# 欧拉角

## 靜態的定義

• ${\displaystyle \alpha }$（进动角）是x-軸與交點線的夾角，
• ${\displaystyle \beta }$（章动角）是z-軸與Z-軸的夾角，
• ${\displaystyle \gamma }$（自旋角）是交點線與X-軸的夾角。

### 角值範圍

• ${\displaystyle \alpha }$${\displaystyle \gamma }$值的範圍為${\displaystyle [0,2\pi )}$弧度
• ${\displaystyle \beta }$值的範圍為${\displaystyle [0,\pi ]}$弧度。

• ${\displaystyle \beta =0}$時，${\displaystyle \alpha }$${\displaystyle \gamma }$之和（在模${\displaystyle 2\pi }$意義下）相等的歐拉角對應同一個取向，如${\displaystyle ({\frac {\pi }{2}},0,0)}$${\displaystyle ({\frac {\pi }{3}},0,{\frac {\pi }{6}})}$${\displaystyle ({\frac {4\pi }{3}},0,{\frac {7\pi }{6}})}$對應的取向相同
• ${\displaystyle \beta =\pi }$時，${\displaystyle \alpha }$${\displaystyle \gamma }$之差相等的歐拉角對應同一個取向，如${\displaystyle ({\frac {\pi }{2}},\pi ,0)}$${\displaystyle (\pi ,\pi ,{\frac {\pi }{2}})}$對應的取向相同

### 旋轉矩陣1

${\displaystyle [\mathbf {R} ]={\begin{bmatrix}\cos \gamma &\sin \gamma &0\\-\sin \gamma &\cos \gamma &0\\0&0&1\end{bmatrix}}{\begin{bmatrix}1&0&0\\0&\cos \beta &\sin \beta \\0&-\sin \beta &\cos \beta \end{bmatrix}}{\begin{bmatrix}\cos \alpha &\sin \alpha &0\\-\sin \alpha &\cos \alpha &0\\0&0&1\end{bmatrix}}}$

${\displaystyle [\mathbf {R} ]={\begin{bmatrix}\cos \alpha \cos \gamma -\cos \beta \sin \alpha \sin \gamma &\sin \alpha \cos \gamma +\cos \beta \cos \alpha \sin \gamma &\sin \beta \sin \gamma \\-\cos \alpha \sin \gamma -\cos \beta \sin \alpha \cos \gamma &-\sin \alpha \sin \gamma +\cos \beta \cos \alpha \cos \gamma &\sin \beta \cos \gamma \\\sin \beta \sin \alpha &-\sin \beta \cos \alpha &\cos \beta \end{bmatrix}}}$

${\displaystyle [\mathbf {R} ]}$逆矩陣是：

${\displaystyle [\mathbf {R} ]^{-1}={\begin{bmatrix}\cos \alpha &-\sin \alpha &0\\\sin \alpha &\cos \alpha &0\\0&0&1\end{bmatrix}}{\begin{bmatrix}1&0&0\\0&\cos \beta &-\sin \beta \\0&\sin \beta &\cos \beta \end{bmatrix}}{\begin{bmatrix}\cos \gamma &-\sin \gamma &0\\\sin \gamma &\cos \gamma &0\\0&0&1\end{bmatrix}}}$
${\displaystyle [\mathbf {R} ]^{-1}={\begin{bmatrix}\cos \alpha \cos \gamma -\cos \beta \sin \alpha \sin \gamma &-\cos \alpha \sin \gamma -\cos \beta \sin \alpha \cos \gamma &\sin \beta \sin \alpha \\\sin \alpha \cos \gamma +\cos \beta \cos \alpha \sin \gamma &-\sin \alpha \sin \gamma +\cos \beta \cos \alpha \cos \gamma &-\sin \beta \cos \alpha \\\sin \beta \sin \gamma &\sin \beta \cos \gamma &\cos \beta \end{bmatrix}}}$

## 動態的定義

• A)繞著XYZ坐標軸旋轉：最初，兩個坐標系統xyz與XYZ的坐標軸都是重疊著的。開始先繞著Z-軸旋轉${\displaystyle \alpha \,}$角值。然後，繞著X-軸旋轉${\displaystyle \beta \,}$角值。最後，繞著Z-軸作角值${\displaystyle \gamma \,}$的旋轉。
• B)繞著xyz坐標軸旋轉：最初，兩個坐標系統xyz與XYZ的坐標軸都是重疊著的。開始先繞著z-軸旋轉${\displaystyle \gamma \,}$角值。然後，繞著x-軸旋轉${\displaystyle \beta \,}$角值。最後，繞著z-軸作角值${\displaystyle \alpha \,}$的旋轉。

${\displaystyle \mathbf {R} _{1}=Z(\gamma )\circ X(\beta )\circ Z(\alpha )\circ \mathbf {r} _{1}\,}$

${\displaystyle Z(\alpha )={\begin{bmatrix}\cos \alpha &\sin \alpha &0\\-\sin \alpha &\cos \alpha &0\\0&0&1\end{bmatrix}}\,}$
${\displaystyle X(\beta )={\begin{bmatrix}1&0&0\\0&\cos \beta &\sin \beta \\0&-\sin \beta &\cos \beta \end{bmatrix}}\,}$
${\displaystyle Z(\gamma )={\begin{bmatrix}\cos \gamma &\sin \gamma &0\\-\sin \gamma &\cos \gamma &0\\0&0&1\end{bmatrix}}\,}$

${\displaystyle \mathbf {r} _{2}=z(\alpha )\circ x(\beta )\circ z(\gamma )\circ \mathbf {R} _{2}\,}$

${\displaystyle z(\alpha )={\begin{bmatrix}\cos \alpha &-\sin \alpha &0\\\sin \alpha &\cos \alpha &0\\0&0&1\end{bmatrix}}\,}$
${\displaystyle x(\beta )={\begin{bmatrix}1&0&0\\0&\cos \beta &-\sin \beta \\0&\sin \beta &\cos \beta \end{bmatrix}}\,}$
${\displaystyle z(\gamma )={\begin{bmatrix}\cos \gamma &-\sin \gamma &0\\\sin \gamma &\cos \gamma &0\\0&0&1\end{bmatrix}}\,}$

${\displaystyle \mathbf {r} _{1}=z(\alpha )\circ x(\beta )\circ z(\gamma )\circ \mathbf {R} _{2}\,}$

${\displaystyle z^{-1}(\gamma )\circ x^{-1}(\beta )\circ z^{-1}(\alpha )\circ \mathbf {r} _{1}=z^{-1}(\gamma )\circ x^{-1}(\beta )\circ z^{-1}(\alpha )\circ z(\alpha )\circ x(\beta )\circ z(\gamma )\circ \mathbf {R} _{2}\,}$

${\displaystyle z^{-1}(\alpha )=Z(\alpha )\,}$
${\displaystyle x^{-1}(\beta )=X(\beta )\,}$
${\displaystyle z^{-1}(\gamma )=Z(\gamma )\,}$

${\displaystyle Z(\gamma )\circ X(\beta )\circ Z(\alpha )\circ \mathbf {r} _{1}=\mathbf {R} _{2}\,}$
${\displaystyle \mathbf {R} _{1}=\mathbf {R} _{2}\,}$

## 參考文獻

• L. C. Biedenharn, J. D. Louck, Angular Momentum in Quantum Physics, Addison-Wesley, Reading, MA, 1981.
• Andrew Gray, A Treatise on Gyrostatics and Rotational Motion, MacMillan, London, 1918.
• M. E. Rose, Elementary Theory of Angular Momentum, John Wiley, New York, NY, 1957.