wrote more introduction

arbeit/bibma.bib

@@ -6,3 +6,19 @@
   title        = "Evolvability as a Quality Criterion for Linear Deformation Representations in Evolutionary Optimization",
   year         = "2016",
 }
+@article{spitzmuller1996bezier,
+  title="Partial derivatives of Bèzier surfaces",
+  author="Spitzmüller, Klaus",
+  journal="Computer-Aided Design",
+  volume="28",
+  number="1",
+  pages="67--72",
+  year="1996",
+  publisher="Elsevier",
+}
+@article{hsu1991dmffd,
+  title={A direct manipulation interface to free-form deformations},
+  author={Hsu, William M},
+  journal={Master's thesis, Brown University},
+  year={1991}
+}

arbeit/files/erklaerung.aux

@@ -22,14 +22,14 @@
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 \setcounter{float@type}{16}
 \setcounter{lstnumber}{1}
-\setcounter{NAT@ctr}{1}
+\setcounter{NAT@ctr}{3}
 \setcounter{AM@survey}{0}
 \setcounter{r@tfl@t}{0}
 \setcounter{subfigure}{0}
 \setcounter{subtable}{0}
-\setcounter{@todonotes@numberoftodonotes}{0}
+\setcounter{@todonotes@numberoftodonotes}{1}
 \setcounter{Item}{0}
-\setcounter{Hfootnote}{0}
+\setcounter{Hfootnote}{2}
 \setcounter{bookmark@seq@number}{16}
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 \setcounter{ALC@unique}{0}

arbeit/ma.md

@@ -30,14 +30,72 @@ quality and potential of such optimisation.
 
 We will replicate the same setup on the same meshes but use \acf{FFD} instead of
 \acf{RBF} to create a deformation and evaluate if the evolution-criteria still
-work as a predictor given the different deformation.
+work as a predictor given the different deformation scheme.
 
 ## What is \acf{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.
+tool for deforming meshes in the first place. For simplicity we only summarize the
+1D-case from \cite{spitzmuller1996bezier} here and go into the extension to the 3D case in chapter \ref{3dffd}.
+
+Given an arbitrary number of points $p_i$ alongside a line, we map a scalar
+value $\tau_i \in [0,1[$ to each point with $\tau_i < \tau_{i+1} \forall i$.
+Given a degree of the target polynomial $d$ we define the curve $N_{i,d,\tau_i}(u)$ as follows:
+
+$$
+\begin{equation}
+\label{ffd1d1}
+N_{i,0,\tau}(u) = \begin{cases} 1, & u \in [\tau_i, \tau_{i+1}[ \\ 0, & \mbox{otherwise} \end{cases}
+\end{equation}
+$$
+and
+$$
+\begin{equation}
+\label{ffd1d2}
+N_{i,d,\tau}(u) = \frac{u-\tau_i}{\tau_{i+d}} N_{i,d-1,\tau}(u) + \frac{\tau_{i+d+1} - u}{\tau_{i+d+1}-\tau_{i+1}} N_{i+1,d-1,\tau}(u)
+\end{equation}
+$$
+
+If we now multiply every $p_i$ with the corresponding $N_{i,d,\tau_i}(u)$ we get the contribution of each
+point $p_i$ to the final curve-point parameterized only by $u \in [0,1[$.
+As can be seen from equation \ref{ffd1d2} we only access points $[i..i+d]$ for any given $i$^[one more for each recursive step.], which
+gives us, in combination with choosing $p_i$ and $\tau_i$ in order, only a local interference of $d+1$ points.
+
+We can even derive this equation straightforward for an arbitrary $N$^[*Warning:* in the case of $d=1$ the recursion-formula yields a $0$ denominator, but $N$ is also $0$. The right solution for this case is a derivative of $0$]:
+
+$$\frac{\partial}{\partial u} N_{i,d,r}(u) = \frac{d}{\tau_{i+d} - \tau_i} N_{i,d-1,\tau}(u) - \frac{d}{\tau_{i+d+1} - \tau_{i+1}} N_{i+1,d-1,\tau}(u)$$
+
+For a B-Spline
+$$s(u) = \sum_{i} N_{i,d,\tau_i}(u) p_i$$
+these derivations yield $\frac{\partial^d}{\partial u} s(u) = 0$.
+
+Another interesting property of these recursive polynomials is that they are continuous (given $d \ge 1$) as every $p_i$ gets
+blended in linearly between $\tau_i$ and $\tau_{i+d}$ and out linearly between $\tau_{i+1}$ and $\tau_{i+d+1}$
+as can bee seen from the two coefficients in every step of the recursion.
+
+### Why is \ac{FFD} a good deformation function?
+
+The usage of \ac{FFD} as a tool for manipulating follows directly from the properties of the polynomials and the correspondence to
+the control points.
+Having only a few control points gives the user a nicer high-level-interface, as she only needs to move these points and the
+model follows in an intuitive manner. The deformation is smooth as the underlying polygon is smooth as well and affects as many
+vertices of the model as needed. Moreover the changes are always local so one risks not any change that a user cannot immediately see.
+
+But there are also disadvantages of this approach. The user loses the ability to directly influence vertices and even seemingly simple tasks as
+creating a plateau can be difficult to achieve\cite[chapter~3.2]{hsu1991dmffd}.
+
+This disadvantages led to the formulation of \acf{DM-FFD}\cite[chapter~3.3]{hsu1991dmffd} in which the user directly interacts with the surface-mesh.
+All interactions will be applied proportionally to the control-points that make up the parametrization of the interaction-point
+itself yielding a smooth deformation of the surface *at* the surface without seemingly arbitrary scattered control-points.
+
+But this approach also has downsides as can be seen in \cite[figure~7]{hsu1991dmffd}\todo{figure hier einfügen?}, as the tessellation of
+the invisible grid has a major impact on the deformation itself.
+
+All in all \ac{FFD} and \ac{DM-FFD} are still good ways to deform a high-polygon mesh albeit the downsides.
+
+## What is evaluational optimization?
+
 
-## Was ist evolutionäre Optimierung?
 
 ## Wieso ist evo-Opt so cool?
 
@@ -48,6 +106,7 @@ tool for deforming meshes in the first place.
 # Hauptteil
 
 ## Was ist FFD?
+\label{3dffd}
 
 - Definition
 - Wieso Newton-Optimierung?

arbeit/ma.tex

@@ -157,15 +157,91 @@ potential of such optimisation.
 We will replicate the same setup on the same meshes but use \acf{FFD}
 instead of \acf{RBF} to create a deformation and evaluate if the
 evolution-criteria still work as a predictor given the different
-deformation.
+deformation scheme.
 
 \section{\texorpdfstring{What is \acf{FFD}?}{What is ?}}\label{what-is}
 
 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.
+good tool for deforming meshes in the first place. For simplicity we
+only summarize the 1D-case from \cite{spitzmuller1996bezier} here and go
+into the extension to the 3D case in chapter \ref{3dffd}.
 
-\section{Was ist evolutionäre
+Given an arbitrary number of points \(p_i\) alongside a line, we map a
-Optimierung?}\label{was-ist-evolutionuxe4re-optimierung}
+scalar value \(\tau_i \in [0,1[\) to each point with
+\(\tau_i < \tau_{i+1} \forall i\). Given a degree of the target
+polynomial \(d\) we define the curve \(N_{i,d,\tau_i}(u)\) as follows:
+
+\[
+\begin{equation}
+\label{ffd1d1}
+N_{i,0,\tau}(u) = \begin{cases} 1, & u \in [\tau_i, \tau_{i+1}[ \\ 0, & \mbox{otherwise} \end{cases}
+\end{equation}
+\] and \[
+\begin{equation}
+\label{ffd1d2}
+N_{i,d,\tau}(u) = \frac{u-\tau_i}{\tau_{i+d}} N_{i,d-1,\tau}(u) + \frac{\tau_{i+d+1} - u}{\tau_{i+d+1}-\tau_{i+1}} N_{i+1,d-1,\tau}(u)
+\end{equation}
+\]
+
+If we now multiply every \(p_i\) with the corresponding
+\(N_{i,d,\tau_i}(u)\) we get the contribution of each point \(p_i\) to
+the final curve-point parameterized only by \(u \in [0,1[\). As can be
+seen from equation \ref{ffd1d2} we only access points \([i..i+d]\) for
+any given \(i\)\footnote{one more for each recursive step.}, which gives
+us, in combination with choosing \(p_i\) and \(\tau_i\) in order, only a
+local interference of \(d+1\) points.
+
+We can even derive this equation straightforward for an arbitrary
+\(N\)\footnote{\emph{Warning:} in the case of \(d=1\) the
+  recursion-formula yields a \(0\) denominator, but \(N\) is also \(0\).
+  The right solution for this case is a derivative of \(0\)}:
+
+\[\frac{\partial}{\partial u} N_{i,d,r}(u) = \frac{d}{\tau_{i+d} - \tau_i} N_{i,d-1,\tau}(u) - \frac{d}{\tau_{i+d+1} - \tau_{i+1}} N_{i+1,d-1,\tau}(u)\]
+
+For a B-Spline \[s(u) = \sum_{i} N_{i,d,\tau_i}(u) p_i\] these
+derivations yield \(\frac{\partial^d}{\partial u} s(u) = 0\).
+
+Another interesting property of these recursive polynomials is that they
+are continuous (given \(d \ge 1\)) as every \(p_i\) gets blended in
+linearly between \(\tau_i\) and \(\tau_{i+d}\) and out linearly between
+\(\tau_{i+1}\) and \(\tau_{i+d+1}\) as can bee seen from the two
+coefficients in every step of the recursion.
+
+\subsection{\texorpdfstring{Why is \ac{FFD} a good deformation
+function?}{Why is  a good deformation function?}}\label{why-is-a-good-deformation-function}
+
+The usage of \ac{FFD} as a tool for manipulating follows directly from
+the properties of the polynomials and the correspondence to the control
+points. Having only a few control points gives the user a nicer
+high-level-interface, as she only needs to move these points and the
+model follows in an intuitive manner. The deformation is smooth as the
+underlying polygon is smooth as well and affects as many vertices of the
+model as needed. Moreover the changes are always local so one risks not
+any change that a user cannot immediately see.
+
+But there are also disadvantages of this approach. The user loses the
+ability to directly influence vertices and even seemingly simple tasks
+as creating a plateau can be difficult to
+achieve\cite[chapter~3.2]{hsu1991dmffd}.
+
+This disadvantages led to the formulation of
+\acf{DM-FFD}\cite[chapter~3.3]{hsu1991dmffd} in which the user directly
+interacts with the surface-mesh. All interactions will be applied
+proportionally to the control-points that make up the parametrization of
+the interaction-point itself yielding a smooth deformation of the
+surface \emph{at} the surface without seemingly arbitrary scattered
+control-points.
+
+But this approach also has downsides as can be seen in
+\cite[figure~7]{hsu1991dmffd}\todo{figure hier einfügen?}, as the
+tessellation of the invisible grid has a major impact on the deformation
+itself.
+
+All in all \ac{FFD} and \ac{DM-FFD} are still good ways to deform a
+high-polygon mesh albeit the downsides.
+
+\section{What is evaluational
+optimization?}\label{what-is-evaluational-optimization}
 
 \section{Wieso ist evo-Opt so cool?}\label{wieso-ist-evo-opt-so-cool}
 
@@ -181,6 +257,8 @@ Optimierung?}\label{was-ist-evolutionuxe4re-optimierung}
 
 \section{Was ist FFD?}\label{was-ist-ffd}
 
+\label{3dffd}
+
 \begin{itemize}
 \tightlist
 \item