\subsection{Extremities}

After computing the mesh around the splines we need to draw the extremities using subdivisions to obtain smoother extremities.
The first step is to calculate the projection of the extremity circle on the correspondent sphere. Then for a subdivision of depth 0 we link this projection with the mesh points along the extrem circle (see Figure \ref{projection}). For one subdivision we take the middle of each segments computed earlier and we project it on the sphere (see Figure \ref{sub}). Finally we made a triangular mesh the way you can see on Figure \ref{mesh}).

The rendering of our extremities on dino.skl for example can be seen on Figures \ref{extremity2} (2 subdivisions) and \ref{extremity10}
 (10 subdivisions).

\begin{figure}[H]
   \begin{center}
   \includegraphics[scale=0.2]{img/projection}
   \caption{\label{projection}Projection of the extrem circle center on the sphere}
   \end{center}
\end{figure}

\begin{figure}[H]
   \begin{center}
   \includegraphics[scale=0.2]{img/Subdivision}
   \caption{\label{sub}One subdivision}
   \end{center}
\end{figure}

\begin{figure}[H]
   \begin{center}
   \includegraphics[scale=0.2]{img/meshExtremity}
   \caption{\label{mesh}The final extremity mesh for one subdivision}
   \end{center}
\end{figure}

\begin{figure}[H]
\begin{minipage}[b]{0.45\linewidth}
\centering
\includegraphics[scale=0.5]{img/2}
\caption{\label{extremity2}The final extremity mesh for 2 subdivisions}
\end{minipage}
\hspace{0.5cm}
\begin{minipage}[b]{0.45\linewidth}
\centering
\includegraphics[scale=0.5]{img/10}
\caption{\label{extremity10}The final extremity mesh for 10 subdivisions}
\end{minipage}
\end{figure}
