Composition of a plane rotation and a reflection ($R_{\alpha }S=R_{\alpha /2}S{R}_{-\alpha /2}$)

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 $S=\mathrm{diag}(1,-1)$. ThenThis 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 $\pi$.