Matrix of rotation of center $y$ and angle $\theta$

Question

Fix a real number $\theta$. Let $s_{y}$ be a rotation of $S^{2}$ with center $y \in S^{2}$ and angle $\theta$.

My question is how can we define the map $s_y$?

My attempt : Let
$s_{y}: \quad S^{2} \longrightarrow S^{2} : x\longrightarrow s_{y}(x)=y+Ax$ with

$$A= \begin{pmatrix}\cos \theta & -\sin \theta & 0 \\ \sin \theta & \cos \theta & 0 \\ 0 & 0 & 1\end{pmatrix}$$ Is my solution true? Any help is appreciated.

Answer 1

$s_y: x \mapsto y + A(x-y)$

First, translate to the center. Then rotate. Then translate back.