Cassini oval

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(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

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 + y²)((x + a)² + y²) = b⁴

Equivalent equations include

(x² + y²)² − 2a²(x² − y²) + a⁴ = b⁴

and

(x² + y² + a²)² − 4a²x² = b⁴

The equivalent polar equation is

r⁴ − 2a²r²cos2θ = b⁴ − a⁴

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 of Bernoulli.

If a = b the curve is rational, but in general the curve is bicircular, meaning it has a pair of double points at infinity in the complex projective plane, at x = ±i, y = 1, z = 0, and no other singularities. It 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

which has j-invariant

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

is constant, rather than the product.

Curves orthogonal to the Cassini ovals: Formed when the foci of the Cassini ovals are the points (a,0) and (-a,0),equilateral hyperbolas centered at (0,0) after a rotation around (0,0) are made to pass through the foci.

Examples

Second lemniscate of Mandelbrot set is Casini oval with equation

References

• J. Dennis Lawrence (1972). A catalog of special plane curves. Dover Publications. pp. 5,153–155. ISBN 0-486-60288-5. 

External links

• MacTutor description
• Weisstein, Eric W., "Cassini Ovals" from MathWorld.
• 2Dcurves.com description
• "Ovale de Cassini" at Encyclopédie des Formes Mathématiques Remarquables (in French)