Cassini Oval Transitions

Cassini Oval Transitions

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.

Calculating Intersections

Let the foci be at (-a, 0) and (a, 0). A point (x, y) on the curve satisfies the equation for a constant product b²:
((x - a)² + y²) ((x + a)² + y²) = b⁴

In polar coordinates, setting x = r cos(θ) and y = r sin(θ), the equation simplifies to:
r⁴ - 2a² r² cos(2θ) + a⁴ - b⁴ = 0

Using the quadratic formula for u = r²:

1. Set a = 1, b = 1.5, θ = 0°.
2. Substitute values: u = 1² cos(0) ± √(1.5⁴ - 1⁴ sin²(0))
3. Calculate discriminant: 1.5⁴ - 0 = 5.0625
4. u = 1 ± √(5.0625) = 1 ± 2.25
5. We get u₁ = 3.25 and u₂ = -1.25.
6. Since r² must be non-negative, the only valid solution is r = √(3.25) ≈ 1.8027756377319946.

Exploring Topologies

Use the Cassini Oval Transitions interactive module to explore how the curve shape depends on a and b.

1. Set Focus half separation to 1 and Product scale to 1.5. Notice how the curve is a Single oval surrounding both foci.
2. Change the Product scale to 1. The curve pinches in the middle to form a Touching lemniscate (the Bernoulli lemniscate), intersecting at the origin.
3. Reduce the Product scale further to 0.5. The curve splits into Two disjoint lobes, each surrounding one focus.
4. Change the Ray degrees to probe different intersection points and observe how the calculated radii change.

Model Limitations

The interactive visualization normalizes trigonometric calculations to handle floating-point roundoff errors around 0 or mathematically exact roots. In a general-purpose implicit equation solver, finding all roots precisely for arbitrary parameters would require sophisticated techniques like Marching Squares, whereas this module analytically finds the roots along explicit probing rays. The fixed angles available in the UI restrict the exploration to specific finite rays.

Primary References

Learn more about Cassini Ovals from these resources:

• Collected ATOs (University of California, Irvine) discusses mathematical curves including Cassini Ovals and provides analytical insights into their topologies.