Questions tagged [riemann-sphere]
For questions about the Riemann sphere, a model of the extended complex plane.
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Stereographic Projection Using a One-Sheeted Hyperboloid
This is in reference to the stereographic projection of a one-sheeted hyperboloid, as detailed on page 199 of this book.
The author visualises the inversive Minkowskian plane by using a stereographic ...
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Feedback on and assistance with this proof about a particular quotient space of $\mathbb{C}P^1$
The goal here is to define the particular equivalence relation I'm attempting to describe, and then provide an equation (in this case, (2)) that can be used to determine whether or not two given ...
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Meromorphic functions are rational
Let $f:\hat{\mathbb{C}} \to \hat{\mathbb{C}}$ be an analytic function, i.e., $f\restriction_{\mathbb{C}}$ and $f(\frac{1}{z})\restriction_{\mathbb{C}}$ are meromorphic or are constant taking the value ...
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How can I adjust the x, y, and z coordinates of inscribed circles on a spinning sphere for different polar angles in an animation?
I am making an animation of a spinning sphere with circles inscribed on it.
I have been successful in rotating it azimuthally. For polar angles, while I can rotate the inscribed circles appropriately, ...
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Holomorphic mappings of Riemann Sphere that preserves each hemisphere
What are some examples of holomorphic functions that satisfy the title?
Especially for Mobius Transformation $f(z) = \frac{az+b}{cz+d}$
Is there a requirement on $ad-bc>0$ ?
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After removing a point from a connected set in sphere, are the connected components still connected when the original point is added back
It can be written as theorem below (It may not true)
If $X$ is a subset of $S^2$ and suppose $X$ is connected, $x_0$ is a point of X, we consider the set $X\setminus\{x_0\}$, if $X\setminus\{x_0\}$}=$\...
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Interpolating points on a sphere between two points
I managed to solve it using the following function:
given a cartesian point A and point B.
the geodesic path on a sphere is defined as:
r(t) = sin(1-t)*A + sin(t)*B, for t=[0, 1]
then normalize r(t)/||...
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How to prove: The stereographic projection is orientation reversing (Complex Analysis by Theodore W. Gamelin 1.3.3)?
This problem is in my book:
Show that as $z$ traverses a small circle in the complex plane in the
positive (counterclockwise) direction, the corresponding point $P$ on
the sphere traverses a small ...
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Understanding continuity in $\hat{\mathbb{C}}$
Let $\hat{\mathbb{C}}$ denote the Riemann sphere. Let $f:B_1(0) \to \hat{\mathbb{C}}$ be continuous. If $f$ is continuous at $z$ and non-zero, then $1/f(z)$ is continuous at $z$ as well. My question ...
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Let $f: \mathbb C \to \mathbb C$ be a proper smooth map with continuous extension $F : \mathbb C P^1 \to \mathbb C P^1$. Must $F$ be smooth?
We know that, if $p : \mathbb C \to \mathbb C$ is a nontrivial polynomial, then the function $P: \mathbb C P^1 \to \mathbb C P^1$ defined as $P([z:1]) = [p(z):1]$ and $P([1:0]) = [1:0]$ is a smooth ...
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Conjugating the Mobius Transformation $\mathcal M(z)=1/(z-2i)$
Consider the Mobius transformation $\mathcal M:\textbf C^*\to\textbf C^*$ (where $\textbf C^*:=\textbf C\cup\{\infty\}$ denotes the extended complex plane/Riemann sphere) defined by:
$$\mathcal M(z):=\...
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Why is conformal mapping important for Riemann Sphere
The mapping of the complex plane to the Riemann sphere is conformal. As a result, many sources contend that it's this property of conformal that makes the concept of Riemann Sphere non-trivial. ...
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Simple connectedness of a horizontal strip
We have the following result in complex analysis.
Let $D$ be a domain (open connected set) in $\mathbb{C}$ and $D_{\infty}$ be the set corresponding to $D$ on the Riemann Sphere $S.$ Then, $D$ is ...
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Why are the Möbius transformations the conformal automorphisms of the Riemann sphere in the sense of Riemannian geometry?
There are two ways, how one can define a conformal structure on an Riemann surface. Either in terms of a complex structure, or in terms of a Riemannian metric.
I understand, that the Möbius ...
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Every degree $2$ holomorphic endomorphism of $\mathbb{CP}^1$ with two ramification points such that $g(a_1) = 0$ and $g(a_2) = \infty$ can be written…
I’d like to show that if $g:\mathbb{CP}^1 \rightarrow \mathbb{CP}^1$ is a holomorphic map of degree 2 with two ramification points $a_1 \neq a_2$ such that $g(a_1) = 0$ and $g(a_2) = \infty$, then $g$ ...
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