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main.tex
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main.tex

@ 110,7 +110,7 @@ After the simulation the properties of the SPH particles needs to be analyzed. F




One of the easiest ways to interpolate a new value between two known values is linear interpolation. \todo{some more text about linear interpolation}




In one dimension linear interpolation is pretty trivial. For example if we assume that we have 20 random points $P$ between 0 and 1 (red and blue in \ref{fig:onediminterpolation}) and have a new point $I$ at $0.4$ for which we want to interpolate (green). Finding the two closest points above and below is trivial as there is only one dimension to compare. Now if for every of these points we have measured a value $f(P)$ a straight line between the two closest values can be drawn (light green) and a interpolated value for $f(I)$ can be found.


In one dimension linear interpolation is pretty trivial. For example if we assume that we have 20 random points $P$ between 0 and 1 (\textcolor{Red}{\textbullet} and \textcolor{Blue}{\textbullet} in \ref{fig:onediminterpolation}) and have a new point $I$ \textcolor{Green}{\textbullet} at $0.4$ for which we want to interpolate . Finding the two closest points \textcolor{Red}{\textbullet} above and below is trivial as there is only one dimension to compare. Now if for every of these points we have measured a value $f(P)$, a straight line \textcolor{LightGreen}{} between the two closest values can be drawn and a interpolated value for $f(I)$ can be found.




\begin{figure}[h] % TODO: h is temporary


\centering



@ 119,11 +119,11 @@ In one dimension linear interpolation is pretty trivial. For example if we assum


\label{fig:onediminterpolation}


\end{figure}




In two dimensions things get more complicated as we now have a set of points with $X$ and $Y$ coordinates (Figure \ref{fig:3dinterpolate1}). One fast way to find the closest points to the point that should be interpolated is using Delaunay triangulation. This separates the space between the points into triangles while trying to maximize their smallest angle. Afterwards the closest three points can be found very quickly by checking the nodes of the surrounding triangle (Figure \ref{fig:3dinterpolate2}).


In two dimensions things get more complicated as we now have a set of points with $X$ and $Y$ coordinates (Figure \ref{fig:3dinterpolate1}). One fast way to find the closest points to the point that should be interpolated is using Delaunay triangulation. This separates the space between the points into triangles while trying to maximize their smallest angle. Afterwards the closest three points can be found very quickly by checking the nodes of the surrounding triangle (Figure \ref{fig:3dinterpolate2}). If we now again have a function $f(X,Y)$ similar to the onedimensional example (Figure \ref{fig:3dinterpolate3}), we can create an unique plain\todo{right word?} through the three points and get the interpolated value for any $X$ and $Y$ on this layer.




\todo[inline]{It might be a better idea (and maybe more correct) to add the green point to the Delaunay list and use it's neighbors as the nearest points instead of the edges of the surrounding triangle.}




\begin{figure*}[h] % also temporary


\begin{figure}[h] % also temporary


\centering


\begin{subfigure}[t]{0.5\textwidth}


\centering



@ 140,7 +140,30 @@ In two dimensions things get more complicated as we now have a set of points wit


\end{subfigure}


\caption{Caption place holder}




\end{figure*}


\end{figure}




\begin{figure}[h] % TODO: h is temporary


\centering


\includegraphics[width=0.8\linewidth]{images/vis2d3.pdf}


\caption{twodimensional example of linear interpolation}


\label{fig:3dinterpolate3}


\end{figure}




This approach has the advantage that it can be extended in more than two dimensions by replacing the triangle in the Delaunay triangulation with a nsimplex in n dimensions. The \texttt{scipy.spatial.Delaunay} python function allows to quickly calculate it thanks to the \texttt{Qhull} library\footnote{\url{http://www.qhull.org/}}




\subsection{Implementation}




\subsection{Results}




\section{Neural Networks}




\subsection{Theory}




\subsection{Implementation}




\subsection{Results}




\subsection{Issues}




\appendix


\chapter{Placeholder}





@ 34,6 +34,7 @@ british, % language of the document




\usepackage{aas_macros} % this imports the aas_macros.sty file that is required to print bibtex exported from ADS




\usepackage[svgnames]{xcolor} % allows defining colors




% Helpful Features 





@ 69,7 +70,7 @@ british, % language of the document


% how the header of the pages looks like


% for something nicer take a look at fancyhdr




\usepackage{xcolor} % allows defining colors


% xcolor is already defined above




% not sure what these two do, but they definitly do something


\colorlet{bluebookmarks}{blue}




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