# 轴角

## 用途

### 例子

$\langle \mathrm{axis}, \mathrm{angle} \rangle = \left( \begin{bmatrix} a_x \\ a_y \\ a_z \end{bmatrix},\theta \right) = \left( \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix},\frac{\pi}{2}\right)$

$\begin{bmatrix} 0 \\ 0 \\ \frac{\pi}{2} \end{bmatrix}$

## 与其他表示的联系

### 从 so(3) 到 SO(3) 的指数映射

$\exp\colon so(3) \to SO(3)$

$R = \exp(\hat{\omega} \theta) = \sum_{k=0}^\infty\frac{(\hat{\omega}\theta)^k}{k!} = I + \hat{\omega} \theta + \frac{1}{2}(\hat{\omega}\theta)^2 + \frac{1}{6}(\hat{\omega}\theta)^3 + \cdots$
$R = I + \hat{\omega}\left(\theta - \frac{\theta^3}{3!} + \frac{\theta^5}{5!} - \cdots\right) + \hat{\omega}^2 \left(\frac{\theta^2}{2!} - \frac{\theta^4}{4!} + \frac{\theta^6}{6!} - \cdots\right)$
$R = I + \hat{\omega} \sin(\theta) + \hat{\omega}^2 (1-\cos(\theta))$

### 从 SO(3) 到 so(3) 的对数映射

$\theta = \arccos\left( \frac{\mathrm{trace}(R) - 1}{2} \right)$

$\omega = \frac{1}{2 \sin(\theta)} \begin{bmatrix} R(3,2)-R(2,3) \\ R(1,3)-R(3,1) \\ R(2,1)-R(1,2) \end{bmatrix}$

### 四元数

$Q = \left(\cos\left(\frac{\theta}{2}\right), \omega \sin\left(\frac{\theta}{2}\right)\right)$

$\theta = 2\,\arccos(q_0)\,$
$\omega = \left\{ \begin{matrix} \frac{q}{ \sin( \theta/2 ) } , & \mathrm{if} \; \theta \neq 0 \\ 0, & \mathrm{otherwise} \end{matrix} \right.$