A点绕坐标轴逆旋转b到B点
设A点坐标\((x,y)\),B点坐标\((x^{\prime},y^{\prime})\)
\(\begin{align*}
x =\ &rcos\alpha\\
y =\ &rsin\alpha\\
x^{\prime} =\ &rcos(\alpha+\beta)\ =\ rcos\alpha cos\beta + rsin\alpha sin\beta\ =\ xcos\beta-ysin\beta\\
y^{\prime} =\ &rsin(\alpha+\beta)\ =\ rsin\alpha cos\beta + rsin\beta cos\alpha\ =\ xsin\beta+ycos\beta
\end{align*}\)
\(\left[
\begin{array}{l}
x^{\prime}\\
y^{\prime}
\end{array}
\right]=\left[\begin{array}{lc}
cos\beta&-sin\beta\\
sin\beta&cos\beta
\end{array}
\right]\cdot
\left[
\begin{array}{l}
x\\
y
\end{array}
\right]
\)
同理可推出顺时针旋转的公式(把\(\beta\)变成\(-\beta\)即可)
\(\left[
\begin{array}{l}
x^{\prime}\\
y^{\prime}
\end{array}
\right]=\left[\begin{array}{cl}
cos\beta&sin\beta\\
-sin\beta&cos\beta
\end{array}
\right]\cdot
\left[
\begin{array}{l}
x\\
y
\end{array}
\right]
\)