Lemniscate of Bernoulli

The lemniscate of Bernoulli came up in a post a few days ago. This shape is a special case of a Cassini oval:

((x + a)² + y²) ((xa)² + y²) = a4.

Here’s another way to arrive at the lemniscate. Draw a hyperbola (blue in the figure below), then draw circles centered at points on the hyperbola and passing through the center of the hyperbola. The figure-eight that is traced out in the middle is a lemniscate.