% A graphical depiction of the central projection between the upper
% semi-sphere and the Euclidean plane: we map each point in the upper half of
% the sphere to the projection of this point in the tangent plane at the north
% pole by drawing a line between this point and the center of the sphere and
% then taking the intersection of this line with the plane.
% (C) Thiago Brevidelli
\begin{tikzpicture}[scale=0.8]
    % The outline of the semi-sphere
    \begin{scope}
      \clip (-3, 0) rectangle (3, 3);
      \draw (0, 0) circle (3);
    \end{scope}

    % The equator
    \begin{scope}
        \clip (-3, 0) rectangle (3, -1);
        \draw ellipse (3 and 1);
    \end{scope}

    % The equator (on the other side of the sphere)
    \draw[dotted] ellipse (3 and 1);

    % The plane
    \draw[dotted] (-4.5, 1.8) --
                  ( 1.5, 1.8) --
                  ( 4.5, 4.8) --
                  (-1.5, 4.8) -- cycle;

    % The center of the sphere
    \filldraw (0, 0) circle (2pt);

    % A line trhough the center of the sphere
    \draw[dotted] (0, 0) -- (1.05, 2.31);
    \draw (1.05, 2.31) -- (1.5, 3.3);

    % The intersection of the line with the plane
    \filldraw (1.5, 3.3) circle (2pt);

    % The intersection of the line and the sphere
    \filldraw (1.05, 2.31) circle (2pt);
\end{tikzpicture}