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Motiejus Jakštys 2021-04-17 20:42:59 +03:00
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IV/mj-msc.tex
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@ -418,14 +418,14 @@ following the rules of the article.
\centering \centering
\begin{subfigure}[b]{.4\textwidth} \begin{subfigure}[b]{.4\textwidth}
\includegraphics[width=\textwidth]{fig6-selfcrossing-before} \includegraphics[width=\textwidth]{fig6-selfcrossing-before}
\caption{Bend's baseline is crossing another bend} \caption{Bend's baseline (dotted) is crossing a neighboring bend}
\end{subfigure} \end{subfigure}
\hfill \hfill
\begin{subfigure}[b]{.4\textwidth} \begin{subfigure}[b]{.4\textwidth}
\includegraphics[width=\textwidth]{fig6-selfcrossing-after} \includegraphics[width=\textwidth]{fig6-selfcrossing-after}
\caption{Self-crossing removed} \caption{Self-crossing removed following the algorithm}
\end{subfigure} \end{subfigure}
\caption{Originally Figure 6: self-line crossing} \caption{Originally Figure 6: simple case of self-line crossing}
\label{fig:fig6-selfcrossing} \label{fig:fig6-selfcrossing}
\end{figure} \end{figure}
@ -438,18 +438,20 @@ figure~\onpage{fig:selfcrossing-1-non-neighbor}.
\centering \centering
\begin{subfigure}[b]{.4\textwidth} \begin{subfigure}[b]{.4\textwidth}
\includegraphics[width=\textwidth]{selfcrossing-1-before} \includegraphics[width=\textwidth]{selfcrossing-1-before}
\caption{Bend's baseline is crossing a non-neighboring bend} \caption{Bend's baseline (dotted) is crossing a non-neighboring bend}
\end{subfigure} \end{subfigure}
\hfill \hfill
\begin{subfigure}[b]{.4\textwidth} \begin{subfigure}[b]{.4\textwidth}
\includegraphics[width=\textwidth]{selfcrossing-1-after} \includegraphics[width=\textwidth]{selfcrossing-1-after}
\caption{Self-crossing removed} \caption{Self-crossing removed following the algorithm}
\end{subfigure} \end{subfigure}
\caption{Self-crossing with non-neighboring bend} \caption{Self-crossing with non-neighboring bend}
\label{fig:selfcrossing-1-non-neighbor} \label{fig:selfcrossing-1-non-neighbor}
\end{figure} \end{figure}
Naively implemented, checking every bend with every bend is costs $O(n^2)$. Naively implemented, checking every bend with every bend is costs $O(n^2)$. In
other words, the time it takes to run the algorithm grows quadratically with
the with the number of vertices.
It is possible to optimize this step and skip checking some of the bends. Only It is possible to optimize this step and skip checking some of the bends. Only
bends whose sum of inner angles is $\pi$ can ever self-cross. If the value is bends whose sum of inner angles is $\pi$ can ever self-cross. If the value is