Stereographic projection

Mar 2017
40
1
chile
Hl! I have problems with this exercises

Let \(\displaystyle \varepsilon\) the Stereographic projection of \(\displaystyle \mathbb{S}^1- \{i \}\), that is


\(\displaystyle \varepsilon : \mathbb{S}^1 - \{i \} \longrightarrow{\mathbb{R}}\)


where


\(\displaystyle \varepsilon(z)=\displaystyle\frac{RE(z)}{1-Im(z)}\)


Find a formula for \(\displaystyle \varepsilon^{-1} :\mathbb{R}\rightarrow{\mathbb{S}^1}- \{i \} \)

I tried to take the line that passes through i and a point on the real axis to see where the circle intersects but I can not find the formula


Thanks
 
Jun 2013
1,144
612
Lebanon
The stereographic projection maps

\(\displaystyle i \to \infty \)

\(\displaystyle -i\to 0\)

\(\displaystyle 1\to 1\)

Therefore its inverse sends

\(\displaystyle \infty \to i\)

\(\displaystyle 0 \to -i\)

\(\displaystyle 1 \to 1\)

it's easy to write down the corresponding Moebius transformation
 
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Mar 2017
40
1
chile
The stereographic projection maps

\(\displaystyle i \to \infty \)

\(\displaystyle -i\to 0\)

\(\displaystyle 1\to 1\)

Therefore its inverse sends

\(\displaystyle \infty \to i\)

\(\displaystyle 0 \to -i\)

\(\displaystyle 1 \to 1\)

it's easy to write down the corresponding Moebius transformation
You can make the first correspondence , please
 
Jun 2013
1,144
612
Lebanon
\(\displaystyle \varepsilon ^{-1}(u)=\frac{i u+1}{u+i}\)

\(\displaystyle u\in \mathbb{R}\)
 
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Mar 2017
40
1
chile
\(\displaystyle \varepsilon ^{-1}(u)=\frac{i u+1}{u+i}\)

\(\displaystyle u\in \mathbb{R}\)
Thanks

but, why the domain is \(\displaystyle \mathbb{R} \)and the codomain is\(\displaystyle \mathbb{S}^1- \{i \}\) ?
 
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