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+\chapter{Isosurfaces}

+

+This assignment was a really interesting one. It introduces a method to implement so called isolines. Such an isoline creates a surface that encapsulates a region

+with a value higher than a given threshold. \\

+

+\section{Description}

+

+An isoline is a line that sort to speak follows a given value. If the values of a field ranges from 0 to 1, a good threshold would be 0.6 for instance. The isoline

+visualizes the points that equal the value of 0.6. \\

+

+Our program is able to define the number of contour-lines and the minimum and maximum values in between the isolines are rendered. \\

+

+\section{Implementation}

+

+The algorithm that implements the isolines follows a structured pattern. In pseudo-code it looks like this: \\

+

+\begin{tabbing}

+  for \= (each cell $ c_i $ of the dataset) \\

+  \{ \\

+  \> for \= (each vertex $ v_j $ of $ c_i $) \\

+  \> \> store inside/outside state of $ v_j $ in bit $ b $ of $ status $; \\

+  \> select the optimized code from the case table using $ status $; \\

+  \> for \= (all cell edges $ e_j $ of the selected case) \\

+  \> \> intersect $ e_j $ with the isovalue; \\

+  \> construct the line segment(s); \\

+  \} \\

+\end{tabbing}

+

+So the algorithm passes through every cell and then checks the four cell vertices \{$ v_0 $, $ v_1 $, $ v_2 $, $ v_3 $\} of that cell. Each vertex has it's own

+value. With that value, the algorithm can check if a vertex is inside ($ v_j \geq threshold $) or outside ($ v_j < threshold $) the isosurface. The inside/outside

+state is then stored in a bit. \\

+

+If a vertex, say $ v_0 $ is inside the isosurface, $ v_0 $ is set to 1, else it's left to 0. This is done for all four vertices which results in a 4-bit status.

+This means there are in total 16 different cases in which the isoline can run through a cell. \\

+

+\begin {center}

+  \includegraphics[width=\textwidth]{marching.png} \\

+  Figure 5: The 16 marching square cases \\

+\end {center}

+

+The above image (figure 5) shows the 16 different cases in the marching squares algorithm. A white vertex indicates the vertex is outside the isosurface and black

+indicates the vertex is inside the isosurface. Every inside or outside case has it's counterpart. So we reduces the number of cases down to 8. In case 0 and 15 for

+instance, no lines have to be rendered, yet they are both very different cases. \\

+

+The cases 5 and 10 are both ambiguous cases as becomes clear from the next image (figure 6). \\

+

+\begin {center}

+  \includegraphics[width=100mm]{ambiguous.png} \\

+  Figure 6: Two ambiguous cases in the marching squares algorithm \\

+\end {center}

+

+\section{Difficulties}

+

+The contouring algorithm is very simple to implement. Just follow the instructions of the method. To only trouble we had with this implementation was the

+interpolation of the intersection. \\