paella/report/subsections/junctions.tex

\subsection{Junctions}

The last step to complete is to mesh the junctions.
A junction is a point of a skeleton where more than 2 branches join. The process that consists in meshing this portion of the skeleton is complex. In fact it needs to be applicable on multiple cases, for instance 3 or 4 branches (see Figure \ref{junction}, and to take into account that the perfect case will not always be there.

\begin{figure}[h!]
   \begin{center}
   \includegraphics[scale=0.5]{img/Junctions3}
   \caption{\label{junction}Junction of three splines}
   \end{center}
\end{figure}

In the figure \ref{junction1}  you can see how must be the theoretical case that we should have with a perfect skeleton. You can see the sphere shared with the three splines that join in this junction, and the three characteristic circles associated. Those circles by pairs join together in one point. 

\begin{figure}[h!]
   \begin{center}
   \includegraphics[scale=0.5]{img/JunctionTheory}
   \caption{\label{junction1}Theory case with perfect characteristic circles}
   \end{center}
\end{figure}

But in most of the practice case it is not like this. This is due to number's approximation on computers. The idea is then to look for the closest points of two consecutives circles and to join them and join every point of the circles in one point upside and one downside.

To be able to do it the first step is to cut the circles in two parts to find the upside and downside of each of them. This can be done by computing the best fitting plane to the set of the circles's center points (figure \ref{junction2}).

\begin{figure}[h!]
   \begin{center}
   \includegraphics[scale=0.5]{img/Junctions2}
   \caption{\label{junction2}Plane that cut the circles in two}
   \end{center}
\end{figure}