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Encyclopedia > Lemniscate

(x2 + y2)2 = a2(x2y2)

Graphing this equation produces a curve similar to . The curve has become a symbol of infinity and is widely used in math. The symbol itself is sometimes referred to as the lemniscate. Its Unicode representation is (`&#8734;`). Jump to: navigation, search Infinity is a term with very distinct, separate meanings which arise in theology, philosophy, mathematics and everyday life. ... In computing, Unicode provides an international standard which has the goal of providing the means to encode the text of every document people want to store on computers. ...

The lemniscate was first described in 1694 by Jakob Bernoulli as a modification of an ellipse, which is the locus of points for which the sum of the distances to each of two fixed focal points is a constant. A lemniscate, by contrast, is the locus of points for which the product of these distances is constant. Bernoulli called it the lemniscus, which is Latin for 'pendant ribbon'. Events February 6 - The colony Quilombo dos Palmares is destroyed. ... Jakob Bernoulli. ... In mathematics, an ellipse (from the Greek for absence) is a plane algebraic curve where the sum of the distances from any point on the curve to two fixed points is constant. ... In mathematics, a locus (plural loci) is a collection of points which share a common property. ... Jump to: navigation, search The distance between two points is the length of a straight line segment between them. ... Jump to: navigation, search A mathematical constant is a quantity, usually a real number or a complex number, that arises naturally in mathematics and does not change. ... Jump to: navigation, search Latin is an Indo-European language originally spoken in the region around Rome called Latium. ...

The lemniscate can be obtained as the inverse transform of a hyperbola, with the inversion circle centered at the center of the hyperbola (bisector of its two foci). In mathematics, inversive geometry is the geometry of circles and the set of transformations that map all circles into circles. ... A graph of a hyperbola, where h = k = 0 and a = b = 2. ... In Euclidean geometry, a circle is the set of all points in a plane at a fixed distance, called the radius, from a fixed point, called the centre. ...

A lemniscate may also be described by the polar equation This article describes some of the common coordinate systems that appear in elementary mathematics. ...

r2 = a2cos2φ

or the bipolar equation In mathematics, bipolar coordinates are a coordinate system where curves are specified as the locus of points in terms of distances to two fixed points. ...

## Arc length and elliptic functions

The determination of the arc length of arcs of the lemniscate leads to elliptic integrals, as was discovered in the eighteenth century. Around 1800, the elliptic functions inverting those integrals were studied by C. F. Gauss (largely unpublished at the time, but allusions in the notes to his Disquisitiones Arithmeticae). The period lattices are of a very special form, being proportional to the Gaussian integers. For this reason the case of elliptic functions with complex multiplication by the square root of minus one is called the lemniscatic case in some sources. For other uses, see Curve (disambiguation). ... In integral calculus, elliptic integrals originally arose in connection with the problem of giving the arc length of an ellipse and were first studied by Fagnano and Leonhard Euler. ... In complex analysis, an elliptic function is, roughly speaking , a function defined on the complex plane which is periodic in two directions. ... Johann Carl Friedrich Gauss Johann Carl Friedrich Gauss (Gauß) (April 30, 1777 _ February 23, 1855) was a legendary German mathematician, astronomer and physicist with a very wide range of contributions; he is considered to be one of the greatest mathematicians of all time. ... The Disquisitiones Arithmeticae is a textbook of number theory written by German mathematician Carl Friedrich Gauss and first published in 1801 when Gauss was 24. ... In mathematics, a fundamental pair of periods is an ordered pair of complex numbers that define a lattice in the complex plane. ... A Gaussian integer is a complex number whose real and imaginary part are both integers. ... In mathematics, complex multiplication is the theory of elliptic curves E that have an endomorphism ring larger than the integers; and also the theory in higher dimensions of abelian varieties A having enough endomorphisms in a certain precise sense (it roughly means that the action on the tangent space at... In mathematics, the imaginary unit i allows the real number system to be extended to the complex number system . ...

• Drawing M-set by escape lines method Results from FactBites:

 Lemniscate - Wikipedia, the free encyclopedia (271 words) The lemniscate was first described in 1694 by Jakob Bernoulli as a modification of an ellipse, which is the locus of points for which the sum of the distances to each of two fixed focal points is a constant. The lemniscate can be obtained as the inverse transform of a hyperbola, with the inversion circle centered at the center of the hyperbola (bisector of its two foci). The determination of the arc length of arcs of the lemniscate leads to elliptic integrals, as was discovered in the eighteenth century.
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