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.
黎曼球面 - 百度百科
但是，黎曼 球面 在几何和解析角度都行为良好，甚至在无穷远点也不例外；它是一个一维复流形，也称黎曼曲面。 它由19世纪数学家 黎曼 而得名。 也称为. 扩充复平面，记为 或者 . 从纯 代数 的角度，复数加上一个无穷远点构成一个数系称为扩充复数。 无穷远点的算数有时和一般的代数规则不符，因此扩充复数不构成一个代数域。 但是，黎曼球面在几何和解析角度都行为良好，甚至在无穷远点也不例外；它是一个一维复流形，也称 黎曼曲面。 复分析 中，黎曼球面对于 亚纯函数 …
Riemann Sphere -- from Wolfram MathWorld
2024年12月18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.
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).
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 …
De nition 6.2. The Riemann sphere is the unit sphere in R3: S2 = f(x;y;z) 2R3 jx2 + y2 + z2 = 1g: To make a correspondence with the sphere and the plane is simply to make a map (a real map not a function). We de ne a function F : S2 f (0;0;1)g! C ˆR3 as follows. Pick a point q = (x;y;z) 2S2, a point of the unit sphere, other than the north ...
The Riemann Sphere - SpringerLink
2024年12月12日 · The Riemann sphere, then, is the canonical compact, simply connected Riemann surface and deserves a careful study in its own right. We will see that stereographic projection has some very nice properties: it is an angle preserving map under which circles on the sphere correspond to generalised circles in the complex plane.
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
2022年6月29日 · The purpose of the Riemann sphere in this paper is to find an invertible map from the (upper hemi)sphere to the plane. Such a map is called a stereographic projection. For this paper, we won’t bother to map the north pole point x = (0,0,1) = e 3, which would go to infinity. Also, technically, the Riemann sphere
The Riemann Sphere - IB Maths Resources from Intermathematics
2014年2月1日 · The Riemann Sphere is a fantastic glimpse of where geometry can take you when you escape from the constraints of Euclidean Geometry – the geometry of circles and lines taught at school. Riemann, the German 19th Century mathematician, devised a way of representing every point on a plane as a point on a sphere.
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