Plane Reflection is Involution

Theorem
Let $M$ be a straight line in the plane passing through the origin.

Let $s_M$ be the reflection of $\R^2$ in $M$.

Then $s_M$ is an involution in the sense that:
 * $s_M \circ s_M = I_{\R^2}$

where $I_{\R^2}$ is the identity mapping on $\R_2$.

That is:
 * $s_M = {s_M}^{-1}$

Proof
Let the angle between $M$ and the $x$-axis be $\alpha$.

Let $P = \tuple {x, y}$ be an arbitrary point in the plane.

Then from Equations defining Plane Reflection:
 * $\map {s_M} P = \tuple {x \cos 2 \alpha + y \sin 2 \alpha, x \sin 2 \alpha - y \cos 2 \alpha}$

Thus:

Hence the result.