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.[8] The name "lemniscate of Booth" for this curve dates to its study by the 19th-century mathematician James Booth.[2]
The lemniscate may be defined as an algebraic curve, the zero set of the quartic polynomial 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.
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 . Bernoulli's brother Jacob Bernoulli also studied the same curve in the same year, and gave it its name, the lemniscate.[9] 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.[10] It is a special case of the hippopede (lemniscate of Booth), with , 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.[2] 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[11]
Another lemniscate, the lemniscate of Gerono or lemniscate of Huygens, is the zero set of the quartic polynomial .[12][13]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.[14]
Other figure-eight shaped algebraic curves include
The Devil's curve, a curve defined by the quartic equation in which one connected component has a figure-eight shape,[15]
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 and has the lemniscate of Bernoulli as a special case.
^ abcdSchappacher, Norbert (1997), "Some milestones of lemniscatomy", Algebraic Geometry (Ankara, 1995), Lecture Notes in Pure and Applied Mathematics, vol. 193, New York: Dekker, pp. 257–290, MR1483331.
^Langer, Joel C.; Singer, David A. (2010), "Reflections on the lemniscate of Bernoulli: the forty-eight faces of a mathematical gem", Milan Journal of Mathematics, 78 (2): 643–682, doi:10.1007/s00032-010-0124-5, MR2781856, S2CID1448521.
^Costa, Luisa Rossi; Marchetti, Elena (2005), "Mathematical and Historical Investigation on Domes and Vaults", in Weber, Ralf; Amann, Matthias Albrecht (eds.), Aesthetics and architectural composition : proceedings of the Dresden International Symposium of Architecture 2004, Mammendorf: Pro Literatur, pp. 73–80.