# Cassini oval

A plane algebraic curve of order four whose equation in Cartesian coordinates has the form:

$$(x^2+y^2)^2-2c^2(x^2-y^2)=a^4-c^4.$$ Figure: c020700a Figure: c020700b Figure: c020700c

A Cassini oval is the set of points (see Fig.) such that the product of the distances from each point to two given points $F_2=(-c,0)$ and $F_1=(c,0)$ (the foci) is constant. When $a\geq c\sqrt2$ the Cassini oval is a convex curve; when $c<a<c\sqrt2$ it is a curve with "waists" (concave parts); when $a=c$ it is a Bernoulli lemniscate; and when $a<c$ it consists of two components. Cassini ovals are related to lemniscates. Cassini ovals were studied by G. Cassini (17th century) in his attempts to determine the Earth's orbit.

How to Cite This Entry:
Cassini oval. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Cassini_oval&oldid=31950
This article was adapted from an original article by D.D. Sokolov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article