When a pendulum or an object attached to a spring swings in an ellipse, the inward acceleration/force is proportional to the distance A = F / m = − − k r . Due to the way vectors add, the component of the force in the x ^ ^ or in the y ^ ^ directions are also proportionate to the respective components of the distances, r x ″ = A x = − − k r x . Hence, the entire analysis can be done separately in these dimensions. This results in the harmonic parabolic equations x = A cos ⁡ ⁡ ( t ) and y = B sin ⁡ ⁡ ( t ) of the ellipse.