Equation of Ovals of Cassini/Cartesian Form

Theorem
Let $P_1$ and $P_2$ be points in the cartesian plane located at $\tuple {a, 0}$ and $\tuple {-a, 0}$ for some constant $a \ne 0$.

Let $b$ be a real constant.

The Cartesian equation:
 * $\paren {x^2 + y^2 + a^2}^2 - 4 a^2 x^2 = b^4$

describes the ovals of Cassini.


 * Ovals-of-Cassini.png

Proof
The ovals of Cassini are the loci of points $M$ in the plane such that:
 * $P_1 M \times P_2 M = b^2$

Let $b$ be chosen.

Let $P = \tuple {x, y}$ be an arbitrary point of $M$.

We have:

Hence the result.

Also see

 * Equation of Ovals of Cassini/Polar Form