diff --git a/mj-msc.tex b/mj-msc.tex index 1358822..eb265e8 100644 --- a/mj-msc.tex +++ b/mj-msc.tex @@ -5,7 +5,9 @@ \usepackage[english]{babel} \usepackage[utf8]{inputenc} \usepackage{a4wide} -\usepackage{csquotes} +%\usepackage{csquotes} +\usepackage [autostyle, english = american]{csquotes} +\MakeOuterQuote{"} \usepackage[maxbibnames=99,style=authoryear]{biblatex} \usepackage[pdfusetitle]{hyperref} \usepackage{enumitem} @@ -270,24 +272,53 @@ purposes) using the following algorithm: \section{Definition of a Bend} \label{sec:definition-of-a-bend} +The original article describes: + +\begin{displayquote}[\cite{wang1998line}][] + A bend can be defined as that part of a line which contains a number of + subsequent vertices, with the inflection angles on all vertices included in + the bend being either positive or negative and the inflection of the bend's + two end vertices being in opposite signs. +\end{displayquote} + +While it gives a good intuitive understanding of what the bend is, some more +technical details would be appreciated. + +Figure~\ref{fig:fig8-definition-of-a-bend} illustrates article's figure 8, +but with bends colored as polygons: each color is a distinctive bend. + \begin{figure}[h] \centering \includegraphics[width=\linewidth]{fig8-definition-of-a-bend} - \caption{Originally Figure 8: detected bends are highlighted} + \caption{Originally figure 8: detected bends are highlighted} \label{fig:fig8-definition-of-a-bend} \end{figure} -End line segments of all lines should also be part of the bend. That way, all -line segments belong to 1 or 2 bends. This characteristic is not obvious when -reading the introductory sections, but becomes unavoidable (there could be no -other way) when reading the following sections in detail. +Once the intuitive definition is established, here are some non-obvious +characteristics that are necessary when writing code to detect the bends: -First and last segments of each bend (except for the two end-line segments) is -also the first vertex of the next bend. This is apparent when looking at the -illustration of the detected bends. +\begin{itemize} + + \item End segments of each line should also belong to bends. That way, all + segments belong to 1 or 2 bends. + + \item First and last segments of each bend (except for the two end-line + segments) is also the first vertex of the next bend. +\end{itemize} + +Properties above may be apparent when looking at illustrations at this article +or reading here, but they are nowhere as such when looking at the original +article. \section{Gentle Inflection at End of a Bend} +The gist of the section is in the original article: + +\begin{displayquote}[\cite{wang1998line}][] + But if the inflection that marks the end of a bend is quite small, people + would not recognize this as the bend point of a bend +\end{displayquote} + Figure~\ref{fig:fig5-gentle-inflection} visualizes original paper's Figure 5, when a single vertex is moved outwards the end of the bend. @@ -302,20 +333,21 @@ when a single vertex is moved outwards the end of the bend. \includegraphics[width=\textwidth]{fig5-gentle-inflection-after} \caption{After applying the inflection rule} \end{subfigure} - \caption{Originally Figure 5: gentle inflections at the ends of the bend} + \caption{Originally figure 5: gentle inflections at the ends of the bend} \label{fig:fig5-gentle-inflection} \end{figure} -The example in this section was clear, but insufficient: it does not specify -how many vertices should be included when calculating the end-of-bend -inflection. We chose the iterative approach -- as long as the angle is "right" +The illustration for this section was clear, but insufficient: it does not +specify how many vertices should be included when calculating the end-of-bend +inflection. We chose the iterative approach --- as long as the angle is "right" and the distance is decreasing, the algorithm should keep re-assigning vertices to different bends; practically not having an upper bound on the number of iterations. -Additional example, not found in the original paper, is illustrated in -figure~\ref{fig:inflection-1-gentle-inflection}, which re-assigns two vertices -to the next bend instead of one. +To prove that the algorithm implementation is correct for multiple vertices, +additional example was created, and illustrated in +figure~\ref{fig:inflection-1-gentle-inflection}: the rule re-assigns two +vertices to the next bend instead of one. \begin{figure}[h] \centering