Lemniscate
In algebraic geometry, a lemniscate (/lɛmˈnɪskɪt/ or /ˈlɛmnɪsˌkeɪt, -kɪt/)[1] is any of several figure-eight or ∞-shaped curves.
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.
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.
[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
[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
[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
