Kleins Model
Tangent in Point $(R,\theta)$:
\[
\begin{pmatrix}
x-R\cos{\theta}\\
y-R\sin{\theta}\\
\end{pmatrix}
\cdot
\begin{pmatrix}
\cos{\theta}\\
\sin{\theta}\\
\end{pmatrix}
=0
\quad \Leftrightarrow
\]
\[
\cos{\theta}x+\sin{\theta}y=R
\]
Tangents in $P \sim (R,\theta_P)$, resp. $Q \sim (R,\theta_Q)$:
\[
\begin{Bmatrix}
\cos{\theta_p}x+\sin{\theta_p}y&=&R\\
\cos{\theta_q}x+\sin{\theta_q}y&=&R\\
\end{Bmatrix}
\]
Cramer:
\[
\begin{pmatrix}
x\\y
\end{pmatrix}
=
\frac{1}
{
\begin{vmatrix}
\cos{\theta_p}&\sin{\theta_p}\\
\cos{\theta_q}&\sin{\theta_q}\\
\end{vmatrix}
}
\begin{pmatrix}
\begin{vmatrix}
R&\sin{\theta_p}\\
R&\sin{\theta_q}\\
\end{vmatrix}
\\\\
\begin{vmatrix}
\cos{\theta_p}&R\\
\cos{\theta_q}&R\\
\end{vmatrix}
\end{pmatrix}
=
\]
\[
\frac{R}
{
\cos{\theta_p}\sin{\theta_q}-\cos{\theta_q}\sin{\theta_p}
}
\begin{pmatrix}
\sin{\theta_q}-\sin{\theta_p}
\\
\cos{\theta_p}-\cos{\theta_q}
\end{pmatrix}
=
\]
\[
\frac{R}
{
\sin{(\theta_q-\theta_p)}
}
\begin{pmatrix}
\sin{\theta_q}-\sin{\theta_p}
\\
\cos{\theta_p}-\cos{\theta_q}
\end{pmatrix}
\]
@Code Fig.tikz.tex height=300px
@Image Fig.svg 300px 300px