Definition:Lemniscate of Bernoulli

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

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

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


 * LemniscateOfBernoulli.png

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

which is the same but for a scale factor.

Also see

 * Equivalence of Definitions of Lemniscate of Bernoulli


 * Length of Lemniscate of Bernoulli