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Difference between revisions of "Bernoulli lemniscate"

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A plane algebraic curve of order four, the equation of which in orthogonal Cartesian coordinates is:
 
A plane algebraic curve of order four, the equation of which in orthogonal Cartesian coordinates is:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015620/b0156201.png" /></td> </tr></table>
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$$(x^2+y^2)^2-2a^2(x^2-y^2)=0;$$
  
 
and in polar coordinates
 
and in polar coordinates
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015620/b0156202.png" /></td> </tr></table>
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$$\rho^2=2a^2\cos2\phi.$$
  
The Bernoulli lemniscate is symmetric about the coordinate origin (Fig.), which is a node with tangents <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015620/b0156203.png" /> and the point of inflection.
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The Bernoulli lemniscate is symmetric about the coordinate origin (Fig.), which is a node with tangents $y=\pm x$ and the point of inflection.
  
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/b015620a.gif" />
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/b015620a.gif" />
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Figure: b015620a
 
Figure: b015620a
  
The product of the distances of any point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015620/b0156204.png" /> to the two given points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015620/b0156205.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015620/b0156206.png" /> is equal to the square of the distance between the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015620/b0156207.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015620/b0156208.png" />. The Bernoulli lemniscate is a special case of the Cassini ovals, the [[Lemniscates|lemniscates]], and the sinusoidal spirals (cf. [[Cassini oval|Cassini oval]]; [[Sinusoidal spiral|Sinusoidal spiral]]).
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The product of the distances of any point $M$ to the two given points $F_1(-a,0)$ and $F_2(a,0)$ is equal to the square of the distance between the points $F_1$ and $F_2$. The Bernoulli lemniscate is a special case of the Cassini ovals, the [[Lemniscates|lemniscates]], and the sinusoidal spirals (cf. [[Cassini oval|Cassini oval]]; [[Sinusoidal spiral|Sinusoidal spiral]]).
  
 
The Bernoulli spiral was named after Jakob Bernoulli, who gave its equation in 1694.
 
The Bernoulli spiral was named after Jakob Bernoulli, who gave its equation in 1694.

Revision as of 18:47, 27 April 2014

A plane algebraic curve of order four, the equation of which in orthogonal Cartesian coordinates is:

$$(x^2+y^2)^2-2a^2(x^2-y^2)=0;$$

and in polar coordinates

$$\rho^2=2a^2\cos2\phi.$$

The Bernoulli lemniscate is symmetric about the coordinate origin (Fig.), which is a node with tangents $y=\pm x$ and the point of inflection.

Figure: b015620a

The product of the distances of any point $M$ to the two given points $F_1(-a,0)$ and $F_2(a,0)$ is equal to the square of the distance between the points $F_1$ and $F_2$. The Bernoulli lemniscate is a special case of the Cassini ovals, the lemniscates, and the sinusoidal spirals (cf. Cassini oval; Sinusoidal spiral).

The Bernoulli spiral was named after Jakob Bernoulli, who gave its equation in 1694.

References

[1] A.A. Savelov, "Planar curves" , Moscow (1960) (In Russian)


Comments

References

[a1] E. Brieskorn, H. Knörrer, "Plane algebraic curves" , Birkhäuser (1986) (Translated from German)
How to Cite This Entry:
Bernoulli lemniscate. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Bernoulli_lemniscate&oldid=12542
This article was adapted from an original article by D.D. Sokolov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article