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:

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