# Definition:Lemniscate of Bernoulli

## Definition

### Geometric Definition

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:

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

### Cartesian Definition

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

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

### Polar Definition

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

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

### Parametric Definition

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

$\begin{cases} x = \dfrac {a \sqrt 2 \cos t} {\sin^2 t + 1} \\ y = \dfrac {a \sqrt 2 \cos t \sin t} {\sin^2 t + 1} \end{cases}$ ### Focus

Each of the two points $P_1$ and $P_2$ can be referred to as a focus of the lemniscate.

### Lobe

Each of the two loops that constitute the lemniscate can be referred to as a lobe of the lemniscate.

### Major Axis

The line $P_1 P_2$ is the major axis of the lemniscate.

### Major Semiaxis

Each of the lines $O P_1$ and $O P_2$ is a major semiaxis of the lemniscate.

## Also see

• Results about lemniscates can be found here.

## Source of Name

This entry was named for Jacob Bernoulli.

## Historical Note

The lemniscate of Bernoulli was investigated in some depth by Jacob Bernoulli, from whom it was given its name.

## Linguistic Note

The word lemniscate comes from the Latin word lemniscus, which means pendant ribbon.

The word may ultimately derive from the Latin lēmniscātus, which means decorated with ribbons.

This may in turn come from the ancient Greek island of Lemnos where ribbons were worn as decorations.