# Lissajous Curves

An interactive demonstration of Lissajous curves.

A Lissajous curve, named after Jules Antoine Lissajous is a graph of the following two parametric equations:

\begin{align} x &= A sin (at + \phi) \\ y &= B sin (bt) \end{align}

$$A$$ and $$B$$ represent amplitudes in the $$x$$ and $$y$$ directions, $$a$$ and $$b$$ are constants, and $$\phi$$ is an phase angle. The user interface above allows you to modify each of these five parameters and see how the graph is affected. The values of $$a$$ and $$b$$ have a particularly strong effect on the shape of the curve.

$$a$$ determines the number of horizontally aligned "lobes" and $$b$$ determines the number of vertically aligned lobes. For example for $$a = 3$$ and $$b = 2$$ you should see 3 horizontal lobes and 2 vertical. You may need to adjust $$\phi$$ to clearly see the lobes as they sometimes overlap for certain values of $$\phi$$. Also note, that the graph with, say, $$a = 6$$ and $$b = 4$$ will be identical to the graph of $$a = 3$$ and $$b = 2$$ as it is specifically the ratio which affects the shape of the graph.