Cassini ovals

PD Image
Three Cassini ovals. The oval with b=1 is a Bernoulli lemniscate.

A Cassini oval is a locus of points determined by two fixed points F₁ and F₂ (the "foci") at a distance 2a 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 F₁ is r₁ and the distance of the same point to F₂ is r₂ then the locus is defined by the product of distances r₁×r₂ being constant and equal to b².

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. 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

$\left[(x-a)^{2}+y^{2}\right]\left[(x+a)^{2}+y^{2}\right]=b^{4}$

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

$\left[(x-1)^{2}+y^{2}\right]^{2}\left[(x+1)^{2}+y^{2}\right]^{2}=b^{4},$

where both sides of the equation were divided by a⁴ and the substitutions

${\frac {x}{a}}\rightarrow x,\quad {\frac {y}{a}}\rightarrow y,\quad {\frac {b}{a}}\rightarrow b$

were made.

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

$(x^{2}+y^{2}+1)^{2}-4x^{2}-b^{4}=0.\;$

Substitution of

$x=r\cos \theta ,\quad y=r\sin \theta ,\quad r^{2}=x^{2}+y^{2},\quad 0\leq \theta \leq 2\pi ,$

gives the form

$(r^{2}+1)^{2}-4r^{2}\cos ^{2}\theta =r^{4}+1-2r^{2}(2\cos ^{2}\theta -1).\;$

Whence the polar equation of the Cassini oval is,

$r^{4}-2r^{2}\cos 2\theta +1-b^{4}=0.\,$

Plotting

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

The polar equation is a quadratic equation in r². It has real roots if the discriminant D is positive or zero,

$D\equiv 4\cos ^{2}2\theta -4+4b^{4}\geq 0\Longrightarrow b^{4}\geq 1-\cos ^{2}2\theta .$

Because 0 ≤ cos² 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 r² may be solved from the polar equation. Since r² 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 x₀ 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 b² < 1)

$x_{0}^{2}=1\pm b^{2}\Longrightarrow -{\sqrt {1+b^{2}}}\leq x\leq -{\sqrt {1-b^{2}}}\quad {\hbox{and}}\quad {\sqrt {1-b^{2}}}\leq x\leq {\sqrt {1+b^{2}}}.$

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.