# On Some Quasi-Curves in Galilean Three-Space

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## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

## 3. Quasi-Frame and Quasi-Equations in ${\mathbb{G}}_{\mathbf{3}}$

**Corollary**

**1.**

**Corollary**

**2.**

## 4. Quasi-Bertrand Curves in ${\mathbb{G}}_{\mathbf{3}}$

**Definition**

**1.**

**Theorem**

**1.**

**Proof.**

**Corollary**

**3.**

**Theorem**

**2.**

**Proof.**

**Corollary**

**4.**

**Proof.**

**Theorem**

**3.**

**Proof.**

## 5. Quasi-Mannheim Curves in ${\mathbb{G}}_{\mathbf{3}}$

**Definition**

**2.**

**Theorem**

**4.**

**Proof.**

**Corollary**

**5.**

**Theorem**

**5.**

**Proof.**

**Corollary**

**6.**

**Proof.**

## 6. Quasi-Involute Curves in ${\mathbb{G}}_{\mathbf{3}}$

**Definition**

**3.**

**Theorem**

**6.**

**Proof.**

**Theorem**

**7.**

**Proof.**

**Corollary**

**7.**

**Theorem**

**8.**

**Proof.**

**Corollary**

**8.**

## 7. Quasi-Evolute Curves in ${\mathbb{G}}_{\mathbf{3}}$

**Definition**

**4.**

**Theorem**

**9.**

**Proof.**

## 8. Quasi-Smarandache Curves in ${\mathbb{G}}_{\mathbf{3}}$

**Definition**

**5.**

#### 8.1. $T{N}_{q}$-Smarandache Curve in ${\mathbb{G}}_{3}$

**Theorem**

**10.**

**Proof.**

**Corollary**

**9.**

**Theorem**

**11.**

**Proof.**

**Corollary**

**10.**

**Corollary**

**11.**

#### 8.2. $T{B}_{q}$-Smarandache Curve in ${\mathbb{G}}_{3}$

**Theorem**

**12.**

**Corollary**

**12.**

**Theorem**

**13.**

**Corollary**

**13.**

**Corollary**

**14.**

#### 8.3. $T{N}_{q}{B}_{q}$-Smarandache Curve in ${\mathbb{G}}_{3}$

**Theorem**

**14.**

**Corollary**

**15.**

**Theorem**

**15.**

**Corollary**

**16.**

**Corollary**

**17.**

**Example**

**1.**

- The $T{N}_{q}$ Smarandache cure is defined as$$\eta \left(s\right)=T+{N}_{q}=(1,coss-sins,sins+coss).$$The quasi-frame of η is$${\left({T}_{q}\right)}_{\eta}=\frac{1}{\sqrt{2}}(0,-sins-coss,coss-sins),$$$${\left({N}_{q}\right)}_{\eta}=\frac{1}{\sqrt{2}}(0,-coss-sins,sins+coss),$$$${\left({B}_{q}\right)}_{\eta}=-{e}_{1}.$$The quasi-curvatures of η are$$\left({\kappa}_{1}\right)\eta =\frac{-1}{\sqrt{2}},$$$$\left({\kappa}_{2}\right)\eta =0,$$$$\left({\kappa}_{3}\right)\eta =0;$$
- The $T{B}_{q}$-Smarandache cure is defined as$$\beta \left(s\right)=T+{B}_{q}=(1,-2sins,2coss).$$The quasi-frame of β is$${\left({T}_{q}\right)}_{\beta}=\frac{1}{\sqrt{2}}(0,-2coss,-2sins),$$$${\left({N}_{q}\right)}_{\beta}=(0,-sins,coss),$$$${\left({B}_{q}\right)}_{\beta}=-{e}_{1}.$$The quasi-curvatures of β are$${\left({\kappa}_{1}\right)}_{\beta}=-1,$$$${\left({\kappa}_{2}\right)}_{\beta}=0,$$$${\left({\kappa}_{3}\right)}_{\beta}=0;$$
- The $T{N}_{q}{B}_{q}$-Smarandache curve is defined as$$\u03f5\left(s\right)=T+{N}_{q}+{B}_{q}=(1,coss-2sins,sins+2coss).$$The quasi-frame of ϵ is$${T}_{\u03f5}=\frac{1}{\sqrt{5}}(0,-sins-2coss,coss-2sins),$$$${\left({N}_{q}\right)}_{\u03f5}=\frac{1}{5}(0,coss-2sins,sins+2coss),$$$${\left({B}_{q}\right)}_{\u03f5}=-{e}_{1}.$$The quasi-curvatures of ϵ are$${\left({\kappa}_{1}\right)}_{\beta}=\frac{-1}{\sqrt{5}},$$$${\left({\kappa}_{2}\right)}_{\beta}=0,$$$${\left({\kappa}_{3}\right)}_{\beta}=0.$$

## 9. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**MDPI and ACS Style**

Elsharkawy, A.; Tashkandy, Y.; Emam, W.; Cesarano, C.; Elsharkawy, N.
On Some Quasi-Curves in Galilean Three-Space. *Axioms* **2023**, *12*, 823.
https://doi.org/10.3390/axioms12090823

**AMA Style**

Elsharkawy A, Tashkandy Y, Emam W, Cesarano C, Elsharkawy N.
On Some Quasi-Curves in Galilean Three-Space. *Axioms*. 2023; 12(9):823.
https://doi.org/10.3390/axioms12090823

**Chicago/Turabian Style**

Elsharkawy, Ayman, Yusra Tashkandy, Walid Emam, Clemente Cesarano, and Noha Elsharkawy.
2023. "On Some Quasi-Curves in Galilean Three-Space" *Axioms* 12, no. 9: 823.
https://doi.org/10.3390/axioms12090823