# Questions tagged [stereographic-projections]

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

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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 ...
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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,...
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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 ...
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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,...
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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 ...
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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 ...
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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 ...
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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 $$\phi(x+iy)=\frac{1}{x^2+y^2+1}(2x,2y,x^2+y^2-1).$$ ...
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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, ...
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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 ...
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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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### 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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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 ...
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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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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