A **Cassini oval** is a plane curve defined as the set of points in the plane with the products of distances to two fixed points (loci) *F*_{1} and *F*_{2} is constant [1]; as a formula, the distance is (*F*_{1}, *F*_{2}) = 2*a* [2].

Cassini ovals are generalizations of lemniscates. The ovals are similar to ellipses, but instead of adding distances to loci, you multiply them.

The curve is symmetric with respect to the x-axis, the y-axis, and the origin.

## Equations

The general equation is (x^{2} + y^{2} + a^{2})^{2} – 4a^{2}x^{2} = b^{4}.

The polar equation is r^{4} − 2a^{2}r^{2}cos 2θ + a^{4}.

## Cassini Oval Shapes

The ovals can take on three different shapes, shown in the following image:

- When a = b, the oval resembles an infinity symbol ∞, which is a
*Leminiscate of Bernoulli*. - When a < b, the oval takes on the shape of an ellipse or peanut shell.
- If a > b, the oval splits into two ellipses shaped like eggs, with the narrow ends of the “eggs” facing each other. The shapes are mirror-images of each other.

The following image shows the various shapes the Cassini oval can take on, in one image:

## Area of Cassini Ovals

The area of a Cassini oval can be found in several ways, including numerical integration and elliptic integrals. What follows is a summary, you can find more detail on pages 235-236 of this PDF.

The oval is symmetric to both the x- and y-axes, so we can find the area of one quarter of an oval, and multiply by 4:

You can also calculate the area with the following elliptic integral, where E(x) is the complete elliptic integral of the second kind:

## History of the Cassini Oval

Cassini ovals were first studied by 15th Century Giovanni Cassini, who used the ovals to model the sun’s orbit around the Earth [3]. The orbits of the planets around the sun, orbits of satellites around a planet, and electron orbits in atoms, can all be modeled by a Cassini Oval. This may come as a surprise; a common misconception is that planetary orbits are highly elliptical [4].

Cassini Ovals have a wide variety of uses, including developing radar and sonar systems, modeling human blood cells, and fuel tank optimization.

## References

“Three different shapes” image created with Desmos.

[1] Mümtaz, K. A Multi Foci Closed Curve: Cassini Oval, Its Properties and Applications. Institutional Archive of the Naval Postgraduate School. 2013. PDF.

[2] Hellmers, J. et al. Simulation of light scattering by

biconcave Cassini ovals using the nullfield method with discrete sources. Journal of

Optics A: Pure and Applied Optics, Pure Appl. Opt. Vol 8. pp. 1-9. 2006.

[3] Gibson, K. The ovals of Cassini. Lecture Notes. 2007. Retrieved February 11, 2012 from: https://mse.redwoods.edu/darnold/math50c/CalcProj/sp07/ken/CalcPres.pdf

[4] Morgado, B. & Soares, V. Kepler’s ellipse, Cassini’s oval and the trajectory of planets. Eur. J. Phys. 35 (2014) 025009.