junctions.tex finished and images added

report/subsections/junctions.tex

@@ -5,7 +5,7 @@ A junction is a point of a skeleton where more than 2 branches join. The process
 
 \begin{figure}[h!]
    \begin{center}
-   \includegraphics[scale=0.5]{img/Junctions3}
+   \includegraphics[scale=0.5]{img/Junctions9}
    \caption{\label{junction}Junction of three splines}
    \end{center}
 \end{figure}
@@ -19,14 +19,45 @@ In the figure \ref{junction1}  you can see how must be the theoretical case that
    \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.
+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 consecutive circles and to join them and join every point of the circles in one point upside and one downside.
+
+We begin by identify the last circles of each spline of the junction at the level of it (see figure \ref{junction3}).
+
+\begin{figure}[h!]
+   \begin{center}
+   \includegraphics[scale=0.5]{img/Junctions3}
+   \caption{\label{junction3}A 3-splines junction}
+   \end{center}
+\end{figure}
 
 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}
+   \includegraphics[scale=0.35]{img/Junctions1}
+   \includegraphics[scale=0.35]{img/Junctions2}
+   \caption{\label{junction2}Best fiting plane to the set of the circles's center points}
    \end{center}
 \end{figure}
 
+Then when points are sorted we connect the up-points of each circle with the up-projection of the sphere's center on itself, idem for the down-points with the down-projection of the sphere's center.
+The result of this process is presented in the figure \ref{junction4}.
+
+\begin{figure}[h!]
+   \begin{center}
+   \includegraphics[scale=0.5]{img/Junctions4}
+   \includegraphics[scale=0.5]{img/Junctions5}
+   \caption{\label{junction4}The edges added and the mesh result}
+   \end{center}
+\end{figure}
+
+To see easily the result of the mesh on a junction, we have made some tests with only the extreme circles that we use. On real skeletons the junction are not easy to watch.
+Here \ref{junction5} is two examples of a junction's mesh for a 3 and a 4-branches junction.
+
+\begin{figure}[h!]
+   \begin{center}
+   \includegraphics[scale=0.5]{img/Junctions10}
+   \includegraphics[scale=0.4]{img/Junctions11}
+   \caption{\label{junction5}Mesh result on a 3 and a 4-branches junction}
+   \end{center}
+\end{figure}