Equations defining Plane Reflection/Matrix

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 $\LL$ reflect an arbitrary point in the plane $P = \tuple {x, y}$ onto $P' = \tuple {x', y'}$.

Then:


 * $\begin {bmatrix} x' \\ y' \end {bmatrix} = \begin {bmatrix} \cos 2 \alpha & \sin 2 \alpha \\ \sin 2 \alpha & -\cos 2 \alpha \end {bmatrix} \begin {bmatrix} x \\ y \end {bmatrix}$

Proof
Let the coordinates of $P'$ be encoded as the elements of a $2 \times 1$ matrix.

We have:

Hence the result.

Also see

 * Determinant of Plane Reflection Matrix
 * Inverse of Plane Reflection Matrix