旋转公式推导

记原坐标为 \((x, y)\)，逆时针旋转 \(\theta\) 后的新坐标为 \((x', y')\)，有：

\[\left\{\begin{matrix} x' = x \cos \theta - y \sin \theta \\ y' = x \sin \theta + y \cos \theta \end{matrix}\right. \]

推导：

\[\left\{\begin{matrix} y = d \sin \alpha \\ x = d \cos \alpha \\ x' = d \cos (\theta + \alpha) \\ y' = d \sin (\theta + \alpha) \end{matrix}\right. \]

\[ x' = d \cdot (\cos \theta \cos \alpha - \sin \theta \sin \alpha) \\ = d \cos \theta \cos \alpha - d \sin \theta \cos \alpha \\ = x \cos \theta - y \sin \theta \\ \]

\[y' = d \cdot (\sin \theta \cos \alpha + \cos \theta \sin \alpha) \\ = d \sin \theta \cos \alpha + d \cos \theta \sin \alpha \\ = x \sin \theta + y \cos \theta \\ \]