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

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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
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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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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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24 views

Inverse of the Stereographic projection $\mathbb{CP}^1 \to S^2$

I've some problems with this exercise: Consider the stereographic projection $$ \varphi \colon S^2 \setminus \{ (0,0,1)\} \to \mathbb{CP}^1\setminus \{[1:0]\}$$ $$ (x,y,z) \mapsto [ x+iy:1-z]$$ and ...
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1answer
34 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
54 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
49 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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136 views

Function of an object that has a shape of circle, square and triangle on 3d projection

What is the function of this kind of object (solid on the bottom right)? I got a lot of material for pondering with keyword cylindrical wedge and hoof, but this is something inverse compared to it. ...
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32 views

Stereographic Projection of sphere through plane $ax+by+cz=d$

I have managed to pullback the equation for the plane to the $uv$-plane, but cannot manipulate it to make it look like a circle in $R^2$. My pullback is as follows: ...
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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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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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20 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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37 views

shortest distance from point to line

Let $L_1$ be the line passing through the point $P_1=(−1, 5, −2)$ with direction vector $\vec{d}=[−3, 2, 3]T$, and let $L_2$ be the line passing through the point $P_2=(2, −5, 5)$ with the same ...
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1answer
48 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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25 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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39 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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419 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
50 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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22 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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59 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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2answers
607 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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34 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?
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1answer
37 views

Looking for reference to a couple of proofs regarding the Stereographic Projection.

I'm looking for a reference to rigorous proofs of the following two claims (if someone is willing to write down a proof that would also be excellent): The Stereographic Projection is a Homeomorphism ...
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1answer
110 views

Ways to project arbitrary Fractals on 2D objects and 3D objects w different dimensions?

I am trying to create a house/texture in 3D and in 2D with fractals, perhaps related. My friend said that fractals can have different dimensions such as 1.74, 1, 4.71111... and pretty much anything. ...
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112 views

Prove that $|PF_{1}|+|PF_{2}|$ is Constant in an Elipse

Given an elipse with two focus $F_{1}$ an $F_{2}$, and $A$ is an arbitrary point at the elipse. Stright line $AF_{1}$ has another intersection point $B$ with the elipse, and $AF_{2}$ has another ...
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2answers
278 views

Equation of a line on a plane…

Hi this question belongs to camera projections but i cannot understand the mathematics... i am not getting how the cross product of two vectors (underlined in red) gives the equation of a ...
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132 views

Diametrically opposite points go to diametrically opposite points under stereographic projection

I asked this question before here but I didn’t get a proper answer. So here I am stating it more clearly : Suppose $P_1$ and $P_2$ are two diametrically opposite points of a circle $C$ in the ...
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0answers
125 views

Change in angle between curves due to stereographic projection

Suppose I have say two curves on the complex plane intersecting at a point $P$. Then is the angle between those curves at $P$ same as the angle between their spherical images on the Riemann Sphere (by ...
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1answer
294 views

Transformation under rotation of Riemann sphere

Suppose the Riemann sphere $S$ is rotated by the angle $\phi$ round the diameter whose end points have $a,-1/\bar{a} $ (which have antipodal preimages) as stereographic projections. Suppose moreover, ...
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2answers
595 views

Radius of the inverse image of a circle under stereographic projection

I need to find the radius of the circle on the Riemann sphere $S$ whose stereographic projection is $C(a;r)$, i.e. the circle with centre $a$ and radius $r$ in the complex plane. I have observed ...