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

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Rank of Jacobian Matrix for the Stereographic Projection

With the definition $S^{n} = \{\ \mathbf{x} \in \mathbb{R}^{n+1}\ | \ ||\mathbf{x}|| = 1\ \}$, and the function $\ f:\mathbb{R}^{n} \to S^{n} \setminus \{ (0,...,0,1) \}$ defined by: $f(\mathbf{u}) ...
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34 views

The identity map from $\mathbb{\bar B}^3$(as a subset of $\mathbb{R}^3)$ into $\mathbb{\bar B}^3$(as a smooth manifold with boundary) is not smooth?

Let $U$ be the open rectangle $(0, \pi) \times (0,2 \pi) \subset \mathbb{R}^2 $ and let $X : U \rightarrow \mathbb{R}^3$ be the following map: $$X(\varphi , \theta)=(\sin \varphi \cos \theta , ...
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23 views

Stereographic projection $S^3 \to \mathbb{P}^2(\mathbb{C})$

I think I can find a stereographic projection $S^2\setminus\{(0,0,1)\} \to \mathbb{P}^1(\mathbb{C})\setminus\{[0,1]\}$ using spherical coordinates: it should be something like this $$(\theta,\phi)\to ...
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1answer
24 views

Projecting a sphere from inside

I am trying to make a renderer for a programming project, and yet I am having trouble projecting the points onto the screen (the way it works so far, the camera can't look down on a face because the ...
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33 views

Conformal mapping from Gaussian grid to rectangular grid

I have data in a Gaussian grid - https://en.wikipedia.org/wiki/Gaussian_grid In this Gaussian grid coordinates will be defined in terms of longitude and latitude). I am going to be transforming this ...
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10 views

Circle homeomorphic to $\hat {]0,1[}$ the compactification of Alexandrov.

I denote $\mathbb S^1=\{x^2+y^2=1\mid x,y\in\mathbb R\}$. So, if I understood well, $$]0,1[\cong \mathbb S^1\backslash \{(0,1)\}$$ (or to $\mathbb S^1\backslash \{(a,b)\}$ where $(a,b)\in\mathbb ...
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1answer
35 views

Doubt with an illustration of algebraic curves and Riemann surfaces

The complex equation $w - z = 0$, $z$, $w \in \mathbb{C}$, represents a complex curve (also called $1$-dimensional complex manifold). This complex curve corresponds to the complex plane $\mathbb{C}$ ...
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11 views

Construction of perspective matrix

Is it possible to construct a perspective matrix by multiplying a perspective transformation matrix with a parallel projection matrix ?
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1answer
32 views

Projectivization and stereographic projection, or: Why nonlinear + nonlinear = linear?

This previous question had me thinking about something I've taken for granted. Consider $\mathrm{SL}_2(\mathbb{C})$ acting on $\mathbb{C}^2$, which descends to an action on the complex projective line ...
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1answer
43 views

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 ...
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50 views

Image of stereographic projection of line

This is homework so dont give a complete answer. I just Dont understand what my professor means by the question: "Determine the images under the stereographic projection of all the lines l[-3,9,a], ...
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33 views

What is the equation of the point on a Sphere in $S^n$\ $P$?

I have been all over the internet but only got surprised in how no one mentions what the general equation(formula) is and how to derive it, for a map $\mathbb{R}^n \rightarrow S^n$ \ $P$where ...
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40 views

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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9 views

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 ...
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88 views

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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30 views

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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142 views

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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1answer
56 views

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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35 views

Show that the preimage of any circle or line in $\mathbb{C}$ under the stereographic projection is a circle on $\mathbb{S}^2$.

So this is an exercise in my complex analysis notes, which has been given the solution as follows (where I write $z = π(x_1,x_2,x_3) = \displaystyle\frac{x_1}{(1-x_3)} + \frac{ix_2}{(1-x_3)}$ for the ...
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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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60 views

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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31 views

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 ...
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1answer
63 views

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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1answer
28 views

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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1answer
414 views

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 ...
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1answer
111 views

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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24 views

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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297 views

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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1answer
60 views

Maps of the sphere!

In the sphere we can introduce two patches that their union covers the whole sphere. Ok, I understand why we need at least two (because of the two poles). The maps in each chart is $\phi_1$ and ...
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1answer
179 views

Is the Riemann sphere conformal equivalent to the 2-sphere?

Today I stumbled across the calculation (mentioned in this post) of the transition maps of the stereographic projections from the 2-sphere to the plane. And I wondered about the result that the last ...
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2answers
75 views

Finding a unique Mobius Transformation

Let $z_1, z_2, z_3$ be three distinct points in $\widetilde{\mathbb{C}}$. (1) show that there is a unique mobius transformation $g$ such that $g(z_1)=0, g(z_2)=1, g(z_3)=\infty$ (2) show ...
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Stereographic projection piecewise smooth at North Pole

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. ...
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1answer
42 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 ...
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80 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 ...
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1answer
35 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 $$ ...
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1answer
203 views

Stereographic projection is conformal — from the line element

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 ...
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1answer
182 views

Is stereographic projection the only way to make a bijection between plane and sphere?

At a math exhibition, I learned the concept of stereographic projection for the first time. However, I am curious about the purpose of the stereographcal projection. I've learned that an area of ...
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0answers
74 views

How do we understand and visualize the meridians of a 3-sphere?

The equator of a n-sphere is (n-1)-sphere. Then the equator of a 3-sphere is a 2-sphere. Does this mean the meridian of a 3-sphere is a hemisphere (half 2-sphere)? In contrast, Wikipedia describes the ...
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1answer
157 views

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, ...
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24 views

find the shortest distance between two lines [duplicate]

Let L1 be the line passing through the point P1=(−9, −7, 14) with direction vector →d1=[−2, −1, 3]T, and let L2 be the line passing through the point P2=(9, −8, 9) with direction vector →d2=[2, −3, ...
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1answer
130 views

Find the image of the following circles on the sphere under stereographic projection

Find the image of the following circles on the sphere under stereographic projection from the north pole onto the equatorial plane: $C = S^2 ∩ \{(x_1, x_2, x_3) \;|\; x_1 = x_2\}$ $C = S^2 ∩ \{(x_1, ...
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69 views

Stereographic Projection proofs(pathagorean triples)

So I recently stumbled upon a proof using stereographic projection to prove Euclid's formula for generating Pythagorean triples: for all $m,n\in \mathbb{N}$, ...
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77 views

Stereographic Projection

Could anyone find a stereographic projection of a $4$-dimensional object with $1$ zero-dimensional hole, $8$ one-dimensional holes, $6$ two-dimensional holes, $8$ three-dimensional holes, and lastly ...
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671 views

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. ...
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1answer
52 views

find f and $d_pf$

Let $$S_1=\{(x,y,z) | x^2+y^2+z^2=1\}-\{N\}$$ $$S_2= \{(x,y,z,0) | x,y\in \Bbb R\}$$ $f:S_1\to S_2$ $f$ is stereographic projection. ,where $\ell$ is a line passing through $N=(0,0,1)$ and $p$ ...
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34 views

Homogeneous matrix to represent rotation out of plane?

How to represent 2D homoheneous projection matrix $M=\left( \begin{array}{ccc} m_{00} & m_{01} & m_{02} \\ m_{10} & m_{11} & m_{12} \\ m_{20} & m_{21} & w \end{array} \right)$ ...
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70 views

Anamorphotic Projections onto Awkward Solid Surfaces

I have been experimenting with some 19th century picture development techniques that involve photochemical image projection: For flat pictures you first mix equal parts of chemicals like ferric ...
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886 views

How to calculate the angle between two vectors, defined by 3 points on the earth?

I want to develop a formula to calculate the angle between two vectors. The vectors will be OX and OY (from point O to X , and Y), where the points are defined by their latitude and longitude values. ...
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36 views

What is the proportion of table from this picture?

I have this picture and I know his height = 75cm Do you know how to find out its proportion like width and lenght using perspective projection?