A點繞坐標軸逆旋轉b到B點
設A點坐標\((x,y)\),B點坐標\((x^{\prime},y^{\prime})\)
\(\begin{align*} x =\ &rcos\alpha\\ y =\ &rsin\alpha\\ x^{\prime} =\ &rcos(\alpha+\beta)\ =\ rcos\alpha cos\beta - rsin\alpha sin\beta\ =\ xcos\beta-ysin\beta\\ y^{\prime} =\ &rsin(\alpha+\beta)\ =\ rsin\alpha cos\beta + rsin\beta cos\alpha\ =\ xsin\beta+ycos\beta \end{align*}\)
\(\left[ \begin{array}{l} x^{\prime}\\ y^{\prime} \end{array} \right]=\left[\begin{array}{lc} cos\beta&-sin\beta\\ sin\beta&cos\beta \end{array} \right]\cdot \left[ \begin{array}{l} x\\ y \end{array} \right] \)
同理可推出順時針旋轉的公式(把\(\beta\)變成\(-\beta\)即可)
\(\left[ \begin{array}{l} x^{\prime}\\ y^{\prime} \end{array} \right]=\left[\begin{array}{cl} cos\beta&sin\beta\\ -sin\beta&cos\beta \end{array} \right]\cdot \left[ \begin{array}{l} x\\ y \end{array} \right] \)