Lissajous

The Family Of The Curve

The Lissajous curves are a series of curves described by the parametric equation shown below. The \(x\) and \(y\) values are independent of each other and both oscillate back and forth at different frequencies and amplitudes in a simple harmonic motion (SHM). Additionally the \(\phi\) term shows a phase difference which can effect the shape of the curve produced. For example when the frequency and amplitude is exactly the same and the phase difference is \(\frac{\pi}{2}\) radians then you get a circle.

$$ x = A sin(\alpha t + \phi), y = B sin(\beta t) $$

The family of curves can be created in a variety of ways. The original manner was to release sand from a pendulum. It should be noted that whilst the period of a pendulum is theoretically only dependent on the length and not on the weight that larger bobs will create increased drag at the pivot point. This will slow the pendulum down faster and changing the weight over time will subtly create a change in the shape of the Lissajous curve. This fascinating change creates complex harmonic motion rather than simple harmonic motion and includes an exponential decay term. Whilst this is interesting I wanted to investigate the SHM family of curves.

With this in mind I chose to create an animation that mimics an oscilloscope making a Lissajous curve. This is where the theme of a green line on a black background originates from. Perhaps a bluer tinge to the background and some flowing shader effects on the green line would produce a more realistic oscilloscope rendering. This is left as an exercise for the reader :)