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+% \subsection{Rotationsmatrizen}
+%
+% \begin{tabular}{ l | l }
+% I & II \\ \hline
+% IV & III \\
+% \end{tabular}
+%
+% \flushleft
+% \hfill
+% \begin{tabular}{l | l l}
+%   Clockwise & & \\ \hline
+%   I & + & + \\
+%   II & + & - \\
+%   III & - & - \\
+%   IV & - & +
+% \end{tabular}
+% \hfill
+% \begin{tabular}{l | l l}
+% Anticlockwise & & \\ \hline
+% I & - & - \\
+% II & - & + \\
+% III & + & + \\
+% IV & + & -
+% \end{tabular}
+% \hfill
+%
+% \subsection{Die Matrizen}
+%
+% \subsubsection{Rotation um die x-Achse}
+%
+% \begin{equation}
+%   R_x (\alpha) =
+%    \begin{pmatrix}
+%       1 & 0 & 0 \\
+%       0 & cos(\alpha) & -sin(\alpha) \\
+%       0 & sin(\alpha) & cos(\alpha)
+%     \end{pmatrix}
+% \end{equation}
+%
+% \subsubsection{Rotation um die y-Achse}
+%
+% \begin{equation}
+%   R_y (\alpha) =
+%    \begin{pmatrix}
+%       cos(\alpha) & 0 & sin(\alpha) \\
+%       0 & 1 & 0 \\
+%       -sin(\alpha) & 0 & cos(\alpha)
+%     \end{pmatrix}
+% \end{equation}
+%
+% \subsubsection{Rotation um die z-Achse}
+%
+% \begin{equation}
+%   R_z (\alpha) =
+%    \begin{pmatrix}
+%       cos(\alpha) & -sin(\alpha) & 0 \\
+%       sin(\alpha) & cos(\alpha) & 0 \\
+%       0 & 0 & 1
+%     \end{pmatrix}
+% \end{equation}
+
+\subsection{Spiralgalaxien}
+
+\subsection{Using Object Oriented Programming (OOP) techniques}
+
+In my case, the objects are galaxies.
+
+\subsubsection{Initialisation}
+
+The galaxy is initialised with the following objects:
+
+\begin{itemize}
+  \item A list storing the coordinates of each star
+  \begin{itemize}
+    \item X Coordinate
+    \item Y Coordinate
+    \item Z Coordinate
+  \end{itemize}
+
+  \item A list storing the individual forces acting on the stars
+  \begin{itemize}
+    \item X Force
+    \item Y Force
+    \item Z Force
+  \end{itemize}
+
+  \item A variable storing the number of stars generated in the galaxy
+
+  \item Newtons gravitational constant
+\end{itemize}
+
+\subsection{Generation of new stars}
+
+The function is given an integer defining the amount of stars that should be
+newly generated.
+
+The newly generated stars are then appended to the list storing the coordinates.
+
+The counter counting the amount of stars in the galaxy is incremented.
+
+\subsection{Printing all the coordinates}
+
+The function cycles through the list storing the star coordinates and prints
+them to the command line.
+
+\subsection{Calculating the Forces acting between the Stars}
+
+The function recieves two star objects and an axis on which the forces should
+be calculated and returns the force acting on the given axis. In case of a
+failture (The two given stars have got the same coordiantes), the function
+just goes on to the next Star.
+
+The Forces can be calculated using the equation (\ref{eq:gravitation_law}).
+
+\subsection{Calculating the forces acting between each star in the galaxy and
+each other star}
+
+To calculate the forces inbetween every star in the galaxy, the function cycles
+through every star, looks if the force that should be calculated hat not been
+calculated yet and calculates it. This includes testing if the force that should
+be calculated is not the force inbetween a star and itself.
+
+The results of the calculations are stored in a list storinf the forces.
+
+\subsection{Printing all the individual forces}
+
+The function is able to print all the forces acting inbetween the stars if no
+argument is given. If an argument n is given, the function print out the nth
+star in the list.
+
+\subsection{Spherical cells}
+
+
+\subsubsection{Testing if a point is inside or outside a sphere}
+
+In order to test is a point is inside a sphere, one just has to test if the
+following conditions are all true:
+
+\begin{equation}
+  \begin{split}
+    S_x - S_r \leq P_x \leq S_x + S_r \\
+    S_y - S_r \leq P_y \leq S_y + S_r \\
+    S_z - S_r \leq P_z \leq S_z + S_r
+  \end{split}
+\end{equation}
+
+\( P_x \) , \( P_y \) and \( P_z \) are the coordinates of the point to be tested,
+\( S_x \) , \( S_y \) and \( S_z \) are the coordinates of the midpoint of the sphere and
+\( S_r \) is the radius of the sphere.
+
+\subsubsection{Testing if a star is inside or outside of a sphere for a whole galaxy}
+
+While testting if a star is inside a sphere or not, because of the alignment
+of the spheres, a point can be in more than one sphere at the same time.
+To get rid of this problem, the software cycles over every star and searches
+for matches within the spheres. If a match is found, the next star is tested.
+This is pretty much as efficient as it can get.
+
+\begin{equation}
+  O(n) = n_{stars} \cdot n_{spheres}
+\end{equation}
+
+\subsubsection{Generate the position of the spheres}
+
+Generating the position of the spheres is accomplished in the following way:
+A 3D-grid is generated and the midpoints of the spheres are positioned on the
+gridpoints.
+
+[Include Graphic]
+
+The distance the spheres have to each other ist defined using the following
+function:
+
+\begin{equation} \label{sphere_distance}
+  \texttt{sphere\_distance} = \frac{\texttt{galaxy\_range}}{\texttt{sampling\_rate}}
+\end{equation}
+
+The higher the \texttt{sampling\_rate} is, the more spheres get generated.
+
+The next goal is to find out the ``sweet spot'' generating the spheres.
+When using a very low \texttt{sampling\_rate}, the reault gets inacurate, but
+when using a high \texttt{sampling\_rate}, the calculations are not affected
+in term of speed and efficiency. The Goal is therefor to find a sampling rate
+that enables the generation of accurate but fast results.
+
+By being able to controll the accuracy anf therefor the time, it is possible to
+teach the system to generate a galaxy in like one hour and it will automatically
+set the sampling rate so low that the simulation will finish perfectly in time.
+
+\subsubsection{The radius of the spheres}
+
+The Radius of the spheres is dynamicly allocated to ensure that the whole galaxy
+is covered.
+
+\begin{equation} \label{sphere_radius}
+  r = \sqrt{\texttt{sphere\_distance}^2 + \texttt{sphere\_distance}^2 + \texttt{sphere\_distance}^2}
+\end{equation}
+
+\begin{equation*}
+  1.7320508075688772 = \sqrt{1^2 + 1^2 + 1^2}
+\end{equation*}
+
+The equation (\ref{sphere_radius}) highly depends on the equation
+(\ref{sphere_distance}) and it's parameters.
+
+\begin{figure}
+  \includegraphics[width=1\textwidth]{figs/sphere_alignment_cc}
+  \caption{The Alignment of the spheres\\
+  Left: The perfect alignment covering the complete space\\
+  Right: A previously generated alignment using small spheres to cover the missing space
+  }
+  \label{sphere_alignment}
+\end{figure}
+
+\subsubsection{Calculate the forces acting on the spheres}
+
+In order to reduce the time that is needed to calculate the forces inbetween
+the stars, the stars are subdivided in different cells, in this case spheres.
+After all the forces acing inside one sphere are calculated, the forces are
+combined and applied to the midpoint of the sphere genrating a new coordinate:
+the mean force. The mean force inbetween all the cells can be calculated.
+
+[Include description binary tree]
+
+[Include graphic binary tree]
+
+\subsubsection{Calculate the forces acting on all the spheres together}
+
+This should be 0.
+
+\subsubsection{Benchmarks}
+
+\begin{tabular}{l | l | l | l}
+  Nr of Stars & Sample rate & Galaxy Range & Time (s) \\\hline
+
+
+  100         & 1           & 100          & 0.0814 \\
+  75          & 1           & 100          & 0.0499 \\
+  50          & 1           & 100          & 0.0295 \\
+  25          & 1           & 100          & 0.0116 \\ \hline
+
+  100         & 1           & 100          & 0.0828 \\
+  100         & 2           & 100          & 0.0909 \\
+  100         & 4           & 100          & 0.1832 \\
+  100         & 8           & 100          & 1.1114 \\
+  100         & 16          & 100          & 7.6944 \\
+  100         & 32          & 100          & 56.5731 \\
+  100         & 64          & 100          & 217.7768 \\ \hline
+
+  100         & 1           & 1            & 0.0809 \\
+  100         & 1           & 2            & 0.0844 \\
+  100         & 1           & 4            & 0.0782 \\
+  100         & 1           & 8            & 0.0758 \\
+  100         & 1           & 16           & 0.0847 \\
+  100         & 1           & 32           & 0.0815 \\
+  100         & 1           & 64           & 0.0770 \\
+
+\end{tabular}
+
+The sample rate is the factor that influences the time the most. Knowing this,
+it (the sample rate) can be used to controll the time in which a galaxy can
+be created.
+This is usefull in particular when using some very powerfull mashine with
+limited time.
+
+\subsection{Calculate the Position of a Star after a timestep}
+
+Not to be considered:
+\begin{itemize}
+  \item drag
+  \item any kind of resistance
+  \item acceleration
+\end{itemize}