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

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:

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