# Tagged Questions

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

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### How to estimate the maximum projection area of a set of spheres?

I have a set of spheres P. The spheres have a known, finite range of radii. It seems that there must be at least one 2 dimensional plane such that the bounding circle around the projection of P onto ...
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### Fisheye equidistant projection mapping to fisheye stereographic projection?

I have a set of images captured by a wide-angle (fisheye) lens camera, and the projection is linear-scaled (equidistant). I would like to remap from this projection to fisheye stereographic, which is ...
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### Pinhole projection of the center of a 3D circle

Consider the pinhole projection of a 3D circle. The projection I am considering is a pinhole camera projection which has a fully known calibration. The projection of a 3D circle will be an ellipse, ...
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### stereographic projection, coordinates

when you want to map a random point on the sphere $S^n\subset\mathbb{R}^{n+1}$. And observe the points $e_\pm=(0,\dotso,0,\pm 1)$, you can use the function $g_+(t)=e_++t(x_1,\dotso, x_n,x_{n+1}-1)$ ...
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### Stereographic projection with de Sitter space and hyperbolic plane

How can we do stereographic projection using de Sitter space $\Bbb S^2_1$ and the hyperbolic plane $\Bbb H^2$, in Lorentz-Minkowski space $\Bbb L^3$. For $\Bbb S^2_1$ it is not clear what point ...
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### benefit of trifocal geometry vs bifocal geometry?

I am at the moment trying to understand what kind of benefit I would have by using three cameras for stereo vision rather than two cameras? I mean, i would only have more constraints related to the ...
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### projection of an ellipsoid on XY plane

The equation of an ellipsoid is $$ax^2+by^2+cz^2+2fyz+2gxz+2hxy+2px+2qy+2rz+d=0$$ The ellipsoid is arbitrary rotated and the orientation angle are given as θ, β and Ѱ and the center is at (x',y',z')....
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### Determining a derivation on the unit sphere of the $\mathbb{R}^3$

Let $S^2 \subseteq \mathbb{R}^3$ be the unit sphere in $\mathbb{R}^3$ (which is a smooth manifold of dimension $2$). Let $\phi = (x_1, x_2): S^2 \backslash \{N\} \to \mathbb{R}^2$ be the ...
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### How to find a function on a 3-dimentional space base on a set of points?

So I have been working on stereographic projections and stuff. I want to input a set of points on the plane z = -1 and find the function on the sphere that it was projecting from. Right now I have ...
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### curvature of an arc in S3, in stereographic projection

$r(t)$ is a unit 4-vector. The derivatives of $r$ are known and well-behaved. I'm interested in images of $r$ in stereographic projection – but (for purposes of this question) I don't yet know where ...
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### Why is $\mu=1$ clear?

I don't understand this reasoning from a solution. Q. Let $S^n=\{(x_0,...,x_n) \in \mathbb{R}^{n+1}:x_0^2+...+x_n^2=1\}$ and $X=S^n-\{1,0,0,...,0\}$ and $Y=\{(y_0,...,y_n):y_0=0\}$. Find $\mu$ ...
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### If $\phi : S^n - \{e_{n+1}\} \to \mathbb{R}^n$ is the stereographic projection, how to compute $\phi^{-1}$?

If $\phi : S^n - \{e_{n+1}\} \to \mathbb{R}^n$ is the stereographic projection, how to compute $\phi^{-1}$? If $$\phi(x) = \frac{1}{1 - x_{n+1}}(x^1,\ldots,x^n),$$ how to compute $\phi^{-1}(y)?$
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### Show that all complex numbers $z \not= -1$ can be written as $z=\frac{1+it}{1-it}$ [duplicate]

Show that for all complex number $z \not= -1$, with $|z| = 1$, can be written as $z=\frac{1+it}{1-it}$, with $t \in \mathbb{R}$. I think this structure seems to be a stereographic projection from the ...
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### Stereographic projection…A more concise explanation?

I have been looking at stereographic projections in books, online but they all seem...I don't know how else to put this, but very pedantic yet skipping the details of calculations. Say, I have a ...
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### geometric interpretation of images of two symmetric points wrt (0,0,1) under stereographic projections

I have Riemann sphere centered at $(0,0,1)$. If I pick a point on it, say $A$ and take $D$ as a point symmetric to it w.r.t. the center of the sphere, what is geometric interpretation of their images ...
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### Chordal Distance (Stereographic Projection)

I was working out Gamelin's Complex Analysis and read through the part where he finds an expression for the chordal distance on the Riemann Sphere corresponding to the stereographic projection w.r.t. ...
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### Image of a Curve under Stereographic Projection $D:\Pi\rightarrow\Sigma-\{N\}$

Let $D:\Sigma-\{N\}\rightarrow\Pi$ denote the stereographic projection or one-to-one mapping of $\Sigma$ onto $\Pi$, where $\Sigma$ is the Riemann Sphere and $N$ is the geometric image of the improper ...
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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*} \...
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### Projection of surfaces in $\mathbb{C}^2, \mathbb{C}P^2, \mathbb{C}H^2$ to $\mathbb{R}^3$

As part of my thesis in Riemannian geometry, I study surfaces in $\mathbb{C}^2$, $\mathbb{C}P^2$ and $\mathbb{C}H^2$. Since visulation is always nice, I was wondering if there existed any "nice" ...
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### Cosine of an area

Let there be a geometric shape $\Omega$ of area $S$ lying in a plane $B$. Let the horizontal plane (the plane $xy$) be $A$. Let the angle between the planes $A$ and $B$ be $\theta$. It could be ...
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### A question about stereographic projection of a plane onto a sphere

I am reading a paper by Christine Bernardi, available here http://epubs.siam.org/doi/pdf/10.1137/0726068, my question relates to page 1237, which I shall elaborate on: In this part we have the unit ...
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### Projection of a sphere's surface vs that of the inner side of the surface

I want to project a catalog of stars on a sphere to the computer screen. As a non-mathematician I was looking for a ready made solution and I found some interesting geographical projections, like ...
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### Connection between Chladni Plates and Projective Geometry?

Does anybody know of a connection between Chladni Plates and Projective Geometry? Any published research on the subject? I am interested to know if Chladni Plates (or surfaces) can be explained/...
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### Is there a locally-distance-preserving map projection?

I'm trying to figure out if there is a family of map projections which preserve local distances: in other words a family of functions $f \in S^2 \rightarrow K, K \subseteq \mathbb R^2$ such that for ...
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### Noneuclidean Geometry - how do I find $(x,y,z)$ in $\mathcal{S}$?

I've been asked to find $(x,y,z)$ in $\mathcal{S}$. I'm stuck on the question attached because although it gives the formula of how to find $\pi_\mathcal{s}$ (stereographic projection), I'm not sure ...
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### Chordal Metric - Showing it is in fact a metric

If I have $f(z_{1},z_{2}) = \displaystyle\frac{|z_{1} - z_{2}|}{\sqrt{1+ |z_{1}|^2} \cdot \sqrt{1 + |z_{2}|^2}}$, for $z_{1}, z_{2} \in \mathbb{C}$, how would I show that $f(z_{1},z_{2})$ is a metric?...
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### an example of two non-commuting projection operators.

Give an example of two non-commuting projection operators in $\mathbb R^2.$ I know in $\mathbb R^2 f(x,y)=(0,y)$ and $g(x,y)=(x,0)$ and $h(x,y)=(x,y)$ are projection but are not commuting.
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### necessary and sufficient conditions for multiplication operator T on L^([a,b]) to be a projection

Let $T$ be a multiplication operator on $L^2([a, b ])$. Find necessary and sufficient conditions for $T$ to be a projection. let g be a fixed function in $L^2([a,b])$, and $T(f(x))=g(x)f(x)$.
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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 $$S^{n}=S^{n}(1) :=\left\{ (x_{1},...,x_{n+1}) \left\vert\,\sum_{i=1}^{n+1} x_{i}^{2}=1\right.\right\}$$ ...
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Let $$\varphi:\mathbb{R}^2 \longrightarrow \mathbb{S}^2-{N} \subset \mathbb{R}^3$$ be the (inverse) stereographic projection from the North pole on the unit sphere centred at the origin. $$\varphi(x,... 1answer 46 views ### Image of a locus via stereographic projections Yesterday evening I was playing around in my head with stereographic projections and I've come up with this idea. Let \gamma(t)=(x(t),y(t)) be a certain curve on a plane. Define a new curve \begin{... 0answers 109 views ### Stereographic projection to show S^n is a submanifold of \Bbb R^{n+1} So S^n in \Bbb R^{n+1} can be described by the equation x_1^2+\ldots+x_{n+1}^2=1. Now consider two subsets U_N:=S^n-\{(0,0,\ldots,1)\} and U_S:=S^n-\{(0,0,\ldots,-1)\}, the sphere less it's ... 1answer 36 views ### Surjectivity of a map D^{2n} \to \mathbb{CP}^n I'm solving an exercise about the complex projective space, and during a step of the solution I'm asked to find a surjective map D^{2n} \to \mathbb{CP}^n. I defined the map in this way$$ (z_0,\dots,...
I'm looking over some fairly basic stuff on complex methods and the book I'm using takes the formula for the stereographic projection: $$z = \cot(\beta/2)e^{i\phi}$$ as well as the line element on ...