diff --git a/arbeit/bibma.bib b/arbeit/bibma.bib index 5b5d2d1..1acc63a 100644 --- a/arbeit/bibma.bib +++ b/arbeit/bibma.bib @@ -5,6 +5,7 @@ publisher = "IEEE", title = "Evolvability as a Quality Criterion for Linear Deformation Representations in Evolutionary Optimization", year = "2016", + note={\url{http://graphics.uni-bielefeld.de/publications/cec16.pdf}, \url{https://pub.uni-bielefeld.de/publication/2902698}}, } @article{spitzmuller1996bezier, title="Partial derivatives of Bèzier surfaces", @@ -15,10 +16,12 @@ pages="67--72", year="1996", publisher="Elsevier", + url={https://doi.org/10.1016/0010-4485(95)00044-5}, } @article{hsu1991dmffd, title={A direct manipulation interface to free-form deformations}, author={Hsu, William M}, journal={Master's thesis, Brown University}, - year={1991} + year={1991}, + url={https://cs.brown.edu/research/pubs/theses/masters/1991/hsu.pdf}, } diff --git a/arbeit/files/erklaerung.aux b/arbeit/files/erklaerung.aux index d93c955..4d28632 100644 --- a/arbeit/files/erklaerung.aux +++ b/arbeit/files/erklaerung.aux @@ -27,7 +27,7 @@ \setcounter{r@tfl@t}{0} \setcounter{subfigure}{0} \setcounter{subtable}{0} -\setcounter{@todonotes@numberoftodonotes}{1} +\setcounter{@todonotes@numberoftodonotes}{3} \setcounter{Item}{0} \setcounter{Hfootnote}{2} \setcounter{bookmark@seq@number}{16} diff --git a/arbeit/ma.md b/arbeit/ma.md index 386b635..877b853 100644 --- a/arbeit/ma.md +++ b/arbeit/ma.md @@ -16,7 +16,7 @@ Unless otherwise noted the following holds: refer to other scalar (real) variables. - lowercase **bold** letters (e.g. $\vec{x},\vec{y}$) refer to 3D coordinates -- uppercase **BOLD** letters (e.g. $D, M$) +- uppercase **BOLD** letters (e.g. $\vec{D}, \vec{M}$) refer to Matrices # Introduction @@ -42,23 +42,19 @@ 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: -$$ -\begin{equation} -\label{ffd1d1} +\begin{equation} \label{eqn:ffd1d1} N_{i,0,\tau}(u) = \begin{cases} 1, & u \in [\tau_i, \tau_{i+1}[ \\ 0, & \mbox{otherwise} \end{cases} \end{equation} -$$ + and -$$ -\begin{equation} -\label{ffd1d2} + +\begin{equation} \label{eqn:ffd1d2} N_{i,d,\tau}(u) = \frac{u-\tau_i}{\tau_{i+d}} N_{i,d-1,\tau}(u) + \frac{\tau_{i+d+1} - u}{\tau_{i+d+1}-\tau_{i+1}} N_{i+1,d-1,\tau}(u) \end{equation} -$$ 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 +As can be seen from \eqref{eqn: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 $0$]: @@ -82,11 +78,12 @@ model follows in an intuitive manner. The deformation is smooth as the underlyin 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}. +creating a plateau can be difficult to achieve\cite[chapter~3.2]{hsu1991dmffd}\todo{cite [24] aus \ref{anrichterEvol}}. 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. +Moreover this increases the efficiency of an evolutionary optimization\todo{cite [25] aus \ref{anrichterEvol}}, which we will use later on. 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. diff --git a/arbeit/ma.pdf b/arbeit/ma.pdf index 3152483..9023b8c 100644 Binary files a/arbeit/ma.pdf and b/arbeit/ma.pdf differ diff --git a/arbeit/ma.tex b/arbeit/ma.tex index 0f79779..3c7648b 100644 --- a/arbeit/ma.tex +++ b/arbeit/ma.tex @@ -140,7 +140,7 @@ Unless otherwise noted the following holds: lowercase \textbf{bold} letters (e.g. \(\vec{x},\vec{y}\))\\ refer to 3D coordinates \item - uppercase \textbf{BOLD} letters (e.g. \(D, M\))\\ + uppercase \textbf{BOLD} letters (e.g. \(\vec{D}, \vec{M}\))\\ refer to Matrices \end{itemize} @@ -171,24 +171,22 @@ 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: -\[ -\begin{equation} -\label{ffd1d1} +\begin{equation} \label{eqn:ffd1d1} N_{i,0,\tau}(u) = \begin{cases} 1, & u \in [\tau_i, \tau_{i+1}[ \\ 0, & \mbox{otherwise} \end{cases} \end{equation} -\] and \[ -\begin{equation} -\label{ffd1d2} + +and + +\begin{equation} \label{eqn:ffd1d2} N_{i,d,\tau}(u) = \frac{u-\tau_i}{\tau_{i+d}} N_{i,d-1,\tau}(u) + \frac{\tau_{i+d+1} - u}{\tau_{i+d+1}-\tau_{i+1}} N_{i+1,d-1,\tau}(u) \end{equation} -\] 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 +seen from \eqref{eqn: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 @@ -222,7 +220,7 @@ 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}. +achieve\cite[chapter~3.2]{hsu1991dmffd}\todo{cite [24] aus \ref{anrichterEvol}}. This disadvantages led to the formulation of \acf{DM-FFD}\cite[chapter~3.3]{hsu1991dmffd} in which the user directly @@ -230,7 +228,9 @@ 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. +control-points. Moreover this increases the efficiency of an +evolutionary optimization\todo{cite [25] aus \ref{anrichterEvol}}, which +we will use later on. But this approach also has downsides as can be seen in \cite[figure~7]{hsu1991dmffd}\todo{figure hier einfügen?}, as the diff --git a/arbeit/settings/commands.tex b/arbeit/settings/commands.tex index fca2ab3..ba7d454 100644 --- a/arbeit/settings/commands.tex +++ b/arbeit/settings/commands.tex @@ -173,3 +173,4 @@ \newcommand\data[1]{\marginpar{\vspace{-35pt}\begin{center} \includegraphics[width=1cm]{img/cd} \end{center}\vspace{-15pt}\centering\footnotesize\texttt{#1}}} +\renewcommand\vec[1]{\textbf{#1}}