Parametrization for the figure '8' curve?

Question

Is there a parametrization for the figure '8' curve, which is self-intersected?

Answer 1

This is an example of a Lissajous figure. (If you've got an oscilloscope with separate $x$ and $y$ inputs and a couple of signal generators you can have hours of fun generating them by applying sine waves of appropriate frequency ratio to the two inputs.)

Answer 2

$x = \frac{a\sqrt{2}\cos(t)}{\sin^2(t) + 1}; \qquad y = \frac{a\sqrt{2}\cos(t)\sin(t)}{\sin^2(t) + 1}$

Check out this source:

https://en.wikipedia.org/wiki/Lemniscate_of_Bernoulli

Answer 3

An easier parameterization of an 8-like figure is $(x,y) = (\sin 2t, \cos t),$ where $0 \leq t \leq 2\pi.$ It can easily be made more 8-like by scaling.