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authorhanemile <emile.hansmaennel@gmail.com>2018-03-07 21:03:53 +0100
committerhanemile <emile.hansmaennel@gmail.com>2018-03-07 21:03:53 +0100
commit38166a2e529de646f8143d02da57ac52d468b421 (patch)
tree1d0c70e4fa621545844e1f4f58bf05d994b8dddd /langfassung
parentafb5c2edf8920ece794dcea2bda4ab40661b4b79 (diff)
langfassung
Diffstat (limited to 'langfassung')
-rw-r--r--langfassung/docs/2_einleitung.aux30
-rw-r--r--langfassung/docs/3_hauptteil.aux67
-rw-r--r--langfassung/docs/4_ergebnisse.aux31
-rw-r--r--langfassung/docs/5_quellen.aux32
-rwxr-xr-xlangfassung/docs/6_abgabe.tex534
-rw-r--r--[-rwxr-xr-x]langfassung/master.aux251
-rw-r--r--[-rwxr-xr-x]langfassung/master.fdb_latexmk27
-rw-r--r--langfassung/master.fls8
-rw-r--r--[-rwxr-xr-x]langfassung/master.log456
-rw-r--r--[-rwxr-xr-x]langfassung/master.out27
-rw-r--r--[-rwxr-xr-x]langfassung/master.pdfbin878421 -> 952660 bytes
-rw-r--r--langfassung/master.synctex.gzbin142055 -> 404757 bytes
-rwxr-xr-xlangfassung/master.tex11
-rw-r--r--[-rwxr-xr-x]langfassung/master.toc131
14 files changed, 1199 insertions, 406 deletions
diff --git a/langfassung/docs/2_einleitung.aux b/langfassung/docs/2_einleitung.aux
deleted file mode 100644
index 44243d1..0000000
--- a/langfassung/docs/2_einleitung.aux
+++ /dev/null
@@ -1,30 +0,0 @@
-\relax 
-\providecommand\hyper@newdestlabel[2]{}
-\@writefile{toc}{\contentsline {subsection}{\numberline {1.1}Themen}{3}{subsection.1.1}}
-\@writefile{toc}{\contentsline {subsection}{\numberline {1.2}Motivation}{3}{subsection.1.2}}
-\@setckpt{docs/2_einleitung}{
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diff --git a/langfassung/docs/3_hauptteil.aux b/langfassung/docs/3_hauptteil.aux
deleted file mode 100644
index f598ad5..0000000
--- a/langfassung/docs/3_hauptteil.aux
+++ /dev/null
@@ -1,67 +0,0 @@
-\relax 
-\providecommand\hyper@newdestlabel[2]{}
-\@writefile{toc}{\contentsline {subsection}{\numberline {2.1}Generierung von elliptischen Galaxien}{5}{subsection.2.1}}
-\@writefile{toc}{\contentsline {subsubsection}{\numberline {2.1.1}Das Navarro-Frenk-White Profil}{5}{subsubsection.2.1.1}}
-\newlabel{eq:NFW_profile}{{1}{5}{Das Navarro-Frenk-White Profil}{equation.2.1}{}}
-\@writefile{toc}{\contentsline {subsubsection}{\numberline {2.1.2}Random Sampling}{5}{subsubsection.2.1.2}}
-\@writefile{lof}{\contentsline {figure}{\numberline {2}{\ignorespaces Eine mit dem NFW-profil und der Random Sampling Methode generierte Galaxie\relax }}{6}{figure.caption.3}}
-\providecommand*\caption@xref[2]{\@setref\relax\@undefined{#1}}
-\newlabel{fig:galaxy}{{2}{6}{Eine mit dem NFW-profil und der Random Sampling Methode generierte Galaxie\relax }{figure.caption.3}{}}
-\@writefile{lof}{\contentsline {figure}{\numberline {3}{\ignorespaces  Die Rho Funktion im Intervall \( [~0~;~10^7 ~] \) geplottet mithilfe von Logarithmischen Achsen.   Die x-Achse beschreibt die Entfernung zum Mittelpunkt der Galaxie   Die y-Achse beschreibt die Warscheinlichkeit das ein Stern generiert wird \relax }}{6}{figure.caption.4}}
-\newlabel{fig:rho}{{3}{6}{Die Rho Funktion im Intervall \( [~0~;~10^7 ~] \) geplottet mithilfe von Logarithmischen Achsen. \\ Die x-Achse beschreibt die Entfernung zum Mittelpunkt der Galaxie \\ Die y-Achse beschreibt die Warscheinlichkeit das ein Stern generiert wird \relax }{figure.caption.4}{}}
-\@writefile{toc}{\contentsline {subsubsection}{\numberline {2.1.3}Lookup Tabellen}{7}{subsubsection.2.1.3}}
-\newlabel{subsec:lookup}{{2.1.3}{7}{Lookup Tabellen}{subsubsection.2.1.3}{}}
-\@writefile{toc}{\contentsline {subsection}{\numberline {2.2}Generierung eines Dunkle-Materie Halos durch Anpassung des NFW-Profils}{7}{subsection.2.2}}
-\newlabel{eq:dark_matter}{{3}{7}{Generierung eines Dunkle-Materie Halos durch Anpassung des NFW-Profils}{equation.2.3}{}}
-\@writefile{toc}{\contentsline {subsection}{\numberline {2.3}Stauchung und Streckung der Galaxie}{7}{subsection.2.3}}
-\@writefile{toc}{\contentsline {subsection}{\numberline {2.4}Rechenaufwand}{8}{subsection.2.4}}
-\newlabel{subsec:big_o}{{2.4}{8}{Rechenaufwand}{subsection.2.4}{}}
-\@writefile{toc}{\contentsline {subsection}{\numberline {2.5}Beschleunigung der Generation}{8}{subsection.2.5}}
-\newlabel{subsec:speeding_things_up}{{2.5}{8}{Beschleunigung der Generation}{subsection.2.5}{}}
-\@writefile{toc}{\contentsline {subsubsection}{\numberline {2.5.1}Lookuptable}{8}{subsubsection.2.5.1}}
-\newlabel{subsec:lookuptable}{{2.5.1}{8}{Lookuptable}{subsubsection.2.5.1}{}}
-\@writefile{toc}{\contentsline {subsubsection}{\numberline {2.5.2}Mehr Rechenleistung!}{9}{subsubsection.2.5.2}}
-\@writefile{toc}{\contentsline {paragraph}{\nonumberline Amazon Web Services}{9}{section*.5}}
-\@writefile{toc}{\contentsline {subsubsection}{\numberline {2.5.3}Nichts in der Konsole ausgeben}{9}{subsubsection.2.5.3}}
-\@writefile{toc}{\contentsline {subsection}{\numberline {2.6}Nutzung eines neuronalen Netzes zum unbeaufsichtigten generieren von Galaxien}{9}{subsection.2.6}}
-\@writefile{toc}{\contentsline {subsubsection}{\numberline {2.6.1}Aufbau des neuronalen Netzes}{9}{subsubsection.2.6.1}}
-\@writefile{toc}{\contentsline {paragraph}{\nonumberline Neuronen und Synapsen}{10}{section*.6}}
-\@writefile{toc}{\contentsline {subsection}{\numberline {2.7}Spiralgalaxien}{11}{subsection.2.7}}
-\@writefile{toc}{\contentsline {subsubsection}{\numberline {2.7.1}Das n-K\IeC {\"o}rper Problem}{11}{subsubsection.2.7.1}}
-\newlabel{eq:n-body-2nd}{{10}{11}{Das n-Körper Problem}{equation.2.10}{}}
-\newlabel{eq:hamilton}{{11}{11}{Das n-Körper Problem}{equation.2.11}{}}
-\@writefile{toc}{\contentsline {subsubsection}{\numberline {2.7.2}Unterteilung des Vektorraumes in verschiedene Zellen}{12}{subsubsection.2.7.2}}
-\@writefile{toc}{\contentsline {subsubsection}{\numberline {2.7.3}Berechnung der wirkenden Kr\IeC {\"a}fte}{12}{subsubsection.2.7.3}}
-\newlabel{eq:gravitation_law}{{16}{12}{Berechnung der wirkenden Kräfte}{equation.2.16}{}}
-\@writefile{toc}{\contentsline {paragraph}{\nonumberline Masse der Sterne}{12}{section*.7}}
-\@writefile{toc}{\contentsline {paragraph}{\nonumberline Abstand der Sterne}{12}{section*.8}}
-\newlabel{eq:pytagoras}{{17}{12}{Abstand der Sterne}{equation.2.17}{}}
-\@writefile{toc}{\contentsline {subsection}{\numberline {2.8}Weiteres}{12}{subsection.2.8}}
-\@writefile{lof}{\contentsline {figure}{\numberline {4}{\ignorespaces Eine Spiralgalaxie generiert mithilfe von Daten aus dem Max-Plank-Institut in Heidelberg\relax }}{13}{figure.caption.9}}
-\newlabel{fig:spiralgalaxy}{{4}{13}{Eine Spiralgalaxie generiert mithilfe von Daten aus dem Max-Plank-Institut in Heidelberg\relax }{figure.caption.9}{}}
-\@setckpt{docs/3_hauptteil}{
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diff --git a/langfassung/docs/4_ergebnisse.aux b/langfassung/docs/4_ergebnisse.aux
deleted file mode 100644
index ec4d7a3..0000000
--- a/langfassung/docs/4_ergebnisse.aux
+++ /dev/null
@@ -1,31 +0,0 @@
-\relax 
-\providecommand\hyper@newdestlabel[2]{}
-\@writefile{toc}{\contentsline {subsection}{\numberline {3.1}Simulations Geschwindigkeit}{15}{subsection.3.1}}
-\@writefile{toc}{\contentsline {subsection}{\numberline {3.2}Lookuptabellen Geschwindigkeit}{15}{subsection.3.2}}
-\@writefile{toc}{\contentsline {subsection}{\numberline {3.3}Fazit}{15}{subsection.3.3}}
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diff --git a/langfassung/docs/5_quellen.aux b/langfassung/docs/5_quellen.aux
deleted file mode 100644
index f7e47e9..0000000
--- a/langfassung/docs/5_quellen.aux
+++ /dev/null
@@ -1,32 +0,0 @@
-\relax 
-\providecommand\hyper@newdestlabel[2]{}
-\@writefile{toc}{\contentsline {paragraph}{\nonumberline Herrn J\IeC {\"o}rg Thar}{17}{section*.11}}
-\@writefile{toc}{\contentsline {paragraph}{\nonumberline Tim Tugendhat}{17}{section*.12}}
-\@writefile{toc}{\contentsline {paragraph}{\nonumberline Konstantin Bosbach}{17}{section*.13}}
-\@writefile{toc}{\contentsline {paragraph}{\nonumberline Tilman Hoffbauer}{17}{section*.14}}
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diff --git a/langfassung/docs/6_abgabe.tex b/langfassung/docs/6_abgabe.tex
index 22fe221..f8112d9 100755
--- a/langfassung/docs/6_abgabe.tex
+++ b/langfassung/docs/6_abgabe.tex
@@ -59,7 +59,7 @@
 %     \end{pmatrix}
 % \end{equation}
 
-\subsection{Spiralgalaxien}
+\subsection{Spiralgalaxies}
 
 \subsection{Using Object Oriented Programming (OOP) techniques}
 
@@ -271,3 +271,535 @@ Not to be considered:
   \item any kind of resistance
   \item acceleration
 \end{itemize}
+
+\subsection{Notes}
+
+\begin{itemize}
+  \item Don't search for spheres very far away!
+\end{itemize}
+
+\begin{equation}
+  % \sum_{lower}^{upper} + \sum_{lower}^{upper} - \sum_{lower}^{upper}
+  \sum A_{fi} + \sum B_{fi} - \sum AB_{fij}
+\end{equation}
+
+\begin{itemize}
+  \item USE dictionaries to store which stars are in wich spheres
+\end{itemize}
+
+\subsection{exec.py}
+
+The exec.py file is used to execute the galaxytools defined in galaxytools.py.
+
+\subsubsection{Importing the galaxytools}
+
+\begin{lstlisting}
+  import galaxytools as galaxytools
+\end{lstlisting}
+
+The complete prgramm is compressed into one object. This Object has to be
+imported in order to be used.
+
+\subsubsection{Generate a new galaxy}
+
+\begin{lstlisting}
+  galaxy = galaxytools.new_galaxy(100)
+\end{lstlisting}
+
+Using the previously imported library, one can start building a galaxy by
+calling the function new\_galaxy(...). The parameter inside the braces defines
+the size of the galaxy.
+
+\subsubsection{Generate new stars in the galaxy}
+
+\begin{lstlisting}
+  galaxy.gen_new_stars(100)
+\end{lstlisting}
+
+The function new\_stars(...) is used to generate in given amount of new stars.
+
+\subsubsection{Print the coordinates of every star in the galaxy relative to
+the origin}
+
+\begin{lstlisting}
+  galaxy.print_stars()
+\end{lstlisting}
+
+Printing the coordinates of every star in the galaxy is useful for debugging:
+It is clearly visible if something is going wrong on the first look. The
+range of the galaxy might be wrong or the whole galaxy might be completely
+wrong scaled.
+
+\subsubsection{Calculate the forces acting inbetween all the stars in the
+galaxy}
+
+\begin{lstlisting}
+  galaxy.calc_all_forces()
+\end{lstlisting}
+
+the function calc\_all\_forces() if used to calculate all the forces acting
+in the selected galaxy. The O notation for this can be calculated using the
+following equation: \( O(n) = n^2 \).
+
+\subsubsection{Print the individual forces acting on the stars}
+
+\begin{lstlisting}
+  galaxy.print_individual_forces()
+\end{lstlisting}
+
+The individual forces (x, y, z) acting on the star can be printed out too!
+Just use the function print\_individual\_forces() and you will recieve the
+individual forces nicley formatted.
+
+\subsubsection{Generate the coordinates of the positions for the spheres}
+
+\begin{lstlisting}
+  galaxy.gen_sphere_positions(2)
+\end{lstlisting}
+
+To generate the sphere positions subdividing the galaxy, the
+gen\_sphere\_positions(...) function is utilized. The Parameter defines how
+many spheres are generated on one axis of the galaxy, so a higher value equals
+more spheres and so a longer time to compute. An infinite high value can be used
+if the value between each star should be calculated (use at own risk!).
+
+\subsubsection{Calculate the forces after 1 time step}
+
+\begin{lstlisting}
+  galaxy.gen_forces_after_t(1)
+\end{lstlisting}
+
+Calculating the new position after one timestep makes it possible to animate
+the galaxy and so visualizing it in an exciting way making people think you've
+done something awesome! This can be acchieved by using the
+gen\_forces\_after\_t(...) function. It uses a timestep as an argument and
+uses it to calculate the new coordinates of the star.
+
+\subsection{galaxytools.py}
+
+Inside this file, pretty much everything for building a galaxy is defined.
+
+\subsubsection{Importing important libraries}
+
+\begin{lstlisting}
+# Import libraries
+import math as math  # general math
+import numpy as np  # advanced math
+# import matplotlib.pyplot as plt  # plotting things
+\end{lstlisting}
+
+This part of the code is used to import libraries which are then used to do
+e.g. advanced math.
+
+\subsubsection{Generating the new\_galaxy class}
+
+\begin{lstlisting}
+# class used to create galaxies
+class new_galaxy(object):
+\end{lstlisting}
+
+The class definition defines the galaxytools classname as new\_galaxy
+
+\subsubsection{Initialisation}
+
+\begin{lstlisting}
+
+    # Initialisation
+    def __init__(self, galaxy_range):
+        print(
+            """>>> Initialising the list storing coordinates, forces and other
+            values"""
+        )
+
+        # list used for storing the coordinates os the stars
+        self.list_coords = []
+
+        # list storing the overall force acting on one star
+        self.list_force_star = []
+
+        # list storing the coordinates of the midpoints of the spheres dividing
+        # the galaxy into equaly big sized cells
+        self.list_sphere_coords = []
+
+        # self.list_sphere_stars = np.array(3, )
+
+        print("\tDone\n")
+        print(">>> Initialising variables and constants")
+
+        # variable storing the number of stars generated
+        self.num_of_stars = 0
+
+        self.galaxy_range = int(galaxy_range)
+
+        # define the universal gravitational constant
+        self.G = 6.67408 * 10e11
+
+        print("\tDone\n")
+
+\end{lstlisting}
+
+\subsubsection{Generating new stars}
+
+\begin{lstlisting}
+
+    # generate n new stars and store the coordinates in list_coords
+    # n = number of stars to be generated
+    # galaxy_range = size of the galaxy
+    def gen_new_stars(self, n):
+        print(">>> Generating Stars...")
+
+        # for a given number of stars
+        for i in range(0, n):
+
+            # generate a temporary random coordinate inside a given range using
+            # numpy
+            self.temp_coord = np.random.uniform(
+                low=0, high=self.galaxy_range, size=(4, ))
+
+            # append the random coordinate to the list storing the coordinates
+            self.list_coords.append(self.temp_coord)
+
+        # increment the generated star counter
+        self.num_of_stars += n
+        print("\tDone")
+        print("\tGenerated " + str(n) + " Stars\n")
+
+\end{lstlisting}
+
+\subsubsection{Print out all the star coordinates}
+
+\begin{lstlisting}
+
+    # print out all the coordinates in list_coords
+    def print_stars(self):
+        print(">>> Listing the coordinates of all stars:")
+        # print the coordinates of every star
+        for value in self.list_coords:
+            print(value)
+
+        print("\tDone\n")
+
+\end{lstlisting}
+
+\subsubsection{Calculate the forces acting inbetween two stars}
+
+\begin{lstlisting}
+
+    # calculate the forces acting between two stars on a specified axis
+    # star1 = coordinates of the first star
+    # star2 = coordinates of the second star
+    # axes = "x", "y" or "z" (CASE SENSITIVE!)
+    def calc_forces(self, star1, star2, axes):
+        if axes == "x":
+            mass = star1[3] * star2[3]
+            distance = math.sqrt(math.pow(star1[0] - star2[0], 2))
+        elif axes == "y":
+            mass = star1[3] * star2[3]
+            distance = math.sqrt(math.pow(star1[1] - star2[1], 2))
+        elif axes == "z":
+            mass = star1[3] * star2[3]
+            distance = math.sqrt(math.pow(star1[2] - star2[2], 2))
+
+        # stop division by zero
+        if distance == 0:
+            pass
+        else:
+            # return the acting force
+            return self.G * mass / math.pow(distance, 2)
+
+\end{lstlisting}
+
+\subsubsection{Calculate all the forces acting in the galaxy}
+
+\begin{lstlisting}
+
+    # calculate all the forces acting in the current galaxy
+    def calc_all_forces(self):
+        print(">>> Calculating all the forces acting inbetween the stars:")
+
+        if (self.num_of_stars <= 5):
+            # print some information above the columns
+            print(">>> Printing the forces acting inbetween every star")
+            print("{:-<60}".format(""))
+            print("\t| {:<3}| {:<3}| ".format("a", "b"))
+            print("\t+{:-<4}+{:-<4}+{:-<60} ".format("", "", ""))
+
+        else:
+            print("\t[W] Too many stars to print out!")
+            print("{:-<60}".format(""))
+
+        # for every star
+        for i in range(0, self.num_of_stars):
+
+            # initialize
+            self.force = 0
+
+            # every other star:
+            for j in range(0, self.num_of_stars):
+
+                # don't calculate the force between a star and and itself
+                if i != j and i < j:
+                    self.arr_force = np.array((0, 0, 0))
+
+                    # calculate the force between the two stars
+                    force_x = self.calc_forces(self.list_coords[i],
+                                               self.list_coords[j], "x")
+                    force_y = self.calc_forces(self.list_coords[i],
+                                               self.list_coords[j], "y")
+                    force_z = self.calc_forces(self.list_coords[i],
+                                               self.list_coords[j], "z")
+
+                    # print("overall force: ", end="")
+                    self.arr_force = np.array((force_x, force_y, force_z))
+
+                    if (self.num_of_stars <= 5):
+                        print("\t| {:<3}| {:<3}| {:<60}".format(
+                            str(i), str(j), str(self.arr_force)))
+                    """
+                    force_x = 42
+                    force_y = 36
+                    force_z = 24
+
+                    (0, 0, 0) --> (42, 36, 24)
+                    """
+
+            # append the variable to the list storing all the forces
+            self.list_force_star.append(self.arr_force)
+
+        print("{:-<60}".format(""))
+        print("\tDone\n")
+
+\end{lstlisting}
+
+\subsubsection{Print the individual forces acting on one star}
+
+\begin{lstlisting}
+
+    # print the individual forces acting on a star
+    def print_individual_forces(self, n=None, print_confirm=False):
+        print(">>> Printing the individual forces acting on every star")
+
+        if self.num_of_stars > 10:
+            print("\t[W] Too many stars to print out!")
+            print("{:-<60}".format(""))
+
+            for i in range(0, 3):
+                print("\t" + str(i) + " " + str(self.list_force_star[i]))
+
+            print("\n\t...\n")
+
+            for i in range(
+                    int(len(self.list_force_star) - 3),
+                    len(self.list_force_star)):
+                print("\t" + str(i) + " " + str(self.list_force_star[i]))
+            print("{:-<60}".format(""))
+
+        else:
+            print("{:-<60}".format(""))
+            if n is None:
+                # for value in self.list_force_star:
+                for i in range(0, len(self.list_force_star)):
+                    print("\t" + str(i) + " " + str(self.list_force_star[i]))
+            else:
+                print(self.list_force_star[n])
+
+            print("{:-<60}".format(""))
+            print("\tDone\n")
+
+\end{lstlisting}
+
+\subsubsection{Find out if a star is inside one sphere}
+
+\begin{lstlisting}
+
+    # star      [x, y, z, mass]
+    # sphere    [x, y, z, radius]
+    def is_star_in_sphere(self, star, sphere):
+
+        # define the sphere values
+        self.sphere_x = sphere[0]
+        self.sphere_y = sphere[1]
+        self.sphere_z = sphere[2]
+        self.sphere_r = sphere[3]
+
+        # define the star coordinates
+        self.star_x = star[0]
+        self.star_y = star[1]
+        self.star_z = star[2]
+
+        # find out the distance between the point and the center of the sphere
+        # if the distance is bigger than the radius of the sphere, the point is
+        # not inside the sphere. Elsewise, the point is inside the sphere
+
+        x = math.pow(self.sphere_x - self.star_x, 2)
+        y = math.pow(self.sphere_y - self.star_y, 2)
+        z = math.pow(self.sphere_z - self.star_z, 2)
+        r = math.sqrt(x + y + z)
+
+        if r > self.sphere_r:
+            return False
+        else:
+            return True
+
+        # self.sphere_x_neg = self.sphere_x - self.sphere_r
+        # self.sphere_x_pos = self.sphere_x + self.sphere_r
+        #
+        # self.sphere_y_neg = self.sphere_y - self.sphere_r
+        # self.sphere_y_pos = self.sphere_y + self.sphere_r
+        #
+        # self.sphere_z_neg = self.sphere_z - self.sphere_r
+        # self.sphere_z_pos = self.sphere_z + self.sphere_r
+        #
+        # # find out if the star is inside the sphere
+        # if self.sphere_x_neg < self.star_x < self.sphere_x_pos:
+        #     if self.sphere_y_neg < self.star_y < self.sphere_y_pos:
+        #         if self.sphere_z_neg < self.star_z < self.sphere_z_pos:
+        #             return True
+        #         else:
+        #             return False
+        #     else:
+        #         return False
+        # else:
+        #     return False
+
+\end{lstlisting}
+
+\subsubsection{Find out which star in in which spheres}
+
+\begin{lstlisting}
+
+    # find out which stars in in which spheres
+    def is_star_in_sphere_all(self):
+
+        # print(self.sphrer_rs)
+
+        print(">>> is_star_in_sphere_all")
+
+        # initialize a temporary counter in order to index the spheres
+        tmp_counter = 0
+
+        # cycle through all the stars
+        for sphere in self.sphere_coords:
+            # print("sphere: " + str(sphere))
+
+            tmp_list = []
+            for star in self.list_coords:
+                # parse the needed values from the sphere list
+
+                # if the star is inside the sphere
+                if (self.is_star_in_sphere(star, sphere) is True):
+                    # print("\nstar: " + str(star))
+
+                    star_x = []
+
+                    for value in star:
+                        # print(value, end=" ")
+                        # print("")
+                        star_x.append(value)
+
+                    # print("star_x :" + str(star_x))
+
+                    tmp_list.append(star_x)
+
+            # print("")
+            # print("tmp_list: " + str(tmp_list))
+            # print("END")
+
+            self.sphere_coords[sphere] = tmp_list
+
+        print("")
+        # print(self.sphere_coords)
+
+        # cycle through the dictionary storing which star is in which cell
+        for value in self.sphere_coords:
+            stars_in_sphere = self.sphere_coords[value]
+
+            # calculate the individual forces in the sphere
+            self.calc_forces_sphere(stars_in_sphere)
+
+            # for value in stars_in_sphere:
+                # print(value)
+
+\end{lstlisting}
+
+\subsubsection{Generate the sphere positions}
+
+\begin{lstlisting}
+
+    # function generating the positions of the sphere cells
+    def gen_sphere_positions(self, sampling_rate):
+
+        print(">>> Generating the sphere positions")
+
+        # initialize a dictionary linking the sphere coordinates to the
+        # coordiantes of the stars in the sphere
+        self.sphere_coords = {}
+
+        # calculate the distance between the midpoints of the spheres
+        sphere_distance = int(round(self.galaxy_range / sampling_rate, 0))
+
+        # define the sphere_radius
+        tmp_var = math.pow(sphere_distance, 2)
+        sphere_radius = math.sqrt(tmp_var + tmp_var + tmp_var)
+
+        # define a sphere counter for "labeling" the spheres
+        tmp_counter = 0
+
+        # cycle through all potential points
+        for i in range(-self.galaxy_range, self.galaxy_range, sphere_distance):
+            for j in range(-self.galaxy_range, self.galaxy_range,
+                           sphere_distance):
+                for k in range(-self.galaxy_range, self.galaxy_range,
+                               sphere_distance):
+
+                    # generate a temporary array combining all values
+                    # temp_arr = np.array((i, j, k, sphere_radius, tmp_counter))
+                    tmp_arr = (i, j, k, sphere_radius, tmp_counter)
+
+                    # append the array to the list storing the sphere infos
+                    # self.list_sphere_coords.append(temp_arr)
+
+                    # print("temp_arr: " + str(temp_arr))
+                    self.sphere_coords[tmp_arr[0:4]] = []
+
+                    # increment the sphere counter
+                    tmp_counter += 1
+
+        # print(self.sphere_coords)
+        print("\tDone\n")
+
+\end{lstlisting}
+
+\subsubsection{Calculate the forces acting inside the sphere}
+
+\begin{lstlisting}
+
+    def calc_forces_sphere(self, stars_in_sphere):
+        print("stars_in_sphere: ", end="")
+        print(stars_in_sphere)
+        # for value in stars_in_sphere:
+        #     self.calc_all_forces(stars_in_sphere)
+        #     print(value)
+
+\end{lstlisting}
+
+\subsubsection{calculate the forces acting in every sphere}
+
+\begin{lstlisting}
+
+    def calc_forces_sphere_all():
+        # for i in range(0, len(num_of_spheres)):
+        #     for star in sphere[i]:
+        #         for star_2 in len(0, num_of_stars_in_sphere[i])
+        #             a = calc_force(star, star[j])
+
+        pass
+
+    def gen_print_forces_after_t(t):
+        pass
+
+    # def all_stars_in_sphere(self, star, se)
+
+\end{lstlisting}
+
+\subsection{GAN}
diff --git a/langfassung/master.aux b/langfassung/master.aux
index 0a6be6c..7ee3365 100755..100644
--- a/langfassung/master.aux
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 \@auxtag{docs/6_abgabe}{%
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+\@writefile{toc}{\contentsline {subsubsection}{\numberline {5.12.13}calculate the forces acting in every sphere}{26}{subsubsection.5.12.13}}
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 \@auxtag{docs/6_abgabe}{%
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index 4204fe5..ce25814 100755..100644
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 # Fdb version 3
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+  "master.toc"
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+  "master.pdf"
diff --git a/langfassung/master.fls b/langfassung/master.fls
index 0ff9eba..7a9c757 100644
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 INPUT master.aux
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 INPUT /usr/share/texmf-dist/fonts/enc/dvips/lm/lm-rm.enc
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 INPUT /usr/share/texmf-dist/fonts/enc/dvips/lm/lm-mathit.enc
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 INPUT /usr/share/texmf-dist/fonts/type1/public/lm/lmr8.pfb
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diff --git a/langfassung/master.log b/langfassung/master.log
index f9b7036..e66875b 100755..100644
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diff --git a/langfassung/master.out b/langfassung/master.out
index 12deeab..21b162f 100755..100644
--- a/langfassung/master.out
+++ b/langfassung/master.out
@@ -26,7 +26,7 @@
 \BOOKMARK [2][-]{subsection.3.3}{Fazit}{section.3}% 26
 \BOOKMARK [1][-]{section.4}{Quellen und Hilfen}{}% 27
 \BOOKMARK [1][-]{section.5}{Nach der Abgabe...}{}% 28
-\BOOKMARK [2][-]{subsection.5.1}{Spiralgalaxien}{section.5}% 29
+\BOOKMARK [2][-]{subsection.5.1}{Spiralgalaxies}{section.5}% 29
 \BOOKMARK [2][-]{subsection.5.2}{Using Object Oriented Programming \(OOP\) techniques}{section.5}% 30
 \BOOKMARK [3][-]{subsubsection.5.2.1}{Initialisation}{subsection.5.2}% 31
 \BOOKMARK [2][-]{subsection.5.3}{Generation of new stars}{section.5}% 32
@@ -43,3 +43,28 @@
 \BOOKMARK [3][-]{subsubsection.5.8.6}{Calculate the forces acting on all the spheres together}{subsection.5.8}% 43
 \BOOKMARK [3][-]{subsubsection.5.8.7}{Benchmarks}{subsection.5.8}% 44
 \BOOKMARK [2][-]{subsection.5.9}{Calculate the Position of a Star after a timestep}{section.5}% 45
+\BOOKMARK [2][-]{subsection.5.10}{Notes}{section.5}% 46
+\BOOKMARK [2][-]{subsection.5.11}{exec.py}{section.5}% 47
+\BOOKMARK [3][-]{subsubsection.5.11.1}{Importing the galaxytools}{subsection.5.11}% 48
+\BOOKMARK [3][-]{subsubsection.5.11.2}{Generate a new galaxy}{subsection.5.11}% 49
+\BOOKMARK [3][-]{subsubsection.5.11.3}{Generate new stars in the galaxy}{subsection.5.11}% 50
+\BOOKMARK [3][-]{subsubsection.5.11.4}{Print the coordinates of every star in the galaxy relative to the origin}{subsection.5.11}% 51
+\BOOKMARK [3][-]{subsubsection.5.11.5}{Calculate the forces acting inbetween all the stars in the galaxy}{subsection.5.11}% 52
+\BOOKMARK [3][-]{subsubsection.5.11.6}{Print the individual forces acting on the stars}{subsection.5.11}% 53
+\BOOKMARK [3][-]{subsubsection.5.11.7}{Generate the coordinates of the positions for the spheres}{subsection.5.11}% 54
+\BOOKMARK [3][-]{subsubsection.5.11.8}{Calculate the forces after 1 time step}{subsection.5.11}% 55
+\BOOKMARK [2][-]{subsection.5.12}{galaxytools.py}{section.5}% 56
+\BOOKMARK [3][-]{subsubsection.5.12.1}{Importing important libraries}{subsection.5.12}% 57
+\BOOKMARK [3][-]{subsubsection.5.12.2}{Generating the new\137galaxy class}{subsection.5.12}% 58
+\BOOKMARK [3][-]{subsubsection.5.12.3}{Initialisation}{subsection.5.12}% 59
+\BOOKMARK [3][-]{subsubsection.5.12.4}{Generating new stars}{subsection.5.12}% 60
+\BOOKMARK [3][-]{subsubsection.5.12.5}{Print out all the star coordinates}{subsection.5.12}% 61
+\BOOKMARK [3][-]{subsubsection.5.12.6}{Calculate the forces acting inbetween two stars}{subsection.5.12}% 62
+\BOOKMARK [3][-]{subsubsection.5.12.7}{Calculate all the forces acting in the galaxy}{subsection.5.12}% 63
+\BOOKMARK [3][-]{subsubsection.5.12.8}{Print the individual forces acting on one star}{subsection.5.12}% 64
+\BOOKMARK [3][-]{subsubsection.5.12.9}{Find out if a star is inside one sphere}{subsection.5.12}% 65
+\BOOKMARK [3][-]{subsubsection.5.12.10}{Find out which star in in which spheres}{subsection.5.12}% 66
+\BOOKMARK [3][-]{subsubsection.5.12.11}{Generate the sphere positions}{subsection.5.12}% 67
+\BOOKMARK [3][-]{subsubsection.5.12.12}{Calculate the forces acting inside the sphere}{subsection.5.12}% 68
+\BOOKMARK [3][-]{subsubsection.5.12.13}{calculate the forces acting in every sphere}{subsection.5.12}% 69
+\BOOKMARK [2][-]{subsection.5.13}{GAN}{section.5}% 70
diff --git a/langfassung/master.pdf b/langfassung/master.pdf
index 350b08d..d17c7a6 100755..100644
--- a/langfassung/master.pdf
+++ b/langfassung/master.pdf
Binary files differdiff --git a/langfassung/master.synctex.gz b/langfassung/master.synctex.gz
index 02b0ae0..19617b2 100644
--- a/langfassung/master.synctex.gz
+++ b/langfassung/master.synctex.gz
Binary files differdiff --git a/langfassung/master.tex b/langfassung/master.tex
index 75e232f..7717ab0 100755
--- a/langfassung/master.tex
+++ b/langfassung/master.tex
@@ -31,7 +31,7 @@
 % ???
 \usepackage{textcomp}
 % \usepackage[onehalfspacing]{setspace}
-% \linespread{0.95}
+\linespread{1}
 
 % Tikz for designing graphics
 \usepackage{tikz}
@@ -42,8 +42,8 @@
 \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
+  % 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
 }
 
 % hyperref / hyperlink settings
@@ -104,27 +104,22 @@ urlcolor=black
 \newpage
 
 \section{Einleitung} \label{Einleitung}
-% \input{docs/2_einleitung}
 \include*{docs/2_einleitung}
 \newpage
 
 \section{Hauptteil} \label{Hauptteil}
-% \input{docs/3_hauptteil}
 \include*{docs/3_hauptteil}
 \newpage
 
 \section{Ergebnisse} \label{ergebnisse}
-% \input{docs/4_ergebnisse}
 \include*{docs/4_ergebnisse}
 \newpage
 
 \section{Quellen und Hilfen} \label{quellen}
-% \input{docs/5_quellen}
 \include*{docs/5_quellen}
 \newpage
 
 \section{Nach der Abgabe...} \label{nach_der_abgabe}
-% \input{docs/6_abgabe}
 \include*{docs/6_abgabe}
 
 \end{document}
diff --git a/langfassung/master.toc b/langfassung/master.toc
index 4a85a0f..cc8faf0 100755..100644
--- a/langfassung/master.toc
+++ b/langfassung/master.toc
@@ -1,54 +1,79 @@
 \select@language {ngerman}
-\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 {section}{\numberline {2}Hauptteil}{3}{section.2}
-\contentsline {subsection}{\numberline {2.1}Generierung von 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 {subsubsection}{\numberline {2.1.3}Lookup Tabellen}{5}{subsubsection.2.1.3}
-\contentsline {subsection}{\numberline {2.2}Generierung eines Dunkle-Materie Halos durch Anpassung des NFW-Profils}{5}{subsection.2.2}
-\contentsline {subsection}{\numberline {2.3}Stauchung und Streckung der Galaxie}{5}{subsection.2.3}
-\contentsline {subsection}{\numberline {2.4}Rechenaufwand}{6}{subsection.2.4}
-\contentsline {subsection}{\numberline {2.5}Beschleunigung der Generation}{6}{subsection.2.5}
-\contentsline {subsubsection}{\numberline {2.5.1}Lookuptable}{6}{subsubsection.2.5.1}
-\contentsline {subsubsection}{\numberline {2.5.2}Mehr Rechenleistung!}{7}{subsubsection.2.5.2}
-\contentsline {paragraph}{\nonumberline Amazon Web Services}{7}{section*.5}
-\contentsline {subsubsection}{\numberline {2.5.3}Nichts in der Konsole ausgeben}{7}{subsubsection.2.5.3}
-\contentsline {subsection}{\numberline {2.6}Nutzung eines neuronalen Netzes zum unbeaufsichtigten generieren von Galaxien}{7}{subsection.2.6}
-\contentsline {subsubsection}{\numberline {2.6.1}Aufbau des neuronalen Netzes}{7}{subsubsection.2.6.1}
-\contentsline {paragraph}{\nonumberline Neuronen und Synapsen}{8}{section*.6}
-\contentsline {subsection}{\numberline {2.7}Spiralgalaxien}{9}{subsection.2.7}
-\contentsline {subsubsection}{\numberline {2.7.1}Das n-K\IeC {\"o}rper Problem}{9}{subsubsection.2.7.1}
-\contentsline {subsubsection}{\numberline {2.7.2}Unterteilung des Vektorraumes in verschiedene Zellen}{10}{subsubsection.2.7.2}
-\contentsline {subsubsection}{\numberline {2.7.3}Berechnung der wirkenden Kr\IeC {\"a}fte}{10}{subsubsection.2.7.3}
-\contentsline {paragraph}{\nonumberline Masse der Sterne}{10}{section*.7}
-\contentsline {paragraph}{\nonumberline Abstand der Sterne}{10}{section*.8}
-\contentsline {subsection}{\numberline {2.8}Weiteres}{10}{subsection.2.8}
-\contentsline {section}{\numberline {3}Ergebnisse}{11}{section.3}
-\contentsline {subsection}{\numberline {3.1}Simulations Geschwindigkeit}{11}{subsection.3.1}
-\contentsline {subsection}{\numberline {3.2}Lookuptabellen Geschwindigkeit}{11}{subsection.3.2}
-\contentsline {subsection}{\numberline {3.3}Fazit}{11}{subsection.3.3}
-\contentsline {section}{\numberline {4}Quellen und Hilfen}{13}{section.4}
-\contentsline {paragraph}{\nonumberline Herrn J\IeC {\"o}rg Thar}{13}{section*.11}
-\contentsline {paragraph}{\nonumberline Tim Tugendhat}{13}{section*.12}
-\contentsline {paragraph}{\nonumberline Konstantin Bosbach}{13}{section*.13}
-\contentsline {paragraph}{\nonumberline Tilman Hoffbauer}{13}{section*.14}
-\contentsline {section}{\numberline {5}Nach der Abgabe...}{14}{section.5}
-\contentsline {subsection}{\numberline {5.1}Spiralgalaxien}{14}{subsection.5.1}
-\contentsline {subsection}{\numberline {5.2}Using Object Oriented Programming (OOP) techniques}{14}{subsection.5.2}
-\contentsline {subsubsection}{\numberline {5.2.1}Initialisation}{14}{subsubsection.5.2.1}
-\contentsline {subsection}{\numberline {5.3}Generation of new stars}{14}{subsection.5.3}
-\contentsline {subsection}{\numberline {5.4}Printing all the coordinates}{14}{subsection.5.4}
-\contentsline {subsection}{\numberline {5.5}Calculating the Forces acting between the Stars}{14}{subsection.5.5}
-\contentsline {subsection}{\numberline {5.6}Calculating the forces acting between each star in the galaxy and each other star}{15}{subsection.5.6}
-\contentsline {subsection}{\numberline {5.7}Printing all the individual forces}{15}{subsection.5.7}
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-\contentsline {subsubsection}{\numberline {5.8.1}Testing if a point is inside or outside a sphere}{15}{subsubsection.5.8.1}
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-\contentsline {subsubsection}{\numberline {5.8.3}Generate the position of the spheres}{15}{subsubsection.5.8.3}
-\contentsline {subsubsection}{\numberline {5.8.4}The radius of the spheres}{16}{subsubsection.5.8.4}
-\contentsline {subsubsection}{\numberline {5.8.5}Calculate the forces acting on the spheres}{16}{subsubsection.5.8.5}
-\contentsline {subsubsection}{\numberline {5.8.6}Calculate the forces acting on all the spheres together}{17}{subsubsection.5.8.6}
-\contentsline {subsubsection}{\numberline {5.8.7}Benchmarks}{17}{subsubsection.5.8.7}
-\contentsline {subsection}{\numberline {5.9}Calculate the Position of a Star after a timestep}{17}{subsection.5.9}
+\contentsline {section}{\numberline {1}Einleitung}{3}{section.1}
+\contentsline {subsection}{\numberline {1.1}Themen}{3}{subsection.1.1}
+\contentsline {subsection}{\numberline {1.2}Motivation}{3}{subsection.1.2}
+\contentsline {section}{\numberline {2}Hauptteil}{4}{section.2}
+\contentsline {subsection}{\numberline {2.1}Generierung von elliptischen Galaxien}{4}{subsection.2.1}
+\contentsline {subsubsection}{\numberline {2.1.1}Das Navarro-Frenk-White Profil}{4}{subsubsection.2.1.1}
+\contentsline {subsubsection}{\numberline {2.1.2}Random Sampling}{4}{subsubsection.2.1.2}
+\contentsline {subsubsection}{\numberline {2.1.3}Lookup Tabellen}{6}{subsubsection.2.1.3}
+\contentsline {subsection}{\numberline {2.2}Generierung eines Dunkle-Materie Halos durch Anpassung des NFW-Profils}{6}{subsection.2.2}
+\contentsline {subsection}{\numberline {2.3}Stauchung und Streckung der Galaxie}{6}{subsection.2.3}
+\contentsline {subsection}{\numberline {2.4}Rechenaufwand}{7}{subsection.2.4}
+\contentsline {subsection}{\numberline {2.5}Beschleunigung der Generation}{7}{subsection.2.5}
+\contentsline {subsubsection}{\numberline {2.5.1}Lookuptable}{7}{subsubsection.2.5.1}
+\contentsline {subsubsection}{\numberline {2.5.2}Mehr Rechenleistung!}{8}{subsubsection.2.5.2}
+\contentsline {paragraph}{\nonumberline Amazon Web Services}{8}{section*.5}
+\contentsline {subsubsection}{\numberline {2.5.3}Nichts in der Konsole ausgeben}{8}{subsubsection.2.5.3}
+\contentsline {subsection}{\numberline {2.6}Nutzung eines neuronalen Netzes zum unbeaufsichtigten generieren von Galaxien}{8}{subsection.2.6}
+\contentsline {subsubsection}{\numberline {2.6.1}Aufbau des neuronalen Netzes}{8}{subsubsection.2.6.1}
+\contentsline {paragraph}{\nonumberline Neuronen und Synapsen}{9}{section*.6}
+\contentsline {subsection}{\numberline {2.7}Spiralgalaxien}{10}{subsection.2.7}
+\contentsline {subsubsection}{\numberline {2.7.1}Das n-K\IeC {\"o}rper Problem}{10}{subsubsection.2.7.1}
+\contentsline {subsubsection}{\numberline {2.7.2}Unterteilung des Vektorraumes in verschiedene Zellen}{11}{subsubsection.2.7.2}
+\contentsline {subsubsection}{\numberline {2.7.3}Berechnung der wirkenden Kr\IeC {\"a}fte}{11}{subsubsection.2.7.3}
+\contentsline {paragraph}{\nonumberline Masse der Sterne}{11}{section*.7}
+\contentsline {paragraph}{\nonumberline Abstand der Sterne}{11}{section*.8}
+\contentsline {subsection}{\numberline {2.8}Weiteres}{11}{subsection.2.8}
+\contentsline {section}{\numberline {3}Ergebnisse}{12}{section.3}
+\contentsline {subsection}{\numberline {3.1}Simulations Geschwindigkeit}{12}{subsection.3.1}
+\contentsline {subsection}{\numberline {3.2}Lookuptabellen Geschwindigkeit}{12}{subsection.3.2}
+\contentsline {subsection}{\numberline {3.3}Fazit}{12}{subsection.3.3}
+\contentsline {section}{\numberline {4}Quellen und Hilfen}{14}{section.4}
+\contentsline {paragraph}{\nonumberline Herrn J\IeC {\"o}rg Thar}{14}{section*.11}
+\contentsline {paragraph}{\nonumberline Tim Tugendhat}{14}{section*.12}
+\contentsline {paragraph}{\nonumberline Konstantin Bosbach}{14}{section*.13}
+\contentsline {paragraph}{\nonumberline Tilman Hoffbauer}{14}{section*.14}
+\contentsline {section}{\numberline {5}Nach der Abgabe...}{15}{section.5}
+\contentsline {subsection}{\numberline {5.1}Spiralgalaxies}{15}{subsection.5.1}
+\contentsline {subsection}{\numberline {5.2}Using Object Oriented Programming (OOP) techniques}{15}{subsection.5.2}
+\contentsline {subsubsection}{\numberline {5.2.1}Initialisation}{15}{subsubsection.5.2.1}
+\contentsline {subsection}{\numberline {5.3}Generation of new stars}{15}{subsection.5.3}
+\contentsline {subsection}{\numberline {5.4}Printing all the coordinates}{15}{subsection.5.4}
+\contentsline {subsection}{\numberline {5.5}Calculating the Forces acting between the Stars}{15}{subsection.5.5}
+\contentsline {subsection}{\numberline {5.6}Calculating the forces acting between each star in the galaxy and each other star}{16}{subsection.5.6}
+\contentsline {subsection}{\numberline {5.7}Printing all the individual forces}{16}{subsection.5.7}
+\contentsline {subsection}{\numberline {5.8}Spherical cells}{16}{subsection.5.8}
+\contentsline {subsubsection}{\numberline {5.8.1}Testing if a point is inside or outside a sphere}{16}{subsubsection.5.8.1}
+\contentsline {subsubsection}{\numberline {5.8.2}Testing if a star is inside or outside of a sphere for a whole galaxy}{16}{subsubsection.5.8.2}
+\contentsline {subsubsection}{\numberline {5.8.3}Generate the position of the spheres}{16}{subsubsection.5.8.3}
+\contentsline {subsubsection}{\numberline {5.8.4}The radius of the spheres}{17}{subsubsection.5.8.4}
+\contentsline {subsubsection}{\numberline {5.8.5}Calculate the forces acting on the spheres}{17}{subsubsection.5.8.5}
+\contentsline {subsubsection}{\numberline {5.8.6}Calculate the forces acting on all the spheres together}{18}{subsubsection.5.8.6}
+\contentsline {subsubsection}{\numberline {5.8.7}Benchmarks}{18}{subsubsection.5.8.7}
+\contentsline {subsection}{\numberline {5.9}Calculate the Position of a Star after a timestep}{18}{subsection.5.9}
+\contentsline {subsection}{\numberline {5.10}Notes}{18}{subsection.5.10}
+\contentsline {subsection}{\numberline {5.11}exec.py}{19}{subsection.5.11}
+\contentsline {subsubsection}{\numberline {5.11.1}Importing the galaxytools}{19}{subsubsection.5.11.1}
+\contentsline {subsubsection}{\numberline {5.11.2}Generate a new galaxy}{19}{subsubsection.5.11.2}
+\contentsline {subsubsection}{\numberline {5.11.3}Generate new stars in the galaxy}{19}{subsubsection.5.11.3}
+\contentsline {subsubsection}{\numberline {5.11.4}Print the coordinates of every star in the galaxy relative to the origin}{19}{subsubsection.5.11.4}
+\contentsline {subsubsection}{\numberline {5.11.5}Calculate the forces acting inbetween all the stars in the galaxy}{19}{subsubsection.5.11.5}
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+\contentsline {subsubsection}{\numberline {5.11.8}Calculate the forces after 1 time step}{20}{subsubsection.5.11.8}
+\contentsline {subsection}{\numberline {5.12}galaxytools.py}{20}{subsection.5.12}
+\contentsline {subsubsection}{\numberline {5.12.1}Importing important libraries}{20}{subsubsection.5.12.1}
+\contentsline {subsubsection}{\numberline {5.12.2}Generating the new\_galaxy class}{20}{subsubsection.5.12.2}
+\contentsline {subsubsection}{\numberline {5.12.3}Initialisation}{20}{subsubsection.5.12.3}
+\contentsline {subsubsection}{\numberline {5.12.4}Generating new stars}{21}{subsubsection.5.12.4}
+\contentsline {subsubsection}{\numberline {5.12.5}Print out all the star coordinates}{21}{subsubsection.5.12.5}
+\contentsline {subsubsection}{\numberline {5.12.6}Calculate the forces acting inbetween two stars}{22}{subsubsection.5.12.6}
+\contentsline {subsubsection}{\numberline {5.12.7}Calculate all the forces acting in the galaxy}{22}{subsubsection.5.12.7}
+\contentsline {subsubsection}{\numberline {5.12.8}Print the individual forces acting on one star}{23}{subsubsection.5.12.8}
+\contentsline {subsubsection}{\numberline {5.12.9}Find out if a star is inside one sphere}{24}{subsubsection.5.12.9}
+\contentsline {subsubsection}{\numberline {5.12.10}Find out which star in in which spheres}{25}{subsubsection.5.12.10}
+\contentsline {subsubsection}{\numberline {5.12.11}Generate the sphere positions}{26}{subsubsection.5.12.11}
+\contentsline {subsubsection}{\numberline {5.12.12}Calculate the forces acting inside the sphere}{26}{subsubsection.5.12.12}
+\contentsline {subsubsection}{\numberline {5.12.13}calculate the forces acting in every sphere}{26}{subsubsection.5.12.13}
+\contentsline {subsection}{\numberline {5.13}GAN}{27}{subsection.5.13}