Cassini Ovals

The Cassini ovals are a family of quartic curves, also called Cassini ellipses, described by a point such that the product of its distances from two fixed points a distance apart is a constant , where . The shape of the curve depends on . If , the curve is a single loop with an oval (left figure above) or dog bone (second figure) shape. The case produces a lemniscate (third figure). If , then the curve consists of two loops (right figure). Cassini ovals are anallagmatic curves.

A series of ovals for values of to 1.5 are illustrated above.

The curve was first investigated by Cassini in 1680 when he was studying the relative motions of the Earth and the Sun. Cassini believed that the Sun traveled around the Earth on one of these ovals, with the Earth at one focus of the oval.

The Cassini ovals are defined in two-center bipolar coordinates by the equation

(1)

with the origin at a focus. Even more incredible curves are produced by the locus of a point the product of whose distances from 3 or more fixed points is a constant.

The Cassini ovals have the Cartesian equation

(2)

or the equivalent form

(3)

and the polar equation

(4)

Solving for using the quadratic equation gives

			(5)
			(6)
			(7)
			(8)
			(9)

Let a torus of tube radius be cut by a plane perpendicular to the plane of the torus's centroid. Call the distance of this plane from the center of the torus hole , let , and consider the intersection of this plane with the torus as is varied. The resulting curves are Cassini ovals, with a lemniscate occurring at . Cassini ovals are therefore toric sections.

If , the curve has area

			(10)
			(11)
			(12)

where is the complete elliptic integral of the second kind with elliptic modulus . If , the curve is given by

			(13)
			(14)

for . This is a lemniscate having area

(15)

(two loops of a curve the linear scale of the usual lemniscate , which has area for each loop). If , the curve becomes two disjoint ovals with equations

(16)

where and

(17)