updated langfassung

3 files changed, 120 insertions, 2 deletions

diff --git a/langfassung/docs/2_einleitung.tex b/langfassung/docs/2_einleitung.tex
index 806c214..bce9ec8 100644
--- a/langfassung/docs/2_einleitung.tex
+++ b/langfassung/docs/2_einleitung.tex
@@ -1,3 +1,4 @@
+op
 Einleitung
 
 \subsection{Themen}
@@ -16,4 +17,50 @@ Einleitung
 
 \subsection{Motivation}
 
+\paragraph{ \( \Phi \) }
+
+\begin{equation}
+  \Phi(r) = - \frac{4\pi G \rho_0 R_s^3}{r} \ln ( 1+ \frac{r}{R_s} )
+\end{equation}
+
+with the limits
+
+\begin{equation}
+  \lim_{r\to \infty} \Phi=0
+\end{equation}
+
+and
+
+\begin{equation}
+  \lim_{r\to 0} \Phi=-4\pi G\rho_0 R_s^2
+\end{equation}
+
+\paragraph{ \( \rho \) }
+
+\begin{equation}
+  \rho(r) = \frac{1}{\sqrt{2 \cdot \pi} \cdot \sigma} \cdot
+  e^{\left( - \frac{(\Phi(r)}{\sigma^{2}} \right)}
+\end{equation}
+
+\paragraph{\( \rho_{new} \rightarrow (deriviation) \) }
+
+\begin{equation}
+  \rho(r) \cdot 1-\frac{1}{(2 \cdot sigma^{2} )} \cdot
+  ( Mxx \cdot x^{2} + 2 \cdot Mxy \cdot xy + Myy \cdot y^{2} ))
+\end{equation}
+
+% def rho_new(x, y, z):
+%     a = (1 - ((1) / (2 * (sigma ** 2))) * ( Mxx * x**2 + 2 * Mxy * x * y + Myy * y**2 ) )
+%     return rho(x, y, z) * a
+%
+% # phi function
+% def phi(x):
+%     if x == 0:
+%         return -4 * pi * f_0 * G * R_s**2
+%
+%     a = - ( 4 * pi * G * f_0 * R_s ** 3 ) / x
+%     b = np.log(1. + (x / R_s) )
+%     c = a * b
+%     return c
+
 Motivations blah
diff --git a/langfassung/docs/3_hauptteil.tex b/langfassung/docs/3_hauptteil.tex
index eab3897..5174f08 100644
--- a/langfassung/docs/3_hauptteil.tex
+++ b/langfassung/docs/3_hauptteil.tex
@@ -43,14 +43,59 @@ Blender is Awesome, Python is Awesome and together they are
 
 \subsection{Making things faster}
 
-Kicking out to many Stars, 1 out of 10000 is just to much...
+\paragraph{ Kicking out to many Stars, 1 out of 10000 is just to much... }
+~\\
+
+\begin{itemize}
+  \item Use a custom Density function for each Axis
+  \begin{itemize}
+    \item \( \phi(r_x)\), \( \phi(r_y) \) and \( \phi(r_z) \)
+    \item more controll
+  \end{itemize}
+\end{itemize}
+
+\begin{tikzpicture}
+\begin{scope}
+
+    \node[draw] (H) at (0,-2)
+        {\( \phi(x_1) > rand_x \)};
+    \node[draw] (I) at (5,-2)
+        {\( \phi(y_1) > rand_y \)};
+    \node[draw] (J) at (10,-2)
+        {\( \phi(z_1) > rand_z \)};
+
+    \node[draw] (K) at (10, -4) {true};
+    \node[draw] (L) at (0, -4) {false};
+
+    \node[draw] (M) at (10, -6) {write data to data.csv};
+
+    \node[draw] (N) at (0, -6) {generate new random coordinates};
+
+\end{scope}
+
+\begin{scope} [every node/.style={fill=white,circle},
+              every edge/.style={draw=red,very thick}]
+
+    \path[->] (H) edge (K);
+    \path[->] (I) edge (K);
+    \path[->] (J) edge (K);
+    \path[->] (H) edge (L);
+    \path[->] (I) edge (L);
+    \path[->] (J) edge (L);
+
+    \path[->] (K) edge (M);
+    \path[->] (L) edge (N);
+
+\end{scope}
+
+\end{tikzpicture}
 
 \subsection{Spiral Galaxies}
 
 The previous Galaxy models where all using a completely spherical model, generating
 a spiral galaxy is just not possible using these models.
 
-\subsubsection{N-Körper problem}
+\subsubsection{N-body problem}
 
 Kurze Beschreibung des N-Körper Problems
 
diff --git a/langfassung/docs/4_ergebnisse.tex b/langfassung/docs/4_ergebnisse.tex
index abd806d..088b13e 100644
--- a/langfassung/docs/4_ergebnisse.tex
+++ b/langfassung/docs/4_ergebnisse.tex
@@ -22,6 +22,32 @@ Pro MegaByte können die Koordinaten von 10000 Sternen gespeichert werden.
 
 Die generierung von Spiralgalaxien gestaltet sich schweiriger als erwartet.
 
+\subsection{Lookup-Table Speed}
+
+\begin{tabular}{l | l | l}
+rho-values  & step  & time (in seconds) \\ \hline\hline
+1500000     & 1     & 8.07  \\
+750000      & 2     & 4.4   \\
+375000      & 4     & 2.26  \\
+187500      & 8     & 1.35  \\
+93750       & 16    & 0.76  \\
+\end{tabular}
+
+The correlation between the number of stars generated and the time needed ist
+clearly linear.
+
+\paragraph{Python script} ~\\
+\lstset{language=Python}
+\begin{lstlisting}[frame = single]
+import matplotlib.pyplot as plt
+
+list_time = [8.07, 4.4, 2.26, 1.35, 0.76]
+list_rho_values = [1500000, 750000, 375000, 187500, 93750]
+
+plt.plot(list_time, list_rho_values, '-ro')
+plt.show()
+\end{lstlisting}
+
 \subsection{Distortion of Galaxies}
 
 Galaxien verformen dinge