Lemniscate of Bernoulli

In geometry, the lemniscate of Bernoulli is a plane curve defined from two given points $F 1$ and $F 2$ , known as foci, at distance $2 c$ from each other as the locus of points $P$ so that $PF 1 \cdot PF 2 = c 2$ . The curve has a shape similar to the numeral 8 and to the ∞ symbol. Its name is from lemniscatus, which is Latin for "decorated with hanging ribbons". It is a special case of the Cassini oval and is a rational algebraic curve of degree 4.

This lemniscate was first described in 1694 by Jakob Bernoulli as a modification of an ellipse, which is the locus of points for which the sum of the distances to each of two fixed focal points is a constant. A Cassini oval, by contrast, is the locus of points for which the product of these distances is constant. In the case where the curve passes through the point midway between the foci, the oval is a lemniscate of Bernoulli.

This curve can be obtained as the inverse transform of a hyperbola, with the inversion circle centered at the center of the hyperbola (bisector of its two foci). It may also be drawn by a mechanical linkage in the form of Watt's linkage, with the lengths of the three bars of the linkage and the distance between its endpoints chosen to form a crossed parallelogram.^[1]

Equations

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 $a = c \sqrt 2$ .

Its Cartesian equation is (up to translation and rotation):
${\begin{aligned}\left(x^{2}+y^{2}\right)^{2}&=a^{2}\left(x^{2}-y^{2}\right)\\&=2c^{2}\left(x^{2}-y^{2}\right)\end{aligned}}$
As a square function of x:
$y^{2}=\left({\sqrt {8x^{2}+a^{2}}}-a\right){\frac {a}{2}}-x^{2}$
As a parametric equation:
$x={\frac {a\cos t}{1+\sin ^{2}t}};\qquad y={\frac {a\sin t\cos t}{1+\sin ^{2}t}}$
A rational parametrization:^[2]
$x=a{\frac {t+t^{3}}{1+t^{4}}};\qquad y=a{\frac {t-t^{3}}{1+t^{4}}}$
In polar coordinates:
$r^{2}=a^{2}\cos {2\theta }$
In the complex plane:
$|z-c||z+c|=c^{2}$
In two-center bipolar coordinates:
$rr'=c^{2}$

Arc length and elliptic functions

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 Around 1800, the elliptic functions inverting those integrals were studied by C. F. Gauss (largely unpublished at the time, but allusions in the notes to his Disquisitiones Arithmeticae).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

\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

{\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 $c$ and $a={\sqrt {2}}c$ are defined as above, $\varpi =2\operatorname {arcsl} {1}$ is the lemniscate constant, $\Gamma$ is the gamma function and $\operatorname {M}$ is the arithmetic–geometric mean.

Angles

Given two distinct points ${\rm {A}}$ and ${\rm {B}}$ , let ${\rm {M}}$ be the midpoint of ${\rm {AB}}$ . Then the lemniscate of diameter ${\rm {AB}}$ can also be defined as the set of points ${\rm {A}}$ , ${\rm {B}}$ , ${\rm {M}}$ , together with the locus of the points ${\rm {P}}$ such that $|{\widehat {\rm {APM}}}-{\widehat {\rm {BPM}}}|$ 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]

F 1

and

F 2

are the foci of the lemniscate,

O

is the midpoint of the line segment

F 1 F 2

and

P

is any point on the lemniscate outside the line connecting

F 1

and

F 2

. The normal

n

of the lemniscate in

P

intersects the line connecting

F 1

and

F 2

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

.

Further properties

The lemniscate is symmetric to the line connecting its foci $F 1$ and $F 2$ and as well to the perpendicular bisector of the line segment $F 1 F 2$ .
The lemniscate is symmetric to the midpoint of the line segment $F 1 F 2$ .
The area enclosed by the lemniscate is $a 2 = 2 c 2$ .
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 $.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}⁠π/4⁠$ with the line connecting $F 1$ and $F 2$ .
The planar cross-section of a standard torus tangent to its inner equator is a lemniscate.
The curvature at $(x,y)$ is ${3 \over a^{2}}{\sqrt {x^{2}+y^{2}}}$ . The maximum curvature, which occurs at $(\pm a,0)$ , is therefore $3/a$ .

Applications

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

Notes

^ Bryant, John; Sangwin, Christopher J. (2008), How round is your circle? Where Engineering and Mathematics Meet, Princeton University Press, pp. 58–59, ISBN 978-0-691-13118-4.
^ Lemmermeyer, Franz (2011). "Parametrizing Algebraic Curves". arXiv:1108.6219 [math.NT].
^ Eymard, Pierre; Lafon, Jean-Pierre (2004). The Number Pi. American Mathematical Society. ISBN 0-8218-3246-8. p. 200
^ Alexander Ostermann, Gerhard Wanner: Geometry by Its History. Springer, 2012, pp. 207-208

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

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)

[1] Bryant, John; Sangwin, Christopher J. (2008), How round is your circle? Where Engineering and Mathematics Meet, Princeton University Press, pp. 58–59, ISBN 978-0-691-13118-4.

[2] Lemmermeyer, Franz (2011). "Parametrizing Algebraic Curves". arXiv:1108.6219 [math.NT].

[3] Eymard, Pierre; Lafon, Jean-Pierre (2004). The Number Pi. American Mathematical Society. ISBN 0-8218-3246-8. p. 200

[4] Alexander Ostermann, Gerhard Wanner: Geometry by Its History. Springer, 2012, pp. 207-208

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