# Bertrand Offsets of Slant Ruled Surfaces in Euclidean 3-Space

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Basic Concepts

**Corollary**

**1.**

**Proof.**

**Definition**

**1.**

## 3. Bertrand Offsets of Slant Ruled Surfaces

#### 3.1. Height Functions

**Proposition**

**1.**

**Proof.**

- (a)
- The osculating circle $\mathcal{S}(\rho ,{\mathfrak{b}}_{0})$ of $\mathfrak{e}\left(u\right)\in {\mathcal{S}}^{2}$ is displayed by$$<{\mathfrak{b}}_{0},\mathfrak{e}>=\rho \left(u\right),\phantom{\rule{4pt}{0ex}}<\mathfrak{e},{\mathfrak{b}}_{0}>=0,<\mathfrak{e},{\mathfrak{b}}_{0}>=0,$$
- (b)
- The curve $\mathfrak{e}\left(u\right)\in {\mathcal{S}}^{2}$ and the osculating circle $\mathcal{S}(\rho ,{\mathfrak{b}}_{0})$ have a touch of at least the fourth order at $\mathfrak{e}\left({u}_{0}\right)$ if ${\chi}^{\prime}=0$, and ${\chi}^{\u2033}\ne 0$.

**Theorem**

**1.**

**Definition**

**2.**

**Theorem**

**2.**

**Theorem**

**3.**

**Proof.**

**Corollary**

**2.**

**Corollary**

**3.**

**Theorem**

**4.**

**Corollary**

**4.**

**Corollary**

**5.**

**Corollary**

**6.**

#### 3.2. Construction of Slant Ruled Surface and Its $\mathcal{BO}$

#### 3.3. Classification of the Slant Ruled Surfaces

**Case 1**. If the striction curve is an asymptotic curve, ${\kappa}_{n}=\lambda -\chi \mu =0$, there are two potential issues:

**Case 2**. If the striction curve is a geodesic curve, we may write

**Case 3**. If the striction curve is a curvature line, we may write $\mu +\chi \lambda =\mathfrak{0}$, and we may have two different cases:

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**MDPI and ACS Style**

Almoneef, A.A.; Abdel-Baky, R.A.
Bertrand Offsets of Slant Ruled Surfaces in Euclidean 3-Space. *Symmetry* **2024**, *16*, 235.
https://doi.org/10.3390/sym16020235

**AMA Style**

Almoneef AA, Abdel-Baky RA.
Bertrand Offsets of Slant Ruled Surfaces in Euclidean 3-Space. *Symmetry*. 2024; 16(2):235.
https://doi.org/10.3390/sym16020235

**Chicago/Turabian Style**

Almoneef, Areej A., and Rashad A. Abdel-Baky.
2024. "Bertrand Offsets of Slant Ruled Surfaces in Euclidean 3-Space" *Symmetry* 16, no. 2: 235.
https://doi.org/10.3390/sym16020235