An ellipse can be defined as the set of points such that the **sum** of the distances to two fixed points, the foci, has a constant value.

A Cassini oval is the set of points such that the **product** of the distances to two foci has a constant value.

You can write down an equation for a Cassini oval for given parameters *a* and *b* as

((*x* + *a*)² + *y*²) ((*x* – *a*)² + *y*²) = *b*².

For some reason, references almost always plot Cassini ovals by fixing *a* and letting *b* vary. When we set *a* = 1 and let *b* = 0.5, 1, 1.5, …, 5 this produces the following plot.

But you could also fix *b* and let *a* vary. Here’s what we get when we set *b* = 1 and let *a* = 0, 0.1, 0.2, …, 1.

Incidentally, the red figure eight in the middle, corresponding to *a* = 1, is known as the lemniscate, or more formally the lemniscate of Bernoulli.