Cassinian Ovals


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.


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.


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: