# Lemniscate of Bernoulli is Special Case of Ovals of Cassini

## Theorem

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

### Proof from Geometric Definition

The ovals of Cassini are defined as follows:

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

The ovals of Cassini are the loci of points $M$ in the plane such that:

$P_1 M \times P_2 M = b^2$

for a real constant $b$.

The lemniscate of Bernoulli is defined geometrically as:

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

The lemniscate of Bernoulli is the locus of points $M$ in the plane such that:

$P_1 M \times P_2 M = a^2$

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

$\blacksquare$

### Proof from Cartesian Definition

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

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

describes the ovals of Cassini.

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

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

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

Setting $b = a$:

 $\displaystyle \paren {x^2 + y^2 + a^2}^2 - 4 a^2 x^2$ $=$ $\displaystyle a^4$ $\displaystyle \leadsto \ \$ $\displaystyle \paren {x^2 + y^2}^2 + 2 \paren {x^2 + y^2} a^2 + a^4 - 4 a^2 x^2$ $=$ $\displaystyle a^4$ $\displaystyle \leadsto \ \$ $\displaystyle \paren {x^2 + y^2}^2$ $=$ $\displaystyle 4 a^2 x^2 - 2 a^2 x^2 - 2 a^2 y^2$ simplifying and rearranging $\displaystyle$ $=$ $\displaystyle 2 a^2 \paren {x^2 - y^2}$ further simplification

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

$\blacksquare$

### Proof from Polar Definition

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 $b = a$:

 $\displaystyle r^4 + a^4 - 2 a^2 r^2 \cos 2 \theta$ $=$ $\displaystyle a^4$ $\displaystyle \leadsto \ \$ $\displaystyle r^4 - 2 a^2 r^2 \cos 2 \theta$ $=$ $\displaystyle 0$ $\displaystyle \leadsto \ \$ $\displaystyle r^2$ $=$ $\displaystyle 2 a^2 \cos 2 \theta$ simplifying and rearranging

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

$\blacksquare$