Equation of Ovals of Cassini/Polar Form

Theorem
Let $P_1$ and $P_2$ be points in a polar coordinate plane located at $\polar {a, 0}$ and $\polar {a, \pi}$ for some constant $a \ne 0$.

Let $b$ be a real constant.

The polar equation:
 * $r^4 + a^4 + 2 a^2 r^2 \cos 2 \theta = b^4$

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 $M$.

Let $d_1 = \size {P_1 P}$ and $d_2 = \size {P_2 P}$.

We have:

Hence the result.

Also see

 * Equation of Ovals of Cassini/Cartesian Form