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.
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

((xa)2 + y2)((x + a)2 + y2) = b4

Equivalent equations include

(x2 + y2)2 − 2a2(x2y2) + a4 = b4

and

(x2 + y2 + a2)2 − 4a2x2 = b4

The equivalent polar equation is

r4 − 2a2r2cos2θ = b4a4

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 \,

which has j-invariant

j = 16 \frac{(b^8-16b^4+16)^3}{b^{16}(1-b^4)}.

Note that the definition of the curve is analogous to that of the ellipse, wherein the sum

\mbox{dist}(q_1, p)+\mbox{dist}(q_2, p)\,

is constant, rather than the product.

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