Some Characteristic Properties of Non-Null Curves in Minkowski 3-Space ${𝔼}_1^3$

Abstract

This paper gives new characteristic properties of non-null spherical and rectifying curves in Minkowski 3-space ${\mathbb{E}}_1^3$. In the light of the causal characteristics, we give some representations of rectifying non-null curves. Additionally, we proved that the tangential function of every non-null curve fulfills a third-order differential equation. Then, a number of well-known characteristic properties of rectifying, Lorentzian, and hyperbolic spherical curves are consequences of this differential equation.

1. Introduction

Locating the properties and descriptions of specific curves is a significant area of research because of their advantages in various different disciplines. In [1], B. Y. Chen defined the rectifying curve as a space curve whose position vector always lies in its rectifying plane, denoted by the symbol Span{$\mathbf{t}$, $\mathbf{b}$}, where $\mathbf{t}$ and $\mathbf{b}$ are the tangent and binormal vector fields of the curve, respectively. As a result, the position vector $\mathit{\alpha }\left(s\right)$ of the curve satisfies the equation

$$\mathit{\alpha }\left(s\right)=\chi \left(s\right)\mathbf{t}\left(s\right)+\lambda \left(s\right)\mathbf{b}\left(s\right),$$

for some differentiable functions $\chi \left(s\right)$ and $\lambda \left(s\right)$ in arc-length parameter s [1]. Chen and Dillen [2] have demonstrated that there is a simple relationship between rectifying curves and centrodes, which play crucial roles in geometry, mechanics, and kinematics. Deshmukh et al. [3] proved the distance function of every Frenet curve in Euclidean 3-space ${\mathbb{E}}^3$ fulfill a fourth order differential equation, and they also distinguished some properties of helices via this equation. In this regard, some well-known curves in Euclidean 3-space ${\mathbb{E}}^3$ were separated as such as specified in [3,4,5,6,7,8,9,10]. The expansions of those works regarding the specific curves in other ambient spaces have been specified in by [7,11,12]. With the growth in the theory of relativity, the helices and spherical, normal, and rectifying curves have been elaborated widely by [6,7,8,13,14,15,16,17,18].

In this paper, we present a new mathematical method for investigating non-null spherical and rectifying curves in Minkowski 3-space ${\mathbb{E}}_1^3$. The necessary and sufficient conditions for spacelike curves to be Lorentzian spherical were obtained. We proved that the height tangential function for any non-null curve fulfills a third-order differential equation. In terms of this differential equation, we discussed many new properties of spherical and rectifying curves on hyperbolic and Lorentzian spheres.

2. Preliminaries

Let ${\mathbb{E}}_1^3$ indicate the Minkowski 3-space ${\mathbb{E}}_1^3$. For vectors $\mathit{\zeta }=\left({\zeta }_1,{\zeta }_2,{\zeta }_3\right)$ and $\mathit{\eta }=({\eta }_1,{\eta }_2,{\eta }_3)$ in ${\mathbb{E}}_1^3$

$$<\mathit{\zeta },\mathit{\eta }>={\zeta }_1{\eta }_1+{\zeta }_2{\eta }_2-{\zeta }_3{\eta }_3,$$

is called the Lorentzian inner product. We also specify a vector

$$\mathit{\zeta }\times \mathit{\eta }=\left|\begin{array}{ccc}\mathbf{i}& \mathbf{j}& -\mathbf{k}\\ {\zeta }_1& {\zeta }_2& {\zeta }_3\\ {\eta }_1& {\eta }_2& {\eta }_3\end{array}\right|=\left(({\zeta }_2{\eta }_3-{\zeta }_3{\eta }_2),({\zeta }_3{\eta }_1-{\zeta }_1{\eta }_3),-({\zeta }_1{\eta }_2-{\zeta }_2{\eta }_1)\right).$$

Since <,> is an indefinite metric, note that a vector $\mathit{\zeta }\in$ ${\mathbb{E}}_1^3$ can have one of three causal characteristics; it can be spacelike if <$\mathit{\zeta },\mathit{\zeta }$> > 0 or $\mathit{\zeta }=\mathbf{0}$, timelike if <$\zeta ,\zeta$> < 0, and null if <$\mathit{\zeta },\mathit{\zeta }$> = 0 and $\mathit{\zeta }\ne \mathbf{0}$. Similarly, a curve $\mathit{\alpha }=\mathit{\alpha }\left(s\right)$ is called timelike, spacelike, or null, if its velocity vector ${\mathit{\alpha }}^{{}^{\prime }}$ is timelike, spacelike, or null, respectively. For $\zeta \in {\mathbb{E}}_1^3$, the norm is specified by $∥\mathit{\zeta }∥=\sqrt{\left|<\mathit{\zeta },\mathit{\zeta }>\right|}$, then $\mathit{\zeta }$ is called a unit vector if $∥\mathit{\zeta }∥=1$ [19,20].

Let $\mathit{\alpha }\left(s\right)$ be a unit speed non-null curve with the Frenet–Serret apparatus $\left\{\kappa \right(s)$, $\tau \left(s\right)$, $\mathbf{t}\left(s\right),$$\mathbf{n}\left(s\right)$, $\mathbf{b}\left(s\right)\}$ in ${\mathbf{E}}_1^3$, where $\mathbf{t}\left(s\right)={\mathit{\alpha }}^{{}^{\prime }}\left(s\right)$, $\mathbf{n}\left(s\right)={\mathit{\alpha }}^{{}^{{}^{\prime \prime }}}\left(s\right)/∥{\mathit{\alpha }}^{{}^{{}^{\prime \prime }}}\left(s\right)∥$ and $\mathbf{b}\left(s\right)=\mathbf{t}\left(s\right)\times \mathbf{n}\left(s\right)$ are called the unit tangent vector, the principal normal vector, and the binormal vector, respectively. The Frenet–Serret equations for $\mathit{\alpha }$ are specified by:

$$\left(\begin{array}{c}{\mathbf{t}}^{\prime }\hfill \\ {\mathbf{n}}^{\prime }\hfill \\ {\mathbf{b}}^{\prime }\hfill \end{array}\right)=\left(\begin{array}{ccc}0\hfill & \kappa \hfill & 0\hfill \\ -{ϵ}_0{ϵ}_1\kappa \hfill & 0\hfill & \tau \hfill \\ 0\hfill & -{ϵ}_1{ϵ}_2\tau \hfill & 0\hfill \end{array}\right)\left(\begin{array}{c}\mathbf{t}\hfill \\ \mathbf{n}\hfill \\ \mathbf{b}\hfill \end{array}\right),\left({}^{\prime }=\frac{d}{ds}\right),$$

(1)

where $\kappa \left(s\right)$ and $\tau \left(s\right)$ are called curvature and torsion of $\mathit{\alpha }\left(s\right)$[13,14,15,20,21]. Then, $\mathit{\alpha }$ is called a Frenet–Serret curve if $\kappa >0$ and $\tau \ne 0$. Further, due to the casual characteristics, there is

$$<\mathbf{t},\mathbf{t}>={ϵ}_0=\pm 1,<\mathbf{n},\mathbf{n}>{ϵ}_1=\pm 1,<\mathbf{b},\mathbf{b}>={ϵ}_2=\pm 1,{ϵ}_0{ϵ}_1{ϵ}_2=-1.$$

Let $\mathbf{p}$ be a fixed point in ${\mathbb{E}}_1^3$. Then, the hyperbolic and Lorentzian unit spheres, respectively, are

$${\mathbb{H}}_{+}^2\left(1\right)=\{\mathbf{x}\in {\mathbb{E}}_1^3\mid {∥\mathbf{x}-\mathbf{p}∥}^2=-1\},$$

(2)

and

$${\mathbb{S}}_1^2\left(1\right)=\{\mathbf{x}\in {\mathbb{E}}_1^3\mid {∥\mathbf{x}-\mathbf{p}∥}^2=1\}.$$

(3)

3. Characteristic Properties of a Non-Null Curve

In this section, we demonstrate that $\tau /\kappa$ of any non-null unit speed Frenet–Serret curve is a non-constant linear function of the pseudo- arc-length s. Thus, we indicate that this monarchy is invariant with respect to of a curve’s causal structure and its rectifying plane.

 Theorem 1. 

Let $\mathit{\alpha }=\mathit{\alpha }\left(s\right)$ be a non-null unit speed Frenet–Serret curve. Then the following statements are equivalents:

 (i) 

There a point $\mathbf{p}\in {\mathbb{E}}_1^3$ such that each timelike or spacelike rectifying plane of $\mathit{\alpha }=\mathit{\alpha }\left(s\right)$ passes through $\mathbf{p}$.

 (ii) 

$\tau /\kappa$ is a non-constant linear function $as+b.$

 (iii) 

There is a fixed point ${\mathbf{p}}_0\in {\mathbb{E}}_1^3$ such that ${∥\mathit{\alpha }\left(s\right)-{\mathbf{p}}_0∥}^2=\left|{ϵ}_0{(s+c)}^2+{ϵ}_2{d}^2\right|$. These constants are connected by

$$a=\frac{{ϵ}_0}{d},b=\frac{{ϵ}_{0}c}{d};d\ne 0,$$

and by the uniqueness of $\mathbf{p}$, $\mathbf{p}$ is equal to ${\mathbf{p}}_0$.

 Proof. 

(i)

Assume that each timelike or spacelike rectifying plane of $\mathit{\alpha }=\mathit{\alpha }\left(s\right)$ passes through a fixed point ${\mathbf{p}}_0\in {\mathbb{E}}_1^3$. The derivative of <$\mathit{\alpha }\left(s\right)-{\mathbf{p}}_0,\mathbf{n}\left(s\right)$> = 0 can be obtained as

$$<\mathit{\alpha }\left(s\right)-{\mathbf{p}}_0,-{ϵ}_2\kappa \mathbf{t}+\tau \mathbf{b}>=0.$$

(4)

It follows that $\mathit{\alpha }\left(s\right)-{\mathbf{p}}_0$ is orthogonal to both $\mathbf{n}$ and $-{ϵ}_2\kappa \mathbf{t}\left(s\right)+\tau \mathbf{b}$. The rectifying plane can thus be represented as

$$\mathit{\alpha }\left(s\right)-{\mathbf{p}}_0=\eta \left(s\right)({ϵ}_0\tau \mathbf{t}-\kappa \mathbf{b}),$$

(5)

for a smooth function $\eta =\eta \left(s\right)$. By derivation of the Equation (4), we attain:

$$-{ϵ}_1\kappa +<\mathit{\alpha }\left(s\right)-{\mathbf{p}}_0,{ϵ}_2{\kappa }^{{}^{\prime }}\mathbf{t}+{\tau }^{{}^{\prime }}\mathbf{b}>=0.$$

(6)

The Equations (5) and (6), leads to

$$\eta =\frac{\kappa {ϵ}_0}{{\tau }^{{}^{\prime }}\kappa -{\kappa }^{{}^{\prime }}\tau }.$$

(7)

In view of Equations (5) and (7), we get

$$\mathit{\alpha }\left(s\right)-{\mathbf{p}}_0=\frac{\kappa \tau }{{\tau }^{{}^{\prime }}\kappa -{\kappa }^{{}^{\prime }}\tau }\mathbf{t}-\frac{{\kappa }^2{ϵ}_0}{{\tau }^{{}^{\prime }}\kappa -{\kappa }^{{}^{\prime }}\tau }\mathbf{b}.$$

(8)

Further, we get

$$\frac{d{\mathbf{p}}_0}{ds}=\left(1-{\left(\frac{\kappa \tau }{{\tau }^{{}^{\prime }}\kappa -{\kappa }^{{}^{\prime }}\tau }\right)}^{{}^{\prime }}\right)\mathbf{t}+{\left(\frac{{\kappa }^2{ϵ}_0}{{\tau }^{{}^{\prime }}\kappa -{\kappa }^{{}^{\prime }}\tau }\right)}^{{}^{\prime }}\mathbf{b}.$$

By the coefficients of $\mathbf{t}$ and $\mathbf{b}$, we attain:

$${\left(\frac{{\kappa }^2{ϵ}_0}{{\tau }^{{}^{\prime }}\kappa -{\kappa }^{{}^{\prime }}\tau }\right)}^{{}^{\prime }}=0,1-{\left(\frac{\kappa \tau }{{\tau }^{{}^{\prime }}\kappa -{\kappa }^{{}^{\prime }}\tau }\right)}^{{}^{\prime }}=0.$$

This means that

$$\frac{\kappa \tau }{{\tau }^{{}^{\prime }}\kappa -{\kappa }^{{}^{\prime }}\tau }=s+c,\frac{{\kappa }^2{ϵ}_0}{{\tau }^{{}^{\prime }}\kappa -{\kappa }^{{}^{\prime }}\tau }=d,c\in \mathbb{R}.$$

(9)

Since $\kappa \left(s\right)\ne 0$, d$\ne 0$. Equations (8) and (9) lead to

$$\left.\begin{array}{c}\mathit{\alpha }\left(s\right)-{\mathbf{p}}_0=(s+c)\mathbf{t}-d\mathbf{b},\hfill \\ \frac{\tau }{\kappa }=as+b;a=\frac{{ϵ}_0}{d},b=\frac{{ϵ}_{0}c}{d},d\ne 0.\hfill \end{array}\right\}$$

(10)

Then, we have

$${∥\mathit{\alpha }\left(s\right)-{\mathbf{p}}_0∥}^2=\left|{ϵ}_0{(s+c)}^2+{ϵ}_2{d}^2\right|.$$

Thus (i)⇒(ii), and (iii). If each timelike or spacelike rectifying plane contains other point ${\mathbf{p}}_0$, let $\gamma (t)$ be the unit speed non-null curve passing through ${\mathbf{p}}_0$ and $\mathbf{p}$. Then, for all $t\in \mathbb{R}$, there are $c\left(t\right)$ and $d\left(t\right)\ne 0$ such that

$$\mathit{\alpha }\left(s\right)-\gamma \left(t\right)=(s+c(t\left)\right)\mathbf{t}\left(s\right)+d\left(t\right)\mathbf{b}\left(s\right).$$

(11)

If dot denotes to derivative with respect to $t,$ then from Equation (11) we have

$$-\stackrel{.}{\gamma }\left(t\right)=\stackrel{.}{c}\left(t\right))\mathbf{t}\left(s\right)+\stackrel{.}{d}\left(t\right)\mathbf{b}\left(s\right).$$

Note that:

$$\frac{\tau }{\kappa }=a\left(t\right)s+b\left(t\right);a\left(t\right)=\frac{{ϵ}_0}{d\left(t\right)},b\left(t\right)=\frac{{ϵ}_{0}c\left(t\right)}{d\left(t\right)}.$$

Thus, we have

$$\stackrel{.}{a}\left(t\right)s+\stackrel{.}{b}\left(t\right)=0.$$

So, we get

$$\stackrel{.}{a}\left(t\right)=\stackrel{.}{b}\left(t\right)=0\Rightarrow \stackrel{.}{\gamma }\left(t\right)=0,\mathbf{p}={\mathbf{p}}_0.$$

(ii)

Assume that $\frac{\tau }{\kappa }=as+b;a\ne 0.$ If we allow

$$\mathbf{p}=\mathit{\alpha }\left(s\right)-(s+\frac{b}{a})\mathbf{t}+\frac{{ϵ}_0}{a}\mathbf{b},$$

then by hypothesis, we get ${\mathbf{p}}^{{}^{\prime }}=\mathbf{0}$. Then, $\mathbf{p}$ is a fixed point in ${\mathbb{E}}_1^3$ and

$$\mathit{\alpha }\left(s\right)-{\mathbf{p}}_0=(s+c)\mathbf{t}+d\mathbf{b},d=\frac{{ϵ}_0}{a},c=\frac{b}{a}.$$

Thus (ii)⇒(i) and (iii).

Now assume that statement (iii) holds true, then

$$<\mathit{\alpha }\left(s\right)-{\mathbf{p}}_0,\mathbf{t}>={ϵ}_0(s+c).$$

(12)

By differentiation of Equation (12), and using the Frenet–Serret formulae, we have

$$\kappa \left(s\right)<\mathit{\alpha }\left(s\right)-{\mathbf{p}}_0,\mathbf{n}>=0;\kappa \left(s\right)\ne 0\Rightarrow <\mathit{\alpha }\left(s\right)-{\mathbf{p}}_0,\mathbf{n}>=0,$$

from which it follows that any rectifying plane of $\mathit{\alpha }\left(s\right)$ passes through a fixed point ${\mathbf{p}}_0\in {\mathbb{E}}_1^3$. Thus, (iii)⇒(i). □

We end this section with several parameterizations of a non-null rectifying curve furnished by its radial projection. Suppose that $\mathit{\alpha }=\mathit{\alpha }\left(s\right)$ is a unit speed spacelike curve with a timelike or spacelike principal normal in ${\mathbb{E}}_1^3$. Then, ${ϵ}_0=1,$${ϵ}_1=-{ϵ}_2=ϵ=\pm 1$. For a fixed point ${\mathbf{p}}_0\in {\mathbb{E}}_1^3$, and by the proof of Theorem 1, assume that

$$\mathit{\beta }\left(s\right)=\frac{1}{r\left(s\right)}(\mathit{\alpha }\left(s\right)-{\mathbf{p}}_0);r\left(s\right)=∥\mathit{\alpha }\left(s\right)-{\mathbf{p}}_0∥=\sqrt{\left|{(s+c)}^2-ϵd^2\right|},$$

(13)

be the radial projection of $\mathit{\alpha }\left(s\right)$ into ${\mathbb{S}}_1^2\left(1\right)$ (resp. ${\mathbb{H}}_{+}^2\left(1\right)$). Then, we have:

$$\left.\begin{array}{c}\mathbf{t}\left(s\right)=r^{\prime }\left(s\right)\mathit{\beta }\left(s\right)+r\left(s\right){\mathit{\beta }}^{\prime }\left(s\right),\hfill \\ \kappa \left(s\right)\mathbf{n}\left(s\right)=r^{\prime \prime }\left(s\right)\mathit{\beta }\left(s\right)+2r^{\prime }\left(s\right){\mathit{\beta }}^{\prime }\left(s\right)+r\left(s\right){\mathit{\beta }}^{\prime \prime }\left(s\right).\hfill \end{array}\right\}$$

(14)

 Theorem 2. 

Let $\mathit{\alpha }=\mathit{\alpha }\left(s\right)$ be a non-null unit speed Frenet–Serret curve. If ${\mathbf{p}}_0\in {\mathbb{E}}_1^3$ is a fixed point, then:

 (i) 

$\mathit{\alpha }\left(s_{\beta }\right)-{\mathbf{p}}_0$ is a spacelike position vector with a spacelike rectifying plane iff, up to a representation, $\mathit{\alpha }\left(s_{\beta }\right)-{\mathbf{p}}_0$ is given by

$$\mathit{\alpha }\left(s_{\beta }\right)-{\mathbf{p}}_0=\frac{d}{cos{s}_{\beta }}\mathit{\beta }\left(s_{\beta }\right),$$

(15)

where $\mathit{\beta }\left(s_{\beta }\right)$ is a unit speed spacelike curve on ${\mathbb{S}}_1^2\left(1\right)$, and

$${\kappa }_{\mathit{\beta }}^2=\frac{r^6}{d^4}{\kappa }^2-1.$$

 (ii) 

$\mathit{\alpha }\left(s_{\beta }\right)-{\mathbf{p}}_0$ is a spacelike position vector with a timelike rectifying plane iff, up to a representation, $\mathit{\alpha }\left(s_{\beta }\right)-{\mathbf{p}}_0$ is given by

$$\mathit{\alpha }\left(s_{\beta }\right)-{\mathbf{p}}_0=\frac{d}{sinh{s}_{\beta }}\mathit{\beta }\left(s_{\beta }\right),$$

(16)

where $\mathit{\beta }\left(s_{\beta }\right)$ is a unit speed timelike curve on ${\mathbb{S}}^2\left(1\right)$, and

$${\kappa }_{\mathit{\beta }}^2=\frac{r^6}{d^4}{\kappa }^2+1.$$

 (iii) 

$\mathit{\alpha }\left(s_{\beta }\right)-{\mathbf{p}}_0$ is a timelike position vector with a timelike rectifying plane iff, up to a representation, $\mathit{\alpha }\left(s_{\beta }\right)-\mathbf{p}$ is given by

$$\mathit{\alpha }\left(s_{\beta }\right)-{\mathbf{p}}_0=\frac{d}{{cosh}_{\mathit{\beta }}}\mathit{\beta }\left(s_{\beta }\right),$$

(17)

where $\mathit{\beta }\left(s_{\beta }\right)$ is a unit speed spacelike curve lying entirely in ${\mathbb{H}}_{+}^2\left(1\right)$, and

$${\kappa }_{\mathit{\beta }}^2=\frac{r^6}{d^4}{\kappa }^2-1.$$

Here ${\kappa }_{\mathit{\beta }}$ is the curvature of $\mathit{\beta }=\mathit{\beta }\left(s_{\mathit{\beta }}\right)$.

 Proof. 

Let $\mathit{\beta }\left(s\right)\in {\mathbb{S}}^2\left(1\right)$. Then, we have

$$\left.\begin{array}{c}\mathit{\beta }\left(s\right)=\frac{1}{r\left(s\right)}(\mathit{\alpha }\left(s\right)-{\mathbf{p}}_0);r^2\left(s\right)={∥\mathit{\alpha }\left(s\right)-{\mathbf{p}}_0∥}^2={(s+c)}^2-ϵd^2,\hfill \\ {∥\mathit{\beta }∥}^2=1,\mathrm{and}<\mathit{\beta },{\mathit{\beta }}^{{}^{\prime }}>=0.\hfill \end{array}\right\}$$

(18)

A simple calculation shows that

$${∥{\mathit{\beta }}^{{}^{\prime }}\left(s\right)∥}^2=-ϵ\frac{d^2}{r^4},$$

(19)

which lead to that $\mathit{\beta }\left(s\right)$ is a timelike or spacelike curve if $ϵ=1$ or $ϵ=-1$, respectively.

(i)

If $ϵ=-1$, the position vector $\mathit{\alpha }\left(s_{\beta }\right)-{\mathbf{p}}_0$ is in a spacelike rectifying plane. If $s_{\mathit{\beta }}$ be the pseudo-arc-length parameter of the curve $\mathit{\beta }\left(s\right)$, then

$$s_{\mathit{\beta }}=\stackrel{s}{\underset{0}{\int }}∥{\mathit{\beta }}^{{}^{\prime }}\left(s\right)∥ds=\left(\frac{d}{{(s+c)}^2+d^2}\right)ds={tan}^{-1}\left(\frac{s+c}{d}\right),$$

(20)

and thus $s+c=dtan{s}_{\mathit{\beta }}$. Using this into Equation (18) with $ϵ=-1$, we find

$$\mathit{\alpha }\left(s_{\mathit{\beta }}\right)-{\mathbf{p}}_0=\frac{d}{cos{s}_{\mathit{\beta }}}\mathit{\beta }\left(s_{\mathit{\beta }}\right).$$

(21)

Conversely, suppose that $\mathit{\alpha }\left(s_{\mathit{\beta }}\right)-{\mathbf{p}}_0$ is a curve specified by Equation (21), where $\mathit{\beta }\left(s_{\mathit{\beta }}\right)$ is a unit speed spacelike curve on ${\mathbb{S}}_1^2\left(1\right)$, that is,

$${∥\mathit{\beta }\left(s\right)∥}^2={∥{\mathit{\beta }}^{{}^{\prime }}\left(s\right)∥}^2=1,\mathrm{and}<\mathit{\beta },{\mathit{\beta }}^{{}^{\prime }}>=0.$$

From the differentiation of the Equation (21) with respect to $s_{\mathit{\beta }}$, we obtain

$${\left(\mathit{\alpha }-{\mathbf{p}}_0\right)}^{\prime }=\frac{d}{{cos}^2{s}_{\mathit{\beta }}}\left(\mathit{\beta }sin{s}_{\mathit{\beta }}+{\mathit{\beta }}^{{}^{\prime }}cos{s}_{\mathit{\beta }}\right).$$

(22)

Therefore, it follows that

$$<{\left(\mathit{\alpha }-{\mathbf{p}}_0\right)}^{\prime },\mathit{\alpha }-{\mathbf{p}}_0>=\frac{d^{2}sin{s}_{\mathit{\beta }}}{{cos}^3{s}_{\mathit{\beta }}},{∥{(\mathit{\alpha }-{\mathbf{p}}_0)}^{\prime }∥}^2=\frac{d^2}{{cos}^4{s}_{\mathit{\beta }}}.$$

(23)

Let us write

$$\mathit{\alpha }\left(s_{\mathit{\beta }}\right)-{\mathbf{p}}_0=\mu \left(s\right){(\mathit{\alpha }\left(s_{\beta }\right)-{\mathbf{p}}_0)}^{\prime }+{(\mathit{\alpha }\left(s_{\beta }\right)-{\mathbf{p}}_0)}^{\perp },$$

for function $\mu \left(s\right)$, where ${(\mathit{\alpha }\left(s_{\mathit{\beta }}\right)-{\mathbf{p}}_0)}^{\perp }$ is the normal component of the position vector $\mathit{\alpha }\left(s_{\mathit{\beta }}\right)-{\mathbf{p}}_0$. Via the last equations, we specified that

$$\mu \left(s\right)=\frac{<{(\mathit{\alpha }-{\mathbf{p}}_0)}^{\prime },\mathit{\alpha }-{\mathbf{p}}_0>}{{∥{(\mathit{\alpha }-{\mathbf{p}}_0)}^{\prime }∥}^2}=sin{s}_{\mathit{\beta }}cos{s}_{\mathit{\beta }}\ne const.$$

Therefore, we have

$${∥{(\mathit{\alpha }\left(s_{\mathit{\beta }}\right)-{\mathbf{p}}_0)}^{\perp }∥}^2={∥(\mathit{\alpha }\left(s_{\beta }\right)-{\mathbf{p}}_0)∥}^2-\frac{<{(\mathit{\alpha }-{\mathbf{p}}_0)}^{\prime },\mathit{\alpha }-{\mathbf{p}}_0{>}^2}{{∥{(\mathit{\alpha }-{\mathbf{p}}_0)}^{\prime }∥}^2}=d^2=const,$$

which lead to that $\mathit{\alpha }\left(s_{\mathit{\beta }}\right)-{\mathbf{p}}_0$ is a rectifying curve.

We will now compute the curvature of $\mathit{\beta }\left(s\right)$. By some calculations, we have:

$$\left.\begin{array}{c}<\mathit{\beta }\left(s\right),{\mathit{\beta }}^{{}^{{}^{\prime \prime }}}\left(s\right)>=-\frac{d^2}{r4},\hfill \\ <{\mathit{\beta }}^{{}^{\prime }}\left(s\right),{\mathit{\beta }}^{{}^{{}^{\prime \prime }}}\left(s\right)>=-\frac{2d^2}{r5}{r}^{{}^{\prime }},\hfill \\ {∥{\mathit{\beta }}^{\prime \prime }\left(s\right)∥}^2=\frac{4d^2{r}^{\prime 2}}{r^6}-\frac{d^4}{r^8}{\kappa }_{\mathit{\beta }}^2.\hfill \end{array}\right\}$$

(24)

When Equations (24) are used to Equations (14) we immediately specified that:

$${\kappa }^2:={∥{\mathit{\alpha }}^{\prime \prime }\left(s\right)∥}^2=\frac{d^4}{r^6}(1+{\kappa }_{\mathit{\beta }}^2).$$

(ii)

If $ϵ=1$, the position vector $\mathit{\alpha }\left(s_{\beta }\right)-{\mathbf{p}}_0$ is a spacelike position vector lying in a timelike rectifying plane. By Equations (18) and (19), we have

$${∥{\mathit{\beta }}^{{}^{\prime }}\left(s\right)∥}^2=-\frac{d^2}{{(s+c)}^2-d^2},\left|s+c\right|>d,$$

(25)

which shows that $\mathit{\beta }\left(s\right)$ is a timelike curve. Then,

$$s_{\mathit{\beta }}=\stackrel{s}{\underset{0}{\int }}\left(\frac{d}{d^2-{(s+c)}^2}\right)ds={coth}^{-1}\left(\frac{s+c}{d}\right)$$

(26)

and thus $s+c=-d{coth}_{\mathit{\beta }}$. Applying this in the Equation (18) with $ϵ=1$, we gain

$$\mathit{\alpha }\left(s_{\mathit{\beta }}\right)-{\mathbf{p}}_0=\frac{d}{sinh{s}_{\mathit{\beta }}}\mathit{\beta }\left(s_{\mathit{\beta }}\right).$$

(27)

Conversely, suppose that $\mathit{\alpha }\left(s_{\mathit{\beta }}\right)$ is a curve realized by Equation (27), where $\mathit{\beta }\left(s_{\beta }\right)$ is a unit speed timelike curve on ${\mathbb{S}}^2\left(1\right)$; that is,

$${∥\mathit{\beta }\left(s\right)∥}^2=-{∥{\mathit{\beta }}^{{}^{\prime }}\left(s\right)∥}^2=1,\mathrm{and}<\mathit{\beta },{\mathit{\beta }}^{{}^{\prime }}>=0.$$

By the differentiation of Equation (27) with respect to $s_{\mathit{\beta }}$, we get

$${\left(\mathit{\alpha }\left(s_{\beta }\right)-{\mathbf{p}}_0\right)}^{\prime }=\frac{d}{{sinh}^2{s}_{\mathit{\beta }}}\left({\mathit{\beta }}^{{}^{\prime }}sinh{s}_{\mathit{\beta }}-\mathit{\beta }cosh{s}_{\mathit{\beta }}\right).$$

(28)

Therefore, it follows that

$$<{\left(\mathit{\alpha }-{\mathbf{p}}_0\right)}^{\prime },\mathit{\alpha }\left(s_{\mathit{\beta }}\right)-{\mathbf{p}}_0>=-\frac{d^{2}cosh{s}_{\mathit{\beta }}}{{sinh}^3{s}_{\mathit{\beta }}},{∥{(\mathit{\alpha }-{\mathbf{p}}_0)}^{\prime }∥}^2=-\frac{d^2}{{sinh}^4{s}_{\mathit{\beta }}}.$$

(29)

Let us put

$$\mathit{\alpha }\left(s_{\mathit{\beta }}\right)-{\mathbf{p}}_0=\mu \left(s\right){(\mathit{\alpha }\left(s_{\beta }\right)-{\mathbf{p}}_0)}^{\prime }+{(\mathit{\alpha }\left(s_{\beta }\right)-{\mathbf{p}}_0)}^{\perp },$$

where $\mu \left(s\right)$ is a differentiable function, and ${(\mathit{\alpha }\left(s_{\mathit{\beta }}\right)-{\mathbf{p}}_0)}^{\perp }$ is the normal component of the position vector $\mathit{\alpha }\left(s_{\mathit{\beta }}\right)-{\mathbf{p}}_0$. Via the latest two equations, we realized that

$$\mu \left(s\right)=\frac{<{(\mathit{\alpha }-{\mathbf{p}}_0)}^{\prime },\mathit{\alpha }-{\mathbf{p}}_0>}{{∥{(\mathit{\alpha }-{\mathbf{p}}_0)}^{\prime }∥}^2}=sinh{s}_{\mathit{\beta }}cosh{s}_{\mathit{\beta }}\ne const.$$

Therefore, we have:

$${∥{(\mathit{\alpha }\left(s_{\mathit{\beta }}\right)-{\mathbf{p}}_0)}^{\perp }∥}^2={∥(\mathit{\alpha }\left(s_{\beta }\right)-{\mathbf{p}}_0)∥}^2-\frac{<{(\mathit{\alpha }-{\mathbf{p}}_0)}^{\prime },\mathit{\alpha }-{\mathbf{p}}_0{>}^2}{{∥{(\mathit{\alpha }-{\mathbf{p}}_0)}^{\prime }∥}^2}=-d^2=const,$$

which lead to that $\mathit{\alpha }\left(s_{\mathit{\beta }}\right)-{\mathbf{p}}_0$ is lying completely in a timelike rectifying plane. In like manner, we can gain:

$$\left.\begin{array}{c}<{\mathit{\beta }}^{\prime }\left(s\right),{\mathit{\beta }}^{{}^{\prime \prime }}\left(s\right)>=\frac{2d^2{r}^{\prime }}{r^5},\hfill \\ <\mathit{\beta }\left(s\right),{\mathit{\beta }}^{{}^{\prime \prime }}\left(s\right)>=\frac{d^2}{r^4},\hfill \\ {∥{\mathit{\beta }}^{\prime \prime }\left(s\right)∥}^2=-\frac{4d^2{r}^{\prime }}{r^6}+\frac{d^4}{r^8}{\kappa }_{\mathit{\beta }}^2,\hfill \end{array}\right\}$$

(30)

so that:

$${\kappa }_{\mathit{\beta }}^2=\frac{r^6}{d^4}{\kappa }^2+1.$$

(iii)

The proof is analogous to the proofs of (i) and (ii).

□

In consent with Case (i) in Theorem 2, we have:

 Proposition 1. 

The pseudo-arc-length of the unit speed spacelike curve $\mathit{\beta }\left(s\right)\in {\mathbb{S}}_1^2\left(1\right)$ is less than $\pi .$

 Proof. 

Let $(s_1,s_2)$ be the domain of $\mathit{\alpha }\left(s\right)$, then the pseudo-arc-length of $\mathit{\beta }\left(s\right)$ satisfies the following:

$$s_{\mathit{\beta }}={\int }_{s_1}^{s_2}∥{\mathit{\beta }}^{{}^{\prime }}\left(s\right)∥ds={tan}^{-1}\left(\frac{s_2+c}{\left|d\right|}\right)-{tan}^{-1}\left(\frac{s_1+c}{\left|d\right|}\right)<\frac{\pi }{2}-(-\frac{\pi }{2})=\pi .$$

□

If we consider timelike curve, then ${ϵ}_0=-1,$${ϵ}_1={ϵ}_2=1$. Thus, we have the next theorem which is analogous to the Theorem 2:

 Theorem 3. 

Let $\mathit{\alpha }=\mathit{\alpha }\left(s\right)$ be a unit speed timelike rectifying curve in ${\mathbb{E}}_1^3$. If ${\mathbf{p}}_0\in {\mathbb{E}}_1^3$ is a fixed point, then:

 (i) 

$\mathit{\alpha }\left(s_{\beta }\right)-{\mathbf{p}}_0$ is a spacelike position vector lying totally on a timelike rectifying plane iff, up to a representation, $\mathit{\alpha }\left(s_{\beta }\right)-{\mathbf{p}}_0$ is specified by

$$\mathit{\alpha }\left(s_{\beta }\right)-{\mathbf{p}}_0=\frac{d}{sinh{s}_{\beta }}\mathit{\beta }\left(s_{\beta }\right),$$

(31)

where $\mathit{\beta }\left(s_{\beta }\right)$ is a unit speed timelike curve on ${\mathbb{S}}_1^2\left(1\right)$, and there holds

$${\kappa }_{\mathit{\beta }}^2=1-\frac{r^6}{d^4}{\kappa }^2.$$

 (ii) 

$\mathit{\alpha }\left(s_{\beta }\right)-{\mathbf{p}}_0$ is a timelike position vector lying totally in a timelike rectifying plane iff, up to a representation, $\mathit{\alpha }\left(s_{\beta }\right)-{\mathbf{p}}_0$ is specified by

$$\mathit{\alpha }\left(s_{\beta }\right)-{\mathbf{p}}_0=\frac{d}{cosh{s}_{\beta }}\mathit{\beta }\left(s_{\beta }\right),$$

(32)

where $\mathit{\beta }\left(s_{\beta }\right)$ is a unit speed spacelike curve on ${\mathbb{H}}_{+}^2\left(1\right),$ and there holds

$${\kappa }_{\mathit{\beta }}^2=1-\frac{r^6}{d^4}{\kappa }^2.$$

4. A Differential Equation for a Non-Null Curve

In this section, we demonstrate that the height-tangential function for every non-null unit speed Frenet–Serret curve satisfies a differential equation of third order. By using this differential equation, we obtain some renowned properties of non-null spherical and rectifying curves in ${\mathbb{E}}_1^3$. Thus, we define a differentiable height-tangential function $h\left(s\right)$, where

$$h\left(s\right)=<\mathit{\alpha }\left(s\right),\mathbf{t}\left(s\right)>.$$

(33)

From now on, we will rarely write the parameter s.

 Proposition 2. 

The height function of any non-null unit speed Frenet–Serret curve $\mathit{\alpha }\left(s\right)$ satisfies the following third-order differential equation

$$\begin{array}{ccc}\hfill & & \rho \sigma h^{{}^{{}^{\prime \prime \prime }}}+\left(2{\rho }^{{}^{\prime }}\sigma +\rho {\sigma }^{{}^{\prime }}\right)h^{{}^{{}^{\prime \prime }}}+\left[{\left(\sigma {\rho }^{{}^{\prime }}\right)}^{{}^{\prime }}+{ϵ}_1\left({ϵ}_0\frac{\sigma }{\rho }+{ϵ}_2\frac{\rho }{\sigma }\right)\right]h^{{}^{\prime }}-{ϵ}_2{\left(\frac{\sigma }{\rho }\right)}^{{}^{\prime }}h\hfill \\ & =& {ϵ}_0{\left(\sigma {\rho }^{{}^{\prime }}\right)}^{{}^{\prime }}-\frac{\rho }{\sigma },\hfill \end{array}$$

(34)

where $\rho \kappa =1$, $\sigma \tau =1$.

 Proof. 

Let $\mathit{\alpha }\left(s\right)$ be a non-null unit speed Frenet–Serret curve. From Equations (1) and (33), it follows that

$$\rho \left(h^{{}^{\prime }}-{ϵ}_0\right)=<\mathit{\alpha }\left(s\right),\mathbf{n}\left(s\right)>,$$

(35)

which, once more on the differentiation, gives

$$\rho \sigma h^{{}^{\prime \prime }}+\sigma {\rho }^{{}^{\prime }}{h}^{{}^{\prime }}-{ϵ}_0\sigma {\rho }^{{}^{\prime }}-{ϵ}_2\frac{\sigma }{\rho }h=<\mathit{\alpha },\mathbf{b}>$$

(36)

In accordance with differentiation of Equation (36) and employing Equation (35), we have

$$\rho \sigma h^{{}^{{}^{\prime \prime \prime }}}+\left(2{\rho }^{{}^{\prime }}\sigma +\rho {\sigma }^{{}^{\prime }}\right)h^{{}^{{}^{\prime \prime }}}+\left[{\left(\sigma {\rho }^{{}^{\prime }}\right)}^{{}^{\prime }}+{ϵ}_1\left({ϵ}_0\frac{\sigma }{\rho }+{ϵ}_2\frac{\rho }{\sigma }\right)\right]h^{{}^{\prime }}-{ϵ}_2{\left(\frac{\sigma }{\rho }\right)}^{{}^{\prime }}h={ϵ}_0{\left(\sigma {\rho }^{{}^{\prime }}\right)}^{{}^{\prime }}-\frac{\rho }{\sigma },$$

as supposed.   □

Applications of Proposition 2

By employing Proposition 2, different well-known properties of non-null spherical and rectifying curves are in the following:

 Corollary 1. 

Any non-null unit speed Frenet–Serret curve $\mathit{\alpha }\left(s\right)$ is lying fully in ${\mathbb{S}}_1^2\left(1\right)$ (resp. ${\mathbb{H}}_{+}^2\left(1\right)$) iff it satisfies

$${ϵ}_0{\left(\sigma {\rho }^{{}^{\prime }}\right)}^2+{\rho }^2=c,{ϵ}_0\in \{1,-1\},$$

(37)

for some constant c.

 Proof. 

Let $\mathit{\alpha }=\mathit{\alpha }\left(s\right)$ be a non-null unit speed curve in $S_1^2\left(1\right)$ (resp. ${\mathbb{H}}_{+}^2\left(1\right)$). Then, we have

$$<\mathit{\alpha }\left(s\right),\mathit{\alpha }\left(s\right)>={ϵ}_0,\mathrm{and}h\left(s\right)=<\mathit{\alpha }\left(s\right),\mathbf{t}\left(s\right)>=0.$$

(38)

Then, the differentiation of Equation (34) reduces to

$${ϵ}_0{\left(\sigma {\rho }^{{}^{\prime }}\right)}^{{}^{\prime }}+\frac{\rho }{\sigma }=0.$$

Multiplying $2\sigma {\rho }^{{}^{\prime }}$ to this equation and integrating gives

$${ϵ}_0{\left(\sigma {\rho }^{{}^{\prime }}\right)}^2+{\rho }^2=c.$$

(39)

Thus $\mathit{\alpha }=\mathit{\alpha }\left(s\right)$ is a Frenet–Serret spherical curve in ${\mathbb{E}}_1^3$. The converse is clear.   □

 Corollary 2. 

Any non-null unit speed Frenet–Serret curve $\mathit{\alpha }\left(s\right)$ is a rectifying curve iff it satisfies

$$\left(s+c\right){\left(\frac{\kappa }{\tau }\right)}^{{}^{\prime }}+\frac{\kappa }{\tau }=0,$$

(40)

for some constant c.

 Proof. 

Let $\mathit{\alpha }=\mathit{\alpha }\left(s\right)$ be a non-null unit speed Frenet–Serret rectifying curve in ${\mathbb{E}}_1^3$. Then, in view of Equation (10), we have

$$h\left(s\right)={ϵ}_0(s+c),$$

(41)

Thus, Equation (34) reduces to

$$\frac{\sigma }{\rho }+(s+c){\left(\frac{\sigma }{\rho }\right)}^{{}^{\prime }}=0,$$

which lead to condition (40).

Conversely, if Equation (40) holds, then by integrating Equation (40), we find $\tau \overline{c}=\left(s+c\right)\kappa$ for a constant $\overline{c}$, which means that $\mathit{\alpha }\left(s\right)$ is a rectifying curve.   □

Since $\mathit{\alpha }\left(s\right)\in {\mathbb{E}}_1^3$ with $\kappa \left(s\right)>0$, and $\left\{\mathbf{t}\right(s),$$\mathbf{n}\left(s\right)$, $\mathbf{b}\left(s\right)\}$ is a Frenet–Serret frame along $\mathit{\alpha }\left(s\right)$, we put:

$$\mathit{\alpha }\left(s\right)=<\mathit{\alpha }\left(s\right),\mathbf{t}\left(s\right)>\mathbf{t}+<\mathit{\alpha },\mathbf{n}>\mathbf{n}+<\mathit{\alpha },\mathbf{b}>\mathbf{b}.$$

(42)

Consequently, the next corollary can be stated:

 Corollary 3. 

Let $\mathit{\alpha }=\mathit{\alpha }\left(s\right)$ be a non-null unit speed Frenet–Serret curve in ${\mathbb{E}}_1^3$. Then,

$${ϵ}_1{\left(<\mathit{\alpha },\mathbf{n}>\right)}^2+{ϵ}_2{\left(<\mathit{\alpha },\mathbf{b}>\right)}^2=a^2=const,$$

(43)

holds for a constant a iff $\mathit{\alpha }\left(s\right)$ is either spherical or a normal curve in ${\mathbb{E}}_1^3$.

 Proof. 

In view of with Equations (42) and (43), we obtain:

$${∥\mathit{\alpha }∥}^2={ϵ}_0<\mathit{\alpha }\left(s\right),\mathbf{t}\left(s\right){>}^2+a^2.$$

(44)

In accordance with differentiation of Equation (44) and employing Equations (33) and (35), we have

$$<\mathit{\alpha },\mathbf{t}>={ϵ}_{0}h{h}^{{}^{\prime }}\Rightarrow h={ϵ}_{0}h{h}^{{}^{\prime }}\Rightarrow h({ϵ}_0{h}^{{}^{\prime }}-1)=0.$$

(45)

Then, either $h=0$, or ${ϵ}_0{h}^{{}^{\prime }}-1=0$. Thus, we have either $h=0$, or $h={ϵ}_0\left(s+c\right)$ for a constant c. Thus, $\mathit{\alpha }\left(s\right)$ is either is a normal or spherical curve. The converse is obvious. □

5. Characterizations of Helices

In this section, various well-known characterizations of helices are consequences of the differential equation in Equation (34). Thus, we give the Minkowski form of the helices which has been realized in ${\mathbb{E}}^3$ as a curve for which <$\mathbf{t},\mathbf{u}$> is a nonzero constant, where $\mathbf{t}$ is the unit tangent vector of the curve $\mathit{\alpha }=\mathit{\alpha }\left(s\right)$ [3,5,15,22].

5.1. Spacelike Helix with a Spacelike Axis

Let $\mathit{\alpha }=\mathit{\alpha }\left(s\right)$ be a non-null unit speed Frenet–Serret curve in ${\mathbb{E}}_1^3$. Then, a spacelike helix with a spacelike axis is specified by

$$<\mathbf{u},\mathbf{t}\left(s\right)>=a=const.,\mathrm{with}{∥\mathbf{u}∥}^2=1,$$

where $\mathbf{u}$ is the constant axis of $\mathit{\alpha }\left(s\right)$

 Theorem 4. 

Let $\mathit{\alpha }=\mathit{\alpha }\left(s\right)$ be a non-null unit speed Frenet–Serret curve in ${\mathbb{E}}_1^3$. Then:

 (i) 

$\mathit{\alpha }\left(s\right)$ is a spacelike helix with a spacelike principal normal and a spacelike axis, iff, with a suitable pseudo-arc-length parameter s, $h\left(s\right)$ satisfies

$${\left(\rho h^{{}^{\prime }}\right)}^{{}^{\prime }}-\left(\frac{\sigma }{\rho }-\frac{\rho }{\sigma }\right)\tau h-{\rho }^{{}^{\prime }}+\frac{s\rho }{{\sigma }^2}=0,$$

(46)

 (ii) 

$\mathit{\alpha }\left(s\right)$ is a spacelike helix with a timelike principal normal and a spacelike axis, iff, with a suitable pseudo-arc-length parameter s, $\left(s\right)$ satisfies

$${\left(\rho h^{{}^{\prime }}\right)}^{{}^{\prime }}-\left(\frac{\sigma }{\rho }+\frac{\rho }{\sigma }\right)\tau h-{\rho }^{{}^{\prime }}+\frac{s\rho }{{\sigma }^2}=0,$$

(47)

 Proof. 

(i)

According to the hypothesis, we have

$$<\mathbf{u},\mathbf{t}\left(s\right)>=cosh\vartheta =const.,<\mathbf{u},\mathbf{n}>=0,\mathrm{with}{∥\mathbf{u}∥}^2=1,$$

(48)

for a given real number $\vartheta$. Consequently, we have

$$\left.\begin{array}{c}\mathbf{u}=cosh\vartheta \mathbf{t}+sinh\vartheta \mathbf{b},\hfill \\ \kappa cosh\vartheta +\tau sinh\vartheta =0.\hfill \end{array}\right\}$$

(49)

Since <$\mathbf{u},\mathit{\alpha }$>${}^{{}^{\prime }}=cosh\vartheta$, we have

$$<\mathbf{u},\mathit{\alpha }>=scosh\vartheta +\overline{c},$$

(50)

for a constant $\overline{c}$. Now, from Equations (49) and (50), we find

$$<\mathbf{t},\mathit{\alpha }>=s+c+<\mathit{\alpha },\mathbf{b}>\frac{\kappa }{\tau },\mathrm{with}c=\frac{\overline{c}}{cosh\vartheta }.$$

(51)

Joining this equation with Equation (33) begins

$$h=s+c+<\mathit{\alpha },\mathbf{b}>\left(\frac{\kappa }{\tau }\right).$$

(52)

By the differentiation of Equation (52) and employing Equation (49), we get

$$\rho \left(h^{{}^{\prime }}-1\right)-<\mathit{\alpha }\left(s\right),\mathbf{n}\left(s\right)>=0,$$

(53)

which once more by differentiation yields

$$\rho h^{{}^{\prime \prime }}+{\rho }^{{}^{\prime }}\left(h^{{}^{\prime }}-1\right)=-\kappa <\mathit{\alpha }\left(s\right),\mathbf{t}\left(s\right)>+\tau <\mathit{\alpha },\mathbf{b}>.$$

(54)

If Equations (51) and (52) are utilized to Equation (54), then

$${\left(\rho h^{{}^{\prime }}\right)}^{{}^{\prime }}+\left(\frac{\sigma }{\rho }-\frac{\rho }{\sigma }\right)\tau h-{\rho }^{{}^{\prime }}+\frac{\rho }{{\sigma }^2}\left(s+c\right)=0,$$

(55)

which, when combined with a suitable pseudo-arc-length parameter s, yields Equation (46).

Conversely, let us assume that $\mathit{\alpha }\left(s\right)$ is a non-null unit speed spacelike helix with a spacelike principal normal that satisfies Equation (46). Thus, by the differentiation of Equation (55) we established

$$\begin{array}{ccc}\hfill & & \rho \sigma h^{{}^{{}^{\prime \prime \prime }}}+\left(2{\rho }^{{}^{\prime }}\sigma +\rho {\sigma }^{{}^{\prime }}\right)h^{{}^{{}^{\prime \prime }}}+\left[{\left(\sigma {\rho }^{{}^{\prime }}\right)}^{{}^{\prime }}+\left(\frac{\sigma }{\rho }-\frac{\rho }{\sigma }\right)\right]h^{{}^{\prime }}+{\left(\frac{\sigma }{\rho }+\frac{\rho }{\sigma }\right)}^{{}^{\prime }}h\hfill \\ & =& {\left(\sigma {\rho }^{{}^{\prime }}\right)}^{{}^{\prime }}-{\left(\frac{\rho }{\sigma }\right)}^{{}^{\prime }}\left(s+c\right)-\frac{\rho }{\sigma }.\hfill \end{array}$$

(56)

By comparing Equation (56) with Equation (34) in Proposition 2, then

$${\left(\frac{\rho }{\sigma }\right)}^{{}^{\prime }}\left(s+c-h\right)=0.$$

(57)

If $s+c-h=0,$ then Equation (55) reduces to

$$\left(\frac{\sigma }{\rho }-\frac{\rho }{\sigma }\right)\left(s+c\right)+\frac{\rho }{\sigma }\left(s+c\right)=0,$$

(58)

from which we have $\sigma =0$, which is not feasible since $\mathit{\alpha }\left(s\right)$ is assumed to be a Frenet–Serret curve. Then, we obtain ${\left(\frac{\rho }{\sigma }\right)}^{{}^{\prime }}=0$ from Equation (57), which leads to $\mathit{\alpha }\left(s\right)$ being a helix.

(ii)

As outlined in the last case, we have

$$<\mathbf{u},\mathbf{t}\left(s\right)>=cos\vartheta =const.,<\mathbf{u},\mathbf{n}>=0,\mathrm{with}{∥\mathbf{u}∥}^2=1,$$

(59)

for a constant angle $\vartheta \ne \pi /2$. Thus, we can write that

$$\left.\begin{array}{c}\mathbf{u}=cos\vartheta \mathbf{t}+sin\vartheta \mathbf{b},\hfill \\ \kappa cos\vartheta +\tau sin\vartheta =0.\hfill \end{array}\right\}$$

(60)

Since <$\mathbf{u},\mathit{\alpha }$>${}^{{}^{\prime }}=cos\vartheta$ holds, we gain

$$<\mathbf{u},\mathit{\alpha }>=scos\vartheta +\overline{c},$$

(61)

for constant $\overline{c}$. Form Equations (60) and (61), we find

$$<\mathbf{t},\mathit{\alpha }>=s+c+<\mathit{\alpha },\mathbf{b}>\frac{\kappa }{\tau },\mathrm{with}c=\frac{\overline{c}}{cos\vartheta }.$$

(62)

Joining this equation with Equation (13) yields

$$h=s+c+<\mathit{\alpha },\mathbf{b}>\left(\frac{\kappa }{\tau }\right)$$

(63)

By the differentiation of Equation (63) and employing Equation (60), we find

$$\rho \left(h^{{}^{\prime }}-1\right)-<\mathit{\alpha }\left(s\right),\mathbf{n}\left(s\right)>=0,$$

(64)

which once more by differentiation leads to

$$\rho h^{{}^{\prime \prime }}+{\rho }^{{}^{\prime }}\left(h^{{}^{\prime }}-1\right)=\kappa <\mathit{\alpha }\left(s\right),\mathbf{t}\left(s\right)>+\tau <\mathit{\alpha },\mathbf{b}>.$$

(65)

We realize that when Equations (62) and (63) are applied to Equation (52), we get:

$${\left(\rho h^{{}^{\prime }}\right)}^{{}^{\prime }}-\left(\frac{\sigma }{\rho }+\frac{\rho }{\sigma }\right)\tau h-{\rho }^{{}^{\prime }}+\frac{\rho }{{\sigma }^2}\left(s+c\right)=0,$$

(66)

which, with a suitable pseudo-arc-length parameter s, lead to Equation (47).

Conversely, let us assume that $\mathit{\alpha }\left(s\right)$ is a non-null unit speed spacelike helix with a timelike principal normal that satisfies Equation (47). Then, by the differentiation of Equation (66) we find that

$$\begin{array}{ccc}\hfill & & \rho \sigma h^{{}^{{}^{\prime \prime \prime }}}+\left(2{\rho }^{{}^{\prime }}\sigma +\rho {\sigma }^{{}^{\prime }}\right)h^{{}^{{}^{\prime \prime }}}+\left[{\left(\sigma {\rho }^{{}^{\prime }}\right)}^{{}^{\prime }}-\left(\frac{\sigma }{\rho }+\frac{\rho }{\sigma }\right)\right]h^{{}^{\prime }}-{\left(\frac{\sigma }{\rho }+\frac{\rho }{\sigma }\right)}^{{}^{\prime }}h\hfill \\ & =& {\left(\sigma {\rho }^{{}^{\prime }}\right)}^{{}^{\prime }}-{\left(\frac{\rho }{\sigma }\right)}^{{}^{\prime }}\left(s+c\right)-\frac{\rho }{\sigma }.\hfill \end{array}$$

(67)

Contrasting Equation (67) with Equation (34) in Proposition 2 leads to

$${\left(\frac{\rho }{\sigma }\right)}^{{}^{\prime }}\left(s+c-h\right)=0.$$

(68)

If $s+c=h$ holds, then Equation (47) becomes

$$-\left(\frac{\sigma }{\rho }+\frac{\rho }{\sigma }\right)\left(s+c\right)+\frac{\rho }{\sigma }\left(s+c\right)=0,$$

(69)

which leads to $\sigma =0$, which is a contradiction, since $\mathit{\alpha }\left(s\right)$ is supposed to be a Frenet–Serret curve. Thus, from Equation (68), we get ${\left(\frac{\rho }{\sigma }\right)}^{{}^{\prime }}=0$, which means that is $\mathit{\alpha }\left(s\right)$ a helix.    □

5.2. Spacelike Helix with a Timelike Axis

Assume that $\mathit{\alpha }=\mathit{\alpha }\left(s\right)$ is a unit speed spacelike curve with a spacelike principal normal and a timelike axis in ${\mathbb{E}}_1^3$. Therefore, a spacelike helix with a timelike axis in ${\mathbb{E}}_1^3$ is specified by

$$<\mathbf{u},\mathbf{t}\left(s\right)>=a=const.,\mathrm{with}{∥\mathbf{u}∥}^2=-1,$$

(70)

where $\mathbf{u}$ is the axis of $\mathit{\alpha }\left(s\right)$.

 Theorem 5. 

If $\mathit{\alpha }\left(s\right)$ is a unit speed spacelike curve such that $\kappa \left(s\right)>0$, then $\mathit{\alpha }\left(s\right)$ is a spacelike helix with a timelike axis and a spacelike principal normal iff, with a suitable pseudo-arc-length parameter s, the function $h\left(s\right)$ satisfies

$${\left(\rho h^{{}^{\prime }}\right)}^{{}^{\prime }}+\left(\frac{\sigma }{\rho }-\frac{\rho }{\sigma }\right)\tau h-{\rho }^{{}^{\prime }}+\frac{s\rho }{{\sigma }^2}=0.$$

(71)

 Proof. 

According to the hypothesis, we have

$$<\mathbf{u},\mathbf{t}\left(s\right)>=sinh\vartheta =const.,<\mathbf{u},\mathbf{n}\left(s\right)>=0,\mathrm{with}{∥\mathbf{u}∥}^2=-1,$$

(72)

for any given real number $\vartheta \ge 0$. Thus, we have

$$\left.\begin{array}{c}\mathbf{u}=sinh\vartheta \mathbf{t}+cosh\vartheta \mathbf{b},\hfill \\ \kappa sinh\vartheta +\tau cosh\vartheta =0.\hfill \end{array}\right\}$$

(73)

Since $<\mathbf{u},\mathit{\alpha }{>}^{{}^{\prime }}=sinh\vartheta$ holds, we have

$$<\mathbf{u},\mathit{\alpha }>=ssinh\vartheta +\overline{c},$$

(74)

for some constant $\overline{c}$. Form Equations (73) and (74), we realize

$$<\mathbf{t},\mathit{\alpha }>=s+c-<\mathit{\alpha },\mathbf{b}>\frac{\kappa }{\tau },\mathrm{with}c=\frac{\overline{c}}{sinh\vartheta }.$$

(75)

Joining this equation with Equation (33) yields

$$h=s+c+<\mathit{\alpha },\mathbf{b}>\left(\frac{\kappa }{\tau }\right).$$

(76)

By the differentiation of the Equation (76) and using Equation (74), we get:

$$\rho \left(h^{{}^{\prime }}-1\right)-<\mathit{\alpha }\left(s\right),\mathbf{n}\left(s\right)>=0,$$

(77)

which again by the differentiation gives us

$$\rho h^{{}^{\prime \prime }}+{\rho }^{{}^{\prime }}\left(h^{{}^{\prime }}-1\right)=-\kappa <\mathit{\alpha }\left(s\right),\mathbf{t}\left(s\right)>+\tau <\mathit{\alpha },\mathbf{b}>.$$

(78)

When Equations (75) and (76) are used with Equation (78), we get:

$${\left(\rho h^{{}^{\prime }}\right)}^{{}^{\prime }}+\left(\frac{\sigma }{\rho }-\frac{\rho }{\sigma }\right)\tau h-{\rho }^{{}^{\prime }}+\frac{\rho }{{\sigma }^2}\left(s+c\right)=0,$$

(79)

which, with a suitable pseudo-arc-length parameter s, leads to Equation (71).

Conversely, suppose that $\mathit{\alpha }\left(s\right)$ is a non-null unit speed spacelike helix with a timelike axis and a timelike principal normal satisfies Equation (71). Then, by differentiation of Equation (79) we obtain

$$\begin{array}{ccc}\hfill & & \rho \sigma h^{{}^{{}^{\prime \prime \prime }}}+\left(2{\rho }^{{}^{\prime }}\sigma +\rho {\sigma }^{{}^{\prime }}\right)h^{{}^{{}^{\prime \prime }}}+\left[{\left(\sigma {\rho }^{{}^{\prime }}\right)}^{{}^{\prime }}+\left(\frac{\sigma }{\rho }-\frac{\rho }{\sigma }\right)\right]h^{{}^{\prime }}+{\left(\frac{\sigma }{\rho }-\frac{\rho }{\sigma }\right)}^{{}^{\prime }}h\hfill \\ & =& -{\left(\sigma {\rho }^{{}^{\prime }}\right)}^{{}^{\prime }}\left(s+c\right)-\frac{\rho }{\sigma }.\hfill \end{array}$$

(80)

Contrasting Equation (80) with Equation (34) in Proposition 2, the result is obvious.   □

6. Conclusions

In order to demonstrate that the ratio of torsion and curvature of any non-null rectifying curve is a non-constant linear function of the arc-length parameter, we have built a novel mathematical framework in Minkowski 3-space ${\mathbb{E}}_1^3$. Then, by taking into consideration a curve’s causal characteristics, we provide some parameterization for rectifying curves. The tangential height function of every non-null curve is then shown to satisfy a third-order differential equation. Then, as a result of this differential equation, various well-known and novel characterizations of spherical, normal, and rectifying curves are obtained. Hopefully, these results will be advantageous in the field of differential geometry and to physicists and others exploring the general relativity theory. There are numerous opportunities for additional work. The Galilean, pseudo-Galilean, isotrobic, and higher dimensional spaces can be considered as a counterpart to the problem presented in the current study.