Equations defining Plane Rotation

Theorem

Cartesian

Let $r_\alpha$ be the rotation of the plane about the origin through an angle of $\alpha$.

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

Then:

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

Polar

Equations defining Plane Rotation/Polar

Matrix

Let $r_\alpha$ be a rotation of the plane about the origin through an angle of $\alpha$.

Let $r_\alpha$ rotate an arbitrary point in the plane $P = \tuple {x, y}$ onto $P' = \tuple {x', y'}$

Then:

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

Examples

Right Angle

Let $r_\Box$ be the rotation of the plane about the origin through a right angle.

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

Then:

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