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# Generalized involute and evolute curves of equiform spacelike curves with a timelike equiform principal normal in \(E_{1}^{3}\)

*Journal of the Egyptian Mathematical Society*
**volume 28**, Article number: 26 (2020)

## Abstract

Equiform geometry is considered as a generalization of the other geometries. In this paper, involute and evolute curves are studied in the case of the curve *α* is an equiform spacelike with a timelike equiform principal normal vector *N*. Furthermore, the equiform frames of the involute and evolute curves are obtained. Also, the equiform curvatures of the involute and evolute curves are obtained in Minkowski 3-space.

## Introduction

In the last two decades, curves in Minkowski space have been studied by many mathematicians such as [1–3]. Specially, involute and evolute curves got an interest from alot of mathematicians in Minkowski 3-space. According to the usual Frenet frame, involute and evolute curves in Minkowski 3-space \(E_{1}^{3}\) were defined and studied in [1, 2, 4, 5]. The equiform geometry was defined in different spaces such as Galilean space [6], pseudo-Galilean space [7], Euclidean space [8], isotropic space [9], and Minkowski space [3, 10, 11].

In this paper, we firstly introduce the equiform parameter, the equiform frame, and the equiform formulas in the case of equiform spacelike curves with a timelike equiform principal normal in Minkowski space \(E_{1}^{3}\). Secondly, we introduce the involute and the evolute of the equiform spacelike curve with a timelike equiform principal normal. Further, the equiform frames for the involute and the evolute curves are obtained. Also, the equiform curvatures of the involute and the evolute curves are obtained.

## Preliminaries

The three-dimensional Lorentzian space, or Minkowski 3-space \(E_{1}^{3}\), is the space *R*^{3} equipped with the metric *g* defined as:

where *U*=(*u*_{1},*u*_{2},*u*_{3}) and *V*=(*v*_{1},*v*_{2},*v*_{3}) are any two vectors in *R*^{3}. The vector *U* in \(E_{1}^{3}\) may be lightlike if *g*(*U*,*U*)=0 and *U*≠0 or spacelike if *g*(*U*,*U*)>0 or *U*=0 or timelike if *g*(*U*,*U*)<0. The norm (length) of the vector *U* is defined by \(||\, U\, ||=\sqrt {|\,g(U,U)\,|}\).

The Lorentzian cross product is given by:

where *U* and \(V\in E_{1}^{3}\) [12, 13].

A differentiable map \(\alpha : I \subset R \rightarrow E_{1}^{3}\) is called smooth curve in Minkowski 3-space, where *I* is an open interval. Suppose that {*t*(*s*),*n*(*s*),*b*(*s*)} be the orthonormal Frenet frame along the curve *α*(*s*), where *t*(*s*),*n*(*s*), and *b*(*s*) are the tangent, the principal normal, and the binormal vectors of the curve *α*, respectively.

Any curve *α* in Minkowski 3-space can be one of the following cases and below the corresponding Frenet formulas:

(1) *α* is a spacelike curve with(i) a spacelike principal normal, then Frenet formulas are given by:

(ii) a timelike principal normal, then Frenet formulas are given by:

(iii) a null (lightlike) principal normal, then Frenet formulas are given by:

(2) *α* is a timelike curve, then Frenet formulas are given by:

[13]

(3) *α* is a lightlike curve, then Frenet formulas are given by:

[13]

The equiform geometry has minor importance related to usual one, and the curves that appear here in equiform geometry can be seen as a generalization of well-known curves from other geometries.

Let *γ*(*s*)=*t*(*s*) be the spherical tangent indicatrix of the curve *α* and *σ* be an arc length parameter of *γ*. We can make a reparameterization of *α* by the parameter *σ*, \(\alpha =\alpha (\sigma):I\rightarrow E_{1}^{3}\), the parameter *σ* is called the equiform parameter, of the curve *α*(*σ*).

Let *σ* be the arc length parameter of spherical tangent indicatrix *ζ*, then we have:

By integration with respect to *s*, we have:

where *ρ* is the radius of curvature of *α* [11].

Let *T*, *N*, and *B* be the orthogonal equiform frame along the curve *α*(*σ*) in Minkowski 3-space, where *T*, *N*, and *B* are the equiform-tangent, the equiform-normal, and the equiform-binomial vectors of the curve *α*(*σ*), respectively. They are given by \(T=\frac {d\alpha }{d\sigma }=\rho t\), *N*=*ρ**n*, *b*=*ρ**b* [10, 11].

The function *K*_{1}:*I*→*R* defined by \(K_{1}=\frac {d\rho }{ds}\) is called the first equiform curvature of *α*(*σ*), and the function *K*_{2}:*I*→*R* defined by \(K_{2}=\frac {\tau }{\kappa }\) is the called second equiform curvature of *α*(*σ*).

###
**Definition 1**

A curve *α*(*σ*) is an equiform spacelike if *g*(*T*,*T*)=*ρ*^{2}>0 or *T*=0, equiform timelike if *g*(*T*,*T*)=−*ρ*^{2}<0, or equiform null if *g*(*T*,*T*)=0 and *T*≠0.

If *α*(*σ*) is an equiform spacelike with a timelike equiform principal normal vector, then the equiform formulas are given in [10] by:

where

###
**Lemma 1**

Suppose that a curve *α* is an equiform spacelike with a timelike equiform principal normal *N*. If *α*(*σ*) is parameterized by the equiform parameter *σ*, then:

###
**Lemma 2**

If a curve *α*(*σ*^{∗}) is an equiform spacelike with a timelike equiform principal normal *N* and *σ*^{∗} is not necessary the equiform parameter of the curve *α*, then:

###
**Lemma 3**

Suppose that a curve *α*is an equiform spacelike with a timelike equiform principal normal *N*. Then, the equiform curvatures are given by:

###
**Definition 2**

A curve *α*(*σ*) is an ordinary helix if the second equiform curvature *K*_{2}=0, and it is a general helix if *K*_{2} is constant.

## The involute of an equiform spacelike curve with a timelike equiform principal normal

In this section, we study the involute curve of the equiform spacelike curve with a timelike equiform principal normal vector *N* in \(E_{1}^{3}\). Also, the equiform frame of the involute curve is introduced. Furthermore, the equiform curvatures of the involute curve are obtained.

###
**Definition 3**

Let *α*(*σ*) be an equiform spacelike curve with a timelike equiform principal normal and a curve *β*(*σ*) be given, then the curve *β* is called an involute of the curve *α*, if the tangent at the point *α*(*σ*)to the curve *α*passes through the tangent at the point *β*(*σ*) to the curve *β*. In the other words, *β*(*σ*)is an involute of *α*(*σ*)if the equation *g*(*T*,*T*^{∗})=0 is satisfied. *β*(*σ*) can be written in terms of the curve *α* as:

Let the equiform frames of the curve *α*(*σ*) and *β*(*σ*) be {*T*,*N*,*B*} and {*T*^{∗},*N*^{∗},*B*^{∗}}, respectively.

###
**Theorem 1**

Let *α*(*σ*) be an equiform spacelike curve with a timelike equiform principal normal and suppose that a curve *β*(*σ*) is the involute of the curve *α*. Then,

where *c* is constant.

###
*Proof*

Suppose that *β*(*σ*) is the involute of *α*(*σ*). Then, we can write *β*(*σ*) as:

By taking the derivative of Eq. (1), with respect to *σ*, we have:

Since *g*(*T*^{∗},*T*)=0, then we have the differential equation:

Hence,

From Eqs. (1) and (2), we obtain:

□

###
**Corollary 1**

The distance between the curve *α*(*σ*) and its involute *β*(*σ*) is | *c*−*s* |.

###
**Theorem 2**

Let *α*(*σ*) be an equiform spacelike curve with a timelike equiform principal normal and suppose that a curve *β* is an involute of the curve *α*, then:

###
*Proof*

By taking the derivative of Eq. (3) with respect to *σ*, we have:

Then,

Let us assume that:

By taking the derivative of Eq. (4) with respect to *σ*, we have:

Hence, we have:

Since \(N^{*}=- \frac {B^{*}\wedge T^{*}}{\rho ^{*}}\), then we obtain:

□

###
**Corollary 2**

If *α*(*σ*) is an equiform spacelike curve with a timelike equiform principal normal vector, then its involutes are equiform timelike curves.

###
**Theorem 3**

Let *β*(*σ*)be an involute of the curve *α*(*σ*), and \(K_{1}^{*}\), \(K_{2}^{*}\) be the first and second equiform curvatures of the curve *β*, respectively. Then, \(K_{1}^{*}\) and \(K_{2}^{*}\) are given respectively by:

###
*Proof*

Since \(T^{*}=\frac {d\beta }{d \sigma ^{*}}=\frac {d\beta }{d \sigma } \frac {d \sigma }{\sigma ^{*}}.\) Using Eq. (4), we obtain:

By taking the derivative Eq. (5) and using Eq. (11), we get:

Therefore, the first equiform curvature \(K_{1}^{*}=\frac {-g(T^{*\prime },T^{*})}{ \rho ^{*2}}\) is given by:

Now, suppose that Eq. (10) is:

where \(a=\frac {\rho ^{*}}{\rho \sqrt {K_{2}^{2}+1}}.\) Taking the derivative of the above equation with respect to *σ*^{∗}, we have:

Thus, the second equiform curvature \(K_{2}^{*}=\frac {g(N^{*\prime },B^{*})}{\rho ^{*2} }\) is given by:

□

###
**Corollary 3**

If *α*(*σ*) is an equiform spacelike curve with a timelike equiform principal normal *N* and *β*(*σ*) is an involute of *α*(*σ*), then:

- 1.
If

*α*(*σ*) is a planar curve, then*β*(*σ*) is also planar. - 2.
If

*α*(*σ*) is an ordinary helix (*K*_{2}=0), then*β*(*σ*) is planar. - 3.
If

*α*(*σ*) is a general helix (*K*_{2}=*c*), then*β*(*σ*) is planar.

###
*Proof*

The proofs come forward from the equation of \(K_{2}^{*}\). □

## The evolute of equiform spacelike curve with a timelike equiform principal normal

In this section, the evolute curves of the equiform spacelike curve with a timelike equiform principal normal *N* are studied in \(E_{1}^{3}\). Moreover, the equiform frame of the evolute curve is introduced. Furthermore, the equiform curvatures of the evolute are computed.

###
**Definition 4**

Let *α*(*σ*) be an equiform spacelike with a timelike equiform principal normal and a curve *γ* with the same interval be given. For ∀*σ*∈*I*, if the tangent at the point *γ*(*σ*) to the curve *γ*(*σ*)passes through the point *α*(*s*) and

then *γ*(*σ*) is called an evolute of the curve *α*(*σ*).

Let the Frenet frame of the curve *α*(*σ*) and *γ*(*σ*) be {*T*,*N*,*B*} and {*T*^{∗},*N*^{∗},*B*^{∗}}, respectively.

###
**Theorem 4**

Let *α*(*σ*) be an equiform spacelike with a timelike equiform principal normal and a curve *γ*(*σ*) be an evolute of *α*, then:

###
*Proof*

Suppose that a curve *γ*(*σ*) be the evolute of the curve *α*(*σ*). Then, the vector *γ*(*σ*)−*α*(*σ*) is perpendicular to the vector *T*(*σ*). Then,

By taking the derivative of Eq. (12) with respect to *σ*, we have:

Then, we get:

Since \(g\left (\frac {d \gamma }{d \sigma },T\right)=0\), then we have:

and hence,

From Eqs. (12) and (14), the vector \(\frac {d \gamma }{d \sigma }\) is parallel to the vector *γ*−*α*, and we have:

Also, we have:

By taking the integration of the last equation, we get:

Hence, we find:

By substituting from Eqs. (13) and (15) into Eq. (12), we have:

Since,

and

moreover, we get:

□

###
**Theorem 5**

Let \(\gamma :I\rightarrow E_{1}^{3}\) be the evolute curve of the equiform spacelike curve \(\alpha :I\rightarrow E_{1}^{3}\). Then, the equiform frame of the curve *γ* is given by:

where \(z=\left (\int K_{2}d\sigma +c\right).\)

###
*Proof*

By similar proof of Theorem 2, we obtain the required. □

###
**Corollary 4**

If the curve *α* is a equiform spacelike curve with a timelike equiform principal normal, then its evolutes are equiform timelike curves.

###
*Proof*

The proof comes forward from Theorem 5. □

###
**Theorem 6**

Let the curve *γ*be an evolute of the curve *α* and let \(K_{1}^{*}\) and \(K_{2}^{*}\) be the first and second equiform curvetures of the curve *γ*. Then,

###
*Proof*

By taking the derivative of *N*^{∗} with respect to *σ*^{∗}, we have:

Then,

Therefore,

Also,

Thus,

Therefore,

□

###
**Corollary 5**

If the curve *α*(*σ*) is planar, then its evolute curve *β*(*σ*) is also planar.

## Availability of data and materials

The datasets used and/or analyzed during the current study are available from the corresponding author on reasonable request.

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Elsharkawy, A. Generalized involute and evolute curves of equiform spacelike curves with a timelike equiform principal normal in \(E_{1}^{3}\).
*J Egypt Math Soc* **28, **26 (2020). https://doi.org/10.1186/s42787-020-00086-4

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### Keywords

- Minkowski 3-space
- Involute
- Evolute
- Equiform geometry
- Equiform curvatures

### AMS Subject Classification

- 53A35
- 53C50