Riemann sphere - Wikipedia
In mathematics, the Riemann sphere, named after Bernhard Riemann, [1] is a model of the extended complex plane (also called the closed complex plane): the complex plane plus one point at infinity. This extended plane represents the extended complex numbers, that is, the complex numbers plus a value for infinity.
Riemann Sphere -- from Wolfram MathWorld
2024年11月5日 · The Riemann sphere, also called the extended complex plane, is a one-dimensional complex manifold C^* (C-star) which is the one-point compactification of the complex numbers C^*=C union {infty^~}, together with two charts.
What is the Riemann Sphere? - Mathematics Stack Exchange
2013年11月29日 · The Riemann Sphere is a unit sphere--a sphere with radius $1$--with its south pole kissing the origin centered around the $z$ axis. Points on the sphere can be associated to points in the plane by projecting from the north pole $N$ through a point $P$ on the surface of the sphere onto the plane at a point $Q$ directly through what is called a line.
Riemann sphere - Encyclopedia of Mathematics
2023年3月17日 · Riemann sphere. A sphere in the Euclidean space $ \mathbf R ^ {3} ( \xi , \eta , t) $ onto which the extended complex plane $ \overline {\mathbf C}\; $ is conformally and one-to-one transformed under stereographic projection. For example, the unit sphere.
Maths in a minute: The Riemann sphere | plus.maths.org
2013年10月1日 · The plane with a point at infinity appended is called the Riemann sphere after the 18th century mathematician Bernhard Riemann (although strictly speaking the Riemann sphere is the complex plane with infinity appended — see here for more on complex numbers).
Riemann Sphere - Complex Analysis
Riemann Sphere. The Point at infinity. For some purposes it is convenient to introduce the point at infinity, denoted by ∞, in addition to the points z ∈ C. We must be careful in doing so, because it can lead to confusion and abuse of the symbol ∞.
Extended Complex Plane -- from Wolfram MathWorld
2024年11月5日 · The extended complex plane is the name given to the complex plane with a point at infinity attached: C union {infty^~}, where infty^~ denotes complex infinity. It is also called the Riemann sphere and is various denoted C^* or C^^.
In terms of the Riemann sphere, a typical M obius transformations is a rotation (although there are other symmetries). Every rotation has an axis and so every rotation has two xed points, where the axis of the rotation meets the sphere. Not every M obius transformations has two xed points. The transformation z ! z + 1
The Riemann sphere and stereographic projection. The initial (and naive) idea of the extended complex plane is that one adjoins to the complex plane a new point, called “1” and decrees that a sequence zn of complex numbers converges to 1 if and only if the real sequence jznj tends to 1 in the usual sense. It may or may not be intuitively ...
2.2: Riemann Sphere - Mathematics LibreTexts
Connect the point \(z\) in the plane with the north pole using a straight line. This line intersects the sphere at the point \(P(z)\). In this way each point \(z=x+iy\) on the complex plane corresponds uniquely to a point \(P(z)\) on the surface of the sphere.