# On Special Kinds of Involute and Evolute Curves in 4-Dimensional Minkowski Space

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## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

## 3. The (0,2)-Involute Curve of a Given Curve in ${\mathit{E}}_{\mathbf{1}}^{\mathbf{4}}$

**Theorem**

**1.**

**Proof.**

**Case 1:**$\phi \ne 0$. In this case, $\nu \ne 0$. $\frac{\mu}{\nu}={t}_{1}$ implies that $\mu ={t}_{1}\nu $ and

**Case 2:**If $\phi =0$, we have the following theorem.

**Theorem**

**2.**

**Remark**

**1.**

## 4. The (1,3)-Evolute Curve of a Given Curve in ${\mathit{E}}_{\mathbf{1}}^{\mathbf{4}}$

**Theorem**

**3.**

**Proof.**

**Case 1:**$t\ne 0$. By differentiating (44) about s and using the Frenet Formula (1), we get

**Case 2:**If $t=0$, we have the following theorem.

**Theorem**

**4.**

**Proof.**

**Remark**

**2.**

## 5. The (1,3)-Evolute Curve of a Cartan Null Curve in ${\mathit{E}}_{\mathbf{1}}^{\mathbf{4}}$

**Theorem**

**5.**

**Proof.**

**Case 1:**$t\ne 0$. By differentiating (75) about s and using the Frenet Formula (2), we get

**Case 2:**For $t=0$, we have the following theorem.

**Theorem**

**6.**

**Proof.**

**Remark**

**3.**

**Condition 2:**

**Theorem**

**7.**

**Proof.**

**Case 1**: $t\ne 0$. By differentiating (106) about s and using the Frenet Formula (2), we get

**Case 2:**For $t=0$, we have the following theorem.

**Theorem**

**8.**

**Proof.**

**Remark**

**4.**

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**MDPI and ACS Style**

Hanif, M.; Hou, Z.H.; Nisar, K.S.
On Special Kinds of Involute and Evolute Curves in 4-Dimensional Minkowski Space. *Symmetry* **2018**, *10*, 317.
https://doi.org/10.3390/sym10080317

**AMA Style**

Hanif M, Hou ZH, Nisar KS.
On Special Kinds of Involute and Evolute Curves in 4-Dimensional Minkowski Space. *Symmetry*. 2018; 10(8):317.
https://doi.org/10.3390/sym10080317

**Chicago/Turabian Style**

Hanif, Muhammad, Zhong Hua Hou, and Kottakkaran Sooppy Nisar.
2018. "On Special Kinds of Involute and Evolute Curves in 4-Dimensional Minkowski Space" *Symmetry* 10, no. 8: 317.
https://doi.org/10.3390/sym10080317