Lemniscate
Encyclopedia
In algebraic geometry
Algebraic geometry
Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...

, a lemniscate refers to any of several figure-eight or shaped curve
Curve
In mathematics, a curve is, generally speaking, an object similar to a line but which is not required to be straight...

s. It may refer to:
• The lemniscate of Bernoulli
Lemniscate of Bernoulli
In geometry, the lemniscate of Bernoulli is a plane curve defined from two given points F1 and F2, known as foci, at distance 2a from each other as the locus of points P so that PF1·PF2 = a2. The curve has a shape similar to the numeral 8 and to the ∞ symbol. Its name is from lemniscus, which is...

, often simply called the lemniscate, the locus of points whose product of distances from two foci equals the square of half the interfocal distance
• The lemniscate of Gerono
Lemniscate of Gerono
In algebraic geometry, the lemniscate of Gerono, or lemnicate of Huygens, or figure-eight curve, is a plane algebraic curve of degree four and genus zero shaped like an \infty symbol, or figure eight...

, the zero set of the quartic polynomial
• The lemniscate of Booth, the zero set of the quartic polynomial with . When this coincides with the lemniscate of Bernoulli.
• The infinity symbol ∞, sometimes called a lemniscate because of its shape

• Cut-point
Cut-point
In topology, a cut-point is a point of a connected space such that its removal causes the resulting space to be disconnected. For example every point of a line is a cut-point, while no point of a circle is a cut-point...

• Lorenz attractor
Lorenz attractor
The Lorenz attractor, named for Edward N. Lorenz, is an example of a non-linear dynamic system corresponding to the long-term behavior of the Lorenz oscillator. The Lorenz oscillator is a 3-dimensional dynamical system that exhibits chaotic flow, noted for its lemniscate shape...

, a three-dimensional dynamic system exhibiting a lemniscate shape
• Polynomial lemniscate
Polynomial lemniscate
In mathematics, a polynomial lemniscate or polynomial level curve is a plane algebraic curve of degree 2n, constructed from a polynomial p with complex coefficients of degree n....

, the set of complex numbers for which a given polynomial has a constant absolute value
Absolute value
In mathematics, the absolute value |a| of a real number a is the numerical value of a without regard to its sign. So, for example, the absolute value of 3 is 3, and the absolute value of -3 is also 3...

• The lemniscate constants
Gauss's constant
In mathematics, Gauss's constant, denoted by G, is defined as the reciprocal of the arithmetic-geometric mean of 1 and the square root of 2:The constant is named after Carl Friedrich Gauss, who on May 30, 1799 discovered thatso that...

, numbers that arise in evaluating the arc length
Arc length
Determining the length of an irregular arc segment is also called rectification of a curve. Historically, many methods were used for specific curves...

of a lemniscate of Bernoulli