Cassini Ovals
A Cassini oval is the locus of points in the plane where the product of distances to two fixed foci equals a constant. Unlike the ellipse (which uses the sum of distances), the Cassini oval undergoes dramatic topological changes: when the constant is small, the curve splits into two separate ovals around each focus. At a critical value, they merge into a lemniscate — Bernoulli’s famous figure-eight. Beyond that, a single smooth oval encloses both foci. Drag the foci and sweep through the constant to watch these transformations unfold.
|P−F1| · |P−F2| = b² · Lemniscate when b = a (half-distance between foci)
Definition and equation
Giovanni Domenico Cassini studied these curves in 1680 while investigating the relative motions of the Earth and Sun. Given two foci F1 and F2 separated by distance 2a, a Cassini oval is the set of all points P such that |PF1| · |PF2| = b² for a positive constant b. In Cartesian coordinates with foci at (±a, 0), the implicit equation is ((x−a)² + y²)((x+a)² + y²) = b&sup4;, which expands to (x² + y²)² − 2a²(x² − y²) = b&sup4; − a&sup4;.
Topological transitions
The shape of the curve depends on the ratio b/a. When b < a, the curve consists of two separate ovals, one around each focus. When b = a exactly, the two ovals pinch together at the midpoint to form a lemniscate of Bernoulli — a figure-eight shape. When b > a, the curve is a single connected oval enclosing both foci. As b grows much larger than a, the oval approaches a circle. This smooth transition between topologically distinct shapes is one of the most beautiful features of Cassini ovals.
Comparison with ellipses
An ellipse is defined by the sum of distances: |PF1| + |PF2| = constant. A Cassini oval uses the product. This seemingly small change produces dramatically different geometry. Ellipses are always convex and always connected. Cassini ovals can be non-convex (with a waist) and can even split into two pieces. The toggle above lets you overlay an ellipse with the same foci for visual comparison. Notice how the Cassini oval bulges more near the foci and pinches in between them, while the ellipse remains smoothly convex.
The lemniscate of Bernoulli
The special case b = a produces the lemniscate, studied by Jacob Bernoulli in 1694. Its polar equation is r² = 2a² cos(2θ), an elegant formula that makes it a favorite in mathematics. The lemniscate has fascinating properties: its arc length involves an elliptic integral (which historically gave this class of integrals its name), and it played a central role in the development of elliptic functions by Euler, Gauss, and Abel. The symbol for infinity (∞) is sometimes said to derive from the lemniscate shape.