A plane algebraic curve of order four, the equation of which in orthogonal Cartesian coordinates is:
and in polar coordinates
The Bernoulli lemniscate is symmetric about the coordinate origin (Fig.), which is a node with tangents $y=\pm x$ and the point of inflection.
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.
|||A.A. Savelov, "Planar curves" , Moscow (1960) (In Russian)|
|[a1]||E. Brieskorn, H. Knörrer, "Plane algebraic curves" , Birkhäuser (1986) (Translated from German)|
Bernoulli lemniscate. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Bernoulli_lemniscate&oldid=31949