moar pages ....

arbeit/bibma.bib

@@ -75,3 +75,15 @@
   organization={Springer},
   url={https://www.lri.fr/~hansen/proceedings/2014/PPSN/papers/8672/86720465.pdf}
 }
+@article{gaussNewton,
+author = {Donald W. Marquardt},
+title = {An Algorithm for Least-Squares Estimation of Nonlinear Parameters},
+journal = {Journal of the Society for Industrial and Applied Mathematics},
+volume = {11},
+number = {2},
+pages = {431-441},
+year = {1963},
+doi = {10.1137/0111030},
+URL = {https://doi.org/10.1137/0111030},
+eprint = {https://doi.org/10.1137/0111030}
+}

arbeit/files/erklaerung.aux

@@ -22,15 +22,15 @@
 \setcounter{ContinuedFloat}{0}
 \setcounter{float@type}{16}
 \setcounter{lstnumber}{1}
-\setcounter{NAT@ctr}{8}
+\setcounter{NAT@ctr}{9}
 \setcounter{AM@survey}{0}
 \setcounter{r@tfl@t}{0}
 \setcounter{subfigure}{0}
 \setcounter{subtable}{0}
-\setcounter{@todonotes@numberoftodonotes}{2}
+\setcounter{@todonotes@numberoftodonotes}{4}
 \setcounter{Item}{0}
 \setcounter{Hfootnote}{2}
-\setcounter{bookmark@seq@number}{19}
+\setcounter{bookmark@seq@number}{21}
 \setcounter{algorithm}{0}
 \setcounter{ALC@unique}{0}
 \setcounter{ALC@line}{0}

arbeit/ma.md

@@ -1,5 +1,5 @@
 ---
-fontsize: 11pt
+fontsize: 12pt
 ---
 
 \chapter*{How to read this Thesis}
@@ -36,6 +36,7 @@ We will replicate the same setup on the same meshes but use \acf{FFD} instead of
 work as a predictor given the different deformation scheme.
 
 ## What is \acf{FFD}?
+\label{sec:intro:ffd}
 
 First of all we have to establish how a \ac{FFD} works and why this is a good
 tool for deforming meshes in the first place. For simplicity we only summarize
@@ -114,11 +115,11 @@ mesh albeit the downsides.
 
 ## What is evolutional optimization?
 
-
+\change[inline]{Write this section}
 
 ## Advantages of evolutional algorithms
 
-\improvement[inline]{Needs citations}
+\change[inline]{Needs citations}
 The main advantage of evolutional algorithms is the ability to find optima of
 general functions just with the help of a given error-function (or
 fitness-function in this domain). This avoids the general pitfalls of
@@ -197,12 +198,116 @@ $\|\vec{G}\|_F = 1$, whereby $\|\cdot\|_F$ denotes the Frobenius-Norm.
 
 # Implementation of \acf{FFD}
 
-## Was ist FFD?
+As general B-Splines have a free parameters $d$ and $\tau$.
+
+As we usually work with regular grids in our \ac{FFD} we define $\tau$
+statically as
+$$\tau_i = \nicefrac{i}{n}$$
+whereby $n$ is the number of control-points in that direction.
+
+$d$ defines the *degree* of the B-Spline-Function (the number of times this
+function is differentiable) and for our purposes we fix $d$ to $3$, but give the
+formulas for the general case so it can be adapted quite freely.
+
+
+## Adaption of \ac{FFD}
+
+As we have established in Chapter \ref{sec:intro:ffd} we can define an
+\ac{FFD}-displacement as
+\begin{equation}
+\Delta_x(u) = \sum_i N_{i,d,\tau_i}(u) \Delta_x c_i
+\end{equation}
+
+Note that we only sum up the $\Delta$-displacements in the control points $c_i$ to get
+the change in position of the point we are interested in.
+
+In this way every deformed vertex is defined by
+$$
+\textrm{Deform}(v_x) = v_x + \Delta_x(u)
+$$
+with $u \in [0..1[$ being the variable that connects the high-detailed
+vertex-mesh to the low-detailed control-grid. To actually calculate the new
+position of the vertex we first have to calculate the $u$-value for each
+vertex. This is achieved by finding out the parametrization of $v$ in terms of
+$c_i$
+$$
+v_x = \sum_i N_{i,d,\tau_i}(u) c_i
+$$
+
+As the B-Spline-functions are smooth and convex we just derive by $u$ yielding
+
+\begin{eqnarray*}
+& \frac{\partial}{\partial u} & v_x - \sum_i N_{i,d,\tau_i}(u) c_i \\
+& = & v_x - \sum_i \left( \frac{d}{\tau_{i+d} - \tau_i} N_{i,d-1,\tau}(u) - \frac{d}{\tau_{i+d+1} - \tau_{i+1}} N_{i+1,d-1,\tau}(u) \right) c_i
+\end{eqnarray*}
+
+and do a gradient-descend to approximate the value of $u$ up to an $\epsilon$ of $0.0001$.
+
+For this we use the Gauss-Newton algorithm\cite{gaussNewton} as the solution to
+this problem may not be deterministic, because we usually have way more vertices
+than control points ($\#v \gg \#c$).
+
+## Adaption of \ac{FFD} for a 3D-Mesh
 \label{3dffd}
 
-- Definition
+This is a straightforward extension of the 1D-method presented in the last
-- Wieso Newton-Optimierung?
+chapter. But this time things get a bit more complicated. As we have a
-  - Was folgt daraus?
+3-dimensional grid we may have a different amount of control-points in each
+direction.
+
+Given $n,m,o$ control points in $x,y,z$-direction each Point on the curve is
+defined by
+$$V(u,v,w) = \sum_{i=0}^{n-d-2} \sum_{j=0}^{m-d-2} \sum_{k=0}^{o-d-2} N_{i,d,\tau_i}(u) N_{j,d,\tau_j}(v) N_{k,d,\tau_k}(w) \cdot C_{ijk}.$$
+
+In this case we have three different B-Splines (one for each dimension) and also
+3 variables $u,v,w$ for each vertex we want to approximate.
+
+Given a target vertex $\vec{p}^*$ and an initial guess $\vec{p}=V(u,v,w)$
+we define the error-function for the gradient-descent as:
+
+$$Err(u,v,w,\vec{p}^{*}) = \vec{p}^{*} - V(u,v,w)$$
+
+And the partial version for just one direction as
+
+$$Err_x(u,v,w,\vec{p}^{*}) = p^{*}_x - \sum_{i=0}^{n-d-2} \sum_{j=0}^{m-d-2} \sum_{k=0}^{o-d-2} {C_{ijk}}_x N_{i,d,\tau_i}(u) N_{j,d,\tau_j}(v) N_{k,d,\tau_k}(w) $$
+
+To solve this we derive partially, like before:
+
+$$
+\begin{array}{rl}
+    \displaystyle \frac{\partial Err_x}{\partial u} & p^{*}_x - \displaystyle \sum_{i=0}^{n-d-2} \sum_{j=0}^{m-d-2} \sum_{k=0}^{o-d-2} {C_{ijk}}_x N_{i,d,\tau_i}(u) N_{j,d,\tau_j}(v) N_{k,d,\tau_k}(w) \\
+  = & \displaystyle - \sum_{i=0}^{n-d-2} \sum_{j=0}^{m-d-2} \sum_{k=0}^{o-d-2} {C_{ijk}}_x N'_i(u) N_{j,d,\tau_j}(v) N_{k,d,\tau_k}(w)
+\end{array}
+$$
+
+The other partial derivatives follow the same pattern yielding the Jacobian:
+
+$$
+J(Err(u,v,w)) = 
+\left(
+\begin{array}{ccc}
+\frac{\partial Err_x}{\partial u} & \frac{\partial Err_x}{\partial v} & \frac{\partial Err_x}{\partial w} \\
+\frac{\partial Err_y}{\partial u} & \frac{\partial Err_y}{\partial v} & \frac{\partial Err_y}{\partial w} \\
+\frac{\partial Err_z}{\partial u} & \frac{\partial Err_z}{\partial v} & \frac{\partial Err_z}{\partial w}
+\end{array}
+\right)
+$$
+
+\unsure[inline]{Should I add an informal complete derivative?\newline
+Like leaving out Sums & $i,j,k$-Indices to make obvious what derivative belongs
+where in what case?}
+
+With the Gauss-Newton algorithm we iterate the formula
+$$J(Err(u,v,w)) \cdot \Delta \left( \begin{array}{c} u \\ v \\ w \end{array} \right) = -Err(u,v,w)$$
+and use Cramers rule for inverting the small Jacobian and solving this system of
+linear equations.
+
+
+## Parametrisierung sinnvoll?
+
+- Nachteile von Parametrisierung
+  - Deformation ist um einen Kontrollpunkt viel direkter zu steuern.
+  - => DM-FFD?
 
 ## Test Scenario: 1D Function Approximation
 

arbeit/ma.tex

@@ -2,7 +2,7 @@
 % abstracton : Abstract mit Ueberschrift
 \documentclass[
 a4paper,				% default
-11pt, 					% default = 11pt
+12pt, 					% default = 11pt
 BCOR6mm,				% Bindungskorrektur bei Klebebindung 6mm, bei Lochen BCOR8.25mm
 twoside,				% default, 2seitig
 titlepage,
@@ -30,6 +30,7 @@ xcolor=dvipsnames,
 %\setlength{\parindent}{0pt} % kein einzug bei absaetzen
 %\setlength{\lineskip}{1ex plus0.5ex minus0.5ex} % dafr abstand zwischen abs<62>zen (funktioniert noch nicht)
 % \renewcommand{\familydefault}{\sfdefault}
+\setstretch{1.44} % 1.5-facher zeilenabstand
 
 %%%%%%%%%%%%%%% Header - Footer %%%%%%%%%%%%%%%
 % ### Fr 2 Seitig (option twopage):
@@ -162,6 +163,8 @@ deformation scheme.
 
 \section{\texorpdfstring{What is \acf{FFD}?}{What is ?}}\label{what-is}
 
+\label{sec:intro:ffd}
+
 First of all we have to establish how a \ac{FFD} works and why this is a
 good tool for deforming meshes in the first place. For simplicity we
 only summarize the 1D-case from \cite{spitzmuller1996bezier} here and go
@@ -243,10 +246,12 @@ high-polygon mesh albeit the downsides.
 \section{What is evolutional
 optimization?}\label{what-is-evolutional-optimization}
 
+\change[inline]{Write this section}
+
 \section{Advantages of evolutional
 algorithms}\label{advantages-of-evolutional-algorithms}
 
-\improvement[inline]{Needs citations} The main advantage of evolutional
+\change[inline]{Needs citations} The main advantage of evolutional
 algorithms is the ability to find optima of general functions just with
 the help of a given error-function (or fitness-function in this domain).
 This avoids the general pitfalls of gradient-based procedures, which
@@ -330,18 +335,125 @@ Frobenius-Norm.
 \chapter{\texorpdfstring{Implementation of
 \acf{FFD}}{Implementation of }}\label{implementation-of}
 
-\section{Was ist FFD?}\label{was-ist-ffd}
+As general B-Splines have a free parameters \(d\) and \(\tau\).
+
+As we usually work with regular grids in our \ac{FFD} we define \(\tau\)
+statically as \[\tau_i = \nicefrac{i}{n}\] whereby \(n\) is the number
+of control-points in that direction.
+
+\(d\) defines the \emph{degree} of the B-Spline-Function (the number of
+times this function is differentiable) and for our purposes we fix \(d\)
+to \(3\), but give the formulas for the general case so it can be
+adapted quite freely.
+
+\section{\texorpdfstring{Adaption of
+\ac{FFD}}{Adaption of }}\label{adaption-of}
+
+As we have established in Chapter \ref{sec:intro:ffd} we can define an
+\ac{FFD}-displacement as
+
+\begin{equation}
+\Delta_x(u) = \sum_i N_{i,d,\tau_i}(u) \Delta_x c_i
+\end{equation}
+
+Note that we only sum up the \(\Delta\)-displacements in the control
+points \(c_i\) to get the change in position of the point we are
+interested in.
+
+In this way every deformed vertex is defined by \[
+\textrm{Deform}(v_x) = v_x + \Delta_x(u)
+\] with \(u \in [0..1[\) being the variable that connects the
+high-detailed vertex-mesh to the low-detailed control-grid. To actually
+calculate the new position of the vertex we first have to calculate the
+\(u\)-value for each vertex. This is achieved by finding out the
+parametrization of \(v\) in terms of \(c_i\) \[
+v_x = \sum_i N_{i,d,\tau_i}(u) c_i
+\]
+
+As the B-Spline-functions are smooth and convex we just derive by \(u\)
+yielding
+
+\begin{eqnarray*}
+& \frac{\partial}{\partial u} & v_x - \sum_i N_{i,d,\tau_i}(u) c_i \\
+& = & v_x - \sum_i \left( \frac{d}{\tau_{i+d} - \tau_i} N_{i,d-1,\tau}(u) - \frac{d}{\tau_{i+d+1} - \tau_{i+1}} N_{i+1,d-1,\tau}(u) \right) c_i
+\end{eqnarray*}
+
+and do a gradient-descend to approximate the value of \(u\) up to an
+\(\epsilon\) of \(0.0001\).
+
+For this we use the Gauss-Newton algorithm\cite{gaussNewton} as the
+solution to this problem may not be deterministic, because we usually
+have way more vertices than control points (\(\#v \gg \#c\)).
+
+\section{\texorpdfstring{Adaption of \ac{FFD} for a
+3D-Mesh}{Adaption of  for a 3D-Mesh}}\label{adaption-of-for-a-3d-mesh}
 
 \label{3dffd}
 
+This is a straightforward extension of the 1D-method presented in the
+last chapter. But this time things get a bit more complicated. As we
+have a 3-dimensional grid we may have a different amount of
+control-points in each direction.
+
+Given \(n,m,o\) control points in \(x,y,z\)-direction each Point on the
+curve is defined by
+\[V(u,v,w) = \sum_{i=0}^{n-d-2} \sum_{j=0}^{m-d-2} \sum_{k=0}^{o-d-2} N_{i,d,\tau_i}(u) N_{j,d,\tau_j}(v) N_{k,d,\tau_k}(w) \cdot C_{ijk}.\]
+
+In this case we have three different B-Splines (one for each dimension)
+and also 3 variables \(u,v,w\) for each vertex we want to approximate.
+
+Given a target vertex \(\vec{p}^*\) and an initial guess
+\(\vec{p}=V(u,v,w)\) we define the error-function for the
+gradient-descent as:
+
+\[Err(u,v,w,\vec{p}^{*}) = \vec{p}^{*} - V(u,v,w)\]
+
+And the partial version for just one direction as
+
+\[Err_x(u,v,w,\vec{p}^{*}) = p^{*}_x - \sum_{i=0}^{n-d-2} \sum_{j=0}^{m-d-2} \sum_{k=0}^{o-d-2} {C_{ijk}}_x N_{i,d,\tau_i}(u) N_{j,d,\tau_j}(v) N_{k,d,\tau_k}(w) \]
+
+To solve this we derive partially, like before:
+
+\[
+\begin{array}{rl}
+    \displaystyle \frac{\partial Err_x}{\partial u} & p^{*}_x - \displaystyle \sum_{i=0}^{n-d-2} \sum_{j=0}^{m-d-2} \sum_{k=0}^{o-d-2} {C_{ijk}}_x N_{i,d,\tau_i}(u) N_{j,d,\tau_j}(v) N_{k,d,\tau_k}(w) \\
+  = & \displaystyle - \sum_{i=0}^{n-d-2} \sum_{j=0}^{m-d-2} \sum_{k=0}^{o-d-2} {C_{ijk}}_x N'_i(u) N_{j,d,\tau_j}(v) N_{k,d,\tau_k}(w)
+\end{array}
+\]
+
+The other partial derivatives follow the same pattern yielding the
+Jacobian:
+
+\[
+J(Err(u,v,w)) = 
+\left(
+\begin{array}{ccc}
+\frac{\partial Err_x}{\partial u} & \frac{\partial Err_x}{\partial v} & \frac{\partial Err_x}{\partial w} \\
+\frac{\partial Err_y}{\partial u} & \frac{\partial Err_y}{\partial v} & \frac{\partial Err_y}{\partial w} \\
+\frac{\partial Err_z}{\partial u} & \frac{\partial Err_z}{\partial v} & \frac{\partial Err_z}{\partial w}
+\end{array}
+\right)
+\]
+
+\unsure[inline]{Should I add an informal complete derivative?\newline
+Like leaving out Sums & $i,j,k$-Indices to make obvious what derivative belongs
+where in what case?}
+
+With the Gauss-Newton algorithm we iterate the formula
+\[J(Err(u,v,w)) \cdot \Delta \left( \begin{array}{c} u \\ v \\ w \end{array} \right) = -Err(u,v,w)\]
+and use Cramers rule for inverting the small Jacobian and solving this
+system of linear equations.
+
+\section{Parametrisierung sinnvoll?}\label{parametrisierung-sinnvoll}
+
 \begin{itemize}
 \tightlist
 \item
-  Definition
+  Nachteile von Parametrisierung
 \item
-  Wieso Newton-Optimierung?
+  Deformation ist um einen Kontrollpunkt viel direkter zu steuern.
 \item
-  Was folgt daraus?
+  =\textgreater{} DM-FFD?
 \end{itemize}
 
 \section{Test Scenario: 1D Function

arbeit/settings/commands.tex

@@ -1,59 +1,59 @@
 \newcommand\exampleend{\hfill$\diamond$}
 % ##### DCJ stuff #####
-\newcommand\dcaj{double cut and join\xspace}
+% \newcommand\dcaj{double cut and join\xspace}
-\newcommand\dcjindel{d_{\DCJ}^{id}}
+% \newcommand\dcjindel{d_{\DCJ}^{id}}
-\newcommand\dcj{d_{\DCJ}}
+% \newcommand\dcj{d_{\DCJ}}
-\renewcommand\gg{\mathcal G}
+% \renewcommand\gg{\mathcal G}
-\newcommand\del{\mathcal A}
+% \newcommand\del{\mathcal A}
-\newcommand\ins{\mathcal B}
+% \newcommand\ins{\mathcal B}
-\newcommand\clean{\varnothing}
+% \newcommand\clean{\varnothing}
-\newcommand\gdel{\mathcal G_{\!A}}
+% \newcommand\gdel{\mathcal G_{\!A}}
-\newcommand\gins{\mathcal G_{\!B}}
+% \newcommand\gins{\mathcal G_{\!B}}
-\newcommand\ag{\AG(A,B)}
+% \newcommand\ag{\AG(A,B)}
-\renewcommand\r{\lambda} % # runs
+% \renewcommand\r{\lambda} % # runs
-\newcommand\R{\Lambda} % # runs after clustering
+% \newcommand\R{\Lambda} % # runs after clustering
-\newcommand\dr{\Delta\r}
+% \newcommand\dr{\Delta\r}
-\newcommand\drr{\Delta\r^{\!\rho}}
+% \newcommand\drr{\Delta\r^{\!\rho}}
-\newcommand\DR{\Delta\R}
+% \newcommand\DR{\Delta\R}
-\newcommand\DRR{\Delta\R^{\!\rho}}
+% \newcommand\DRR{\Delta\R^{\!\rho}}
-%\newcommand\lab[1]{\lambda(#1)}
+% %\newcommand\lab[1]{\lambda(#1)}
-\newcommand\redu[1]{\left.#1\right|_\mathcal G}
+% \newcommand\redu[1]{\left.#1\right|_\mathcal G}
-\newcommand\h[1]{#1^h}
+% \newcommand\h[1]{#1^h}
-\renewcommand\t[1]{#1^t}
+% \renewcommand\t[1]{#1^t}
-\renewcommand\a{$A$\xspace}
+% \renewcommand\a{$A$\xspace}
-\renewcommand\b{$B$\xspace}
+% \renewcommand\b{$B$\xspace}
-\renewcommand\O{\mathcal O}
+% \renewcommand\O{\mathcal O}
-\renewcommand\AA{$A\!A$\xspace}
+% \renewcommand\AA{$A\!A$\xspace}
-\newcommand\AB{$A\!B$\xspace}
+% \newcommand\AB{$A\!B$\xspace}
-\newcommand\BA{$B\!A$\xspace}
+% \newcommand\BA{$B\!A$\xspace}
-\newcommand\BB{$B\!B$\xspace}
+% \newcommand\BB{$B\!B$\xspace}
-\newcommand\ab{\del\ins}
+% \newcommand\ab{\del\ins}
-\newcommand\ba{\ins\del}
+% \newcommand\ba{\ins\del}
-\def\aa{A\!A}
+% \def\aa{A\!A}
-\def\bb{B\!B}
+% \def\bb{B\!B}
-\def\AAab{A\!A_{\!\del\!\ins}}
+% \def\AAab{A\!A_{\!\del\!\ins}}
-\def\AAa{A\!A_{\!\del}}
+% \def\AAa{A\!A_{\!\del}}
-\def\AAb{A\!A_{\ins}}
+% \def\AAb{A\!A_{\ins}}
-\def\BBab{B\!B_{\!\!\del\!\ins}}
+% \def\BBab{B\!B_{\!\!\del\!\ins}}
-\def\BBa{B\!B_{\!\!\del}}
+% \def\BBa{B\!B_{\!\!\del}}
-\def\BBb{B\!B_{\ins}}
+% \def\BBb{B\!B_{\ins}}
-\def\ABab{A\!B_{\!\!\del\!\ins}}
+% \def\ABab{A\!B_{\!\!\del\!\ins}}
-\def\ABba{A\!B_{\ins\!\del}}
+% \def\ABba{A\!B_{\ins\!\del}}
-\def\ABa{A\!B_{\!\!\del}}
+% \def\ABa{A\!B_{\!\!\del}}
-\def\ABb{A\!B_\ins}
+% \def\ABb{A\!B_\ins}
-\def\ABm{A\!B_\bullet}
+% \def\ABm{A\!B_\bullet}
-\def\AAo{A\!A_{\!\varnothing}}
+% \def\AAo{A\!A_{\!\varnothing}}
-\def\ABo{A\!B_{\!\varnothing}}
+% \def\ABo{A\!B_{\!\varnothing}}
-\def\BBo{B\!B_{\!\varnothing}}
+% \def\BBo{B\!B_{\!\varnothing}}
-\def\xx{A\!B_\times}
+% \def\xx{A\!B_\times}
-\def\aba{\del\!\gr{\ins\!\del}}
+% \def\aba{\del\!\gr{\ins\!\del}}
-\def\bab{\ins\!\gr{\del\!\ins}}
+% \def\bab{\ins\!\gr{\del\!\ins}}
-\newcommand\clusterize{accumulate\xspace}
+% \newcommand\clusterize{accumulate\xspace}
-\newcommand\clustering{accumulating\xspace} %aggregate, unite/unifying
+% \newcommand\clustering{accumulating\xspace} %aggregate, unite/unifying
-\newcommand\clusterized{accumulated\xspace}
+% \newcommand\clusterized{accumulated\xspace}
-\newcommand\clusterization{accumulation\xspace}
+% \newcommand\clusterization{accumulation\xspace}
-\renewcommand\v[1]{V(#1)}
+% \renewcommand\v[1]{V(#1)}
-\newcommand\vset[2]{\v{#1}=\set{#2}}
+% \newcommand\vset[2]{\v{#1}=\set{#2}}
 
 % ##### math #####
 \DeclareMathOperator\AG{AG}
@@ -174,8 +174,8 @@
                      \includegraphics[width=1cm]{img/cd}
                     \end{center}\vspace{-15pt}\centering\footnotesize\texttt{#1}}}
 \renewcommand\vec[1]{\textbf{#1}}
-\newcommandx{\unsure}[2][1=]{\todo[linecolor=red,backgroundcolor=red!25,bordercolor=red,#1]{#2}}
+\newcommandx{\unsure}[2][1=]{\todo[linecolor=red,backgroundcolor=red!25,bordercolor=red,#1]{\textbf{Unsure:} #2}}
-\newcommandx{\change}[2][1=]{\todo[linecolor=blue,backgroundcolor=blue!25,bordercolor=blue,#1]{#2}}
+\newcommandx{\change}[2][1=]{\todo[linecolor=blue,backgroundcolor=blue!25,bordercolor=blue,#1]{\textbf{Change:} #2}}
-\newcommandx{\info}[2][1=]{\todo[linecolor=OliveGreen,backgroundcolor=OliveGreen!25,bordercolor=OliveGreen,#1]{#2}}
+\newcommandx{\info}[2][1=]{\todo[linecolor=OliveGreen,backgroundcolor=OliveGreen!25,bordercolor=OliveGreen,#1]{\textbf{Info:} #2}}
 \newcommandx{\improvement}[2][1=]{\todo[linecolor=violet,backgroundcolor=violet!25,bordercolor=violet,#1]{#2}}
 \newcommandx{\thiswillnotshow}[2][1=]{\todo[disable,#1]{#2}}

arbeit/settings/packages.tex

@@ -15,7 +15,7 @@
 \usepackage{color}						%\colorbox
 \usepackage{dsfont}						%\mathds
 \usepackage{draftwatermark}
-\SetWatermarkLightness{0.9} % default: 0.8
+\SetWatermarkLightness{0.95} % default: 0.8
 \usepackage{epigraph}
 % \usepackage{euler}  					% euler: uni, eucal: baake, ohne: standard
 \usepackage{eucal}						% euler calligraphy

arbeit/template.tex

@@ -30,6 +30,7 @@ xcolor=dvipsnames,
 %\setlength{\parindent}{0pt} % kein einzug bei absaetzen
 %\setlength{\lineskip}{1ex plus0.5ex minus0.5ex} % dafr abstand zwischen abs<62>zen (funktioniert noch nicht)
 % \renewcommand{\familydefault}{\sfdefault}
+\setstretch{1.44} % 1.5-facher zeilenabstand
 
 %%%%%%%%%%%%%%% Header - Footer %%%%%%%%%%%%%%%
 % ### Fr 2 Seitig (option twopage):