When an object is subjected to two simultaneous periodic forces perpendicular to each other, it traces a line, ellipse or circle. The relative amplitudes and the phase difference between the motions in two perpendicular directions determines the path.
The phase for each direction depends on the initial position, initial speed and angular frequency as
$$tan(\phi) = - \frac{v(0)}{\omega x (0)}$$
| Phase difference | Relative amplitudes | comments | |
|---|---|---|---|
| line | 0o | Can be anything. | Relative amplitude determines the angle made by the path of motion wrt to x axis. |
| ellipse | 90o | a ≠ b | This is only a special case. The phase difference > 0 produces an ellipse |
| circle | 90o | a = b | - |