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Questions tagged [stereographic-projections]

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

In geometry, the stereographic projection is a particular mapping that projects a sphere onto a plane. The projection is defined on the entire sphere, except at one point: the projection point. Where it is defined, the mapping is smooth and bijective. It is conformal, meaning that it preserves angles. It is neither isometric nor area-preserving: that is, it preserves neither distances nor the areas of figures.

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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 $$\phi(x+iy)=\frac{1}{x^2+y^2+1}(2x,2y,x^2+y^2-1).$$ Then if $z=x+iy, z'=x'+iy'$,…
Math101
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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 $T(f(x))=g(x)f(x)$.
saeid sahragard
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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. $$\varphi(x,y) = \left(…
ZahaMan
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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?
Majlik
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How to project a portion of a sphere to a plane preserving distance?

Stereographic projection preserves angles but not distances. I've been lead to believe that it isn't possible to go from a sphere to a plane preserving distances between points, but is it possible to do so with a segment of a sphere? For context,…
Kieran Edraney
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Construct the Stereographic projection formulation

I want to know how to construct the Stereographic projection formulation. I search google and can't find any good resource. Even Wikipedia just give the formula below: In Cartesian coordinates $(x, y, z)$ on the sphere and $(X, Y)$ on the plane,…
Dr.Antidode
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Formula for Stereographic Projection of Sphere and its Inverse

There must be some error in my formulas for the stereographic projection of the sphere and the inverse projection. However, I can't find the error. Here's what I have: Let $S_K^n$ be the sphere with sectional curvature $K$, then the stereographic…
ndrizza
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Formula for Stereographic Projection of Ball of Radius $R$

Let the following be the set of points of radius $R$: $$ B^n_R:=\{x\in\mathbb{R}^{n+1}\,\,| \,\,||x||_2^2=R^2\} $$ What is the formula to stereographically project $B_R^n$ to the $n$-dimensional hyperplane $E^n$? $$ E^n:=\{x\in\mathbb{R}^{n+1}\,\,|…
ndrizza
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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 object (plane?). In his examples he placed the…
Dando18
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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 easily proved that $AB \cos \theta$ is the length of…
marmistrz
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