# Category:Lemniscates

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This category contains results about **Lemniscates**.

Definitions specific to this category can be found in Definitions/Lemniscates.

### 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:

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

## Subcategories

This category has only the following subcategory.

## Pages in category "Lemniscates"

The following 5 pages are in this category, out of 5 total.