Equation of Ovals of Cassini

Theorem

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

Let $b$ be a real constant.

Cartesian Form

The Cartesian equation:

$\paren {x^2 + y^2 + a^2}^2 - 4 a^2 x^2 = b^4$

describes the ovals of Cassini.

Polar Form

The polar equation:

$r^4 + a^4 - 2 a^2 r^2 \cos 2 \theta = b^4$

describes the ovals of Cassini.

Sources

• Weisstein, Eric W. "Cassini Ovals." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/CassiniOvals.html