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Casper Steinmann added results.tex
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\section{Results}
Figure~\ref{fig:sysred} shows the reduction in size of the system of linear equations when solving for the unknown charges, $\mathbf{q}$, from eq~\ref{eqn:linear} as the scale of the van der Waal radius is increased. We observe for both the PCM and MK surfaces, that the system size is reduced for increasing van der Waal radii. This is expected since the increasing radius will make the final surface appear more and more uniform and thus all atoms will contribute equally in the construction of $\mathbf{A}$ (see also eq~\ref{eqn:amatrix}). For a very low scaling of the vdw-radius, the PCM-240 surface and the MK-4 surface sees little to no reduction in the size of the system of linear equations (77 and 75, respectively). As the scaling is increased, the system size of the PCM surface drops off notably faster than the MK surface. This, we can explain by the fact that even though the PCM surface has more points which is very good for reproducing a sphere and yields an appropriate surface at low scaling factors, but at larger scaling factors, the whole surface will more quickly approach a uniform sphere when standing in any atom with resulting linear dependencies as a consequence. Contrary, the MK surface is quite orientation dependent and is much less uniform.
