Bernoulli lemniscate
From Encyclopedia of Mathematics
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 
and the point of inflection.
Figure: b015620a
The product of the distances of any point 
to the two given points 
and 
is equal to the square of the distance between the points 
and 
. 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=31949
Bernoulli lemniscate. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Bernoulli_lemniscate&oldid=31949
This article was adapted from an original article by D.D. Sokolov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article