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