Lemniscate of Bernoulli

• 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 square function of x:

  ${\displaystyle y^{2}=\left({\sqrt {8x^{2}+a^{2}}}-a\right){\frac {a}{2}}-x^{2}}$

• 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
  :

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

• In the complex plane:

  ${\displaystyle |z-c||z+c|=c^{2}}$

• In two-center bipolar coordinates:

  ${\displaystyle rr'=c^{2}}$

The determination of the

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

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

where ${\displaystyle c}$ and ${\displaystyle a={\sqrt {2}}c}$ are defined as above, ${\displaystyle \varpi =2\operatorname {arcsl} {1}}$ is the lemniscate constant, ${\displaystyle \Gamma }$ 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.

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
• Lemniscate constant
• Lemniscatic elliptic function
• Cassini oval

References

• 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
  .

External links

Wikimedia Commons has media related to Lemniscate of Bernoulli.

• 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)