Quantum redactiones paginae "Lemniscus (mathematica)" differant

 

[[Mathematicus|Mathematici]] multi studium in hanc figuram dabant, sicut [[Iacobus Bernoulli]], [[Camillus-Christophorus Gerono]], et [[Booth]]{{dubsig}}. Ex his, lemniscus Bernoulli notissimus.

 

==Lemniscus Iacobi Bernoulli==

[[Image:Lemniscate.png|thumb|201px|Lemniscus Bernoulli]]

Lemniscus Bernoulli primum descriptus est [[1694]]. Eius forma describitur ab [[aequatio]]ne [[Renatus Cartesius|Cartesiana]]:

:<math>(x^2 + y^2)^2 = a^2 (x^2 - y^2)\,</math>

 

Hic est locus punctorum quo ''productum'' distantiarum est constans.

 

===Other equations===

A lemniscate may also be described by the [[polar coordinates|polar]] equation

:<math>r^2 = a^2 \cos 2\theta\,</math>

 

 

===Arc length and elliptic functions===

The determination of the [[arc length]] of arcs of the lemniscate leads to [[elliptic integral]]s, as was discovered in the eighteenth century. Around 1800, the [[elliptic function]]s inverting those integrals were studied by [[C. F. Gauss]] (largely unpublished at the time, but allusions in the notes to his ''[[Disquisitiones Arithmeticae]]'').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 lattice]]s are of a very special form, being proportional to the [[Gaussian integer]]s. 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.

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Lemniscus Gerono est [[curvus]] [[algebra|algebraicus]] a [[Camillus-Christophorus Gerono|Camillo-Christophoro Gerono]] studebatur, grado [[4 (numerus)|4]], genere [[Cifra|0]], in forma 8 horizontaliter. Aequatio eius est:

:<math>x^4-x^2+y^2 = 0.</math>

 

Qui curvus est 0 genere, parametrizatur functionibus rationalibus, sicut:

:<math>x = \frac{t^2-1}{t^2+1},\ y = \frac{2t(t^2-1)}{(t^2+1)^2}.</math>

 

==Lemniscus Booth==

[[Image:Lemniscate of Booth.png|thumb|Lemniscus Booth; c = 0.25, 0.5, 0.75, and 1]]

 

:<math>(x^2+y^2)^2 + 4y^2 = 4c(x^2+y^2).</math>

 

==Vide etiam==

 

[[Categoria:Algebra]]

 

[[af:Lemniskaat]]