iff {\widehat{\mathbf{e}}}_0={\widehat{a}}_1{\widehat{\mathbf{z}}}_1+{\widehat{a}}_3{\widehat{\mathbf{z}}}_3;$$

for some dual numbers ${\widehat{a}}_1,{\widehat{a}}_3\in \mathcal{D},$ and ${\widehat{a}}_1^2-{\widehat{a}}_3^2=1$.

ii. 

$\widehat{\mathrm{\Omega }}$ will be invariant in the second evaluation if ${\widehat{\mathbf{e}}}_0$ is an ${\widehat{\mathbf{e}}}_2$ evolute of ${\widehat{\mathbf{e}}}_0\in {\mathcal{H}}_{+}^2$, that is,

$${\widehat{\mathrm{\Omega }}}^{\prime }={\widehat{\mathrm{\Omega }}}^{\prime \prime }=0\iff {\widehat{\mathbf{e}}}_0=\pm \widehat{\mathbf{e}}.$$

iii. 

$\widehat{\mathrm{\Omega }}$ will be invariant in the third evaluation if ${\widehat{\mathbf{e}}}_0$ is an ${\widehat{\mathbf{e}}}_3$ evolute of ${\widehat{\mathbf{e}}}_0\in {\mathcal{H}}_{+}^2$, that is,

$${\widehat{\mathrm{\Omega }}}^{\prime }={\widehat{\mathrm{\Omega }}}^{\prime \prime }={\widehat{\mathrm{\Omega }}}^{\prime \prime \prime }=0\iff {\widehat{\mathbf{e}}}_0=\pm \widehat{\mathbf{e}},\mathrm{and}{\widehat{ϵ}}^{\prime }\ne 0.$$

iv. 

$\widehat{\mathrm{\Omega }}$ will be invariant in the fourth evaluation if ${\widehat{\mathbf{e}}}_0$ is an ${\widehat{\mathbf{e}}}_4$ evolute of ${\widehat{\mathbf{e}}}_0\in {\mathcal{H}}_{+}^2$, that is,

$${\widehat{\mathrm{\Omega }}}^{\prime }={\widehat{\mathrm{\Omega }}}^{\prime \prime }={\widehat{\mathrm{\Omega }}}^{\prime \prime \prime }={\widehat{\mathrm{\Omega }}}^{iv}=0\iff {\widehat{\mathbf{e}}}_0=\pm \widehat{\mathbf{e}},{\widehat{ϵ}}^{\prime }=0,\mathrm{and}{\widehat{ϵ}}^{\prime \prime }\ne 0.$$

Proof. 

For the first derivation of $\widehat{\mathrm{\Omega }}$ we obtain

$${\widehat{\mathrm{\Omega }}}^{\prime }=<{\widehat{\mathbf{z}}}_1^{\prime },{\widehat{\mathbf{b}}}_0>.$$

(32)

So, we obtain

$${\widehat{\mathrm{\Omega }}}^{\prime }=0\iff <{\widehat{\mathbf{z}}}_2,{\widehat{\mathbf{e}}}_0>=0\iff {\widehat{\mathbf{e}}}_0={\widehat{a}}_1{\widehat{\mathbf{z}}}_1+{\widehat{a}}_3{\widehat{\mathbf{z}}}_3;$$

(33)

for some dual numbers ${\widehat{a}}_1,$${\widehat{a}}_3\in \mathcal{D},$ and ${\widehat{a}}_1^2-{\widehat{a}}_3^2=1$, the outcome is evident.

2- The derivation of Equation (32) gives

$${\widehat{\mathrm{\Omega }}}^{\prime \prime }=<{\widehat{\mathbf{z}}}_1^{\prime \prime },{\widehat{\mathbf{e}}}_0>=<{\widehat{\mathbf{z}}}_1+\widehat{ϵ}{\widehat{\mathbf{z}}}_3,{\widehat{\mathbf{e}}}_0>.$$

(34)

By Equations (33) and (34), we have

$${\widehat{\mathrm{\Omega }}}^{\prime }={\widehat{\mathrm{\Omega }}}^{\prime \prime }=0\iff <{\widehat{\mathbf{z}}}_1^{\prime },{\widehat{\mathbf{e}}}_0>=<{\widehat{\mathbf{z}}}_1^{\prime \prime },{\widehat{\mathbf{e}}}_0>=0\iff {\widehat{\mathbf{e}}}_0=\pm \frac{{\widehat{\mathbf{z}}}_1^{\prime }\times {\widehat{\mathbf{z}}}_1^{\prime \prime }}{∥{\widehat{\mathbf{z}}}_1^{\prime }\times {\widehat{\mathbf{z}}}_1^{\prime \prime }∥}=\pm \widehat{\mathbf{e}}.$$

3- The derivation of Equation (34) gives

$${\widehat{\mathrm{\Omega }}}^{\prime \prime \prime }=<{\widehat{\mathbf{z}}}^{\prime \prime \prime },{\widehat{\mathbf{e}}}_{\mathbf{0}}>=\left(1-{\widehat{ϵ}}^2\right)<{\widehat{\mathbf{z}}}_2,{\widehat{\mathbf{e}}}_0>+{\widehat{ϵ}}^{\prime }<{\widehat{\mathbf{z}}}_3,{\widehat{\mathbf{e}}}_0>.$$

Then, we possess

$${\widehat{\mathrm{\Omega }}}^{\prime }={\widehat{\mathrm{\Omega }}}^{\prime \prime }={\widehat{\mathrm{\Omega }}}^{\prime \prime \prime }=0\iff {\widehat{\mathbf{e}}}_0=\pm \widehat{\mathbf{e}},\mathrm{and}{\widehat{ϵ}}^{\prime }\ne 0.$$

4- By the comparable arguments, we can also possess

$${\widehat{\mathrm{\Omega }}}^{\prime }={\widehat{\mathrm{\Omega }}}^{\prime \prime }={\widehat{\mathrm{\Omega }}}^{\prime \prime \prime }={\widehat{\mathrm{\Omega }}}^{\prime \prime \prime \prime }=0\iff {\widehat{\mathbf{e}}}_0=\pm \widehat{\mathbf{e}},{\widehat{ϵ}}^{\prime }=0,\mathrm{and}{\widehat{ϵ}}^{\prime \prime }\ne 0.$$

The proof is unmitigated. □

In view of Proposition 2, we possess the following:

(a) The osculating circle $\mathcal{S}(\widehat{\rho },{\widehat{\mathbf{e}}}_0)$ of $\widehat{\mathbf{z}}\left(\widehat{\phi }\right)\in {\mathcal{H}}_{+}^2$ is attained by

$$<{\widehat{\mathbf{e}}}_0,\widehat{\mathbf{z}}>=\widehat{\rho }\left(\widehat{\phi }\right),<{\widehat{\mathbf{z}}}^{\prime },{\widehat{\mathbf{e}}}_0>=0,<{\widehat{\mathbf{z}}}^{\prime \prime },{\widehat{\mathbf{e}}}_0>=0,$$

which are identified via the condition that the osculating circle must have osculate of at least the third order at $\widehat{\mathbf{z}}\left({\widehat{\phi }}_0\right)$ if ${\widehat{ϵ}}^{\prime }\ne 0$.

(b) The osculating circle $\mathcal{S}(\widehat{\rho },{\widehat{\mathbf{e}}}_0)$ and the curve $\widehat{\mathbf{z}}\left(\widehat{\phi }\right)\in {\mathcal{H}}_{+}^2$ are at least the fourth order at $\widehat{\mathbf{z}}\left({\phi }_0\right)$ if ${\widehat{ϵ}}^{\prime }=0$ and ${\widehat{ϵ}}^{\prime \prime }\ne 0$.

In this mode, by considering the evolutes of $\widehat{\mathbf{z}}\left(\widehat{\phi }\right)\in {\mathcal{H}}_{+}^2$, we can obtain a concatenation of evolutes ${\widehat{\mathbf{e}}}_2$, ${\widehat{\mathbf{e}}}_3$,..., ${\widehat{\mathbf{e}}}_n$. The properties and the joint relationships among these evolutes and their involutes are extremely interestings problems. For the model, it is easy to show that when ${\widehat{\mathbf{e}}}_0=\pm \widehat{\mathbf{e}}$ and ${\widehat{ϵ}}^{\prime }=0$, $\widehat{\mathbf{z}}\left(\widehat{\phi }\right)$ is standing at $\widehat{\phi }$ and is immutable relative to ${\widehat{\mathbf{e}}}_0$. In this issue, the $\mathcal{TLDA}$ is invariant up to the second order, and the line $\widehat{\mathbf{z}}$ proceeds over it with immutable pitch. Thus, the ($\widehat{z}$) with immutable $\mathcal{TLDA}$ is formed by $\mathcal{TL}$ line $\widehat{\mathbf{z}}$ existing at an immutable distance ${\omega }^{*}$ and fixed angle $\omega$ with respect to the $\mathcal{TLDA}$, that is,

$$\widehat{ϵ}\left(\widehat{\phi }\right):=ϵ+\epsilon \left(\mathrm{\Gamma }-ϵ\mu \right)=coth\widehat{\omega }=\widehat{c},$$

where $\widehat{c}=c+\epsilon c^{*}\in \mathcal{D}$.

Theorem 1. 

A non-developable $\mathcal{TLRS}$$\left(\widehat{z}\right)$ is a $\mathcal{TL}$ constant axis $\mathcal{RS}$ if $ϵ\left(\phi \right)=$ constant and $\mathrm{\Gamma }-ϵ\mu =$ constant.

3.4. Construction of the $\mathcal{TL}$ Constant Axis $\mathcal{RS}$ Family

In this subsection, we make a construction of the $\mathcal{TL}$ constant axis $\mathcal{RS}$ family. In view of Equation (30) and since $\widehat{ϵ}\left(\widehat{\phi }\right)$ is constant, we find the ODE, ${\widehat{\mathbf{z}}}^{\prime \prime \prime }+{\widehat{\kappa }}^2{\widehat{\mathbf{z}}}^{\prime }=\mathbf{0}$. Via several algebraic manipulations, the general solution of this equation is

$$\widehat{\mathbf{z}}\left(\widehat{\varkappa }\right)=\left(cosh\widehat{\omega },sinh\widehat{\omega }cos\widehat{\varkappa },sinh\widehat{\omega }sin\widehat{\varkappa }\right),$$

(35)

Here, $\widehat{\kappa }\widehat{\phi }:=\widehat{\varkappa }=\varkappa +\epsilon {\varkappa }^{*}$, where $0\le \varkappa$ and ${\varkappa }^{*}\in \mathbb{R}$; where $\omega$ and ${\omega }^{*}$ are constants. Then,

$$d\widehat{\phi }:=(1+\epsilon \mu )d\phi =sinh\widehat{\omega }d\widehat{\varkappa },$$

(36)

and

$$\mu \left(\varkappa \right)={\omega }^{*}coth\omega +{\varkappa }^{{*}^{\prime }},{(}^{\prime }=\frac{d}{d\varkappa }).$$

(37)

In view of Equations (31) and (37), we acquire

$$\mathrm{\Gamma }\left(\varkappa \right)={\varkappa }^{{*}^{\prime }}coth\omega +{\omega }^{*}.$$

(38)

Then,

$${\varkappa }^{*}\left(\varkappa \right)={\varkappa }_0^{*}+[(\stackrel{\varkappa }{\underset{0}{\int }}\mathrm{\Gamma }d\varkappa )cosh\omega -(\stackrel{\varkappa }{\underset{0}{\int }}\mu d\varkappa )sinh\omega ]sinh\omega ,\mathrm{with}{\varkappa }_0^{*}={\varkappa }^{*}\left(0\right).$$

(39)

From the real and dual parts of Equation (35), respecitvely, we have

$$\mathbf{z}\left(\varkappa \right)=\left(cosh\omega ,sinh\omega cos\varkappa ,sinh\omega sin\varkappa \right),$$

(40)

and

$${\mathbf{z}}^{*}\left(\varkappa \right)=\left[\begin{array}{c}{z}_{11}^{*}\\ z_{12}^{*}\\ z_{13}^{*}\end{array}\right]=\left[\begin{array}{c}{\omega }^{*}sinh\omega \hfill \\ {\omega }^{*}cosh\omega cos\varkappa -{\varkappa }^{*}sin\varkappa sinh\omega \hfill \\ {\omega }^{*}cosh\omega sin\varkappa +{\varkappa }^{*}cos\varkappa sinh\omega \hfill \end{array}\right].$$

Let $\mathbf{r}(r_1,r_2,r_3)$ be a point on $\widehat{\mathbf{z}}$. From $\mathbf{r}\times \mathbf{z}={\mathbf{z}}^{*}$, we acquire the arrangement of linear equations in $r_1,$$r_2,$ and $r_3$:

$$\left.\begin{array}{c}\hfill (-r_{2}sin\varkappa +r_{3}cos\varkappa )sinh\omega =z_{11}^{*},\\ \hfill -r_{1}sinh\omega sin\varkappa +r_{3}cosh\omega =z_{12}^{*},\\ \hfill r_{1}sinh\omega cos\varkappa -r_{2}cosh\omega =z_{13}^{*}.\end{array}\right\}$$

The matrix of coefficients of unknowns, $r_1,$$r_2,$ and $r_3$, is

$$\mathcal{A}=\left[\begin{array}{ccc}0\hfill & -sin\varkappa sinh\omega \hfill & cos\varkappa sinh\omega \hfill \\ -sinh\omega sin\varkappa \hfill & 0\hfill & cosh\omega \hfill \\ sinh\omega cos\varkappa \hfill & -cosh\omega \hfill & 0\hfill \end{array}\right].$$

It is evident that $rank\left(\mathcal{A}\right)$ = 2, where $\varkappa \ne p\pi$ (p is an integer) and $\omega \ne 0$. The $rank$ of the augmented matrix,

$$\left[\begin{array}{cccc}0\hfill & -sin\varkappa sinh\omega \hfill & cos\varkappa sinh\omega \hfill & z_{11}^{*}\\ -sinh\omega sin\varkappa \hfill & 0\hfill & cosh\omega \hfill & z_{12}^{*}\\ sinh\omega cos\varkappa \hfill & -cosh\omega \hfill & 0\hfill & z_{13}^{*}\end{array}\right],$$

is two. Then, this set has infinitely numerous solutions displaced with

$$\begin{array}{c}\hfill r_2=-{\omega }^{*}sin\varkappa +\left(r_1-{\varkappa }^{*}\right)tanh\omega cos\varkappa ,\\ \hfill r_3={\omega }^{*}cos\varkappa +\left(r_1-{\varkappa }^{*}\right)tanh\omega sin\varkappa ,\\ \hfill -r_{2}sin\varkappa +r_{3}cos\varkappa ={\omega }^{*}.\end{array}$$

(41)

Since $r_1$ is assumed at random, then we may let $r_1-{\varkappa }^{*}=0$. In this situation, Equation (40) reads as

$$r_1={\varkappa }^{*},r_2=-{\omega }^{*}sin\varkappa ,r_3={\omega }^{*}cos\varkappa .$$

Then,

$$\mathbf{c}\left(\varkappa \right)=\left({\varkappa }^{*},-{\omega }^{*}sin\varkappa ,{\omega }^{*}cos\varkappa \right).$$

(42)

By considering Equation (27) with Equations (40) and (42), we achieve

$$\left(\widehat{z}\right):\mathbf{z}(\varkappa ,r)=\left[\begin{array}{c}{\varkappa }^{*}+rcosh\omega \\ -{\omega }^{*}sin\varkappa +rsinh\omega cos\varkappa \\ {\omega }^{*}cos\varkappa +rsinh\omega sin\varkappa \end{array}\right],$$

(43)

where ${\varkappa }^{*}$ is located by Equation (39). $\omega$and ${\omega }^{*}$ can control the shape of $\left(\widehat{z}\right)$. For some values of $\mathrm{\Gamma }\left(\varkappa \right)$ and $\mu \left(\varkappa \right)$, we confer some epitomes for $\left(\widehat{z}\right)$, where we contemplate ${\varkappa }_0^{*}=0$, ${\omega }^{*}=\pm 1$, and $\omega =1.5$. Hence, any two of the $\mathcal{TL}$ constant axis $\mathcal{RS}$ family are reciprocal of one another.

(1) $\mathcal{TL}$ Archimedes helicoids with $\mathrm{\Gamma }\left(\varkappa \right)=\mu \left(\varkappa \right)=1$, $-1\le r\le 1$, and $0\le \varkappa \le 2\pi$ (Figure 4 and Figure 5).

Figure 4. $\mathcal{TL}$ Archimedes helicoid with ${\omega }^{*}=1$ and $\mathrm{\Gamma }\left(\varkappa \right)=\mu \left(\varkappa \right)=1$.

Figure 5. $\mathcal{TL}$ Archimedes helicoid with ${\omega }^{*}=-1$ and $\mathrm{\Gamma }\left(\varkappa \right)=\mu \left(\varkappa \right)=1$.

(2) $\mathcal{TL}$ Archimedes helicoid with $\mathrm{\Gamma }\left(\varkappa \right)=\mu \left(\varkappa \right)=\varkappa$, $-4\le v\le 4$, and $0\le \varkappa \le 2\pi$ (Figure 6 and Figure 7).

Figure 6. $\mathcal{TL}$ Archimedes helicoid with ${\omega }^{*}=1$ and $\mathrm{\Gamma }\left(\varkappa \right)=\mu \left(\varkappa \right)=\varkappa$.

Figure 7. $\mathcal{TL}$ Archimedes helicoid with ${\omega }^{*}=-1$ and $\mathrm{\Gamma }\left(\varkappa \right)=\mu \left(\varkappa \right)=\varkappa$.

3.5. Organizations among a $\mathcal{TL}$ Constant Axis $\mathcal{RS}$ and Its Striction Curve

The major geometrical organizations of a $\mathcal{TL}$ constant axis $\mathcal{RS}$ and its striction curve can be described as follows:

(a) Since $\mu \left(\varkappa \right)=0$, then $\left(\widehat{z}\right)$ is a tangential developable. In this situation, from Equations (39) and (43), we attain

$${\omega }^{*}coth\omega +{\varkappa }^{{*}^{\prime }}=0\Rightarrow {\varkappa }^{*}=-\varkappa {\omega }^{*}coth\omega ,$$

and

$$\left.\begin{array}{c}\hfill \mathbf{c}\left(\varkappa \right)=\left(-\varkappa {\omega }^{*}coth\omega ,-{\omega }^{*}sin\varkappa ,{\omega }^{*}cos\varkappa \right),\\ \hfill {\mathbf{c}}^{\prime }\left(\varkappa \right)=\left(-{\omega }^{*}coth\omega ,-{\omega }^{*}cos\varkappa ,-{\omega }^{*}sin\varkappa \right),\\ \hfill {\mathbf{c}}^{\prime \prime }\left(\varkappa \right)=\left(0,{\omega }^{*}sin\varkappa ,-{\omega }^{*}cos\varkappa \right),\\ \hfill {\mathbf{c}}^{\prime \prime \prime }\left(\varkappa \right)=\left(0,{\omega }^{*}cos\varkappa ,{\omega }^{*}sin\varkappa \right).\end{array}\right\}$$

Since

$$<{\mathbf{c}}^{\prime },{\mathbf{z}}^{\prime }>=0\mathrm{and}{∥{\mathbf{c}}^{\prime }∥}^2=-\frac{{\omega }^{*2}}{{sinh}^2\omega }<0;{(}^{\prime }=\frac{d}{d\varkappa }),$$

then the $\mathcal{SC}$$\mathbf{c}\left(\varkappa \right)$ is of type $\mathcal{TL}$ curve of ($\widehat{z}$). The curvature ${\kappa }_c\left(\varkappa \right)$ and the torsion ${\tau }_c\left(\varkappa \right)$, respectively, are

$${\kappa }_c\left(\varkappa \right):=\frac{∥{\mathbf{c}}^{\prime }\times {\mathbf{c}}^{\prime \prime }∥}{{∥{\mathbf{c}}^{\prime }∥}^3}=\frac{{sinh}^2\omega }{\left|{\omega }^{*}\right|}\mathrm{and}{\tau }_c\left(\varkappa \right):=\frac{det({\mathbf{c}}^{\prime },{\mathbf{c}}^{\prime \prime },{\mathbf{c}}^{\prime \prime \prime })}{{∥{\mathbf{c}}^{\prime }\times {\mathbf{c}}^{\prime \prime }∥}^2}=\frac{sinh2\omega }{2{\omega }^{*}}.$$

It is clear that $\mathbf{c}\left(\varkappa \right)$ is a cylindrical helix with the $\mathcal{DA}$ as the cylindrical axis. Furthermore, we possess

$$\left(\widehat{z}\right):\mathbf{z}(\varkappa ,r)=\left[\begin{array}{c}-\varkappa {\omega }^{*}coth\omega +rcosh\omega \\ -{\omega }^{*}sin\varkappa +rsinh\omega cos\varkappa \\ {\omega }^{*}cos\varkappa +rsinh\omega sin\varkappa \end{array}\right],$$

which is a family of two-parameter $\mathcal{TL}$ tangential surfaces; for ${\omega }^{*}=\pm 1$, $\omega =1.5$, $-1\le r\le 1$, and $0\le \varkappa \le 2\pi$ (Figure 8 and Figure 9).

Figure 8. $\mathcal{TL}$ tangential surface with ${\omega }^{*}=1$ and $\mu \left(\varkappa \right)=0.$

Figure 9. $\mathcal{TL}$ tangential surface with ${\omega }^{*}=-1$ and $\mu \left(\varkappa \right)=0.$

(b) Since $\mathrm{\Gamma }=0$, then $\left(\widehat{z}\right)$ is a $\mathcal{TL}$ binormal surface. From Equation (39), we attain

$${\varkappa }^{{*}^{\prime }}coth\omega +{\omega }^{*}=0\Rightarrow {\varkappa }^{*}=-\varkappa {\omega }^{*}tanh\omega ,$$

and then Equation (42) is

$$\mathbf{c}\left(\varkappa \right)=\left(-\varkappa {\omega }^{*}tanh\omega ,-{\omega }^{*}sin\varkappa ,{\omega }^{*}cos\varkappa \right).$$

In a similar manner, we find

$$<{\mathbf{c}}^{\prime },{\mathbf{z}}^{\prime }>=0\mathrm{and}{∥{\mathbf{c}}^{\prime }∥}^2=\frac{{\omega }^{*2}}{{cosh}^2\omega }>0;{(}^{\prime }=\frac{d}{d\varkappa }).$$

Thus, the $\mathcal{SC}$$\mathbf{c}\left(\varkappa \right)$ is a type of $\mathcal{SL}$ curve of ($\widehat{z}$). The curvature ${\kappa }_c\left(\varkappa \right)$ and the torsion ${\tau }_c\left(\varkappa \right)$, respectively, are

$${\kappa }_c\left(\varkappa \right)=\frac{{cosh}^2\omega }{\left|{\omega }^{*}\right|}\mathrm{and}{\tau }_c\left(\varkappa \right)=\frac{cosh2\omega }{2{\omega }^{*}}.$$

It is clear that $\mathbf{c}\left(\varkappa \right)$ is a $\mathcal{SL}$ cylindrical helix with the $\mathcal{DA}$ as the cylindrical axis. Further, we have

$$\left(\widehat{z}\right):\mathbf{z}(\varkappa ,r)=\left[\begin{array}{c}-\varkappa {\omega }^{*}tanh\omega +rcosh\omega \\ -{\omega }^{*}sin\varkappa +rsinh\omega cos\varkappa \\ {\omega }^{*}cos\varkappa +rsinh\omega sin\varkappa \end{array}\right],$$

which is a family of two-parameter $\mathcal{TL}$ binormal surfaces; for ${\omega }^{*}=\pm 1$, $\omega =1.5$, $-3\le r\le 3$, and $0\le \varkappa \le 2\pi$ (Figure 10 and Figure 11).

Figure 10. $\mathcal{TL}$ binormal surface with ${\omega }^{*}=1$ and $\mathrm{\Gamma }\left(\varkappa \right)=0.$

Figure 11. $\mathcal{TL}$ binormal surface with ${\omega }^{*}=-1$ and $\mathrm{\Gamma }\left(\varkappa \right)=0.$

(c) Since $\left(\widehat{z}\right)$ is a $\mathcal{TL}$ cone, that is, $\mu \left(\varkappa \right)=\mathrm{\Gamma }\left(\varkappa \right)=0$, and ${\mathbf{c}}^{\prime }\left(\varkappa \right)=0$, then

$${\mathbf{c}}^{\prime }\left(\varkappa \right)=\mathbf{0}\iff \mathbf{0}=\left(\mathrm{\Gamma }cosh\omega -\mu sinh\omega )sinh\omega ,-{\omega }^{*}cos\varkappa ,-{\omega }^{*}sin\varkappa \right),$$

(44)

or

$$-{\omega }^{*}cos\varkappa =0,-{\omega }^{*}sin\varkappa =0,(\mathrm{\Gamma }cosh\omega -\mu sinh\omega )sinh\omega =0,$$

(45)

This leads to ${\omega }^{*}=0$ and $\varkappa \ne 0$. Then, the $\mathcal{TLRS}$$\left(\widehat{z}\right)$ is locally isometric to

$$\left(\widehat{z}\right):\mathbf{z}(\varkappa ,r)=r(cosh\omega ,sinh\omega cos\varkappa ,sinh\omega sin\varkappa ),$$

(46)

which is a one-parameter family of $\mathcal{TL}$ cones for $\omega =1.5$, $-4\le r\le 4$, and $0\le \varkappa \le 2\pi$ (Figure 12).

Figure 12. $\mathcal{TL}$ cone.

(d) A cylindrical surface is produced by the one-parameter locomotion of an oriented line that maintains a fixed direction in space. We found previously from Equation (24) that, in the case of $d\phi =0$, all rulings are parallel and $\left(\widehat{z}\right)$ is a $\mathcal{TL}$ cylindrical surface. In this case, the $\mathcal{BF}$ is not specified since there is no discerned central point. Then, at a cylindrical surface, we performed it in the other direction: we appointed any transverse ${\mathcal{C}}^2$-curve: $\mathbf{c}\in I\to$ ${\mathcal{E}}^3$ on $\left(\widehat{z}\right)$ as $\mathcal{SC}$, that is, the orthogonal section realized by the equation $\mathbf{c}\left(\phi \right)=\mathbf{z}\times {\mathbf{z}}^{*}$. For a $\mathcal{TL}$ cylindrical surface, any two infinitesimally spaced rulings $\mathbf{z}\left(\phi \right)$ and $\mathbf{z}(\phi +d\phi )$ are parallel to each other, so $d\widehat{\phi }=\epsilon d{\phi }^{*}$(replace $d{\phi }^{*}$ with $d\phi$ for short) is the arc length of $\mathcal{SC}$$\mathbf{c}\left(\phi \right)$. In this manner and by definition, $\mathbf{c}\left(\phi \right)$ is continuously orthogonal to $\mathbf{z}$ and its end point traces out a planar curve, that is, $<\mathbf{z},\mathbf{c}>=0$. This means that the $\mathcal{SC}$ of the $\mathcal{TL}$ cylindrical surface is unique for an orthogonal section. Differentiating $<\mathbf{z},\mathbf{c}>=0$ and noting that $\mathbf{z}$ is steady vector gives $<\mathbf{z},{\mathbf{c}}^{\prime }>=0$; ${(}^{\prime }=\frac{d}{d\phi }$), which designates that the tangent vector of the $\mathcal{SC}$ is orthogonal to the $\mathcal{TL}$ rulings. However, the limit position of the mutual orthogonal to any two infinitesimally spaced rulings $\mathbf{z}\left(\phi \right)$ and $\mathbf{z}(\phi +d\phi )$ is undefined, that is, the central tangent ${\widehat{\mathbf{z}}}_3\left(\phi \right)$ does not exist in that sense. It is feasible to explain the $\mathcal{BF}$ as follows. The central tangent is explained as

$${\widehat{\mathbf{z}}}_3\left(\phi \right)={\mathbf{z}}_3\left(\phi \right)+\epsilon {\mathbf{z}}_3^{*}\left(\phi \right);{\mathbf{z}}_3^{*}=\mathbf{c}\times {\mathbf{z}}_3,\mathrm{with}{\mathbf{c}}^{\prime }\left(\phi \right)={\mathbf{z}}_3.$$

(47)

Note that the central tangent ${\mathbf{z}}_3$ is tangent to the $\mathcal{SC}$. Also, the $\mathcal{SL}$ central normal is specified by

$${\widehat{\mathbf{z}}}_2\left(\phi \right)={\mathbf{z}}_2\left(\phi \right)+\epsilon {\mathbf{z}}_2^{*}\left(\phi \right)=-{\widehat{\mathbf{z}}}_1\left(\phi \right)\times {\widehat{\mathbf{z}}}_3\left(\phi \right).$$

(48)

Further, we can find the variation in the following form:

$${\widehat{\mathbf{z}}}_1^{\prime }\left(\phi \right)=\frac{d}{d\phi }[{\mathbf{z}}_1\left(\phi \right)+\epsilon {\mathbf{z}}_1^{*}\left(\phi \right)]=0+\epsilon \frac{d}{d\phi }(\mathbf{c}\times {\mathbf{z}}_1)={\mathbf{z}}_3\times {\mathbf{z}}_1=\epsilon {\mathbf{z}}_2\left(\phi \right).$$

(49)

To derive the variation of ${\widehat{\mathbf{z}}}_3\left(\phi \right)$, we first locate the second derivative of the striction curve. A variation of ${\mathbf{c}}^{\prime }\left(\phi \right)$ gives ${\mathbf{c}}^{\prime \prime }\left(\phi \right)={\mathbf{z}}_3^{\prime }\left(\phi \right)$. As we mentioned above, the $\mathcal{SC}$ is planar and constantly columnar to the $\mathcal{TL}$ constant ruling of the cylindrical surface. From the theory of planar curve, the second derivative ${\mathbf{c}}^{\prime \prime }\left(\phi \right)$ may be reported by ${\mathbf{c}}^{\prime \prime }\left(\phi \right)=\kappa \left(\phi \right){\mathbf{z}}_2\left(\phi \right)$, where $\kappa \left(\phi \right)$ is the curvature function of $\mathbf{c}\left(\phi \right)$. Therefore, we acquire

$${\widehat{\mathbf{z}}}_3^{\prime }\left(\phi \right)=\frac{d}{d\phi }[{\mathbf{z}}_3\left(\phi \right)+\epsilon {\mathbf{z}}_3^{*}\left(\phi \right)]=\kappa \left(\phi \right){\mathbf{z}}_2\left(\phi \right).$$

(50)

So, we acquire

$${\widehat{\mathbf{z}}}_2^{\prime }\left(\phi \right)={[{\mathbf{z}}_3\left(\phi \right)\times {\mathbf{z}}_1\left(\phi \right)+\epsilon \mathbf{c}\left(\phi \right)\times ({\mathbf{z}}_3\left(u\right)\times {\mathbf{z}}_1\left(u\right))]}^{\prime }=-\kappa \left(\phi \right){\mathbf{z}}_3\left(\phi \right)+\epsilon {\mathbf{z}}_1\left(\phi \right).$$

(51)

Then,

$$\left[\begin{array}{c}{\widehat{\mathbf{z}}}_1^{\prime }\hfill \\ {\widehat{\mathbf{z}}}_2^{\prime }\hfill \\ {\widehat{\mathbf{z}}}_3^{\prime }\hfill \end{array}\right]=\left[\begin{array}{ccc}0\hfill & \epsilon \hfill & 0\hfill \\ \epsilon \hfill & 0\hfill & -\kappa \hfill \\ 0\hfill & \kappa \hfill & 0\hfill \end{array}\right]\left[\begin{array}{c}{\widehat{\mathbf{z}}}_1\hfill \\ {\widehat{\mathbf{z}}}_2\hfill \\ {\widehat{\mathbf{z}}}_3\hfill \end{array}\right]=\widehat{\omega }\times \left[\begin{array}{c}{\widehat{\mathbf{z}}}_1\hfill \\ {\widehat{\mathbf{z}}}_2\hfill \\ {\widehat{\mathbf{z}}}_3\hfill \end{array}\right],$$

(52)

where $\widehat{\omega }\left(\phi \right)=\kappa \left(\phi \right){\widehat{\mathbf{z}}}_1\left(\phi \right)+\epsilon {\widehat{\mathbf{z}}}_3\left(\phi \right)$. Therefore,

$$\widehat{\mathbf{e}}\left(\phi \right):=\frac{\widehat{\omega }\left(\phi \right)}{∥\widehat{\omega }\left(\phi \right)∥}={\widehat{\mathbf{z}}}_1\left(\phi \right)+\epsilon \rho \left(\phi \right){\widehat{\mathbf{z}}}_3\left(\phi \right).$$

(53)

Furthermore, we attain

$$\widehat{\mathbf{p}}\left(\phi \right):=\widehat{\mathbf{e}}\times \widehat{\mathbf{t}}={\widehat{\mathbf{z}}}_3\left(\phi \right)+\epsilon \rho {\widehat{\mathbf{z}}}_1\left(\phi \right).$$

(54)

Hence, the $\mathcal{SFF}$ of cylindrical surface is

$$\left[\begin{array}{c}\widehat{\mathbf{t}}\hfill \\ \widehat{\mathbf{p}}\hfill \\ \widehat{\mathbf{e}}\hfill \end{array}\right]=\left[\begin{array}{ccc}0\hfill & 1\hfill & 0\hfill \\ \epsilon \rho \hfill & 0\hfill & 1\hfill \\ 1\hfill & 0\hfill & \epsilon \rho \hfill \end{array}\right]\left[\begin{array}{c}{\widehat{\mathbf{z}}}_1\hfill \\ {\widehat{\mathbf{z}}}_2\hfill \\ {\widehat{\mathbf{z}}}_3\hfill \end{array}\right].$$

(55)

From Equations (28) and (55), we attain $\omega =0$ and ${\omega }^{*}=\rho$. Then,

$$\left(\widehat{z}\right):\mathbf{z}(\varkappa ,r)=(r,-\rho sin\varkappa ,\rho cos\varkappa ),$$

(56)

which clarifies a family of two-parameters of $\mathcal{TL}$ cylinders, for $\rho =1$, $0\le r\le 3$, and $0\le \varkappa \le 2\pi$ (Figure 13).

Figure 13. $\mathcal{TL}$ cylinder with $\rho =1.$

4. Conclusions

In this paper, based on the constant angle properties of surfaces and using the Blaschke framework, we study and classify $\mathcal{TL}$ ruled surfaces in Minkowski 3-space ${\mathcal{E}}_1^3$. Then, we gain some new constant axis ruled surfaces in Lorentzian line space and their geometrical illustrations. We also earn several organizations among a constant axis ruled surface and its striction curve. For future research, we will deal with integrating the study of singularity theory and submanifold theory, as in [15,16,17], with the consequences of this study being to search in a novel manner for further theorems on this topic related to symmetric possessions.