Cassini oval

A Cassini oval is a quartic plane curve defined as the set (or locus) of points in the plane such that the product of the distances to two fixed points is constant. This may be contrasted to an ellipse, for which the sum of the distances is constant, rather than the product. Cassini ovals are the special case of polynomial lemniscates when the polynomial used has degree 2.

Cassini ovals are named after the astronomer Giovanni Domenico Cassini who studied them in 1680. Cassini believed that the Sun traveled around the Earth on one of these ovals, with the Earth at one focus of the oval. Other names include Cassinian ovals, Cassinian curves and ovals of Cassini.

Let q₁ and q₂ be two fixed points in the plane and let b be a constant. Then a Cassini oval with foci q₁ and q₂ is defined to be the locus of points p so that the product of the distance from p to q₁ and the distance from p to q₂ is b². That is, if we define the function dist(x,y) to be the distance from a point x to a point y, then all points p on a Cassini oval satisfy the equation

If the foci are (a, 0) and (−a, 0), then the equation of the curve is

When expanded this becomes

The equivalent polar equation is

The curve depends, up to similarity, on e = b/a. When e < 1, the curve consists of two disconnected loops, each of which contains a focus. When e = 1, the curve is the lemniscate of Bernoulli having the shape of a sideways figure eight with a double point (specifically, a crunode) at the origin. When e > 1, the curve is a single, connected loop enclosing both foci. It is peanut-shaped for ${\displaystyle 1<e<{\sqrt {2}}}$ and convex for ${\displaystyle e\geq {\sqrt {2}}}$. The limiting case of a → 0 (hence e → ${\displaystyle \infty }$), in which case the foci coincide with each other, is a circle.

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