# Quaternionic Shape Operator and Rotation Matrix on Ruled Surfaces

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

**Definition 1**

**Definition 2**

**Definition 3**

**Lemma 1**

**Definition 4**

**Theorem 1**

**Theorem 2**

**Theorem 3**

**Corollary 1**

## 3. Quaternion Shape Operator and Rotation Matrix on Ruled Surface

**Theorem 4.**

**Proof.**

**Theorem 5.**

**Proof.**

**Theorem 6.**

**Proof.**

**Corollary 2.**

**Theorem 7.**

**Proof.**

## 4. Quaternion Shape Operator and Rotation Matrices on Special Ruled Surfaces

#### 4.1. The Ruled Surface Drawn by Tangent Vector

**Theorem 8.**

**Proof.**

**Theorem 9.**

**Proof.**

**Corollary 3.**

**Theorem 10.**

**Proof.**

#### 4.2. The Ruled Surface Drawn by Principal Normal Vector

**Theorem 11.**

**Proof.**

**Corollary 4.**

**Theorem 12.**

**Proof.**

#### 4.3. The Ruled Surface Drawn by Binormal Vector

**Theorem 13.**

**Proof.**

**Corollary 5.**

**Theorem 14.**

**Proof.**

**Example 1.**

**Example 2.**

## 5. Conclusions and Discussion

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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Li, Y.; Çalışkan, A.
Quaternionic Shape Operator and Rotation Matrix on Ruled Surfaces. *Axioms* **2023**, *12*, 486.
https://doi.org/10.3390/axioms12050486

**AMA Style**

Li Y, Çalışkan A.
Quaternionic Shape Operator and Rotation Matrix on Ruled Surfaces. *Axioms*. 2023; 12(5):486.
https://doi.org/10.3390/axioms12050486

**Chicago/Turabian Style**

Li, Yanlin, and Abdussamet Çalışkan.
2023. "Quaternionic Shape Operator and Rotation Matrix on Ruled Surfaces" *Axioms* 12, no. 5: 486.
https://doi.org/10.3390/axioms12050486