doc/klein-to-poincare.tkz.tex

\begin{figure}[!hbt]
  \centering
  \begin{tikzpicture}[scale=1.85]
    \coordinate (O) at (0, 0) {};
    
    % Klein model points
    \coordinate (P) at (-1.42, .04) {};
    \coordinate (Q) at (-0.933,-0.48) {};

    % Projections onto sphere
    \coordinate (Ps) at (-1.42,-.85) {};
    \coordinate (Qs) at (-0.93,-1.3) {};

    % Sphere poles
    \coordinate [label=above:$N$] (N) at (0, 2) {};
    \coordinate (S) at (0, -2) {};

    % Disk control points
    \coordinate [] (L) at ($ (O)-(2,0) $) {};
    \coordinate [] (R) at ($ (O)+(2,0) $) {};

    % Projection lines
    \draw (P)--(Ps);
    \coordinate (Pp)
                at (intersection of Ps--N and O--P);                
    \draw (Ps)--(Pp);
    \draw (Q)--(Qs);
    \coordinate (Qp)
                at (intersection of Qs--N and O--Q);                
    \draw (Qs)--(Qp);

 
    % Fade lower hemisphere                 
    \path [fill=white, opacity=.5]
                (L) to[out=270,in=180] (S)
                    to[out=0,in=270] (R)
                    (O) ellipse(2 and 1);

     % Klein disk
    \draw [opacity=.3, pattern=north east lines]
               (O) ellipse(2 and 1);
    \draw [thick] (O) ellipse(2 and 1);

    % Chord
    \coordinate [dot] (X) at (-1.8,0.4358) {};
    \coordinate [dot] (Y) at (-0.464,-0.97) {};
    \draw[] (X)--(Y);
    \draw[] (X) to[out=-15,in=135,looseness=.9] (Pp)
                to [out=315,in=120,looseness=.7] (Qp)
                to[out=300,in=65,looseness=.8] (Y);
                    
    % Visible projection lines
    \draw (Pp)--(N);
    \draw (P)--(O);
    \draw (Qp)--(N);
    \draw (Q)--(O);

    % Sphere
    \draw[thick] (O) circle(2);

    % labels placed last
    \begin{scope}[inner sep=2pt]
      \node [label=below:$X$] at (X){};
      \node [label=below:$Y$] at (Y){};
      \node [label=right:$O$] at (O){};
      \node [label=left:$P$] at (P){};
      \node [label=left:$Q$] at (Q){};
      \node [label=above:$P^\prime$] at (Pp){};
      \node [inner sep=4pt,label=right:$Q^\prime$] at (Qp){};
      \node [label=below:$P_s$] at (Ps){};
      \node [label=below:$Q_s$] at (Qs){};
    \end{scope}
    
    % dot marks
    \foreach \p in {O,P,Pp,Q,Qp,N} \draw node[dot] at (\p) {};
    \node[dot,color=gray] at (Ps) {};
    \node[dot,color=gray] at (Qs) {};
  \end{tikzpicture}
  \caption[Связь моделей Клейна и Пуанкаре]{Связь моделей Клейна и
    Пуанкаре: $\xi(P) = P^\prime,\: \xi(Q)=Q^\prime$}
  \label{fig:klein-to-poincare}
\end{figure}