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Questions tagged [stereographic-projections]

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

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How to write an equation which mapping a square into a sphere?

Let say the curved cells on that sphere actually create a perfect square grid in shadow. Is it possible to write an equation which will mapping the square in 2D into the curved cell in 3D? The ...
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Showing $\bigg( \frac{x_1}{1-x_{n+1}},…, \frac{x_n}{1-x_{n+1}} \bigg)=y$ as onto?

Am I allowed to use an index of $y \in \mathbb{R}^n$ as constant, when showing onto ($\exists$ s.t. $f(x)=y$)? Particularly, I'm trying to show $\bigg( \frac{x_1}{1-x_{n+1}},..., \frac{x_n}{1-x_{n+...
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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 ...
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How to change the center of a stereographic projection?

Right now I am plotting a streographic projection with a center at the z axis by getting a P vector (Px, Py, Pz) from different directional indices in a unit sphere. The following process converts ...
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240 views

What is the intersection of a plane and a sphere? Is it necessarily a circle? Can it be an ellipse?

A First Course in Complex Analysis by Matthias Beck, Gerald Marchesi, Dennis Pixton, and Lucas Sabalka Exer 3.27,Exer 3.28 (Exer 3.27) Consider the plane $H$ determined by the equation $x + y -z = ...
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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 $[0,1]$ (...
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Plotting a subset of the 4-sphere in the stereographic projection

I'm looking for a program, library or function in a math language (SAGE will be fantastic) that allows me to plot in 3D a subset of $\mathbb{S}^4$ through the canonical stereographic projection that ...
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Composition of stereographic projection and inverse Hopf map

There are several definitions of the Hopf map $H \colon S^3 \to S^2$. The one I use is $$ H(p) = \begin{pmatrix} 2(p_1 p_3 + p_2 p_4) \\ 2(p_1 p_2 - p_3 p_4) \\ -p_1^2 + p_2^2 + p_3^2 - p_4^2 \\ \end{...
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62 views

Why image of a line in complex plane is a circle in the Riemann sphere?

Let $ax+by+c=0$ be a line in complex plane. If $\pi$ be the stereographic progection, then since $$\pi^{-1}(x_1,x_2,x_3)=\left(\frac{2x_1}{2-x_3},\frac{2x_2}{2-x_3}\right)$$ we have $$a\left(\frac{...
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Computing the surface integral of the octant of a sphere with polar coordinate substitution

Let me first describe where I start: $$\iint_Sz^2\,dS$$ We want to compute the surface integral of the octant of a sphere $S$. The radius = 1. The sphere is centered at the origin. $$S=x^2+y^2+z^...
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How to describe the relative positions?

There is a problem in my complex variable textbook as follows: Discribe the relative positions of the images of $z$, $-z$ and $\bar z$ on the Riemann sphere. But I don't understand what does this ...
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Smoothness of streographic projection

Let $\pi:S^2\to \mathbb{R}^2$ be the streographic projection from the north pole $(0, 0, 1)$. i.e., $$\pi(x_1,x_2,x_3)= \left(\frac{x_1}{1-x_3},\frac{x_2}{1-x_3}\right)$$ for any $x=(x_1,x_2,x_3)\in S^...
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Bijection between $\mathbb{P}^{1}(\mathbb{C})$ and the unit sphere

I have the following complex analysis problem that I am completely lost on. Let $(z_{1},z_{2})$ be a non-zero vector in $\mathbb{C}^{2}$ and define $F:\mathbb{C}^{2}\to\mathbb{R}^{3}$ by $$F(z_{1},...
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Distance induced in $\hat{\mathbb{C}}$ by stereographic projection

I have the $\phi(z) = \frac{1}{1+|z|^2}\cdot(2\cdot Re(z), 2\cdot Im(z), |z|^2-1)$ I have to show that $\hat{d}(z, w)= ||\phi(z) - \phi(w)|| = \frac{2\cdot |w - z|}{(1+|z|^2)^\frac{1}{2}\cdot (1+|w|^...
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How does the stereographic map from $S^{2}$ to $\mathbb R^2$ “induce” a metric on $\mathbb R^2$

Let $S^2\subset\mathbb{E}^3$ be a sphere with radius $1$ and center $(0,0,1)$ in cartesian coordinates. The northpole is point $(0,0,2)$ on the sphere and $Oxy$ is the $xy$-plane. Let $\pi:S^2\...
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94 views

How to project 5D to 3D?

When I have a 4D polyhedron inscribed in a sphere, I project it to 3D with the stereographic projection. But now, if I have a 5D polyhedron inscribed in the sphere, how could I project it to the 3D ...
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Math software to generate stereographic projection of polyhedra

Is there any software to generate the stereographic projection of polyhedra? Note that since the stereographic projection is infinite, it should take a viewing window as an input. (In my particular ...
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Question on nomenclature of … mappings?

I would like to understand "in english", what this sentence is saying here: I understand what $R^3$ means, but I am not sure I understand the rest... Thanks! EDIT: The image is from this paper.
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49 views

finding two charts to cover the manifold $S^n$

i have a problem in stereography of Manifold Sphere $S^n$ for finding two charts to cover it , still stereography is hard for me to understand but the problem is how can i find these 2 charts to cover ...
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Complex exponentials and the math behind Mercator's projection

I came across this post by David Bau, in which he reproduces the most widespread Mercator projection as the plot of the complex function $$f(z)=\exp \mathrm i z.$$ The result is familiar: ...
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Understanding the formula for stereographic projection of a point.

I was wondering about the equation of line I can write which can help me finding the coordinates of Point $P'$ in relation with coordinates of points on the sphere that is $P$. Let the $P'(X,Y)$, ...
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diffeomorphism between the complex line and the sphere

I want to write the diffeomorphism between the complex line and the sphere. $$\mathbb{C}P^1 = \{<(z_0,z_1)>\ \vert\ (z_0,z_1) \ne 0\} \\S^2 = \{(x,y,z)\ \vert\ x^2+y^2+z^2 = 1\}$$ I get that ...
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$p_n(z)=z^n$ is a smooth map on $S^1$

I have to solve this one: Consider local coordinates on $S^n$ given by the atlas $\{(S^n\setminus\{N\},\varphi_N),(S^n\setminus\{S\},\varphi_S)\}$, where $\varphi_N$ and $\varphi_S$ are the ...
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$\gamma'_z(t)=(ie^{it}z_1,ie^{it}z_2)$ in stereographic coordinates is non-vanishing for all $t$

I have to solve this one: Consider $S^3$ as a subset of $\mathbb{C}^2$. For all $(z_1,z_2)\in S^3$, consider $\gamma_z:\mathbb{R}\rightarrow S^3$ defined by $\gamma_z(t)=(e^{it}z_1,e^{it}z_2)$. ...
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Local coordinates and angle function

I'm stuck with this problem: "Let us consider, on the sphere, the local coordinates associated with the differential structure determined by the atlas $\{(U_1=S^n\setminus\{N\},\varphi_N),(U_2=S^n\...
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Stereographic projection of real numbers

I have the next question about the stereographic projection: How would you find the point $(x_{0},y_{0})$ which corresponds, by the stereographic projection, with the real number $-\frac{1}{4}$? (for ...
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Real projective space and the stereographic projection

What is the relationship between the real projective plane $\mathbb RP^2$ and the stereographic projection i.e. a plane $\mathbb C$ with a point at infinity $\infty$, i.e. a one point compactification ...
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Is there any difference between a Riemann sphere and a normal stereographic projection of a complex plane?

As how I know, the Riemann sphere is a stereographic projection of the complex plane. Assuming that anyone who is familiar with projective geometry can have an intuition to do the same with complex ...
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Stereographic projection of Hopf map

The Hopf fibration $h:S^{3}\rightarrow S^{2}$ is given by $h(a,b,c,d)=(a^{2}+b^{2}-c^{2}-d^{2},2(ad+bc),2(bd-ac))$. A stereographic projection is a map $s:S^{3}\backslash (1,0,0,0)\rightarrow \mathbb{...
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Is there a hidden connection between RH and the golden ratio?

I realized today that, considering the circle $ \Gamma_{\Delta} $ on the Riemann sphere whose image through the stereographic projection is the critical line $ \Delta $, the affixes of the images of ...
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Is there a metric on the sphere a geodesic thereof gives rise to the critical line?

Let $ \Delta : =\{z\in\mathbb{C},\Re(z)=1/2\} $ be the critical line that appears in the Riemann Hypothesis, and let $\Gamma_{\Delta}$ the circle on the Riemann sphere $S^{2}$ that transforms into $\...
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Transform n-dimensional standard normal data to a uniform distribution on the unit n-sphere

While thinking of some algorithms related to machine learning, I went on a tangent and eventually asked myself if I could transform a standard normal distribution into a uniform distribution on the ...
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Compactness of the sphere in $\mathbb{R}^3$

Let us consider the plane $P = \{(x, y, -1) \in \mathbb{R}^3\}$ in three dimensional space. It is a closed set, but because it isn't bounded, it isn't compact. Now take the sphere without a pole: $$S^...
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114 views

Area of spherical circle under stereographic projection

Let $w=\frac1{\sqrt3}(1,-1,1)$. Find the area of the spherical circle C($w, \frac{\pi}6$) under the stereographic projection $\phi(x,y,z)=\left(\frac{x}{1-z}, \frac{y}{1-z}\right)$. Unfortunately, I ...
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283 views

Projection of 4D sphere to 3D - focus point

I just finished watching this interesting/funny Numberphile video on stereographic projection. In the video he demonstrated that one could draw an $n$-dimensional sphere onto an $(n-1)$-dimensional ...
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Image of intersection of a plane with S$^{2}$ under stereographic projection

I have the plane H and its equation is x+y-z = 0. I want to find the image of H$\cap\mathbb{S}^{2}$ under stereographic projection. I basically said that if the point (x,y,z) gets mapped to say (p,q,...
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Why is the stereographic projection a map from $S^2\to\mathbb{R}^2$?

How can I understand that the stereographic projection $$X=\cot\left(\frac{\theta}{2}\right)\cos\phi,\hspace{0.5cm}Y=\cot\left(\frac{\theta}{2}\right)\sin\phi\tag{1}$$ is a map from the surface of a ...
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276 views

2D Mercator map onto globe method

Preface to the question: I came up with a method of transferring a 2d projection map of the earth onto a globe and made a webapp to help me do that here: http://codepen.io/vez/full/YVWLRm/ (press ...
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296 views

Stereographic projection

If F is the stereographic projection of a sphere $ S\in S^2 $ to a plane $P$ (here take the plane as passing through the centre of the sphere S) then how can we prove that if $\ell$ is a great circle ...
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Stereographic projection calculating the Transition function

I am trying to understand how one can calculate the transition function in general by calculating it from the stereographic projection. If we have $n=2$ and the two maps $$\phi_1^{-1}:u\rightarrow ...
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Inverse Stereographic Projection of Circle

I understand how given a point in $\mathbb{C}$ you can map to a point on the unit sphere using the inverse of stereographic projection with the following formula: $\pi^{-1}(x+iy)=\frac{2x,2y,x^2+y^2+...
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The inverse map of the stereographic projection

Let $\mathbb S^1$ the unit circle centered at the origin and the pole $p=(0,1)$. The stereographic projection is the homeomorphism $\varphi:\mathbb S^1\setminus \{p\}\to \mathbb R^1$. In order to find ...
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254 views

Projecting the line y=x onto the Riemann Sphere

I am trying to project the line "y=x" on the complex plane including the point at infinity to the Riemann Sphere. I know the projection is a circle, but I want to understand how to find the radius of ...
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2 dimensional sphere

For $a,b,c >0$ let, $E(a,b,c) = \{ (x,y,z) \in \mathbb {R^3} : \frac {x^2}{a^2} +\frac {y^2}{b^2} + \frac {z^2}{c^2} =1 \}$ I am asked to Show that $E(a, b, c)$ is diffeomorphic to the 2-...
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63 views

Atiyah-Singer theorem and the projection of a non-compact space on a compact one

Although this question is related to physics, its origin is purely mathematical. Suppose the formal expression $$ \tag 1 \text{det}(iD), \quad iD \equiv i\gamma^{\mu}D_{\mu} = \partial_{\mu} - iA_{\...
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Calculate real length of object from perspective image- width of building [duplicate]

Please refer to the following image: I have the real height of the building which is $12.5 \text{ m}$ (red line). I have the blue lines in the image (pixels) as well as the red line. I have the ...
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488 views

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 ...
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98 views

Understanding Charles Sanders Peirce's cartography

Charles Sanders Peirce wrote$^\dagger$ about an orthomorphic or conform projection formed by transforming the stereographic projection, with a pole at infinity, by means of an elliptic function. ("...
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409 views

A planar graph can be embedded in the plane in such a way that any vertex is on the exterior face

Given a planar graph $G$ and a vertex $v$, there exists an embedding of $G$ in the plane such that $v$ is on the exterior face of the embedded graph. Can someone explain how this works? I tried to ...
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151 views

How to find the center of projection?

So I have a transformation matrix that performs a stereographic projection and I need to find the center of projection and the plane to which it maps the transformed points. $P = \begin{bmatrix}0&...