Definition:Ovals of Cassini/Shape

Definition
Let $P_1$ and $P_2$ be points in the plane such that $P_1 P_2 = 2 a$ for some constant $a$.

Let the ovals of Cassini be defined as the loci of points $M$ in the plane such that:
 * $P_1 M \times P_2 M = b^2$

for real constant $b$.

When $b > a$, $M$ is in one continuous piece, either oval or bone-shaped.

When $b < a$, $M$ is in two separate pieces, each surrounding one of the foci of $M$.

When $b = a$, $M$ degenerates into the lemniscate of Bernoulli.


 * Ovals-of-Cassini.png

Also see

 * Lemniscate of Bernoulli is Special Case of Ovals of Cassini