Questions tagged [stereographic-projections]

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

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Question about projection. [closed]

Let $S$ be a sphere $x^2 + y^2 + z^2 = 1$. $\forall$ point $\vec{p}=(x,y,z) \in S$, consider the line $L_{\vec{p}}$ that passes through the pole $(0,0,1) \in S$ and by the point $\vec{p}$. Let $\vec{q}...
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Limits involving points at infinity in the extended complex plane

Im trying to understand the idea of infinity in imaginary numbers. In the book "Complex variables and applications" (8 edition, brown/churchill, section 17) to explain this concept the ...
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Inverse of stereographic projection of a circle on a line

I am curious about the stereographic projection of the unit cycle centered at $0$, from the ''north pole'' $(0,1)$ on the line $y = -1$. We know: $$\phi: S_1/ \{(0,1)\} \to \mathbb{R}$$ can be given ...
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Prove Stereographic projection is a homeomorphism from Sphere except North Pole to Complex Plane

I'm self-studying exercises of Jones in Reference and I'm stuck at 1D which is the following; 1D My question The (standard) proof that I know (for instance Q4 of this note) does not seem to involve ...
Rowing0914's user avatar
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Stereographic projection maps lines and circles on the plane into circles in the sphere

From pages $19$ and $20$ of Ahlfors' Complex Analysis: Why is the converse true? Why does any line or circle in the plane mapped into a circle in the sphere?
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sterographic projection of complex plane

How author arrives at last step in following picture? How author arrives at $t=\frac{|z|^2}{1+|z|^2}$. My attempt: since we see in pic that $(1-t)^2|z|^2=t(1-t)$ from this we conclude that, $t=(1-t)|z|...
Akash Patalwanshi's user avatar
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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 ...
Cuhrazatee's user avatar
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Do Schläfli symbols unambiguously represent gemetric shapes?

According to Wikipedia, the tesseract is the four-dimensional analogue of the cube and has the Schläfli symbol {4,3,3}, and Wikipedia features the following visualization: However, when looking at it,...
HelloGoodbye's user avatar
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How do we know that there is a line connecting the point at infinity, a point on the Riemann sphere, and its stereographic projection point?

I am reading through Ahlfor's Complex Analysis and I came upon an item from his book which confused me (page 19). As a background, the Riemann Sphere is the unit sphere in $R^3$: $x_1^2 + x_2^2 + x_3^...
Bryan Ex-Academic's user avatar
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Find a basis on the $n$-sphere's tangent space $\mathcal{T}_s\mathbb{S}^n$

Let $\mathbb{S}^n$ be the $n$-sphere and $\mathcal{T}_s\mathbb{S}^n$ the tangent space at $\mathbf{s}\in\mathbb{S}^n$. I am looking for a basis for the tangent $\mathcal{T}_s\mathbb{S}^n$, but I got ...
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Construct a circle with a specific radius on a sphere, under stereographic projection.

Given three poins $v_1,v_2,v_3$ on the sphere $S^2$, it is obvious that one can construct the circle with radius $|v_1v_2|$ around the point $v_3$, as seen in the following cartoon image: In this ...
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Stereographic projections preserve angles

It seems to be a dumb question (according to my research), but I do not see (in algebric way) why stereographic projections preserve angles. There is a really good paper here that gives a geometric ...
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Stereographic projection of $\mathbb{R}^4$ on $S^4$ and the explicit relation between the measures

For the stereographic projection of the plane on the $2-$sphere $S^2$, we have the following formula between the meassures: \begin{equation} dA=\frac{4}{(1+X^2+Y^2)^2}dXdY \end{equation} where $dA$ is ...
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On projecting a sphere onto a cylinder, and preserving horizontal arcs

You take a sphere, and inscribe it in a cylinder (so the edges are touching). You project the sphere onto the surface of the cylinder through rays emanating from the axis of the cylinder outward, ...
Waleed Dahshan's user avatar
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Legendre's proof of Euler's formula (page-189, Visual Differential Geometry)

Legendre presented an ingenious proof in 1794, 2 the first step of which was to project the polyhedron onto a sphere. The specific way in which he carried out this projection (described in Ex. 26) ...
tryst with freedom's user avatar
4 votes
1 answer
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Calculating the intersection of $u^2 - v^3$ with a 3-sphere

For context, I'm coding a 3D visualisation of the Milnor fibration of a Trefoil knot. I've found some code https://www.unf.edu/~ddreibel/research/milnor/milnor-fibers.nb that calculates the ...
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How do you get the stereographic projection on a circle of radius $r$ if you know it on $S^1$?

I know the formulas for stereographic projection on $S^1$, we have two functions, $\varphi_N : S^1 \setminus \{(0,1)\}\to \mathbb{R}$, $\varphi_N(x^1, x^2)=\frac{x^1}{1-x^2}$ and $\varphi_S : S^1 \...
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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 ...
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Inverting a function of 3 variables [duplicate]

We can define a stereographic projection by $$f(x_1,x_2,x_3)=\frac{x_1+ix_2}{1-x_3}$$ It is said the inverse of f can be computed to receive the general point where the line intersects the sphere. ...
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How can I prove that the inverse stereographic projection function is continuous?

Suppose we have the function $f(x, y) = (\frac{2x}{x^2 + y^2 + 1}, \frac{2y}{x^2 + y^2 + 1}, \frac{x^2 + y^2 - 1}{x^2 + y^2 + 1})$ where $f:\mathbb{R}^2 \to S^2 \setminus \{N\}$ where $N$ is the north ...
Adam Swearingen's user avatar
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Confusion about complex operations (stereographic projection)

Hi I am reading the solution for the problem below and understand the general idea except the part where the write equates $X' to -X$. Why did the $|z|^{2}$ disappeared and why did $|\frac{1}{\bar{z}}|...
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Mapping a Unit Sphere onto $\mathbb R^2$

Q: Place a unit sphere in the $xy$-plane centered at the origin; then draw a line through the North pole $N$ and some point $(x,y,z)$ on the sphere. The line will also cross the $xy$-plane at $(X,Y).$ ...
kelsey ball's user avatar
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1 answer
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How do I apply the Stereographic projection to generate the Star Finder templates at different latitudes?

I am trying to create my own Star Finder based on the templates from the 2120-D model. I started with a semi-sphere expressed as circles of latitude and semi circles of longitude, rotated along the X ...
PolAndre's user avatar
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Prove the existence of a special diffeomorphism from $\mathbb{R}^n \mapsto \mathcal{S}^n$

I have the following exercise which I cannot solve: Show that there exists a smooth map $\mathbb{R}^m \mapsto \mathcal{S}^m$ onto the m-sphere such that the open ball $\{x \in \mathbb{R}^m | \: \| x\|...
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Project a half-sphere onto a disk to get to cartesian coordinates

I guess this is an age-old problem related to making maps of the world BUT If I have latitude and longitudes of points on a hemisphere and want to get cartesian coordinates on the 2D plane out of that,...
Charles's user avatar
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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\...
TheAssistant's user avatar
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Show that any mapping from the extended complex plane to the Riemann Sphere back to the extended complex plane is a Möbius Transformation [duplicate]

Definitions: Extended Complex Plane $\mathbb{C}^\infty = \mathbb{C} \cup \{\infty\}$. Stereographic projection: A mapping from a sphere in $\mathbb{R}^3$ to the extended complex plane. Möbius ...
fieke_2000's user avatar
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1 answer
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Where do the circles go when you stereographically project the dodecahedron?

I'm going to ask what I think is surely a silly and simple question but I can't for the life of me work this out. I am interested in stereographically projecting a dodecahedron to the plane (in an ...
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Continuity of $f(x) = \left(\frac{2x}{1+x^2},\frac{1-x^2}{1+x^2}\right)$

I am looking at a proof that shows $S^1$ is a topological manifold, where the author defines this function: $$ f_1^{-1}: \mathbb R \to S^1 \setminus \{(0,-1)\} $$ by $$ f_1^{-1}(x) = \biggl( \frac{2x}{...
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How to calculate 3D depth reconstruction of an image using a set of 3 similar images?

I have tried asking this in computer science SE, but sadly get no answers. Since this is fundamentally mathematical, I hope someone could give pointers to an answer for this, please. From a set of 2 ...
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n-sphere stereographic projection in hyperspherical coords

I need atlases for working with $n$-spheres, specifically $S^1$, $S^3$, and $S^5$. Stereographic projection charts seem like the most straightforward and economical way to make these. However, I want ...
Jim Eshelman's user avatar
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0 answers
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How can I find the arc length of some interval on this parametric object?

I want to find the arclength some interval bounded by points $P$ and $Q$ on a parametric object. This parametric object is defined by the 2D stereographic function: $f(x, y) = (tx, 1 + t(y-1))$ . If ...
Adam Swearingen's user avatar
1 vote
2 answers
253 views

Suggestions about Stereographic projection

I have to teach a 1 hr class about the stereographic projection in the complex plane and i am looking for sources or some interesting fact about this. The best I have found is in the Alhfors of ...
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Stereographic Projection Through South Pole

I am trying to find the map: $$\psi_2(u,w,v):\mathbb{S}^2\smallsetminus\{0,0,-1\}\rightarrow\mathbb{C}$$ So this is what I have done, let $(u,v,w)$ be a point on the sphere, and let $(0,0,-1)$ be the ...
Chris's user avatar
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stereographic coordinates of a sum

Let $S=\{(x_1,x_2,x_3):x_1^2+x_2^2+x_3^2=1\}$ be the unit sphere in ${\mathbb R}^3$, and $\phi: {\mathbb C}\rightarrow S$ the stereographic map $$\phi(x+iy)=\frac{1}{x^2+y^2+1}(2x,2y,x^2+y^2-1).$$ ...
Math101's user avatar
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3 votes
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Rotation around the diameter in Riemann sphere

Consider Riemann sphere And consider the following projection : The plane $\zeta =0 $ here is the complex plane, and consider the following map: Each point (except the north pole) of the sphere, ...
FreeZe's user avatar
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Derivation of polar stereographic projection

I have tried to reproduce the transformation rules when one does polar stereographic projection from Mathworld web. Similar equations appear in Snyder (1987). When I do the calculation, I get (for the ...
atapaka's user avatar
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Confusion about Riemann sphere

I'm confused about the following. Consider a circle $C$ on the Riemann sphere with the point at infinity in its interior but not at its center. Let's say $C_1$ is its center, and $C_2$ is the center ...
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The stereographic projection is a conformal transformation.

The stereographic projection $\sigma :\mathbb S^n(R) \setminus\{N\} \to \mathbb R^n$ is a conformal transformation . To prove this theorem, we should show that for all $v\in T_q\mathbb R^n$ ($q\in \...
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Non-Euclidean Geometry: Correspondence of inversion in spherical lines and reflections in great circles under stereographic projection

I am trying to prove the statement: Reflection in a spherical line (i.e. the image of a great circle under stereographic projection) in the extended complex plane corresponds (via stereographic ...
Oybbor99's user avatar
4 votes
1 answer
254 views

Synthetic geometry: stereographic projections of $\mathbb{C}$ on Riemann sphere $\Sigma$ are inversions in sphere $K$ centered on $\infty$ of $\Sigma$

The complete statement is the following: Show that if $K$ is the sphere of radius $\sqrt{2}$ centered at the north pole ($N=\infty$) of the Riemann sphere $\Sigma$ s.t. $K$ intersects $\Sigma$ about ...
shintuku's user avatar
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Vector field on $S^2$

Given the two charts for the stereographic projection from the north and south poles respectively: $$ \phi_N(x,y,z)= \frac{1}{1-z}(u,v)$$ $$ \phi_S(x,y,z)= \frac{1}{1+z}(u,v)$$ and the changes in ...
MicrosoftBruh's user avatar
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1 answer
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Complex Structure on S^6 using Stereographic projection. Why does not work?

Consider $S^6$ and do a Stereographic projection over $\mathbb{R}^6$ (https://en.wikipedia.org/wiki/Stereographic_projection). Give to $\mathbb{R}^6$ the natural complex structure, so it is $\mathbb{C}...
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Is the stereographic projection a homomorphism between the sphere and the plane?

I am taking a course on complex analysis, and we define the stereographic projection. Isn't this an onto and $1-1$ continuous mapping from the sphere to the plane ? Meaning that it exist a ...
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question about stereographic projection

Suppose that $S^1 = \{(x,y)\in \mathbb{R}^2|x^2+y^2=1\}$, and let $x \in S^1$. How can I go about finding a homeomorphism from $S^1 \to S^1$ that sends $x$ to $(0,1)$. I thought about defining the ...
Joey's user avatar
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South Pole and Stereographic Projection

I'm trying to parameterize a sphere through the stereographic projection. This projection fails on the south pole, once I'm not working with $\infty$ as an element of the ring $\mathbb{R}$. How can I ...
hugh_maths's user avatar
1 vote
1 answer
508 views

Stereographic projection: line element

We assign coordinates ($\rho, \phi$) to each point in the surface of a sphere, where $\rho$ is the distance from the south pole of the sphere to the point where a straight line passing through both ...
Jorge Casajus's user avatar
1 vote
0 answers
128 views

Why is the Stereographic Projection not used for Navigation

The mercators map is most often used to map a rhumb line (path of constant bearing) between two points due to its conformity, however, the stereographic projection is also conformal, so why do we use ...
swang's user avatar
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3 votes
2 answers
312 views

Stereographic Projection: Cartography Applications

Compared to the Mercator's, which is also conformal, how does the Stereographic projection help in areas such as navigation? Or any application besides simply mapping polar areas, although I would ...
swang's user avatar
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Stereographic Projection: Does Conformality Imply Circle Preservation?

This may be a silly question. I have learned to prove that circles are preserved at the infinitesimal scale, however, does this ALONE imply that circles are mapped as circles for the stereographic ...
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