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-rw-r--r--langfassung/docs/1_kurzfassung.tex13
-rw-r--r--langfassung/docs/2_einleitung.tex90
-rw-r--r--langfassung/docs/3_hauptteil.tex318
-rw-r--r--langfassung/docs/5_quellen.tex12
-rw-r--r--langfassung/figs/2017-12-05-164935_374x65_scrot.pngbin0 -> 8460 bytes
-rw-r--r--langfassung/master.aux62
-rw-r--r--langfassung/master.fdb_latexmk37
-rw-r--r--langfassung/master.fls25
-rw-r--r--langfassung/master.log351
-rw-r--r--langfassung/master.out42
-rw-r--r--langfassung/master.pdfbin223308 -> 288544 bytes
-rw-r--r--langfassung/master.pyg13
-rw-r--r--langfassung/master.synctex.gzbin49962 -> 74231 bytes
-rw-r--r--langfassung/master.tex80
-rw-r--r--langfassung/master.toc56
15 files changed, 647 insertions, 452 deletions
diff --git a/langfassung/docs/1_kurzfassung.tex b/langfassung/docs/1_kurzfassung.tex
index b4aee7d..e9aea85 100644
--- a/langfassung/docs/1_kurzfassung.tex
+++ b/langfassung/docs/1_kurzfassung.tex
@@ -1 +1,12 @@
-Kurzfassung
+\begin{abstract}
+\large
+Ziel meines Projektes war es, Galaxien zu generieren und diese mit den Modellen
+von echten Galaxien zu vergleichen um mit der Zeit immer bessere Simulationen
+zu generieren. Dies gelang mir im Verlauf des letzten Jahres immer besser, wobei
+ich zurzeit (\today) auf meinem Laptop ca. 1000 neue Sterne in 1.5 Stunden
+generieren kann.
+Zu dem Projekt bin ich während meines Praktikum im Zentrum für Astronomie in
+Heidelberg (ZAH) gekommen bei dem ich von Tim Tugendhat eine Einführung in das
+NFW-Profil bekommen habe, welches genutzt wird um Galaxien zu generieren.
+
+\end{abstract}
diff --git a/langfassung/docs/2_einleitung.tex b/langfassung/docs/2_einleitung.tex
index bce9ec8..a204783 100644
--- a/langfassung/docs/2_einleitung.tex
+++ b/langfassung/docs/2_einleitung.tex
@@ -1,66 +1,40 @@
-op
-Einleitung
+Nach meinem letzten Jugend-Forscht Projekt ergab sich die Möglichkeit ein
+Praktikum im Zentrum für Astronomie in Heidelberg zu absolvieren. Über die
+social-media Platform Reddit stellte ich den kontakt mit Tim Tugendkat her
+der zurzeit seinen PhD. in Physik an der Universität in Heidelberg macht.
+Dieser ermöglichte es mir, die Physikalische Fakultät an einer Uni mal genauer
+zu sehen und das täglich leben eines Physikers mitzuerleben.
+\par
+Während des Praktikums stellte ich fest das ich die im letzten Jahr erlerne Fähigkeit mit
+Python\footnote{Programmiersprache} zu Programmieren und mit
+Blender\footnote{3D Software Suite} umzugehen nutzen konnte um Galaxien
+darzustellen.
+Dies war insgesamt unglaublich Interessant und zeigte mir zum wiederholten mal:
+Projekte sind sehr dazu geeignet um sich in neues einzuarbeiten oder neues
+zu lernen und bieten einem ein Ziel welches man erreichen möchte was einem
+immer genügend motivation bietet weiterzumachen.
+\par
+Eine frage die ich mir öfters gestellt habe war warum man eigentlich Galaxien
+simuliert? Wäre es nicht einfacher einfach in den Himmel zu gucken und
+die bereits bestehenden Galaxien zu beobachten?
+Nach kurzer recherche lag die Antwort auf der Hand: Galaxien brauchen mehrere
+Millionen Jahre um sich zu entwickeln, also kann man ihre Entwicklung als
+normaler Mensch nicht in dem Umfang beobachten, um dann daraus schlüsse zu
+ziehen. Daher simuliert man die Galaxien und kann dann somit vorhersagen oder
+herrausfinden wie die Galaxien entstanden sind bzw. was mit ihnen passieren
+wird.
 
 \subsection{Themen}
 
 \begin{itemize}
-  \item Heidelberg bla
-  \item Galaxy Foo bla
-  \item Stauchen / Strecken bla
-  \item Problem: Geschwindigkeit bla
-  \item Benchmarks:
-  \begin{itemize}
-    \item 10000 Sterne - 1 Stern
-    \item ...
-  \end{itemize}
+  \item Generierung von Elliptischen Galaxien
+  \item Generierung von einem Dark-Matter Halo um die Elliptische Galaxie
+  \item Stauchung und Streckung des Dark-Matter mit beinflussung der eigentlichen Galaxie
+  \item Beschleunigung des generierungsprozesses mithilfe einer sogennanten ''lookup-table``
+  \item Aufbau eines neuronalen Netzes für die unbeaufsichtigte Generation von Galaxien
+  \item Generation von Spiralgalaxien
 \end{itemize}
 
 \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
+Ich habs einfach mal getan...
diff --git a/langfassung/docs/3_hauptteil.tex b/langfassung/docs/3_hauptteil.tex
index 5174f08..617e4f9 100644
--- a/langfassung/docs/3_hauptteil.tex
+++ b/langfassung/docs/3_hauptteil.tex
@@ -1,107 +1,303 @@
-\subsection{Navarro–Frenk–White profile}
+
+% \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
+%
+
+
+\subsection{Generierung der Elliptischen Galaxien}
+\subsubsection{Das Navarro-Frenk-White Profil}
 
 Das Navarro-Frenk-White profil (NFW-profil) ist im grunde genommen eine Funktion
 die einem die Warscheinlichkeit das ein Stern an einer bestimmten position ist
 liefert.
 Die Funktion ist im allgemeinen wie folgt aufgebaut:
 
-\begin{equation}
-  \rho = \frac{ 1 }{ \sqrt{ 2 \pi } \cdot \sigma } \cdot
+\begin{equation} \label{eq:NFW_profile}
+  \rho_{NFW}(r) = \frac{ 1 }{ \sqrt{ 2 \pi } \cdot \sigma } \cdot
   \exp \left( \frac{ -\phi(r) }{ \sigma^{ 2 } } \right)
 \end{equation}
 
-\begin{equation}
-  \phi_{NFW}(r) = \frac{ 4\pi \cdot G \cdot f_{0} \cdot R_{s}^3 }{ r } \cdot
+\begin{equation*}
+  \phi(r) = \frac{ 4\pi \cdot G \cdot f_{0} \cdot R_{s}^3 }{ r } \cdot
   ln{ \left( 1 + \frac{ r }{ R_{s} } \right) }
-\end{equation}
+\end{equation*}
 
-Sieht kompliziert aus, ist es aber nicht: Um zu gucken ob ein zufälliger Stern
+Um die Formel (\ref{eq:NFW_profile}) einfach zu beschreiben kann man sie sich
+wie folgt vorstellen:
+Um zu gucken ob ein zufälliger Stern
 bei \( x_1 \), \( y_1 \) und \( z_1 \) generiert werden kann wird wie folgt
 vorgegangen: Aus den Koordinaten wird der Wert \( r \) mithilfe des Satz des
-Phtargoras berechnet, dieser gibt
+Pytargoras berechnet ( \( r = \sqrt{{x_1}^2 + {x_2}^2 + {x_3}^2} \) ) , dieser gibt
 an wie weit der jeweilige Stern vom Zentrum der Galaxie entfernt ist. Um zu
 prüfen ob der Stern generiert wird, wird dieser \( r \)-wert in die Funktion
-\( \phi \) eingesetzt. Der entstehende Wert gibt an wie warscheinlich es ist,
+\( \rho_{NFW} \) eingesetzt. Der entstehende Wert gibt an wie warscheinlich es ist,
 das ein Stern in der Entfernung zum Ursprung generiert wird.
-\par
-Um herrauszufinden ob der Stern generiert wird, wird ein weiterer zufälliger
-Wert \( x \) im bereich \( [\phi_{max}; \phi_{min}] \) generiert. Liegt dieser
-Wert über dem Wert aus der Funktion \( \phi \) wird kein Stern generiert.
-Liegt dieser Stern jedoch unter dem wert aus der \( \phi \) funktion wird
+
+\subsubsection{Random Sampling}
+
+Die sogennante ''Random Sampling`` Methode wird genutzt um herrauszufinden ob
+ein Stern generiert wird oder nicht.Es wird dazu ein zufälliger
+Wert \( x \) im bereich \( [~\rho_{max}~;~\rho_{min}~] \) generiert. Liegt dieser
+Wert über dem Wert aus der Funktion \( \rho \) wird kein Stern generiert.
+Liegt dieser Stern jedoch unter dem wert aus der \( \rho \) Funktion wird
 ein Stern an den Koordinaten \( x_1 \), \( y_1 \) und \( z_1 \) generiert.
 
-\subsection{Einasto profile}
+Um das generieren zu Beschleunigen wird eine sogenneante ''lookuptable``
+verwendet. (\( \rightarrow \) \ref{subsec:lookuptable})
+
+Generiert man ein paar Sterne mithilfe des NFW-Profils hat man theoretisch
+schon eine Galaxie, jedoch ist diese nicht klar definiert. Um eine klare
+definition zu erreichen müssen mehrere hundert Sterne generiert werden.
+
+% \subsubsection{Das Einasto Profil}
+%
+% \begin{equation}
+%   \gamma(r) = \frac{ d \ln(\rho(r)) }{ d \ln(\rho) } \propto r^{\alpha}
+% \end{equation}
+
+% \subsubsection{Blender + Python}
+%
+% Blender is Awesome, Python is Awesome and together they are
+% \bold{SUPER AWESOME!!!}
+%
+% \begin{enumerate}
+%   \item Generate the galaxy-data using the NFW-Profile or the Einasto-profile
+%   \item Display the data in Blender and create an image using the OpenGL-renderer
+%   \item Train a Neural Network (NN) to classify galaxies
+%   \item Let the NN modify the galaxy to generate a perfect galaxy
+% \end{enumerate}
+
+
+\subsection{Generierung eines Dunkle-Materie Halos}
+
+Das sogennannte ''Dunkle-Materie Halo`` ist eine art Kugel die eine Galaxie
+umspannt: Duch dieses Halo ist die Dichte der Dunklen Materie welches sich um die
+Galaxie herum befindet definiert. Problematisch ist jedoch, dass wir dieses
+Halo nicht sehen können weshalb wir nur aufgrund anderer phänomäne welche durch
+die Halos verursacht werden auf die Eigenschaften des Halos schließen können.
+\par
+Um diese Halos darzustellen wird das NFW-Profil~(\ref{eq:NFW_profile})
+abgewandelt und quasi mit dem Profil für Elliptische Galaxien verbunden.
+
+...
+
+\subsubsection{Anpassung des NFW-Profils}
 
 \begin{equation}
-  \gamma(r) = \frac{ d \ln(\rho(r)) }{ d \ln(\rho) } \propto r^{\alpha}
+  \rho(r) = \frac{1}{\sqrt{2 \cdot \pi} \cdot \sigma} \cdot
+  e^{\left( - \frac{(\Phi(r)}{\sigma^{2}} \right)}
 \end{equation}
 
-\subsection{Blender + Python}
+\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}
 
-Blender is Awesome, Python is Awesome and together they are
-\bold{SUPER AWESOME!!!}
+\begin{lstlisting}
 
-\subsection{Making things faster}
+# new rho function
+def rho_new(x, y, z):
+  a = (1 - ((1) / (2 * (sigma ** 2)))
+  b = ( Mxx * x**2 + 2 * Mxy * x * y + Myy * y**2 ) )
+  c = a * b
+return rho(x, y, z) * c
 
-\paragraph{ Kicking out to many Stars, 1 out of 10000 is just to much... }
-~\\
+# phi function
+def phi(x):
+  if x == 0:
+    return -4 * pi * f_0 * G * R_s**2
 
-\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}
+  a = - ( 4 * pi * G * f_0 * R_s ** 3 ) / x
+  b = np.log(1. + (x / R_s) )
+  c = a * b
+  return c
 
-\begin{tikzpicture}
-\begin{scope}
+\end{lstlisting}
 
-    \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};
+\subsection{Stauchung und Streckung der Galaxie}
 
-    \node[draw] (M) at (10, -6) {write data to data.csv};
+Wird eine Galaxie gestreckt oder gestaucht kann das an der umliegenden Dunklen
+Materie liegen. Um solch eine Streckung darzustellen wird wie folgt vorgegangen:
+Die Position eines Sternes an einer Achse muss mit einem Skalar multipliziert
+bzw. dividiert werden.
+Dies ist relativ einfach machbar da die Koordinaten der jeweiligen Sterne
+in einer Datei nach dem Format \( [x, y, z] \) gespeichert sind.
+Um die Galaxie vertikal zu strecken wird z.B. für jeden Stern die z-Koordinate
+mit dem skalar \( s \) multipliziert. Da gestaucht werden soll liegt dieser
+Wert im Intervall \( 0 < s < 1 \). Die neue Koordinate für einen Stern ist also
+\( [x, y, z \cdot s] \). Möchte man die Galaxie strecken muss das Skalar \( s \)
+im Intervall \( 1 < s < \infty \) liegen.
 
-    \node[draw] (N) at (0, -6) {generate new random coordinates};
+\subsection{Beschleunigung der Generierung}
 
-\end{scope}
+Die Sterne schnell zu generieren ist natürlich energieeffizienter aber auch
+wichtig damit das neuronale netzt in unserer lebzeit fertig wird.
 
-\begin{scope} [every node/.style={fill=white,circle},
-              every edge/.style={draw=red,very thick}]
+Es gibt ein paar Aktionen die umgebaut werden können um das generieren zu
+beschleunigen:
 
-    \path[->] (H) edge (K);
-    \path[->] (I) edge (K);
-    \path[->] (J) edge (K);
-    \path[->] (H) edge (L);
-    \path[->] (I) edge (L);
-    \path[->] (J) edge (L);
+\subsubsection{n-Sterne}
 
-    \path[->] (K) edge (M);
-    \path[->] (L) edge (N);
+Statt am Anfang mehrere Millionen Sterne zu generieren wird wenn eine
+neue Koordinate benötigt wird eine neue erstellt. So erstellt man auf keinen
+Fall zu viele Koordinaten was Zeit spaart.
+Dem programm kann also gesagt werden, dass es genau \( n_1 \) Sterne aus
+\( m_1 \) potentiellen Sternen generieren soll, andernfalls werden \( n_2 \)
+Sterne aus \( m_2 > m_1 \) potentiellen Sternen generiert.
 
-\end{scope}
+\subsubsection{Lookuptable} \label{subsec:lookuptable}
 
-\end{tikzpicture}
+Eine Weitere Möglichkeit für meherere Berechnungen Zeit zu Spaaren ist, den
+Entsprechenden Wert aus dem NFW-Profil (Formel \ref{eq:NFW_profile}) vorher zu
+berechnen und in eine Tabelle zu schreiben.
+Dies kann für z.B. \( 2e8 \) Werte getan werden was zwar eine 6 GB große Datei
+erzeugt, diese kann jedoch innerhalb weniger Sekunden eingelesen werden.
+
+\subsubsection{Weitere Optimierungen}
+
+\paragraph{Nichts in der Konsole ausgeben:}
+
+Eine Vorgang der erstaunlicherweise sehr viel Rechenleistung erfordert, ist
+der Vorgang beim ausgeben von Text in die Konsole. Gibt man jede potentielle
+Koordinate in die Konsole aus, stürtzt das Programm aufgund von Überlast ab.
+Um dies zu umgehen kann z.B. nur jeder 100.000 Wert in die Konsole ausgegeben
+werden.
+
+\paragraph{...}
+
+\newpage
+\subsection{Nutzung eines Neuronalen Netzes zum unbeaufsichtigeten generieren}
+\subsubsection{Aufbau des Neuronalen Netzes}
+
+Ein Neuronales Netz ist wie folgt aufgebaut:
 
-\subsection{Spiral Galaxies}
+\bigskip
 
-The previous Galaxy models where all using a completely spherical model, generating
-a spiral galaxy is just not possible using these models.
+\hrule
 
-\subsubsection{N-body problem}
+\bigskip
 
-Kurze Beschreibung des N-Körper Problems
+\tikzset{%
+  every neuron/.style={
+    circle,
+    draw,
+    minimum size=1cm
+  },
+  neuron missing/.style={
+    draw=none,
+    scale=2,
+    text height=0.333cm,
+    execute at begin node=\color{black}$\vdots$
+  },
+}
 
-\subsubsection{Hilbert Spiral}
+\begin{center}
+  \begin{tikzpicture}[x=2cm, y=1.5cm, >=stealth]
+
+  \foreach \m/\l [count=\y] in {1,2,3,missing,4}
+    \node [every neuron/.try, neuron \m/.try] (input-\m) at (0,2.5-\y) {};
+
+  \foreach \m [count=\y] in {1,missing,2}
+    \node [every neuron/.try, neuron \m/.try ] (hidden-\m) at (2,2-\y*1.25) {};
+
+  \foreach \m [count=\y] in {1,missing,2}
+    \node [every neuron/.try, neuron \m/.try ] (output-\m) at (4,1.5-\y) {};
+
+  \foreach \l [count=\i] in {1,2,3,n}
+    \draw [<-] (input-\i) -- ++(-1,0)
+      node [above, midway] {$I_\l$};
+
+  \foreach \l [count=\i] in {1,n}
+    \node [above] at (hidden-\i.north) {$H_\l$};
+
+  \foreach \l [count=\i] in {1,n}
+    \draw [->] (output-\i) -- ++(1,0)
+      node [above, midway] {$O_\l$};
+
+  \foreach \i in {1,...,4}
+    \foreach \j in {1,...,2}
+      \draw [->] (input-\i) -- (hidden-\j);
+
+  \foreach \i in {1,...,2}
+    \foreach \j in {1,...,2}
+      \draw [->] (hidden-\i) -- (output-\j);
+
+  \foreach \l [count=\x from 0] in {Eingabe, Versteckte, Ausgabe}
+    \node [align=center, above] at (\x*2,2) {\l \\ ebene};
+
+  \end{tikzpicture}
+\end{center}
+\bigskip
+
+\hrule
+
+\bigskip
+
+Das \textbf{Neuronale Netz} besitze mehrere Ebenen: die \textbf{Eingabe ebene},
+die \textbf{Versteckte Ebene(n)} und die \textbf{Ausgabe Ebene}.
+Diese Ebenen bestehen aus sogennanten \textbf{Neuronen} die wie im Menschlichen
+Gehirn Informationen aufnehmen und weitergeben. Die Eingabe kann verschieden
+gewichtet sein, es kann also sein das eine Eingabe eine Gewichtung von
+\( 10\% \) hat und eine andere eine Gewichtung von \( 90\% \).
+Die Eingabe Ebene ist dazu da eine Eingabe inform einer Matrix an die
+verschiedenen Neuronen in der Versteckten Ebene weiterzuleiten.
+Die Versteckte Ebene verarbeitet die Information aus der Matrix und leitet
+diese an die Ausgabe Ebene weiter die die Information ausgibt.
+\par
+Das sogennante ''Trainieren`` ist der Prozess, bei dem die Gewichtung der
+Neuronen so Verändert wird, damit ein gewünschtes Ergebnis herrauskommt.
+Beispiel: man möchte ein Neuronales Netz darauf Trainieren eine Galaxie zu
+Identifizieren, dann werden ganz viele positiv Beispiele durch das Netz gejagt
+welche die Gewichtung immer weiter anpassen. In der Ausgangs Ebene wird dann
+mithilfe zweier Neuronen entweder dargestellt das das eingegebene Bild eine
+Galaxie ist oder das das eingegebene Bild eben keine Galaxie ist.
+
+\subsubsection{Nutzung eines Neuronalen Netztes zur verbesserung von Galaxien Simulationen}
+
+Möchte man mithilfe eines Neuronalen Netztes vorhandene Galaxiensimulationen
+verbessern, wird wie im folgenden Diagramm zusehen vorgegangen:
+
+\begin{tikzpicture}
+[node distance = 4cm, auto, ->, on grid]
+
+\node [draw, minimum width=3cm, text depth = 1cm] (galaxy) {Galaxie};
+\node [draw, right of=galaxy] (neural_net) {Neuralonales Netz};
+
+\node [draw] (yes) [right of=neural_net] {Ja}
+node [right=3cm of yes, align=center] {Galaxie ist eine Galaxie};
+
+\node [draw] (no) [below=2cm of neural_net] {Nein}
+node [right=3.5cm of no, align=center] {Galaxie ist keine Galaxie \\
+\( \rightarrow \) ändere parameter und \\generiere eine neue Galaxie};
+
+
+\draw[->, line width=0.25mm] (galaxy) -- (neural_net)
+node [above=1cm of neural_net, align=center] {Testet ob die Eingabe \\eine
+Galaxie ist oder nicht};
+
+\node[draw, yshift=5mm] (paramter) at (galaxy.south) {paramter};
+
+\draw[->, line width=0.25mm] (neural_net) -- (yes);
+\draw[->, line width=0.25mm] (neural_net) -- (no);
+
+\path[line width=0.25mm] (no) edge [bend left] node {} (paramter);
+
+\end{tikzpicture}
 
-Beschreibung der Hilbert Spirale
+\subsection{Spiralgalaxien}
+\subsubsection{Das n-Körper Problem}
 
 \subsection{Größeneinheiten}
 
diff --git a/langfassung/docs/5_quellen.tex b/langfassung/docs/5_quellen.tex
index 17bb702..f928fa3 100644
--- a/langfassung/docs/5_quellen.tex
+++ b/langfassung/docs/5_quellen.tex
@@ -5,20 +5,16 @@ Quellen
   Das Python-Programm sowie die Blender Darstellungen wurden vollständig ohne fremde Hilfe selber erstellt.
  }
 \end{center}
-\par Einen Großteil der Formeln fand ich durch eine Wikipedia Recherche, jedoch wurden auch Informationen aus dem 'SPACETRACK REPORT' von Felix R. Hoots und Ronald L. Roerich,
-No. 3 (Dec. 1980) entnommen.
-\par Das Programieren in der Programiersprache Python habe ich wärend des Projektes mithilfe der Python-Documentation gelernt. Mit dem Umgang des 3D-Programms Blender
+\par Einen Großteil der Formeln fand ich durch eine Wikipedia Recherche.
+\par Das Programieren in der Programiersprache Python habe ich während meines Jugen-Forscht Projektes im letztem Jahr (2017) gelernt. Mit dem Umgang des 3D-Programms Blender
 bin ich schon vertraut gewesen. Die Grundlagen für \LaTeX, in dem diese Langfassung geschrieben wurde, erlernte ich durch das Studieren diverser Beiträge in Foren und der Einsicht
-in das Jugend Forscht Projekt von Konstantin Bosbach, Tilman Hoffbauer und Steffen Ritsche aus dem vorherigem Jahr (2016, Underwater Accoustic Communication).
-\par Da es das erste Mal war, dass ich Python mit Blender kombiniert habe, musste ich mich in diesen Bereich ebenfalls einarbeiten.
-Dabei war die Blender API Documentation (www.blender.org/api) von großer Hilfe.
-
+in das Jugend Forscht Projekt von Konstantin Bosbach, Tilman Hoffbauer und Steffen Ritsche (2016, Underwater Accoustic Communication).
 
 \raggedleft
 \section*{Dank gilt...}
 \paragraph{Herrn Jörg Thar} meinem Betreuer
 \paragraph{Konstantin Bosbach} welcher mir eine Möglichkeit gab für 2 Wochen in Heidelberg zu wohnen.
-\paragraph{Tilman Hoffbauer}
+\paragraph{Tilman Hoffbauer} der bei problemen bereit war Licht ins Dunkle zu bringen.
 
 \centering
 \vspace{0.5cm} \textbf{Außerdem gilt mein Dank allen, die mich auf jede nur erdenkliche Weise unterstützt haben.}
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new file mode 100644
index 0000000..9a95146
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index a073a11..abb2010 100644
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index e3a6179..67b7a5e 100644
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diff --git a/langfassung/master.out b/langfassung/master.out
index c2573b4..3248c60 100644
--- a/langfassung/master.out
+++ b/langfassung/master.out
@@ -2,23 +2,25 @@
 \BOOKMARK [2][-]{subsection.1.1}{Themen}{section.1}% 2
 \BOOKMARK [2][-]{subsection.1.2}{Motivation}{section.1}% 3
 \BOOKMARK [1][-]{section.2}{Hauptteil}{}% 4
-\BOOKMARK [2][-]{subsection.2.1}{Notes}{section.2}% 5
-\BOOKMARK [3][-]{subsubsection.2.1.1}{Todo}{subsection.2.1}% 6
-\BOOKMARK [3][-]{subsubsection.2.1.2}{Generally building a galaxy}{subsection.2.1}% 7
-\BOOKMARK [3][-]{subsubsection.2.1.3}{Spiral Galaxy}{subsection.2.1}% 8
-\BOOKMARK [3][-]{subsubsection.2.1.4}{Important aspects of spiral galaxies}{subsection.2.1}% 9
-\BOOKMARK [3][-]{subsubsection.2.1.5}{Speed}{subsection.2.1}% 10
-\BOOKMARK [2][-]{subsection.2.2}{Navarro\205Frenk\205White profile}{section.2}% 11
-\BOOKMARK [2][-]{subsection.2.3}{Einasto profile}{section.2}% 12
-\BOOKMARK [2][-]{subsection.2.4}{Blender + Python}{section.2}% 13
-\BOOKMARK [2][-]{subsection.2.5}{Making things faster}{section.2}% 14
-\BOOKMARK [2][-]{subsection.2.6}{Spiral Galaxies}{section.2}% 15
-\BOOKMARK [3][-]{subsubsection.2.6.1}{N-body problem}{subsection.2.6}% 16
-\BOOKMARK [3][-]{subsubsection.2.6.2}{Hilbert Spiral}{subsection.2.6}% 17
-\BOOKMARK [2][-]{subsection.2.7}{Gr\366\337eneinheiten}{section.2}% 18
-\BOOKMARK [1][-]{section.3}{Ergebnisse}{}% 19
-\BOOKMARK [2][-]{subsection.3.1}{Simulation Speed}{section.3}% 20
-\BOOKMARK [2][-]{subsection.3.2}{Spiral Galaxies}{section.3}% 21
-\BOOKMARK [2][-]{subsection.3.3}{Lookup-Table Speed}{section.3}% 22
-\BOOKMARK [2][-]{subsection.3.4}{Distortion of Galaxies}{section.3}% 23
-\BOOKMARK [1][-]{section.4}{Quellen und Hilfen}{}% 24
+\BOOKMARK [2][-]{subsection.2.1}{Generierung der Elliptischen Galaxien}{section.2}% 5
+\BOOKMARK [3][-]{subsubsection.2.1.1}{Das Navarro-Frenk-White Profil}{subsection.2.1}% 6
+\BOOKMARK [3][-]{subsubsection.2.1.2}{Random Sampling}{subsection.2.1}% 7
+\BOOKMARK [2][-]{subsection.2.2}{Generierung eines Dunkle-Materie Halos}{section.2}% 8
+\BOOKMARK [3][-]{subsubsection.2.2.1}{Anpassung des NFW-Profils}{subsection.2.2}% 9
+\BOOKMARK [2][-]{subsection.2.3}{Stauchung und Streckung der Galaxie}{section.2}% 10
+\BOOKMARK [2][-]{subsection.2.4}{Beschleunigung der Generierung}{section.2}% 11
+\BOOKMARK [3][-]{subsubsection.2.4.1}{n-Sterne}{subsection.2.4}% 12
+\BOOKMARK [3][-]{subsubsection.2.4.2}{Lookuptable}{subsection.2.4}% 13
+\BOOKMARK [3][-]{subsubsection.2.4.3}{Weitere Optimierungen}{subsection.2.4}% 14
+\BOOKMARK [2][-]{subsection.2.5}{Nutzung eines Neuronalen Netzes zum unbeaufsichtigeten generieren}{section.2}% 15
+\BOOKMARK [3][-]{subsubsection.2.5.1}{Aufbau des Neuronalen Netzes}{subsection.2.5}% 16
+\BOOKMARK [3][-]{subsubsection.2.5.2}{Nutzung eines Neuronalen Netztes zur verbesserung von Galaxien Simulationen}{subsection.2.5}% 17
+\BOOKMARK [2][-]{subsection.2.6}{Spiralgalaxien}{section.2}% 18
+\BOOKMARK [3][-]{subsubsection.2.6.1}{Das n-K\366rper Problem}{subsection.2.6}% 19
+\BOOKMARK [2][-]{subsection.2.7}{Gr\366\337eneinheiten}{section.2}% 20
+\BOOKMARK [1][-]{section.3}{Ergebnisse}{}% 21
+\BOOKMARK [2][-]{subsection.3.1}{Simulation Speed}{section.3}% 22
+\BOOKMARK [2][-]{subsection.3.2}{Spiral Galaxies}{section.3}% 23
+\BOOKMARK [2][-]{subsection.3.3}{Lookup-Table Speed}{section.3}% 24
+\BOOKMARK [2][-]{subsection.3.4}{Distortion of Galaxies}{section.3}% 25
+\BOOKMARK [1][-]{section.4}{Quellen und Hilfen}{}% 26
diff --git a/langfassung/master.pdf b/langfassung/master.pdf
index d61dfc8..f8fc36e 100644
--- a/langfassung/master.pdf
+++ b/langfassung/master.pdf
Binary files differdiff --git a/langfassung/master.pyg b/langfassung/master.pyg
new file mode 100644
index 0000000..05dd500
--- /dev/null
+++ b/langfassung/master.pyg
@@ -0,0 +1,13 @@
+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
diff --git a/langfassung/master.synctex.gz b/langfassung/master.synctex.gz
index f9ed4ad..5baf9cf 100644
--- a/langfassung/master.synctex.gz
+++ b/langfassung/master.synctex.gz
Binary files differdiff --git a/langfassung/master.tex b/langfassung/master.tex
index b248c7e..c3ed027 100644
--- a/langfassung/master.tex
+++ b/langfassung/master.tex
@@ -10,9 +10,18 @@
 \usepackage{caption}
 \usepackage{lmodern}
 \usepackage{textcomp}
+\usepackage[onehalfspacing]{setspace}
 
 \usepackage{tikz}
+\usetikzlibrary{positioning}
+
 \usepackage{listings}
+\lstset{ %
+  frame=single,	                   % adds a frame around the code
+  language=Python,                 % the language of the code
+  numbers=left,                    % where to put the line-numbers; possible values are (none, left, right)
+  stepnumber=1,                    % the step between two line-numbers. If it's 1, each line will be numbered
+}
 
 \usepackage{hyperref}
 \hypersetup{
@@ -36,7 +45,7 @@ urlcolor=black
 
 \begin{document}
 
-\title{Galaxy Stuff}
+\title{Galaxy Generation}
 \subtitle{Jugend Forscht \the\year}
 \author{ Emile Hansmaennel \texttt{ emile.hansmaennel@gmail.com }}
 \date{\today}
@@ -58,75 +67,6 @@ urlcolor=black
 \newpage
 
 \section{Hauptteil} \label{Hauptteil}
-\subsection{Notes}
-
-\subsubsection{Todo}
-
-\begin{itemize}
-  \item Find a better way to kick out stars using a more efficient method
-  \item Find a way to generate a spiral with a curvature y of the arms
-  \item Compare the generated galaxies with real galaxies to find out if they
-  are legit
-  \item Generate the dark-matter, link the galaxies to the dark matter and watch
-  how the dark matter affects the galaxy while deforming the dark matter.
-\end{itemize}
-
-\subsubsection{Generally building a galaxy}
-
-\begin{enumerate}
-  \item Generate the galaxy-data using the NFW-Profile or the Einasto-profile
-  \item Display the data in Blender and create an image using the OpenGL-renderer
-  \item Train a Neural Network (NN) to classify galaxies
-  \item Let the NN modify the galaxy to generate a perfect galaxy
-\end{enumerate}
-
-\subsubsection{Spiral Galaxy}
-
-\begin{itemize}
-  \item Generate the coordinates independently from each other:
-
-  \item Generate the spiral on the x/y plane:
-  \begin{itemize}
-    \item x / y: generate a spiral using the Polar coordinate system
-    \item z: generate the according z-value inddependently using the pythargoras
-    theorem to calculate the distance of a possible star from the center of the
-    galaxy, then continue by using the generated value inside of the NFW-profile
-    to find out if the star should be generated or not
-  \end{itemize}
-  \item Use the function to adapt the arms to real conditions
-\end{itemize}
-
-\subsubsection{Important aspects of spiral galaxies}
-
-\begin{itemize}
-  \item Thickness of the galaxy at the bulge in the middle
-  \item Radius
-  \begin{itemize}
-    \item of the entire galaxy
-    \item of the bulge in the middle
-  \end{itemize}
-  \item Number of arms
-  \item Density
-  \begin{itemize}
-    \item of the spiral arms
-    \item of the bulge in the middle
-  \end{itemize}
-\end{itemize}
-
-\subsubsection{Speed}
-
-\begin{itemize}
-  \item Manipulat the NFW-Profile (don't kick out so many stars!)
-  \item lookup-table for the random numbers
-  \item Amazon-Web-Services (AWS)?
-  \item PyPy
-  \item generally try to generate less galaxies further away ftom the center
-  so kicking out as many stars doesn't become problematic anymore.
-  (start with a linear function above the NFW-profile and kick out the remaining
-  stars to enhance the profile)
-\end{itemize}
-
-
 \input{docs/3_hauptteil}
 \newpage
 
diff --git a/langfassung/master.toc b/langfassung/master.toc
index dce2f87..0c43803 100644
--- a/langfassung/master.toc
+++ b/langfassung/master.toc
@@ -2,32 +2,32 @@
 \contentsline {section}{\numberline {1}Einleitung}{2}{section.1}
 \contentsline {subsection}{\numberline {1.1}Themen}{2}{subsection.1.1}
 \contentsline {subsection}{\numberline {1.2}Motivation}{2}{subsection.1.2}
-\contentsline {paragraph}{\nonumberline \( \Phi \)}{2}{section*.2}
-\contentsline {paragraph}{\nonumberline \( \rho \)}{2}{section*.3}
-\contentsline {paragraph}{\nonumberline \( \rho _{new} \rightarrow (deriviation) \)}{2}{section*.4}
 \contentsline {section}{\numberline {2}Hauptteil}{3}{section.2}
-\contentsline {subsection}{\numberline {2.1}Notes}{3}{subsection.2.1}
-\contentsline {subsubsection}{\numberline {2.1.1}Todo}{3}{subsubsection.2.1.1}
-\contentsline {subsubsection}{\numberline {2.1.2}Generally building a galaxy}{3}{subsubsection.2.1.2}
-\contentsline {subsubsection}{\numberline {2.1.3}Spiral Galaxy}{3}{subsubsection.2.1.3}
-\contentsline {subsubsection}{\numberline {2.1.4}Important aspects of spiral galaxies}{3}{subsubsection.2.1.4}
-\contentsline {subsubsection}{\numberline {2.1.5}Speed}{4}{subsubsection.2.1.5}
-\contentsline {subsection}{\numberline {2.2}Navarro\IeC {\textendash }Frenk\IeC {\textendash }White profile}{4}{subsection.2.2}
-\contentsline {subsection}{\numberline {2.3}Einasto profile}{4}{subsection.2.3}
-\contentsline {subsection}{\numberline {2.4}Blender + Python}{4}{subsection.2.4}
-\contentsline {subsection}{\numberline {2.5}Making things faster}{5}{subsection.2.5}
-\contentsline {paragraph}{\nonumberline Kicking out to many Stars, 1 out of 10000 is just to much...}{5}{section*.5}
-\contentsline {subsection}{\numberline {2.6}Spiral Galaxies}{5}{subsection.2.6}
-\contentsline {subsubsection}{\numberline {2.6.1}N-body problem}{5}{subsubsection.2.6.1}
-\contentsline {subsubsection}{\numberline {2.6.2}Hilbert Spiral}{5}{subsubsection.2.6.2}
-\contentsline {subsection}{\numberline {2.7}Gr\IeC {\"o}\IeC {\ss }eneinheiten}{5}{subsection.2.7}
-\contentsline {section}{\numberline {3}Ergebnisse}{6}{section.3}
-\contentsline {subsection}{\numberline {3.1}Simulation Speed}{6}{subsection.3.1}
-\contentsline {subsection}{\numberline {3.2}Spiral Galaxies}{6}{subsection.3.2}
-\contentsline {subsection}{\numberline {3.3}Lookup-Table Speed}{6}{subsection.3.3}
-\contentsline {paragraph}{\nonumberline Python script}{6}{section*.6}
-\contentsline {subsection}{\numberline {3.4}Distortion of Galaxies}{7}{subsection.3.4}
-\contentsline {section}{\numberline {4}Quellen und Hilfen}{8}{section.4}
-\contentsline {paragraph}{\nonumberline Herrn J\IeC {\"o}rg Thar}{8}{section*.8}
-\contentsline {paragraph}{\nonumberline Konstantin Bosbach}{8}{section*.9}
-\contentsline {paragraph}{\nonumberline Tilman Hoffbauer}{8}{section*.10}
+\contentsline {subsection}{\numberline {2.1}Generierung der Elliptischen Galaxien}{3}{subsection.2.1}
+\contentsline {subsubsection}{\numberline {2.1.1}Das Navarro-Frenk-White Profil}{3}{subsubsection.2.1.1}
+\contentsline {subsubsection}{\numberline {2.1.2}Random Sampling}{3}{subsubsection.2.1.2}
+\contentsline {subsection}{\numberline {2.2}Generierung eines Dunkle-Materie Halos}{3}{subsection.2.2}
+\contentsline {subsubsection}{\numberline {2.2.1}Anpassung des NFW-Profils}{4}{subsubsection.2.2.1}
+\contentsline {subsection}{\numberline {2.3}Stauchung und Streckung der Galaxie}{4}{subsection.2.3}
+\contentsline {subsection}{\numberline {2.4}Beschleunigung der Generierung}{5}{subsection.2.4}
+\contentsline {subsubsection}{\numberline {2.4.1}n-Sterne}{5}{subsubsection.2.4.1}
+\contentsline {subsubsection}{\numberline {2.4.2}Lookuptable}{5}{subsubsection.2.4.2}
+\contentsline {subsubsection}{\numberline {2.4.3}Weitere Optimierungen}{5}{subsubsection.2.4.3}
+\contentsline {paragraph}{\nonumberline Nichts in der Konsole ausgeben:}{5}{section*.2}
+\contentsline {paragraph}{\nonumberline ...}{5}{section*.3}
+\contentsline {subsection}{\numberline {2.5}Nutzung eines Neuronalen Netzes zum unbeaufsichtigeten generieren}{6}{subsection.2.5}
+\contentsline {subsubsection}{\numberline {2.5.1}Aufbau des Neuronalen Netzes}{6}{subsubsection.2.5.1}
+\contentsline {subsubsection}{\numberline {2.5.2}Nutzung eines Neuronalen Netztes zur verbesserung von Galaxien Simulationen}{6}{subsubsection.2.5.2}
+\contentsline {subsection}{\numberline {2.6}Spiralgalaxien}{7}{subsection.2.6}
+\contentsline {subsubsection}{\numberline {2.6.1}Das n-K\IeC {\"o}rper Problem}{7}{subsubsection.2.6.1}
+\contentsline {subsection}{\numberline {2.7}Gr\IeC {\"o}\IeC {\ss }eneinheiten}{7}{subsection.2.7}
+\contentsline {section}{\numberline {3}Ergebnisse}{8}{section.3}
+\contentsline {subsection}{\numberline {3.1}Simulation Speed}{8}{subsection.3.1}
+\contentsline {subsection}{\numberline {3.2}Spiral Galaxies}{8}{subsection.3.2}
+\contentsline {subsection}{\numberline {3.3}Lookup-Table Speed}{8}{subsection.3.3}
+\contentsline {paragraph}{\nonumberline Python script}{8}{section*.4}
+\contentsline {subsection}{\numberline {3.4}Distortion of Galaxies}{9}{subsection.3.4}
+\contentsline {section}{\numberline {4}Quellen und Hilfen}{10}{section.4}
+\contentsline {paragraph}{\nonumberline Herrn J\IeC {\"o}rg Thar}{10}{section*.6}
+\contentsline {paragraph}{\nonumberline Konstantin Bosbach}{10}{section*.7}
+\contentsline {paragraph}{\nonumberline Tilman Hoffbauer}{10}{section*.8}