Lemniscate

Lemniscate of Booth

The consideration of curves with a figure-eight shape can be traced back to Proclus, a Greek Neoplatonist philosopher and mathematician who lived in the 5th century AD. Proclus considered the cross-sections of a torus by a plane parallel to the axis of the torus. As he observed, for most such sections the cross section consists of either one or two ovals; however, when the plane is tangent to the inner surface of the torus, the cross-section takes on a figure-eight shape, which Proclus called a horse fetter (a device for holding two feet of a horse together), or "hippopede" in Greek.[7] The name "lemniscate of Booth" for this curve dates to its study by the 19th-century mathematician James Booth.[1]

The lemniscate may be defined as an algebraic curve, the zero set of the quartic polynomial ${\displaystyle (x^{2}+y^{2})^{2}-cx^{2}-dy^{2}}$ when the parameter d is negative (or zero for the special case where the lemniscate becomes a pair of externally tangent circles). For positive values of d one instead obtains the oval of Booth.

Lemniscate of Bernoulli

In 1680, Cassini studied a family of curves, now called the Cassini oval, defined as follows: the locus of all points, the product of whose distances from two fixed points, the curves' foci, is a constant. Under very particular circumstances (when the half-distance between the points is equal to the square root of the constant) this gives rise to a lemniscate.

In 1694, Johann Bernoulli studied the lemniscate case of the Cassini oval, now known as the lemniscate of Bernoulli (shown above), in connection with a problem of "isochrones" that had been posed earlier by Leibniz. Like the hippopede, it is an algebraic curve, the zero set of the polynomial ${\displaystyle (x^{2}+y^{2})^{2}-a^{2}(x^{2}-y^{2})}$. Bernoulli's brother Jacob Bernoulli also studied the same curve in the same year, and gave it its name, the lemniscate.[8] It may also be defined geometrically as the locus of points whose product of distances from two foci equals the square of half the interfocal distance.[9] It is a special case of the hippopede (lemniscate of Booth), with ${\displaystyle d=-c}$, and may be formed as a cross-section of a torus whose inner hole and circular cross-sections have the same diameter as each other.[1] The lemniscatic elliptic functions are analogues of trigonometric functions for the lemniscate of Bernoulli, and the lemniscate constants arise in evaluating the arc length of this lemniscate.

Lemniscate of Gerono: solution set of x⁴ − x² + y² = 0[10]

Another lemniscate, the lemniscate of Gerono or lemniscate of Huygens, is the zero set of the quartic polynomial ${\displaystyle y^{2}-x^{2}(a^{2}-x^{2})}$.[11][12] Viviani's curve, a three-dimensional curve formed by intersecting a sphere with a cylinder, also has a figure eight shape, and has the lemniscate of Gerono as its planar projection.[13]

Other figure-eight shaped algebraic curves include

• The Devil's curve, a curve defined by the quartic equation ${\displaystyle y^{2}(y^{2}-a^{2})=x^{2}(x^{2}-b^{2})}$ in which one connected component has a figure-eight shape,[14]
• Watt's curve, a figure-eight shaped curve formed by a mechanical linkage. Watt's curve is the zero set of the degree-six polynomial equation ${\displaystyle (x^{2}+y^{2})(x^{2}+y^{2}-d^{2})^{2}+4a^{2}y^{2}(x^{2}+y^{2}-b^{2})=0}$ and has the lemniscate of Bernoulli as a special case.