Equations defining Plane Rotation
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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:
- $\map {r_\Box} P = \tuple {y, -x}$