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Riemann sphere

Index Riemann sphere

In mathematics, the Riemann sphere, named after Bernhard Riemann, is a model of the extended complex plane (also called the closed complex plane): the complex plane plus one point at infinity. [1]

Table of Contents

  1. 101 relations: Addition, Additive inverse, Algebraic curve, Algebraic geometry, Atlas (topology), Automorphism, Azimuth, Bernhard Riemann, Biholomorphism, Bloch sphere, Branches of physics, Cartesian coordinate system, Celestial sphere, Closed manifold, Codomain, Compact space, Compactification (mathematics), Complex analysis, Complex line, Complex manifold, Complex number, Complex plane, Complex projective plane, Complex projective space, Conformal geometry, Constant curvature, Continuous function, Cross-ratio, Dessin d'enfant, Determinant, Diffeomorphism, Differentiable manifold, Directed infinity, Division (mathematics), Division by zero, Douglas N. Arnold, Equivalence class, Field (mathematics), Fubini–Study metric, Gauss–Bonnet theorem, Gaussian curvature, General relativity, Geometry, Group (mathematics), Holomorphic function, Homogeneous coordinates, Homography, Hopf fibration, Hyperbolic space, Indeterminate form, ... Expand index (51 more) »

  2. Bernhard Riemann
  3. Spheres

Addition

Addition (usually signified by the plus symbol) is one of the four basic operations of arithmetic, the other three being subtraction, multiplication and division.

See Riemann sphere and Addition

Additive inverse

In mathematics, the additive inverse of a number (sometimes called the opposite of) is the number that, when added to, yields zero.

See Riemann sphere and Additive inverse

Algebraic curve

In mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables.

See Riemann sphere and Algebraic curve

Algebraic geometry

Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometrical problems.

See Riemann sphere and Algebraic geometry

Atlas (topology)

In mathematics, particularly topology, an atlas is a concept used to describe a manifold.

See Riemann sphere and Atlas (topology)

Automorphism

In mathematics, an automorphism is an isomorphism from a mathematical object to itself.

See Riemann sphere and Automorphism

Azimuth

An azimuth (from the directions) is the horizontal angle from a cardinal direction, most commonly north, in a local or observer-centric spherical coordinate system.

See Riemann sphere and Azimuth

Bernhard Riemann

Georg Friedrich Bernhard Riemann (17 September 1826 – 20 July 1866) was a German mathematician who made profound contributions to analysis, number theory, and differential geometry.

See Riemann sphere and Bernhard Riemann

Biholomorphism

In the mathematical theory of functions of one or more complex variables, and also in complex algebraic geometry, a biholomorphism or biholomorphic function is a bijective holomorphic function whose inverse is also holomorphic.

See Riemann sphere and Biholomorphism

Bloch sphere

In quantum mechanics and computing, the Bloch sphere is a geometrical representation of the pure state space of a two-level quantum mechanical system (qubit), named after the physicist Felix Bloch. Riemann sphere and Bloch sphere are projective geometry.

See Riemann sphere and Bloch sphere

Branches of physics

Physics is a scientific discipline that seeks to construct and experimentally test theories of the physical universe.

See Riemann sphere and Branches of physics

Cartesian coordinate system

In geometry, a Cartesian coordinate system in a plane is a coordinate system that specifies each point uniquely by a pair of real numbers called coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, called coordinate lines, coordinate axes or just axes (plural of axis) of the system.

See Riemann sphere and Cartesian coordinate system

Celestial sphere

In astronomy and navigation, the celestial sphere is an abstract sphere that has an arbitrarily large radius and is concentric to Earth. Riemann sphere and celestial sphere are spheres.

See Riemann sphere and Celestial sphere

Closed manifold

In mathematics, a closed manifold is a manifold without boundary that is compact.

See Riemann sphere and Closed manifold

Codomain

In mathematics, a codomain or set of destination of a function is a set into which all of the output of the function is constrained to fall.

See Riemann sphere and Codomain

Compact space

In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space.

See Riemann sphere and Compact space

Compactification (mathematics)

In mathematics, in general topology, compactification is the process or result of making a topological space into a compact space.

See Riemann sphere and Compactification (mathematics)

Complex analysis

Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers.

See Riemann sphere and Complex analysis

Complex line

In mathematics, a complex line is a one-dimensional affine subspace of a vector space over the complex numbers.

See Riemann sphere and Complex line

Complex manifold

In differential geometry and complex geometry, a complex manifold is a manifold with an atlas of charts to the open unit disc in the complex coordinate space \mathbb^n, such that the transition maps are holomorphic.

See Riemann sphere and Complex manifold

Complex number

In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted, called the imaginary unit and satisfying the equation i^.

See Riemann sphere and Complex number

Complex plane

In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the horizontal -axis, called the real axis, is formed by the real numbers, and the vertical -axis, called the imaginary axis, is formed by the imaginary numbers.

See Riemann sphere and Complex plane

Complex projective plane

In mathematics, the complex projective plane, usually denoted P2(C) or CP2, is the two-dimensional complex projective space. Riemann sphere and complex projective plane are projective geometry.

See Riemann sphere and Complex projective plane

Complex projective space

In mathematics, complex projective space is the projective space with respect to the field of complex numbers. Riemann sphere and complex projective space are projective geometry.

See Riemann sphere and Complex projective space

Conformal geometry

In mathematics, conformal geometry is the study of the set of angle-preserving (conformal) transformations on a space.

See Riemann sphere and Conformal geometry

Constant curvature

In mathematics, constant curvature is a concept from differential geometry.

See Riemann sphere and Constant curvature

Continuous function

In mathematics, a continuous function is a function such that a small variation of the argument induces a small variation of the value of the function.

See Riemann sphere and Continuous function

Cross-ratio

In geometry, the cross-ratio, also called the double ratio and anharmonic ratio, is a number associated with a list of four collinear points, particularly points on a projective line. Riemann sphere and cross-ratio are projective geometry.

See Riemann sphere and Cross-ratio

Dessin d'enfant

In mathematics, a dessin d'enfant is a type of graph embedding used to study Riemann surfaces and to provide combinatorial invariants for the action of the absolute Galois group of the rational numbers.

See Riemann sphere and Dessin d'enfant

Determinant

In mathematics, the determinant is a scalar-valued function of the entries of a square matrix.

See Riemann sphere and Determinant

Diffeomorphism

In mathematics, a diffeomorphism is an isomorphism of differentiable manifolds.

See Riemann sphere and Diffeomorphism

Differentiable manifold

In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus.

See Riemann sphere and Differentiable manifold

Directed infinity

A directed infinity is a type of infinity in the complex plane that has a defined complex argument θ but an infinite absolute value r.

See Riemann sphere and Directed infinity

Division (mathematics)

Division is one of the four basic operations of arithmetic.

See Riemann sphere and Division (mathematics)

Division by zero

In mathematics, division by zero, division where the divisor (denominator) is zero, is a unique and problematic special case.

See Riemann sphere and Division by zero

Douglas N. Arnold

Douglas Norman "Doug" Arnold is a mathematician whose research focuses on the numerical analysis of partial differential equations with applications in mechanics and other fields in physics.

See Riemann sphere and Douglas N. Arnold

Equivalence class

In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S into equivalence classes.

See Riemann sphere and Equivalence class

Field (mathematics)

In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined and behave as the corresponding operations on rational and real numbers.

See Riemann sphere and Field (mathematics)

Fubini–Study metric

In mathematics, the Fubini–Study metric (IPA: /fubini-ʃtuːdi/) is a Kähler metric on a complex projective space CPn endowed with a Hermitian form. Riemann sphere and Fubini–Study metric are projective geometry.

See Riemann sphere and Fubini–Study metric

Gauss–Bonnet theorem

In the mathematical field of differential geometry, the Gauss–Bonnet theorem (or Gauss–Bonnet formula) is a fundamental formula which links the curvature of a surface to its underlying topology. Riemann sphere and Gauss–Bonnet theorem are Riemann surfaces.

See Riemann sphere and Gauss–Bonnet theorem

Gaussian curvature

In differential geometry, the Gaussian curvature or Gauss curvature of a smooth surface in three-dimensional space at a point is the product of the principal curvatures, and, at the given point: K.

See Riemann sphere and Gaussian curvature

General relativity

General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics.

See Riemann sphere and General relativity

Geometry

Geometry is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures.

See Riemann sphere and Geometry

Group (mathematics)

In mathematics, a group is a set with an operation that associates an element of the set to every pair of elements of the set (as does every binary operation) and satisfies the following constraints: the operation is associative, it has an identity element, and every element of the set has an inverse element.

See Riemann sphere and Group (mathematics)

Holomorphic function

In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space.

See Riemann sphere and Holomorphic function

Homogeneous coordinates

In mathematics, homogeneous coordinates or projective coordinates, introduced by August Ferdinand Möbius in his 1827 work Der barycentrische Calcul, are a system of coordinates used in projective geometry, just as Cartesian coordinates are used in Euclidean geometry. Riemann sphere and homogeneous coordinates are projective geometry.

See Riemann sphere and Homogeneous coordinates

Homography

In projective geometry, a homography is an isomorphism of projective spaces, induced by an isomorphism of the vector spaces from which the projective spaces derive. Riemann sphere and homography are projective geometry.

See Riemann sphere and Homography

Hopf fibration

In the mathematical field of differential topology, the Hopf fibration (also known as the Hopf bundle or Hopf map) describes a 3-sphere (a hypersphere in four-dimensional space) in terms of circles and an ordinary sphere.

See Riemann sphere and Hopf fibration

Hyperbolic space

In mathematics, hyperbolic space of dimension n is the unique simply connected, n-dimensional Riemannian manifold of constant sectional curvature equal to −1.

See Riemann sphere and Hyperbolic space

Indeterminate form

In calculus, it is usually possible to compute the limit of the sum, difference, product, quotient or power of two functions by taking the corresponding combination of the separate limits of each respective function.

See Riemann sphere and Indeterminate form

Infinity

Infinity is something which is boundless, endless, or larger than any natural number.

See Riemann sphere and Infinity

Isometry

In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective.

See Riemann sphere and Isometry

Limit of a function

Although the function is not defined at zero, as becomes closer and closer to zero, becomes arbitrarily close to 1.

See Riemann sphere and Limit of a function

Manifold

In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point.

See Riemann sphere and Manifold

Mass

Mass is an intrinsic property of a body.

See Riemann sphere and Mass

Mathematical model

A mathematical model is an abstract description of a concrete system using mathematical concepts and language.

See Riemann sphere and Mathematical model

Mathematics

Mathematics is a field of study that discovers and organizes abstract objects, methods, theories and theorems that are developed and proved for the needs of empirical sciences and mathematics itself.

See Riemann sphere and Mathematics

Möbius plane

In mathematics, the classical Möbius plane (named after August Ferdinand Möbius) is the Euclidean plane supplemented by a single point at infinity.

See Riemann sphere and Möbius plane

Möbius transformation

In geometry and complex analysis, a Möbius transformation of the complex plane is a rational function of the form f(z). Riemann sphere and Möbius transformation are projective geometry and Riemann surfaces.

See Riemann sphere and Möbius transformation

Meromorphic function

In the mathematical field of complex analysis, a meromorphic function on an open subset D of the complex plane is a function that is holomorphic on all of D except for a set of isolated points, which are poles of the function.

See Riemann sphere and Meromorphic function

Multiplication

Multiplication (often denoted by the cross symbol, by the mid-line dot operator, by juxtaposition, or, on computers, by an asterisk) is one of the four elementary mathematical operations of arithmetic, with the other ones being addition, subtraction, and division.

See Riemann sphere and Multiplication

Multiplicative inverse

In mathematics, a multiplicative inverse or reciprocal for a number x, denoted by 1/x or x−1, is a number which when multiplied by x yields the multiplicative identity, 1.

See Riemann sphere and Multiplicative inverse

Orientability

In mathematics, orientability is a property of some topological spaces such as real vector spaces, Euclidean spaces, surfaces, and more generally manifolds that allows a consistent definition of "clockwise" and "anticlockwise".

See Riemann sphere and Orientability

Orthogonal group

In mathematics, the orthogonal group in dimension, denoted, is the group of distance-preserving transformations of a Euclidean space of dimension that preserve a fixed point, where the group operation is given by composing transformations.

See Riemann sphere and Orthogonal group

Oxford University Press

Oxford University Press (OUP) is the publishing house of the University of Oxford.

See Riemann sphere and Oxford University Press

Pathological (mathematics)

In mathematics, when a mathematical phenomenon runs counter to some intuition, then the phenomenon is sometimes called pathological.

See Riemann sphere and Pathological (mathematics)

Photon

A photon is an elementary particle that is a quantum of the electromagnetic field, including electromagnetic radiation such as light and radio waves, and the force carrier for the electromagnetic force.

See Riemann sphere and Photon

Photon polarization

Photon polarization is the quantum mechanical description of the classical polarized sinusoidal plane electromagnetic wave.

See Riemann sphere and Photon polarization

Point at infinity

In geometry, a point at infinity or ideal point is an idealized limiting point at the "end" of each line. Riemann sphere and point at infinity are projective geometry.

See Riemann sphere and Point at infinity

Projective geometry

In mathematics, projective geometry is the study of geometric properties that are invariant with respect to projective transformations.

See Riemann sphere and Projective geometry

Projective line

In mathematics, a projective line is, roughly speaking, the extension of a usual line by a point called a point at infinity. Riemann sphere and projective line are projective geometry.

See Riemann sphere and Projective line

Projective linear group

In mathematics, especially in the group theoretic area of algebra, the projective linear group (also known as the projective general linear group or PGL) is the induced action of the general linear group of a vector space V on the associated projective space P(V). Riemann sphere and projective linear group are projective geometry.

See Riemann sphere and Projective linear group

Projective space

In mathematics, the concept of a projective space originated from the visual effect of perspective, where parallel lines seem to meet at infinity. Riemann sphere and projective space are projective geometry.

See Riemann sphere and Projective space

Projectively extended real line

In real analysis, the projectively extended real line (also called the one-point compactification of the real line), is the extension of the set of the real numbers, \mathbb, by a point denoted. Riemann sphere and projectively extended real line are projective geometry.

See Riemann sphere and Projectively extended real line

Quantum mechanics

Quantum mechanics is a fundamental theory that describes the behavior of nature at and below the scale of atoms.

See Riemann sphere and Quantum mechanics

Qubit

In quantum computing, a qubit or quantum bit is a basic unit of quantum information—the quantum version of the classic binary bit physically realized with a two-state device.

See Riemann sphere and Qubit

Rational function

In mathematics, a rational function is any function that can be defined by a rational fraction, which is an algebraic fraction such that both the numerator and the denominator are polynomials.

See Riemann sphere and Rational function

Riemann surface

In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. Riemann sphere and Riemann surface are Bernhard Riemann and Riemann surfaces.

See Riemann sphere and Riemann surface

Riemannian manifold

In differential geometry, a Riemannian manifold is a geometric space on which many geometric notions such as distance, angles, length, volume, and curvature are defined.

See Riemann sphere and Riemannian manifold

Rotation

Rotation or rotational motion is the circular movement of an object around a central line, known as axis of rotation.

See Riemann sphere and Rotation

Scaling (geometry)

In affine geometry, uniform scaling (or isotropic scaling) is a linear transformation that enlarges (increases) or shrinks (diminishes) objects by a scale factor that is the same in all directions.

See Riemann sphere and Scaling (geometry)

Simply connected space

In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected) if it is path-connected and every path between two points can be continuously transformed into any other such path while preserving the two endpoints in question.

See Riemann sphere and Simply connected space

Smith chart

The Smith chart (sometimes also called Smith diagram, Mizuhashi chart (水橋チャート), Mizuhashi–Smith chart (水橋スミスチャート), Volpert–Smith chart (Диаграмма Вольперта—Смита) or Mizuhashi–Volpert–Smith chart), is a graphical calculator or nomogram designed for electrical and electronics engineers specializing in radio frequency (RF) engineering to assist in solving problems with transmission lines and matching circuits.

See Riemann sphere and Smith chart

Smoothness

In mathematical analysis, the smoothness of a function is a property measured by the number, called differentiability class, of continuous derivatives it has over its domain.

See Riemann sphere and Smoothness

Spherical coordinate system

In mathematics, a spherical coordinate system is a coordinate system for three-dimensional space where the position of a given point in space is specified by three numbers, (r, θ, φ): the radial distance of the radial line r connecting the point to the fixed point of origin (which is located on a fixed polar axis, or zenith direction axis, or z-axis); the polar angle θ of the radial line r; and the azimuthal angle φ of the radial line r.

See Riemann sphere and Spherical coordinate system

Spin (physics)

Spin is an intrinsic form of angular momentum carried by elementary particles, and thus by composite particles such as hadrons, atomic nuclei, and atoms.

See Riemann sphere and Spin (physics)

Stereographic projection

In mathematics, a stereographic projection is a perspective projection of the sphere, through a specific point on the sphere (the pole or center of projection), onto a plane (the projection plane) perpendicular to the diameter through the point. Riemann sphere and stereographic projection are projective geometry.

See Riemann sphere and Stereographic projection

String theory

In physics, string theory is a theoretical framework in which the point-like particles of particle physics are replaced by one-dimensional objects called strings.

See Riemann sphere and String theory

Subatomic particle

In physics, a subatomic particle is a particle smaller than an atom.

See Riemann sphere and Subatomic particle

The Road to Reality

The Road to Reality: A Complete Guide to the Laws of the Universe is a book on modern physics by the British mathematical physicist Roger Penrose, published in 2004.

See Riemann sphere and The Road to Reality

Topological manifold

In topology, a branch of mathematics, a topological manifold is a topological space that locally resembles real n-dimensional Euclidean space.

See Riemann sphere and Topological manifold

Topology

Topology (from the Greek words, and) is the branch of mathematics concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing holes, opening holes, tearing, gluing, or passing through itself.

See Riemann sphere and Topology

Translation (geometry)

In Euclidean geometry, a translation is a geometric transformation that moves every point of a figure, shape or space by the same distance in a given direction.

See Riemann sphere and Translation (geometry)

Twistor theory

In theoretical physics, twistor theory was proposed by Roger Penrose in 1967 as a possible path to quantum gravity and has evolved into a widely studied branch of theoretical and mathematical physics.

See Riemann sphere and Twistor theory

Uniformization theorem

In mathematics, the uniformization theorem states that every simply connected Riemann surface is conformally equivalent to one of three Riemann surfaces: the open unit disk, the complex plane, or the Riemann sphere. Riemann sphere and uniformization theorem are Riemann surfaces.

See Riemann sphere and Uniformization theorem

Wheel theory

A wheel is a type of algebra (in the sense of universal algebra) where division is always defined.

See Riemann sphere and Wheel theory

William Goldman (mathematician)

William Mark Goldman (born 1955 in Kansas City, Missouri) is a professor of mathematics at the University of Maryland, College Park (since 1986).

See Riemann sphere and William Goldman (mathematician)

Worldsheet

In string theory, a worldsheet is a two-dimensional manifold which describes the embedding of a string in spacetime.

See Riemann sphere and Worldsheet

Zenith

The zenith is an imaginary point directly "above" a particular location, on the celestial sphere.

See Riemann sphere and Zenith

Zeros and poles

In complex analysis (a branch of mathematics), a pole is a certain type of singularity of a complex-valued function of a complex variable.

See Riemann sphere and Zeros and poles

3D rotation group

In mechanics and geometry, the 3D rotation group, often denoted SO(3), is the group of all rotations about the origin of three-dimensional Euclidean space \R^3 under the operation of composition.

See Riemann sphere and 3D rotation group

See also

Bernhard Riemann

Spheres

References

[1] https://en.wikipedia.org/wiki/Riemann_sphere

Also known as C-*, Complex closed curve, Complex projective line, Complex sphere, Extended complex number, Extended complex numbers, Extended complex plane, Lewis sphere, Rieman sphere, Riemann's sphere, Sphere of Riemann.

, Infinity, Isometry, Limit of a function, Manifold, Mass, Mathematical model, Mathematics, Möbius plane, Möbius transformation, Meromorphic function, Multiplication, Multiplicative inverse, Orientability, Orthogonal group, Oxford University Press, Pathological (mathematics), Photon, Photon polarization, Point at infinity, Projective geometry, Projective line, Projective linear group, Projective space, Projectively extended real line, Quantum mechanics, Qubit, Rational function, Riemann surface, Riemannian manifold, Rotation, Scaling (geometry), Simply connected space, Smith chart, Smoothness, Spherical coordinate system, Spin (physics), Stereographic projection, String theory, Subatomic particle, The Road to Reality, Topological manifold, Topology, Translation (geometry), Twistor theory, Uniformization theorem, Wheel theory, William Goldman (mathematician), Worldsheet, Zenith, Zeros and poles, 3D rotation group.