Questions tagged [stereographic-projections]

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

Filter by
Sorted by
Tagged with
0
votes
0answers
21 views

How to actually draw an epipolar line on a $2D$ image without OpenCV

Suppose I have the Fundamental Matrix $F$, a point correspondence $x$ $(1,2)$ and an image $(\text{height},\text{weight},3)$. The epipolar line is now given by $l'= Fx$ (with $x$ being converted to ...
1
vote
0answers
32 views

Show that any mapping from the extended complex plane to the Riemann Sphere back to the extended complex plane is a Möbius Transformation

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 ...
3
votes
1answer
49 views

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 ...
1
vote
1answer
45 views

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}{...
0
votes
0answers
10 views

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 ...
0
votes
0answers
77 views

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 ...
0
votes
0answers
33 views

Stereographic projection - problem deriving a formula for a metric in the projected space and strategies to deal with unwieldy terms

This question isn't so much about the stereographic projection itself than it is about deriving a metric of the image under this projection, but there is no tag for 'term manipulation' ... The example ...
1
vote
0answers
67 views

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 ...
1
vote
2answers
67 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 ...
1
vote
1answer
187 views

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 ...
1
vote
0answers
60 views

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 $$\phi(x+iy)=\frac{1}{x^2+y^2+1}(2x,2y,x^2+y^2-1).$$ ...
3
votes
1answer
82 views

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, ...
0
votes
0answers
16 views

why stereographic projection of the circle passing through -1 is not the i axis

https://www.youtube.com/watch?v=d4EgbgTm0Bg&t=677s time stamp 15:32 "so most points on the actual i axis, like 0, 2i, and 3i,are completely invisible to Felix." I can't really imagine 2i,...
1
vote
0answers
25 views

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 ...
0
votes
0answers
39 views

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 ...
2
votes
0answers
68 views

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 \...
0
votes
0answers
24 views

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 ...
4
votes
1answer
134 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 ...
1
vote
0answers
51 views

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 ...
1
vote
1answer
65 views

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}...
0
votes
1answer
97 views

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 ...
0
votes
1answer
41 views

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 ...
1
vote
1answer
76 views

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 ...
1
vote
1answer
122 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 ...
1
vote
0answers
58 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 ...
3
votes
2answers
85 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 ...
2
votes
1answer
64 views

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 ...
5
votes
0answers
57 views

Find geometric derivation of $\rho(a,b)=\frac{2|a-b|}{\sqrt{1+|a|^2}\sqrt{1+|b^2|}}$ for stereographic projection.

When the complex plane is projected to the spherical surface, we can brute force the formula for the the distance between the two image points $a,b$ on the sphere $d(a,b)=\frac{2|a-b|}{\sqrt{1+|a|^2}\...
1
vote
2answers
421 views

How to prove that the stereographic projection of the $S^2$ is conformal

I read on Wikipedia that the stereographic projection of the $S^2$ sphere is conformal. However, there was no proof provided. How can one show it?
1
vote
1answer
75 views

Stereographic projection and complex polynomial. (Do Carmo 2-3-16)

$\newcommand{\set}[1]{\left\lbrace #1 \right\rbrace} \newcommand{\suchthat}{\;\big\vert\;}$ Let $ \mathbb{R}^{2} = \set{(x,y,z)\in\mathbb{R}^{3} \suchthat z=-1} $ be identified with the complex plane ...
2
votes
1answer
63 views

Inverse of Multivariable Functions on Manifolds

Consider the unit circle, $S=${$(x,y) | x^2 + y^2 =1$} with the stereographic charts $(U_N,\phi_N)$, $(U_S,\phi_S)$ i.e. $U_N=S$ \ {$(0,1)$}, $\phi_N((x,y))=\frac{x}{1-y}$ $U_S=S$ \ {$(0,-1)$}, $\...
1
vote
1answer
267 views

The stereographic projection is bijective

I want to prove that the stereographic projection:$f:\{\ (x,y):x^{2}+y^{2} = 1 \}\ \rightarrow \mathbb{R}$ where $f$ is defined as: $$ f(x,y) = \frac{x}{1-y}$$ is bijective to the real number. I have ...
2
votes
1answer
51 views

Showing that $D^2\setminus S^1$ is diffeomorphic to $\mathbb{R}^2$

I'm studying some manifolds and I'm stuck at the exercise 1.5c of Tu's Introduction To Manifolds. There he asks us to verify that the function $h \colon D^2 \setminus S^1 \to \mathbb{R}^2$ given by $$...
1
vote
1answer
70 views

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 ...
1
vote
0answers
437 views

Stereographic Projection and Diffeomorphism

I am recently studying about stereographic projection and it is stated that the punctured 2-sphere $S^2\backslash \left\{0,0,1\right\}$ is diffeomorphic to $\mathbb{R}^2$. I want to prove this ...
2
votes
1answer
134 views

Bijective projection from a unit disk to a unit sphere

Is there any proper method that makes conformal one-to-one mapping (preserving the angles) from a unit disk to a unit sphere? I know with simple stereographic projection you can project a unit disk to ...
1
vote
0answers
37 views

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,...
0
votes
0answers
20 views

Transform the problem of finding the smallest enclosing circle on the surface of sphere to smallest circle in carthesian space

I have points on the surface of a sphere and I need to find the cutting-plane that cuts off all points and also cuts off the minimum possible volume to do so. There is the https://people.inf.ethz.ch/...
2
votes
2answers
111 views

Formula for the intersection of a sphere with regards to stereographic projection

So I have this question for an assignment and am just completely lost. Let $S^n$ be the unit sphere with centre at $0$ in the space $R^{n+1}$. Let $N=(0,...,0,1)$ in such a space. Define the ...
0
votes
1answer
147 views

One point compactification of $\mathbb{R}^n$ proof

I was recently reading about one-point compactification, and a proof of how the one-point compactification of $\mathbb{R}^n$ is homeomorphic to $\mathbb{S}^n$.The proof is example 4.1 at https://...
1
vote
1answer
206 views

Stereographic projection of a disc.

Stereographic projection maps circles of the unit sphere, which do not contain the north pole, to circles in the complex plane. My question is what happens to the center of the circle in the complex ...
-1
votes
2answers
173 views

Inverse of stereographic projection

I am trying to compute the inverse of $$f(\theta,\phi) = \left(\frac{\cos\theta \sin\phi}{1-\cos\phi}, \frac{\sin\theta \sin\phi}{1-\cos\phi}\right)$$ but I lack some basic knowledge on how I can do ...
0
votes
1answer
293 views

Stereographic Projection: Proof that circles from the plane goes to circles in the sphere

I'm trying to prove algebraically that for the stereographic projection $p:S^2\to \mathbb{R}^2\cup\{\infty\}$ $$ p(x,y,z)=\left( \dfrac{x}{1-z},\dfrac{y}{1-z}\right) $$ $$ p^{-1}(a,b)=\left( \dfrac{...
2
votes
0answers
28 views

Understanding the distance-from-the-origin formula in the Poincaré disk

The Poincare disk is $\{ z \in \mathbb{C} : |z|<1 \}$ and the hyperbolic distance from the origin is given by $\rho = 2 \tanh^{-1}(r)$ where $r$ is the Euclidean distance between the point and the ...
-1
votes
1answer
77 views

Plotting a stereographic projection [duplicate]

I need to sketch the image under the stereographical projection of sphere where the spherical cap $A < X < 1,$ with center lying on the equator, for fixed $ A. $ Separating the cases according ...
2
votes
1answer
58 views

Find the rate at which distances decrease in stereographic projection

I want to map a 3D space onto the inside's surface of a sphere. The 3D space is represented by points (x,y,z) where the z axis is the height. The first thing I did was to use the following equation ...
0
votes
2answers
41 views

Help to understand the claim with the stereographic projection

I don't understand the following claim: We consider $\Bbb S^2$ as a subspace of $\Bbb R^3$. We have the stereographic projection $\rho: \Bbb S^2 \setminus \{(0,0,1)\} \to \Bbb R^2$, $\rho (x,y,z)=\...
-1
votes
1answer
170 views

What maps to a plane in a stereographic projection of a 3-sphere to $\mathbb{R}^3$?

Any straight line in a stereographic projecton from a 2-sphere to $\mathbb{R}^2$ maps back to a circle passing through the projection point. That I can see why: the rays between the projection point ...
1
vote
1answer
104 views

Projecting a point from a hyperplane onto a sphere

Assuming I have a point $P=(a_0,...,a_{n})\in\mathbb{R}^n$, and a vector $\vec{v}=(v_0,...,v_{n+1})\in\vec{\mathbb{R}}^{n+1}$. I would like to project the point $P$ onto $S^n$ by doing an inverse ...
0
votes
0answers
26 views

Any circle in sphere is intersection of the sphere and a unique plane.

I would like to prove the existence and uniqueness of such a plane $P$, the truth is I can't think of a relatively simple way to prove it. Let $\Lambda$ be a circle lying in $S$. Then there is a ...