Lissajous figures are the plots of the curves in two dimensions with
x = A sin (w₁ t + Ф) and y = B sin w₂ t
These two equations represent signals. The plot tells how one signal is varying with respect to another. In each direction x and y, the motion of a particle is in SHM and on the whole, it is a complex harmonic motion. As frequency is time varying.
These figures form closed curves for some combinations of w₁ and w₂. Otherwise, they may be open curves. The shape of the curve depends on w₁/w₂.
1. On an oscilloscope in a physics or electrical laboratory, we see one voltage vs another voltage relation on an oscilloscope. Or, one current signal vs another in different parts of the circuit.
The oscilloscope takes in both analog signals and using a complex circuit, it plots the lissajous figures on the screen of oscilloscope. The graph gives the relation between the frequencies and the phase difference. The plot is usually seen in X-Y mode of oscilloscope. In the normal mode each signal is shown versus time.
2. Generally in transmission of stereophonic audio signals the left and the right signals are analysed for the phase difference between the two. They are mixed using advanced mixers with a choice of parametric mixing of them. The equipment (oscilloscope) to do this utilizes Lissajous figures.
3. On laptops and desktops, the screen saver applications use Lissajous figures with continuously varying w₁ / w₂ ratio.
4. In science fiction movies and TV shows, these figures are used.
5. They are used to design new Logos.
6. Space stations and satellites use Lissajous orbits.