# Lemniscate of Bernoulli is Special Case of Ovals of Cassini

## Contents

## 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:

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

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:

The Cartesian equation:

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

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 $b = a$.

$\blacksquare$

## Sources

- 1968: Murray R. Spiegel:
*Mathematical Handbook of Formulas and Tables*... (previous) ... (next): $\S 11$: Special Plane Curves: Ovals of Cassini: $11.31$

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