Lemniscates

Lying eight is a forwarding to this article. For the so-called power plug, see device plug # C7 .

A lemniscate (from Greek λημνίσκος lēmnískos , loop ') is a loop-shaped geometric curve in the shape of a lying figure eight. Mostly the lemniscate of Bernoulli is meant with "lemniscate" .

Lemniscate by Bernoulli

Lemniscate by Bernoulli

→ Main article : Bernoulli's lemniscate

The lemniscate of Bernoulli (after Jacob Bernoulli ) is an algebraic curve of degree 4, it has the equation

${\displaystyle (x^{2}+y^{2})^{2}=2a^{2}(x^{2}-y^{2})}$

with one parameter . In polar coordinates it is given by the equation ${\displaystyle a\in \mathbb {R} }$

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

described.

Lemniscate from Booth

Lemniscate from Booth

A Booth lemniscate (after James Booth ) is an algebraic curve of degree 4, it has the equation

${\displaystyle (x^{2}+y^{2})^{2}=cx^{2}+dy^{2}}$

with . ${\displaystyle c>0>d}$

For you get a lemniscate from Bernoulli. ${\displaystyle d=-c}$

It is a special case of the hippopede of Proclus ( o. B. D. A. applies and ): ${\displaystyle c>0}$ ${\displaystyle c>d}$

${\displaystyle (x^{2}+y^{2})^{2}=cx^{2}+dy^{2}}$

just in case . For one has oval-shaped closed curves, which is why they are called Ovals by Booth in this case. The name Hippopede comes from the Greek and has its origin in the fact that they are reminiscent of an ankle cuff for horses. They are special cases of the spirals of Perseus, which result as parallel sections through a torus , the planes being perpendicular to the axis in the plane of the torus. The lemniscate results when the plane just touches the inner ring in the torus. ${\displaystyle d<0}$ ${\displaystyle d>0}$

Lemniscate from Gerono

Gerono lemniscate: solution set of x ⁴ −x ² + y ² = 0

The Gerono lemniscate named after Camille-Christophe Gerono is an algebraic curve of degree 4 and gender 0, it has the equation

${\displaystyle x^{4}-x^{2}+y^{2}=0.}$

As a curve of gender 0 , it can be parameterized by rational functions , for example by:

${\displaystyle x={\frac {t^{2}-1}{t^{2}+1}}}$

${\displaystyle y={\frac {2t(t^{2}-1)}{(t^{2}+1)^{2}}}}$

A simpler parameterization is the parameterization as a Lissajous figure :

${\displaystyle x=\cos \varphi }$

${\displaystyle y=\sin \varphi \,\cos \varphi =\sin(2\varphi )/2}$

Lawrence gives the somewhat more general equation:

${\displaystyle z^{4}=a^{2}\cdot (z^{2}-w^{2})}$

This has the parameter representation:

${\displaystyle z=a\cos(t)}$

${\displaystyle w=a\sin(t)\cdot \cos(t)}$

with . ${\displaystyle -\pi \leq t\leq \pi }$

It is also known as an eight knot .

The curve was already Grégoire de Saint-Vincent (Opus geometricum quadraturae circuli et sectionum coni, 1647, as parabolis virtualis ), Christiaan Huygens (letter to Gottfried Wilhelm Leibniz March 16, 1691, with the designation Lemniskate) and Gabriel Cramer (1750, the she called double sack). Jules Antoine Lissajous treated it, parameterized by trigonometric functions, in 1857. The curve was named after Gerono at the end of the 19th century (for example Gabriel-Marie , Exercices de géométrie descriptive, 1900).

literature

• JD Lawrence: A Catalog of Special Plane Curves. Dover 1972. ISBN 0-486-60288-5 .

Web links

Commons : Lemniscate  - collection of images, videos and audio files

Wiktionary: Lemniskate  - explanations of meanings, word origins, synonyms, translations

Individual evidence

1. ^ French website on Booth's lemniscate
2. ↑ figure eight curve.
3. ↑ Lawrence: A catalog of special plane curves. Dover 1972, p. 124.
4. ↑ discussion in mathoverflow