Cassini oval

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Some Cassini ovals. The foci are (-1, 0) and (1, 0). The curves are annotated with the value of b².

In mathematics, a Cassini oval is a set (or locus) of points in the plane such that each point p on the oval bears a special relation to two other, fixed points q₁ and q₂: the product of the distance from p to q₁ and the distance from p to q₂ is constant. That is, if we define the function dist(a,b) to be the distance from a point a to a point b, then all points on a Cassini oval satisfy the equation

$\mbox{dist}(q_1, p)\mbox{dist}(q_2, p)=b^2\,$

where b is a constant.

The points q₁ and q₂ are called the foci of the oval.

Cassini ovals are named after the astronomer Giovanni Domenico Cassini. Other names include Cassinian ovals and ovals of Cassini.

Suppose q₁ is the point (a,0), and q₂ is the point (-a,0). Then the points on the curve satisfy the equation

((x - a) 2 + y 2)((x + a) 2 + y 2) = b 4

Equivalent equations include

(x 2 + y 2) 2 - 2 a 2 (x 2 - y 2) + a 4 = b 4

and

(x 2 + y 2 + a 2) 2 - 4 a 2 x 2 = b 4

The equivalent polar equation is

r 4 - 2 a 2 r 2 cos2θ = b 4 - a 4

The shape of the oval depends on the ratio b/a. When b/a is greater than 1, the locus is a single, connected loop. When b/a is less than 1, the locus comprises two disconnected loops. When b/a is equal to 1, the locus is a lemniscate.

If a = b the curve is rational, but in general the curve has a pair of double points at infinity in the complex projective plane, at x = ±i, y = 1, z = 0 and no other singularities, and is a plane algebraic curve of genus one, and hence birationally equivalent to an elliptic curve.

Rescaling by substituting ax for x and ay for y, we obtain a one-parameter family

$(x^2+y^2+1)^2-4x^2=b^4 \,$