Lemniscate

{{#invoke:Hatnote|hatnote}} In algebraic geometry, a lemniscate may refer to any of several figure-eight or -shaped curves. The word comes from the Latin "lēmniscātus" meaning "decorated with ribbons", which in turn may come from the ancient Greek island of Lemnos where ribbons were worn as decorations, or alternatively may refer to the wool from which the ribbons were made.

Although the name "lemniscate" dates to the late 17th century, 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). The Greek phrase for a horse fetter became the word hippopede, the name for this figure-eight shaped curve, which is also called the lemniscate of Booth. It may be defined algebraically as the zero set of the quartic polynomial (x2 + y2)2cx2dy2 when the parameter d is negative. For positive values of d one instead obtains an oval-shaped curve, the oval of Booth.

In 1694, Johann Bernoulli studied the zero set of the polynomial equation $(x^{2}+y^{2})^{2}=2a^{2}(x^{2}-y^{2})$ in connection with a problem of "isochrones" that had been posed earlier by Leibniz. Bernoulli's brother Jacob Bernoulli also studied the same curve in the same year, and gave it its name, the lemniscate. 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. It is a special case of the hippopede, with $d=-c$ , 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. 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.

Another lemniscate, the lemniscate of Gerono or lemniscate of Huygens, is the zero set of the quartic polynomial equation $y^{2}=x^{2}(a^{2}-x^{2})$ . 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.

Other figure-eight shaped algebraic curves include