The equations can be stated in terms of the focal distance c or the half-width a of a lemniscate. These parameters are related as a = c√2.

• Its Cartesian equation is (up to translation and rotation):

  ${\displaystyle {\begin{aligned}\left(x^{2}+y^{2}\right)^{2}&=a^{2}\left(x^{2}-y^{2}\right)\\&=2c^{2}\left(x^{2}-y^{2}\right)\end{aligned}}}$ 

• As a parametric equation:

  ${\displaystyle x={\frac {a\cos t}{1+\sin ^{2}t}};\qquad y={\frac {a\sin t\cos t}{1+\sin ^{2}t}}}$ 

• A rational parametrization:^[2]

  ${\displaystyle x=a{\frac {t+t^{3}}{1+t^{4}}};\qquad y=a{\frac {t-t^{3}}{1+t^{4}}}}$ 

• In polar coordinates:

  ${\displaystyle r^{2}=a^{2}\cos 2\theta }$ 

• Its equation in the complex plane is:

  ${\displaystyle |z-c||z+c|=c^{2}}$ 

• In two-center bipolar coordinates:

  ${\displaystyle rr'=c^{2}}$ 

• In rational polar coordinates:

  ${\displaystyle Q=2s-1}$ 

The determination of the arc length of arcs of the lemniscate leads to elliptic integrals, as was discovered in the eighteenth century. Around 1800, the elliptic functions inverting those integrals were studied by C. F. Gauss (largely unpublished at the time, but allusions in the notes to his Disquisitiones Arithmeticae). The period lattices are of a very special form, being proportional to the Gaussian integers. For this reason the case of elliptic functions with complex multiplication by √−1 is called the lemniscatic case in some sources.

Using the elliptic integral

${\displaystyle \operatorname {arcsl} x{\stackrel {\text{def}}{{}={}}}\int _{0}^{x}{\frac {dt}{\sqrt {1-t^{4}}}}}$ 

the formula of the arc length L can be given as

${\displaystyle {\begin{aligned}L&=4{\sqrt {2}}\,c\int _{0}^{1}{\frac {dt}{\sqrt {1-t^{4}}}}=4{\sqrt {2}}\,c\,\operatorname {arcsl} 1\\[6pt]&={\frac {\Gamma (1/4)^{2}}{\sqrt {\pi }}}\,c={\frac {2\pi }{\operatorname {M} (1,1/{\sqrt {2}})}}c\approx 7{.}416\cdot c\end{aligned}}}$ 

where ${\displaystyle \Gamma }$  is the gamma function and ${\displaystyle \operatorname {M} }$  is the arithmetic–geometric mean.

Given two distinct points ${\displaystyle {\rm {A}}}$  and ${\displaystyle {\rm {B}}}$ , let ${\displaystyle {\rm {M}}}$  be the midpoint of ${\displaystyle {\rm {AB}}}$ . Then the lemniscate of diameter ${\displaystyle {\rm {AB}}}$  can also be defined as the set of points ${\displaystyle {\rm {A}}}$ , ${\displaystyle {\rm {B}}}$ , ${\displaystyle {\rm {M}}}$ , together with the locus of the points ${\displaystyle {\rm {P}}}$  such that ${\displaystyle |{\widehat {\rm {APM}}}-{\widehat {\rm {BPM}}}|}$  is a right angle (cf. Thales' theorem and its converse).^[3]

The following theorem about angles occurring in the lemniscate is due to German mathematician Gerhard Christoph Hermann Vechtmann, who described it 1843 in his dissertation on lemniscates.^[4]

F₁ and F₂ are the foci of the lemniscate, O is the midpoint of the line segment F₁F₂ and P is any point on the lemniscate outside the line connecting F₁ and F₂. The normal n of the lemniscate in P intersects the line connecting F₁ and F₂ in R. Now the interior angle of the triangle OPR at O is one third of the triangle's exterior angle at R (see also angle trisection). In addition the interior angle at P is twice the interior angle at O.

• The lemniscate is symmetric to the line connecting its foci F₁ and F₂ and as well to the perpendicular bisector of the line segment F₁F₂.
• The lemniscate is symmetric to the midpoint of the line segment F₁F₂.
• The area enclosed by the lemniscate is a² = 2c².
• The lemniscate is the circle inversion of a hyperbola and vice versa.
• The two tangents at the midpoint O are perpendicular, and each of them forms an angle of π/4 with the line connecting F₁ and F₂.
• The planar cross-section of a standard torus tangent to its inner equator is a lemniscate.
• The curvature at ${\displaystyle (x,y)}$  is ${\displaystyle {3 \over a^{2}}{\sqrt {x^{2}+y^{2}}}}$ . The maximum curvature, which occurs at ${\displaystyle (\pm a,0)}$ , is therefore ${\displaystyle 3/a}$ .

Dynamics on this curve and its more generalized versions are studied in quasi-one-dimensional models.

• Lemniscate of Booth
• Lemniscate of Gerono
• Gauss's constant
• Lemniscatic elliptic function
• Cassini oval

• J. Dennis Lawrence (1972). A catalog of special plane curves. Dover Publications. pp. 4–5, 121–123, 145, 151, 184. ISBN 0-486-60288-5.

• Weisstein, Eric W. "Lemniscate". MathWorld.
• "Lemniscate of Bernoulli" at The MacTutor History of Mathematics archive
• "Lemniscate of Bernoulli" at MathCurve.
• Coup d'œil sur la lemniscate de Bernoulli (in French)