Lemniscate of Bernoulli

In mathematics, the lemniscate of Bernoulli is an algebraic curve described by a Cartesian equation of the form:

:$(x^2\; +\; y^2)^2\; =\; 2a^2\; (x^2\; -\; y^2).,$

The curve has a shape similar to the numeral 8 and to the $infty$ symbol.

The 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.

Bernoulli called it the "lemniscus", which is Latin for "pendant ribbon".

The lemniscate 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).

Other equations

A lemniscate may also be described by the polar equation

:$r^2\; =\; a^2\; cos\; 2\; heta,$

or the bipolar equation

:$rr\text{'}\; =\; frac\{a^2\}\{2\}$

Derivatives

Each first derivative below was calculated using implicit differentiation.

With $y$ as a function of $x$

:

:

With $x$ as a function of $y$

:

:

Curvature

Once the first two derivatives are known, curvature is easily calculated:

:$kappa\; =\; pm3(x^2\; +\; y^2)^\{1/2\}a^\{-2\}\; ,$

the sign being chosen according to the direction of motion along the curve. The lemniscate has the property that the magnitude of the curvature at any point is proportional to that point's distance from the origin.

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 Around 1800, the elliptic functions inverting those integrals were studied by C. F. Gauss (largely unpublished at the time, but allusions in the notes to his "Disquisitiones Arithmeticae").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 the square root of minus one is called the "lemniscatic case" in some sources.

See also

*Lemniscate of Booth

References

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