Rotation-operator-toy-example

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Rotation-operator-toy-example

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Implementation of Rotation operator (SO(3)) without Library

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# Rotation operator toy example
[![License: MIT](https://img.shields.io/badge/License-MIT-yellow.svg)](https://opensource.org/licenses/MIT)  
This project implements rotating a point cloud by randomly generating a rotation angle and unit vector.  
I have implemented a rotation matrix and a quaternion, respectively, and visualized the results.

## Preliminary knowledge

Given a rotation angle $\phi$ and a rotation vector (unit vector) $\vec{u}$ as input,  
the following formulas can be utilized to implement a rotation matrix and a quaternion, respectively.

Given input point cloud of $x$, applying the rotation operator to each of them gives the following representation.

### Rotation matrix  
$$
R = exp(\[ \phi \cdot \vec{u} \]_{\times})
$$

$$
x' = R x
$$

### Quaternion  
$$
q = exp( \phi \cdot \vec{u} / 2)
$$

$$
x' = q \otimes x \otimes q^*
$$

## How to start

Refer to `requirements.txt` to setup the dependencies and execute the code like below.  
I tested this on ubuntu 18.04.

```
python main.py --input_path ./bunny.ply
```

## Result

### Rotation matrix result

### Quaternion result

## Reference  
- Sola, Joan. "Quaternion kinematics for the error-state Kalman filter." arXiv preprint arXiv:1711.02508 (2017).  
- Tricks to do annotation : https://github.com/isl-org/Open3D/issues/2

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