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.
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 -- from Wolfram MathWorld
5 天之前 · 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.
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
Riemann sphere - Encyclopedia of Mathematics
2023年3月17日 · 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. $$ S _ {2} = \ { { ( \xi , \eta , t) \in \mathbf R ^ {3} } : { \xi ^ {2} + \eta ^ {2} + t ^ {2} = 1 } \} $$
Riemann sphere - Simple English Wikipedia, the free encyclopedia
The Riemann sphere is a mathematical model of the extended complex plane. The model is named after Bernhard Riemann. The extended complex plane is the plane of the complex numbers, plus a point at infinity. With this model, this point at infinity is "close" to very large numbers in the same way zero is "close" to very small ones.
Stereographic projection gives us a more concrete way of identifying ̃C with a sphere, one which, moreover, yields a lot of geometrical insight. We shall take the Riemann sphere to be the usual unit sphere S f x; y; t 2 R3 : x2 y2 t2 1g, but we shall identify the real x; y -plane f x; y; t ; t 0g with C, setting z x iy.
2022年6月29日 · This paper introduces the Riemann sphere and shows how to map points in the unit upper-half sphere to points in the xy-plane and then back again. Basic knowledge of 3D coordinates and vectors is assumed. The discussion here is basic for the topic.
In mathematics, the Riemann sphere, named after the 19th century mathematician Bernhard Riemann, is a model of the extended complex plane, the complex plane plus a point at infinity. This extended plane represents the extended complex numbers, that is, the complex numbers plus a value ∞ for infinity.
The Riemann Sphere - SpringerLink
2024年12月12日 · The sphere with this complex structure is called the Riemann Sphere and is an example of a Riemann Surface. Now that we have a complex structure, we can do complex analysis on the sphere: rather than working on the sphere …
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