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:

\(\ds \begin {bmatrix} x' \\ y' \end {bmatrix}\)

\(=\)

\(\ds P'\)

\(\ds \)

\(=\)

\(\ds \phi_\alpha\)

Definition of Plane Reflection

\(\ds \)

\(=\)

\(\ds \begin {bmatrix} x \cos 2 \alpha + y \sin 2 \alpha \\ x \sin 2 \alpha - y \cos 2 \alpha \end {bmatrix}\)

Equation defining Plane Reflection

\(\ds \)

\(=\)

\(\ds \begin {bmatrix} \cos 2 \alpha & \sin 2 \alpha \\ \sin 2 \alpha & -\cos 2 \alpha \end {bmatrix} \begin {bmatrix} x \\ y \end {bmatrix}\)

Definition of Matrix Product (Conventional)

Hence the result.

$\blacksquare$

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Also see

• Determinant of Plane Reflection Matrix
• Inverse of Plane Reflection Matrix