The [[Affine Transformations|Affine]] rotation is the same as a [[Rotation Transformation]] with $\vec b = \vec 0$. #### Rotation around a point To rotate around a point $P$ by an angle $\theta$, you can [[Affine Transformations|Translate]] the space to bring the point $P$ to the origin $\mathcal O$ , apply the [[Rotation Transformation|Rotation]], then translate back $\mathcal O \to P$. $\huge R_{P \, \op{by} \, \theta} = T_{\ang{P}} \circ R_{\theta} \circ T_{\ang{P}}^{-1} $ >[!example]- > $\let T: \Rn2 \to \Rn 2$ be a rotation by $90\degree$ around $(3, 1)$. >$\huge A_{\circlearrowleft90\degree} = \mat{0 & -1 \\ 1 & 0} $ >Find a fixed point of $T$. >$\huge >\begin{align*} >\vec x_{0}&= \pa{3, 1}\\ >A\vec x_0 + \vec b &= \vec x_0 \\ >\vec b &= \vec x_{0}- A\vec x_0 >\end{align*}$ >$\huge\begin{align*} >\vec b =&\mat{3\\1} - \mat{0&-1\\1&0}\mat{3\\1} \\ >\vec b &= \mat{4 \\ -2 } \\ >\therefore T(\vec x) &: \mat{0&-1\\1&0}\vec x + \mat{4\\-2} >\end{align*}$ >$\huge \begin{align*} >\vec b &= \vec x_{0}- A \vec x_{0}\\ >\vec b &= (I - A)\vec x _{0}\\ >\vec b &= -(A - I)\vec x_{0}\\ >\end{align*}$