Cassini Oval Transitions
Change a fixed distance product to split or join two lobes.
Free theory articleChange a fixed distance product to split or join two lobes.
Concept & interpretation
A Cassini oval is the locus of points in the plane such that the product of the distances to two fixed points (foci) is a constant. This differs from an ellipse, where the sum of the distances is constant. By varying the distance between the foci and the product value, the curve transitions between different topologies.
Exploring Topologies
Use the Cassini Oval Transitions interactive module to explore how the curve shape depends on a and b.
- Set Focus half separation to 1 and Product scale to 1.5. Notice how the curve is a Single oval surrounding both foci.
- Change the Product scale to 1. The curve pinches in the middle to form a Touching lemniscate (the Bernoulli lemniscate), intersecting at the origin.
- Reduce the Product scale further to 0.5. The curve splits into Two disjoint lobes, each surrounding one focus.
- Change the Ray degrees to probe different intersection points and observe how the calculated radii change.