Equations defining Plane Reflection/Cartesian

Theorem
Let $\LL$ be a straight line through the origin $O$ of a cartesian plane.

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

Let $\phi_\alpha$ denote the reflection in the plane whose axis is $\LL$.

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

Then:
 * $\map {\phi_\alpha} P = \tuple {x \cos 2 \alpha + y \sin 2 \alpha, x \sin 2 \alpha - y \cos 2 \alpha}$

Proof

 * Reflection-equations-origin.png

Let $\LL$ reflect $P = \tuple {x, y}$ onto $P' = \tuple {x', y'}$.

Let $OP$ form an angle $\theta$ with the $x$-axis.

We have:
 * $OP = OP'$

Thus:

Then:

and:

The result follows.