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{\bf Question}

Write down an explicit formula for the stereographic projection
map from the Euclidean circle in ${\bf C}$ with center $3+2i$ and
radius $3$ to the horizontal Euclidean line through $3 + 5i$
(union $\{\infty\}$).

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{\bf Answer}

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(remember to project from the point on the circle $C$ opposite
from the point of tangency of $C$ and $L$).

equation of line through $N,\ Z$:

\begin{itemize}
\item
$m=\ds\frac{-1-\rm{Im}(z)}{3-\rm{Re}(z)}$

\item
equation:

$\begin{array} {ccl} y+1 & = & m(x-3)\\ y+1 & = &
\ds\frac{-1-\rm{Im}(z)}{{3-\rm Re}(z)}(x-3) \end{array}$

\item
Set $y=5$ (to get the intersection with $L$) and solve for $x$:

$-6 \cdot \ds\frac{{3-\rm Re}(z)}{1+\rm{Im}(z)}=x-3$

$x=-6 \cdot \ds\frac{{3-\rm Re}(z)}{1+\rm{Im}(z)}+3$.
\end{itemize}
So $\xi(z)=\left\{\begin{array} {lc} -6 \cdot \ds\frac{{3-\rm
Re}(z)}{1+\rm{Im}(z)}+3+5i & z \ne N\\ \infty & z=N
\end{array}\right.$
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