% This drawing represents the correspondance between conjugation by pure
% unitary quaternions and rotations in the 3-dimensional euclidian space: the
% coordinates of a unitary quaternion p number with zero real cofficient induce
% a line through the origin in the 3-dimensional enclidian space, and
% conjugation by cos t + p sin t acts as rotation by 2 t around this axis.
% (C) Thiago Brevidelli
\begin{tikzpicture}
  % The rotation axis
  \begin{scope}[rotate=-60]
    % The axis
    \draw[->] (0, 0) -- (0, 2.5) node[right]{$p$};

    % The rotation
    \draw[->] (0.43, 2) node[right]{$2 \theta$} arc (30:360:0.5 and 0.25);

    % The origin and the intersection of ratation axis with the unit sphere
    \filldraw (0, 0) circle (1pt) (0, 1) circle (1pt);
  \end{scope}

  % The circunference
  \draw (0, 0) circle (1.5);

  % The equator
  \begin{scope}
    \clip (-1.5, 0) rectangle (1.5, -1.5);
    \draw ellipse (1.5 and 0.5);
  \end{scope}

  % The equator (on the other side of the sphere)
  \begin{scope}
    \clip (-1.5, 0) rectangle (1.5, 1.5);
    \draw[dotted] ellipse (1.5 and 0.5);
  \end{scope}

  % Greenwhich
  \begin{scope}
    \clip (-1.5, -1.5) rectangle (0, 1.5);
    \draw ellipse (0.5 and 1.5);
  \end{scope}

  % Greenwhich (on the other side of the sphere)
  \begin{scope}
    \clip (1.5, -1.5) rectangle (0, 1.5);
    \draw[dotted] ellipse (0.5 and 1.5);
  \end{scope}
\end{tikzpicture}
