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Bernoulli lemniscate

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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
This article was adapted from an original article by D.D. Sokolov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article