The function that draws a figure eight

Question

I'm trying to describe a counterexample for a theorem which includes the figure eight or "infinity" symbol, but I'm having trouble finding a good piecewise function to draw it. I need it to be the symbol, except at the "crossing point" the function jumps (not continuous) so that we still have a manifold.

Answer 1

You can use the function $$t\in(-\tfrac12\pi,\tfrac32\pi)\mapsto(\cos t,\sin t\cos t)\in\mathbb R^2.$$

The resulting curve is (I'm omitting a little bit on the left and a little bit on the right from the domain in this picture):

Answer 2

There are no functions that describe the "lemniscate", but there are parametric and polar equations for the lemniscate of Bernoulli and the lemniscate of Gerono, to name two of the more famous lemniscates.

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Might as well share this, since the question already has an accepted answer. Here is my favorite way of generating the lemniscate of Bernoulli, as an envelope of circles centered on a rectangular hyperbola, and passing through the origin:

Answer 3

Is the lemniscate what you want? I don't know what you need as a jump at the crossing point, but maybe you can get that with a trigonometric parameterization.