\begin{definition}[Definition:Lemniscate of Bernoulli/Cartesian Definition/Also defined as]
Some sources define the Cartesian equation for the '''lemniscate of Bernoulli''' as:
:$\paren {x^2 + y^2}^2 = a^2 \paren {x^2 - y^2}$
which is the same but for a scale factor:
:620px
Its associated parametric equation is:
:$\begin {cases} x = \dfrac {a \cos t} {\sin^2 t + 1} \\ y = \dfrac {a \cos t \sin t} {\sin^2 t + 1} \end{cases}$
\end{definition}