# Cassini ovals

A **Cassini oval** is a locus of points determined by two fixed points F_{1} and F_{2} (the "foci") at a distance 2*a* apart (in the figure the foci are on the *x*-axis at F_{1,2} = ±1). If the distance of a certain point in the plane to F_{1} is *r*_{1} and the distance of the same point to F_{2} is *r*_{2} then the locus is defined by the product of distances *r*_{1}×*r*_{2} being constant and equal to *b*^{2}.

The shape of the curves depends on the ratio *b* **:** *a*. If *b* > *a*, the curve is a single loop (red curve in the figure)—a dog bone. The case *a* = *b* produces a lemniscate of Bernoulli (green curve). The case *b* < *a* gives two disconnected ovals (eggs).

The Cassini ovals are a family of quartic curves, also called Cassini ellipses. They were introduced by Jean-Dominique Cassini (1625–1712), a French-Italian astronomer, who studied them as a possible alternative for the Kepler elliptic planetary orbits.^{[1]}
They did not appear in print until Cassini's son Jacques (1677-1756) published the *Eléments d'Astronomie* in 1740.

## Equations

If one takes the foci on the *x*-axis at ±*a*, the family of loci is given by

Without loss of generality one may take *a* as a new unit of distance (rescaling). Then the equation may be written as

where both sides of the equation were divided by *a*^{4} and the substitutions

were made.

The equation may be cast in a slightly different form, the *Cartesian equation of the Cassini oval*,

Substitution of

gives the form

Whence the *polar equation of the Cassini oval* is,

### Plotting

A few words about plotting the Cassini ovals may be useful.

The polar equation is a quadratic equation in *r*^{2}. It has real roots if the discriminant *D* is positive or zero,

Because 0 ≤ cos^{2} 2θ ≤ 1, this condition is satisfied for all θ if *b* ≥ 1 (= *a*). Hence in the case of the lemniscate (*b* = 1) and the dog bone (*b* > 1) the quantity *r*^{2} may be solved from the polar equation. Since *r*^{2} must be positive, one of the two roots (the one with −*D*) can be discarded. Now *r* > 0 follows by taking a square root and the Cassini ovals for *b* ≥ *a* (=1) can be drawn as polar plots, as is shown in the figure.

The case *b* < 1 requires a different approach. If one looks at the figure (at the "eggs") one sees that there is range of *x*-values on the positive axis and a mirror range on the negative *x*-axis where the function is defined. The boundary points *x*_{0} of the ranges are given by *y* = 0 (where the ovals cross the *x*-axis). Introducing *y* = 0 into the Cartesian form one derives easily (and remember *b*^{2} < 1)

If one restricts *x* to these two intervals, a positive and negative *y*-value can be solved for given *x* from the Cartesian equation. In this way one obtains *y* as function of *x* and a graph may be plotted.

## Reference

- ↑ J. Sivardiere,
*Kepler ellipse or Cassini oval?*, European Journal of Physics, vol.**15**, pp. 62-64 (1994)