# 轴角

（重定向自軸角

## 用途

### 例子

${\displaystyle \langle \mathrm {axis} ,\mathrm {angle} \rangle =\left({\begin{bmatrix}a_{x}\\a_{y}\\a_{z}\end{bmatrix}},\theta \right)=\left({\begin{bmatrix}0\\0\\1\end{bmatrix}},{\frac {\pi }{2}}\right)}$

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

## 与其他表示的联系

### 从 so(3) 到 SO(3) 的指数映射

${\displaystyle \exp \colon so(3)\to SO(3)}$

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

### 从 SO(3) 到 so(3) 的对数映射

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

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

### 四元数

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

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