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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.

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    $\begingroup$ It would rather be $s_y(x)=y+A(x-y)$ if I'm not mistaken. $\endgroup$ Mar 2, 2022 at 14:11
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    $\begingroup$ There isn't a unique rotation with center $y$ and angle $\theta$. So your definition of $s_y$ is incomplete. Or you should say "Let $s_y$ be $\color{red}{\text{a}}$ rotation ..." $\endgroup$
    – jjagmath
    Mar 2, 2022 at 14:14
  • $\begingroup$ thank you very much. $\endgroup$ Mar 2, 2022 at 14:33

1 Answer 1

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

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

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  • $\begingroup$ thank you very much for your answer, I understood . $\endgroup$ Mar 2, 2022 at 14:32

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