Lemniscate of Bernoulli is Special Case of Ovals of Cassini/Polar Proof

Theorem

The lemniscate of Bernoulli is a special case of the ovals of Cassini.

Proof

The ovals of Cassini can be defined by a polar equation as follows:

The polar equation:

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

describes the ovals of Cassini.

The lemniscate of Bernoulli can be defined by a polar equation as follows:

The lemniscate of Bernoulli is the curve defined by the polar equation:

$r^2 = 2 a^2 \cos 2 \theta$

Setting $b = a$:

\(\ds r^4 + a^4 - 2 a^2 r^2 \cos 2 \theta\)

\(=\)

\(\ds a^4\)

\(\ds \leadsto \ \ \)

\(\ds r^4 - 2 a^2 r^2 \cos 2 \theta\)

\(=\)

\(\ds 0\)

\(\ds \leadsto \ \ \)

\(\ds r^2\)

\(=\)

\(\ds 2 a^2 \cos 2 \theta\)

simplifying and rearranging

It follows that the lemniscate of Bernoulli is an oval of Cassini where $b = a$.

$\blacksquare$