The lemniscate of Bernoulli

The lemniscate of Bernoulli is a figure eight lying on its side, the curve behind the infinity symbol ∞. It is the set of points whose distances to two fixed points multiply to a constant.

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The lemniscate (x² + y²)² = 4(x² − y²) with its foci at (±√2, 0).

Equations to type

Cartesian

(x^2 + y^2)^2 = 4(x^2 - y^2)

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Polar

r = 2sqrt(cos(2theta))

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At a glance

Cartesian equation: (x² + y²)² = a²(x² − y²)
Polar equation: r² = a² cos 2θ
Foci: (±a/√2, 0)
Total area: a²

The product of distances

An ellipse is the set of points whose distances to two foci add up to a constant. Jakob Bernoulli’s lemniscate (1694) multiplies them instead: with foci at (±c, 0), the points whose distances multiply to c² form a figure eight through the origin. With c = a/√2 this gives the equation above.

In polar coordinates

Substituting x = r cos θ and y = r sin θ turns the equation into r² = a² cos 2θ. r is real only where cos 2θ ≥ 0, that is for −π/4 ≤ θ ≤ π/4 (the right lobe) and 3π/4 ≤ θ ≤ 5π/4 (the left lobe). Between those angles there is no curve, which leaves the two lobes crossing at the origin at right angles.

Area and length

The two lobes together enclose area a², a neat result from ½ ∫ a² cos 2θ dθ. The arc length is not elementary: it is given by the lemniscate constant, and the study of that integral led Gauss and others to elliptic functions.

One member of the Cassini ovals

Fix the foci at (±c, 0) and require the product of the distances to be b² for any b. These curves are the Cassini ovals, studied by the astronomer Giovanni Cassini in 1680. When b < c the curve is two separate ovals around the foci; when b > c it is a single peanut-shaped loop; and exactly at b = c the two ovals touch at the origin, which is the lemniscate.