Equations defining Plane Reflection

Theorem

Cartesian

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:

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

Polar

Equations defining Plane Reflection/Polar

Matrix

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:

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

Examples

$x$-Axis

Let $\phi_x$ denote the reflection in the plane whose axis is the $x$-axis.

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

Then:

$\map {\phi_x} P = \tuple {x, -y}$

$y$-Axis

Let $\phi_y$ denote the reflection in the plane whose axis is the $y$-axis.

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

Then:

$\map {\phi_y} P = \tuple {-x, y}$