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Questions tagged [stereographic-projections]

For question about stereographic projection, a particular mapping that projects a sphere onto a plane.

In geometry, the stereographic projection is a particular mapping that projects a sphere onto a plane. The projection is defined on the entire sphere, except at one point: the projection point. Where it is defined, the mapping is smooth and bijective. It is conformal, meaning that it preserves angles. It is neither isometric nor area-preserving: that is, it preserves neither distances nor the areas of figures.

226 questions
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Showing that stereographic projection is a homeomorphism

For any $n\geq 0$, the unit $n$-sphere is the space $S^{n}\subset \mathbb{R}^{n+1}$ defined by $$ S^{n}=S^{n}(1) := \left\{ (x_{1}, \dots, x_{n+1}) \;\middle\vert\; \sum_{i=1}^{n+1} x_{i}^{2} = 1 \right\} $$ with the subspace topology. The point…
user202406
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What is the metric on the $n$-sphere in stereographic projection coordinates?

The metric on the $n$-sphere is the metric induced from the ambient Euclidean metric. Find the metric, $d\Omega^2_n$, on the $n$-sphere and the volume form, $\Omega_{S_n}$ , of $S^n$ in terms of the stereographic coordinates on $U_N =S^n − (0, . . .…
Okazaki
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Mapping The Unit Disc To The Hemisphere?

Question: Can a disc drawn in the Euclidean plane be mapped to the surface of a hemisphere in Euclidean space ? If $U$ is the unit disc drawn in the Euclidean plane is there a map, $\pi$, which sends the points of $U$ to the surface of a…
Anthony
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How to construct three mutually orthogonal circles in stereographic projection?

I'm new to spherical geometry and I enjoy doing ruler-and-compass constructions, so I'm trying to teach myself to do them in stereographic projection. I'm finding it challenging, to put it mildly. The following picture shows three mutually…
helveticat
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Prove that $|PF_{1}|+|PF_{2}|$ is Constant in an Elipse

Given an elipse with two focus $F_{1}$ an $F_{2}$, and $A$ is an arbitrary point at the elipse. Stright line $AF_{1}$ has another intersection point $B$ with the elipse, and $AF_{2}$ has another intersection point $C$ with the elipse. And point $P$…
easymath3
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Stereographic projection (Theorem that circles on the sphere get mapped to circles on the plane)

I'm trying to understand the proof of the theorem (given in the link) that states "Stereographic projection maps circles of the unit sphere, which do not contain the north pole, to circles in the complex plane." Link to the proof In the proof it…
Kevin
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Explicit homeomorphism between $\mathbb{S^2}$ and $\mathbb{P^1(C)}$

I know that $\mathbb{P^1(C)} \cong \mathbb{P^1(C)} \cup \{N\} $, where $N$ is the north-pole of the sphere, is homeomorphic to the sphere $S^2$ thanks to the stereographic projection, but I am not sure if the explicit projection could be the…
Phi_24
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Intersection of random line segments in the plane

Let a point on the plane be randomly chosen via $(\sqrt{\frac{t}{1-t}}\cos(2\pi\theta),\sqrt{\frac{t}{1-t}}\sin(2\pi\theta))$, where $t$ and $\theta$ are uniformly randomly chosen on $[0,1]$ (equivalently, choose a point uniformly randomly on the…
Feryll
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Why does stereographic projection appear here?

I'm working on some calc III problems, and found that the unit tangent vector of $$\langle t + \dfrac{1}{t}, 2\ln(t)\rangle $$ is $$\left\langle \dfrac{t^2 - 1}{t^2 + 1}, \dfrac{2t}{t^2 + 1} \right\rangle.$$ This is weird to me! I recognize this…
TheAssistant
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Explicit fractional linear transformations which rotate the Riemann sphere about the $x$-axis

Background: Let $S^2$ denote the unit sphere in $\mathbb{R}^3$. By "stereographic projection", I mean the mapping from $S^2$ (remove the north pole) to the complex plane which sends \begin{align*} \begin{bmatrix} x \\ y \\ t \end{bmatrix} \in S^2 …
Mike F
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Seeking a good book or site describing the 3-sphere

would you be able to recommend a good book/chapter or a web site on visualization / structural elements / projection of the 3-sphere. I am trying to locate a good information source on this subject, which will show the math derivations,…
Jay
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A stereographic projection for the Chebyshev polynomials

This question may be too vague for the MSE crowd, if so, please feel free to ask clarifying questions or just remove. The Chebyshev polynomials are a family of orthogonal polynomials typically defined over the domain $x\in[-1,1]$. The nth Chebyshev…
Cuhrazatee
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Proving geometrically that stereographic projection conserves circles

I am aware of a few analytical calculations showing that the stereographic projection sends circles on the sphere to circles on the equatorial plane. There are related questions here. What about a geometrical proof? I have found one somewhere that…
fffred
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Path of the sun across the sky in a 4D world

Someone asked a question on worldbuilding about navigating by the stars on a 4D planet. In thinking about it I came up with a question that seems appropriate to ask here, as it's purely a maths question. Suppose you're at a point on the surface of a…
N. Virgo
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Find geometric derivation of $\rho(a,b)=\frac{2|a-b|}{\sqrt{1+|a|^2}\sqrt{1+|b^2|}}$ for stereographic projection.

When the complex plane is projected to the spherical surface, we can brute force the formula for the the distance between the two image points $a,b$ on the sphere $d(a,b)=\frac{2|a-b|}{\sqrt{1+|a|^2}\sqrt{1+|b^2|}}$ In retrospect, we see the…
ps1
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