# One-Parameter Lorentzian Dual Spherical Movements and Invariants of the Axodes

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## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

**Definition**

**1.**

- (i)
- If $\widehat{\mathbf{x}}$ and $\widehat{\mathbf{y}}$ are spacelike dual vectors, then
- In case they span a spacelike dual plane, there exists a unique dual number $\widehat{\phi}=\phi +\epsilon {\phi}^{*}$; $0\le \phi \le \pi $, and ${\phi}^{*}\in \mathbb{R}$ in which $<\widehat{\mathbf{x}},\widehat{\mathbf{y}}>=\u2225\widehat{\mathbf{x}}\u2225\u2225\widehat{\mathbf{y}}\u2225cos\widehat{\phi}$. Such number is named the spacelike dual angle between $\widehat{\mathbf{x}}$ and $\widehat{\mathbf{y}}$.
- In case they span a timelike dual plane, then there exists a unique dual number $\widehat{\phi}=\phi +\epsilon {\phi}^{*}\ge 0$ in which $<\widehat{\mathbf{x}},\widehat{\mathbf{y}}>=\u03f5\u2225\widehat{\mathbf{x}}\u2225\u2225\widehat{\mathbf{y}}\u2225cosh\widehat{\phi}$, such that $\u03f5=+1$ or $\u03f5=-1$ based to $sign\left({\widehat{\mathbf{x}}}_{2}\right)=sign\left({\widehat{\mathbf{y}}}_{2}\right)$ or $sign\left({\widehat{\mathbf{x}}}_{2}\right)\ne sign\left({\widehat{\mathbf{y}}}_{2}\right)$, respectively. Such number is named the central dual angle between $\widehat{\mathbf{x}}$ and $\widehat{\mathbf{y}}$.

- (ii)
- If $\widehat{\mathbf{x}}$ and $\widehat{\mathbf{y}}$ are timelike dual vectors, then there exists a unique dual number $\widehat{\phi}=\phi +\epsilon {\phi}^{*}\ge 0$ in which $<\widehat{\mathbf{x}},\widehat{\mathbf{y}}>=\u03f5\u2225\widehat{\mathbf{x}}\u2225\u2225\widehat{\mathbf{y}}\u2225cosh\widehat{\phi}$, such that $\u03f5=-1$ or $\u03f5=+1$ based on $\widehat{\mathbf{x}}$ and $\widehat{\mathbf{y}}$ having the same time-orientation or different time-orientation, respectively. This dual number is named the Lorentzian timelike dual angle between $\widehat{\mathbf{x}}$ and $\widehat{\mathbf{y}}$.
- (iii)
- Let $\widehat{\mathbf{x}}$ be spacelike dual, and $\widehat{\mathbf{y}}$ be timelike dual, then there exists a unique $\widehat{\phi}=\phi +\epsilon {\phi}^{*}\ge $ 0 in which $<\widehat{\mathbf{x}},\widehat{\mathbf{y}}>=\u03f5\u2225\widehat{x}\u2225\u2225\widehat{\mathbf{y}}\u2225sinh\widehat{\phi}$, such that $\u03f5=+1$ or $\u03f5=-1$ based on whether $sign\left({\widehat{\mathbf{x}}}_{2}\right)=sign\left({\widehat{y}}_{1}\right)$ or $sign\left({\widehat{\mathbf{x}}}_{2}\right)\ne sign\left({\widehat{\mathbf{y}}}_{1}\right)$. Such number is named the Lorentzian timelike dual angle between $\widehat{\mathbf{x}}$ and $\widehat{\mathbf{y}}$.

## 3. One-Parameter Lorentzian Dual Spherical Movements

**Theorem**

**1.**

#### 3.1. Lorentzian Spatial Kinematics and Invariants of the Axodes

**Corollary**

**1.**

**Proof.**

**Corollary**

**2.**

#### 3.2. Euler–Savary Formula for the Spacelike Axodes

**$\widehat{r}$**${}_{i}-\widehat{p}{\widehat{\mathbf{g}}}_{i}$, and

**$\widehat{r}$**${}_{i}$ and ${\widehat{\mathbf{b}}}_{i}$, then we obtain

#### Velocity and Acceleration for a Spacelike Line Trajectory

#### 3.3. Disteli Formulae for a Spacelike Line Trajectory

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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Li, Y.; Alluhaibi, N.; Abdel-Baky, R.A.
One-Parameter Lorentzian Dual Spherical Movements and Invariants of the Axodes. *Symmetry* **2022**, *14*, 1930.
https://doi.org/10.3390/sym14091930

**AMA Style**

Li Y, Alluhaibi N, Abdel-Baky RA.
One-Parameter Lorentzian Dual Spherical Movements and Invariants of the Axodes. *Symmetry*. 2022; 14(9):1930.
https://doi.org/10.3390/sym14091930

**Chicago/Turabian Style**

Li, Yanlin, Nadia Alluhaibi, and Rashad A. Abdel-Baky.
2022. "One-Parameter Lorentzian Dual Spherical Movements and Invariants of the Axodes" *Symmetry* 14, no. 9: 1930.
https://doi.org/10.3390/sym14091930