View source for Bernoulli lemniscate

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>

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>

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.

<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/b015620a.gif" />

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]]).

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

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.A. Savelov,   "Planar curves" , Moscow  (1960)  (In Russian)</TD></TR></table>



====Comments====


====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  E. Brieskorn,   H. Knörrer,   "Plane algebraic curves" , Birkhäuser  (1986)  (Translated from German)</TD></TR></table>