Cassinian Ovals

Introduction:

The ovals of Cassini are famous generalizations of some curves, including the lemniscate of Bernoulli and the circle. They vary continuously as the parameter a changes, and can be stacked to form a three-dimensional object, as shown below.

Ovals of Cassini at a = 1.9, 2, and 2.1
Stacked ovals of Cassini from a = 0 to a = 4

 

The lemniscate of Bernoulli is formed when a = b, and the circle when a = 0.

Definition:

The ovals of Cassini are defined implicitly by

Note the similarity to the definition of a circle (when a = 0) and the lemniscate of Bernoulli (when a = b). The ovals are defined geometrically by taking the set of points P such that the product of the distances from P to two other points A and B is constant. In the equations above, this constant is b2, and the distance between the two foci is 2a.

Properties:

It is difficult to compute formulas for the velocity and acceleration of the Cassinian ovals, except in special cases. Because the formulas are not particularly instructive, they have been omitted.

Arc Length: