Lemniscate

Figure eight shaped curve
figure eight shaped curves in algebraic geometry

In algebraic geometry, a lemniscate (/l//ɛ//m/'/n//ɪ//s//k//ɪ//t/ or '/l//ɛ//m//n//ɪ//s/,/k//eɪ//t//, // //k//ɪ//t/) is any of several figure eight or ∞ shaped curves. from the Greek , meaning "ribbon", or which alternatively may refer to the wool from which the ribbons were made. The name "lemniscate of Booth" for this curve dates to its study by the 19th century mathematician James Booth. 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. 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 (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.]] Another lemniscate, the lemniscate of Gerono or lemniscate of Huygens, is the zero set of the quartic polynomial 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.