Lemniscate of Bernoulli

A lemniscate of Bernoulli and its two foci Template:Math and Template:Math

The lemniscate of Bernoulli is the pedal curve of a rectangular hyperbola

Template:Sinusoidal spirals.svg In geometry, the lemniscate of Bernoulli is a plane curve defined from two given points Template:Math and Template:Math, known as foci, at distance Template:Math from each other as the locus of points Template:Math so that Template:Math. The curve has a shape similar to the numeral 8 and to the ∞ symbol. Its name is from Template:Wikt-lang, 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.

Equations

The equations can be stated in terms of the focal distance Template:Mvar or the half-width Template:Mvar of a lemniscate. These parameters are related as Template:Math.

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

  $\begin{array}{cc}{(x^2+y^2)}^2& =a^2(x^2-y^2)\\ & =2c^2(x^2-y^2)\end{array}$

• As a square function of x:

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

• As a parametric equation:

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

• A rational parametrization:

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

• 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:

  $r{r}^{\prime }=c^2$

Arc length and elliptic functions

The lemniscate sine and cosine relate the arc length of an arc of the lemniscate to the distance of one endpoint from the origin.

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 [[square root of minus one|Template:Sqrt]] is called the lemniscatic case in some sources.

Using the elliptic integral

$\mathrm{arcsl}x\stackrel{\text{def}}{=}\underset{0}{\overset{x}{\int }}\frac{dt}{\sqrt{1-t^4}}$

the formula of the arc length Template:Mvar can be given as

$\begin{array}{cc}L& =4a\underset{0}{\overset{1}{\int }}\frac{dt}{\sqrt{1-t^4}}=4a\mathrm{arcsl}1=2\varpi a\\ & =\frac{\mathrm{\Gamma }(1/4{)}^2}{\sqrt{\pi }}c=\frac{2\pi }{\mathrm{M}(1,1/\sqrt{2})}c\approx 7.416\cdot c\end{array}$

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

Angles

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

relation between angles at Bernoulli's lemniscate

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.

Template:Math and Template:Math are the foci of the lemniscate, Template:Math is the midpoint of the line segment Template:Math and Template:Math is any point on the lemniscate outside the line connecting Template:Math and Template:Math. The normal Template:Math of the lemniscate in Template:Math intersects the line connecting Template:Math and Template:Math in Template:Math. Now the interior angle of the triangle Template:Math at Template:Math is one third of the triangle's exterior angle at Template:Math (see also angle trisection). In addition the interior angle at Template:Math is twice the interior angle at Template:Math.

Further properties

The inversion of hyperbola yields a lemniscate

• The lemniscate is symmetric to the line connecting its foci Template:Math and Template:Math and as well to the perpendicular bisector of the line segment Template:Math.
• The lemniscate is symmetric to the midpoint of the line segment Template:Math.
• The area enclosed by the lemniscate is Template:Math.
• The lemniscate is the circle inversion of a hyperbola and vice versa.
• The two tangents at the midpoint Template:Math are perpendicular, and each of them forms an angle of Template:Math with the line connecting Template:Math and Template:Math.
• The planar cross-section of a standard torus tangent to its inner equator is a lemniscate.
• The curvature at $(x,y)$ is $\frac{3}{a^2}\sqrt{x^2+y^2}$. 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.

See also

• Lemniscate of Booth
• Lemniscate of Gerono
• Lemniscate constant
• Lemniscatic elliptic function
• Cassini oval

References

• Template:Cite book

External links

• Template: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)