= Composition of a plane rotation and a reflection
{title2=$R_\alpha S=R_{\alpha/2}SR_{-\alpha/2}$}
A <planar rotation> about a point on the axis of a <Euclidean reflection>, composed after that <Euclidean reflection>, is a <Euclidean reflection> whose axis has been rotated by half the rotation angle. In coordinates centred at the fixed point, with the original axis horizontal, write $S=\operatorname{diag}(1,-1)$. Then
$$
R_\alpha S=\begin{pmatrix}\cos\alpha&\sin\alpha\\\sin\alpha&-\cos\alpha\end{pmatrix}=R_{\alpha/2}SR_{-\alpha/2}.
$$
This <matrix> fixes the direction $(\cos(\alpha/2),\sin(\alpha/2))$ and reverses its perpendicular direction. The axis is an unoriented line, so its angle is defined modulo $\pi$.
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