Timelike Constant Axis Ruled Surface Family in Minkowski 3-Space

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 Then, Equation (5) can be formed as Here, $∥\widehat{\varpi }∥:=\varpi \left(v\right)+\epsilon {\varpi }^{*}\left(v\right)$ is the angular speed of the $\mathcal{BF}$ movement; Therefore, the $\mathcal{TLDA}$ is the instantaneous screw-axis of the $\mathcal{BF}$.

In fact, we find the following:

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

If the $\mathcal{BF}$- movement is a pure turnover, that is, $p\left(v\right)=0$, then 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, where From the Equations (7), (12), and (16), we attain Equation (17) is a Lorentzian version of the Hamilton and Mannheim formulae (compared with [1,2,3]).

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 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 Note that it is a third-order polynomial at the coordinates $x,y$, and z. Also, we have So, the two limits of $\left(\widehat{e}\right)$ are demonstrated by which represents two isotropic torsal $\mathcal{SL}$ planes, each of them containing one isotropic $\mathcal{SL}$ torsal line $\mathcal{L}$.

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:

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.

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, From the Equations (4) and (23), we have 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 Then, 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 where Comparably, we possess where 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.

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 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, and In view of Equations (31) and (37), we acquire Then, From the real and dual parts of Equation (35), respecitvely, we have and 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,$$r_2,$ and $r_3$:The matrix of coefficients of unknowns, $r_1,$$r_2,$ and $r_3$, is 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, is two. Then, this set has infinitely numerous solutions displaced with Since $r_1$ is assumed at random, then we may let $r_1-{\varkappa }^{*}=0$. In this situation, Equation (40) reads as Then, By considering Equation (27) with Equations (40) and (42), we achieve 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).

(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).

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

(b) Since $\mathrm{\Gamma }=0$, then $\left(\widehat{z}\right)$ is a $\mathcal{TL}$ binormal surface. From Equation (39), we attain and then Equation (42) is In a similar manner, we find 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 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 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).