Lemniscate

Source: Wikipedia, the free encyclopedia.
This article is about figure-eight shaped curves in algebraic geometry. For other uses, see Lemniscate (disambiguation).
The lemniscate of Bernoulli and its two foci

In

ribbons were made.[2]

Curves that have been called a lemniscate include three

ancient Greek mathematics, but the term "lemniscate" for curves of this type comes from the work of Jacob Bernoulli
in the late 17th century.

History and examples

Lemniscate of Booth

Lemniscate of Booth
Main article: Hippopede

The consideration of curves with a figure-eight shape can be traced back to

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

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
.

Lemniscate of Bernoulli

Lemniscate of Bernoulli
Main article: 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,

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.
lemniscate constants arise in evaluating the arc length
of this lemniscate.

Lemniscate of Gerono

Main article: Lemniscate of Gerono
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]

Others

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.

See also

  • Analemma, the figure-eight shaped curve traced by the noontime positions of the sun in the sky over the course of a year
  • Infinity symbol
  • Lemniscates as generalized conics
  • Lorenz attractor
    , a three-dimensional dynamic system exhibiting a lemniscate shape
  • Polynomial lemniscate, a level set of the absolute value of a complex polynomial

References

  1. ^ "lemniscate". Dictionary.com Unabridged (Online). n.d.
  2. ^
    MR 1483331
    .
  3. ^
    ISBN 9780883855768
    .
  4. Perseus Project
    .
  5. ^ Harper, Douglas. "lemniscus". Online Etymology Dictionary.
  6. Perseus Project
    .
  7. Perseus Project
    .
  8. ^ ἱπποπέδη in Liddell and Scott.
  9. MR 0774250
    .
  10. S2CID 1448521
    .
  11. ^ Köller, Jürgen. "Acht-Kurve". www.mathematische-basteleien.de. Retrieved 2017-11-26.
  12. ^ Basset, Alfred Barnard (1901), "The Lemniscate of Gerono", An elementary treatise on cubic and quartic curves, Deighton, Bell, pp. 171–172.
  13. ISBN 9780198526759
    .
  14. ^ 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.
  15. ISBN 9780471667001
    .

External links

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