# 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.

## Sources

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