2D Rotation

Example of a 2D rotation through an angle φ where the coordinates x, y go into x', y'. Note that φ is positive for a counterclockwise rotation and that the rotation is about the origin (0, 0).

Derivation of the formula for rotation

(old coordinates are (x, y) and the new coordinates are (x', y'))

θ = initial angle, φ = angle of rotation.

x = r cos θ
y = r sin θ

x' = r cos ( θ + φ ) = r cos θ cos φ - r sin θ sin φ
y' = r sin ( θ + φ ) = r sin θ cos φ + r cos θ sin φ

hence:

x' = x cos φ - y sin φ
y' = y cos φ + x sin φ