Definition:Lemniscate of Bernoulli

Definition
The lemniscate of Bernoulli is the curve defined by the Cartesian equation:
 * $\left({x^2 + y^2}\right)^2 = 2 a^2 \left({x^2 - y^2}\right)$

or the polar equation:
 * $r^2 = 2a^2 \cos 2\theta$

or the parametric equation:
 * $\begin{cases}x = \dfrac {a \sqrt 2 \cos \left({t}\right)} {\sin \left({t}\right)^2 + 1} \\ y = \dfrac {a \sqrt 2 \cos \left({t}\right) \sin \left({t}\right)} {\sin \left({t}\right)^2 + 1}\end{cases}$


 * LemniscateOfBernoulli.png

Also defined as
Some sources give this as:
 * $\left({x^2 + y^2}\right)^2 = a^2 \left({x^2 - y^2}\right)$

which is the same but for a scale factor.

Also see

 * Equivalence of Definitions of Lemniscate of Bernoulli


 * Length of Lemniscate of Bernoulli

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.