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Robin Clark 2010-08-22 09:41:10 +01:00
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eulerg/eulerg.tex
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@ -134,6 +134,12 @@ contour can be both pair-wisely intersected and enclosed with the same contour.
Also enclosure, is transitive. That is to say if B encloses A, and A encloses C Also enclosure, is transitive. That is to say if B encloses A, and A encloses C
then B encloses C, see figure \ref{fig:eulerg_enc}. then B encloses C, see figure \ref{fig:eulerg_enc}.
\begin{definition}
\label{def:mutex}
No contour can be both pair-wisely intersected and enclosed with the same contour.
\end{definition}
\begin{figure}[h] \begin{figure}[h]
\centering \centering
@ -155,7 +161,8 @@ then B encloses C, see figure \ref{fig:eulerg_enc}.
$$ B {\enc} A \wedge A {\enc} C \implies B {\enc} C $$ $$ B {\enc} A \wedge A {\enc} C \implies B {\enc} C $$
\begin{definition} \begin{definition}
Enlcosure relationships are transitive \label{def:enctrans}
Enlcosure relationships are transitive.
\end{definition} \end{definition}
\section{Representing Euler Diagrams \\ as sets of relationships} \section{Representing Euler Diagrams \\ as sets of relationships}
@ -198,13 +205,14 @@ Figure \ref{fig:eulerg_pic} shows a {\pic} consisting of contours $M,N,O,P$ and
%\textbf{rule:} %\textbf{rule:}
\begin{definition} \begin{definition}
\label{def:encpic}
If any contour in a {\pic} is enclosed by any contour not belonging to the chain, If any contour in a {\pic} is enclosed by any contour not belonging to the chain,
all the countours within the all the countours within the
{\pic} will be enclosed by it. {\pic} will be enclosed by it.
\end{definition} \end{definition}
This is because a contour Note that any contour
enclosing which bisects another contour in a {\pic} which bisects another contour in a {\pic}
becomes part of the pair-wise~intersection~chain. becomes part of the pair-wise~intersection~chain.
% Hmmmm thats true but a better way to say it ???? % Hmmmm thats true but a better way to say it ????
%The diagram in figure \ref{fig:eulerg_enc} can be represented by the following relationships. %The diagram in figure \ref{fig:eulerg_enc} can be represented by the following relationships.
@ -272,13 +280,14 @@ shows contour A enclosing all elements in $PIC1$
\label{fig:eulerg_pic_g_a} \label{fig:eulerg_pic_g_a}
\end{figure} \end{figure}
\pagebreak[0] \pagebreak[1]
\subsection{Reducing clutter in the graph}
Because we know that a contour enclosing a contour within a {\pic} but not belonging Because we know that a contour enclosing a contour within a {\pic} but not belonging
to it, encloses all elements of the {\pic}, we can draw this in a less cluttered way to it, encloses all elements of the {\pic}, see definition \ref{def:encpic}, we can draw this in a less cluttered way
see figure \ref{fig:eulerg_pic_g_a_unc}. see figure \ref{fig:eulerg_pic_g_a_unc}.
\pagebreak[0]
\subsection{Reducing clutter in the graph}
\begin{figure}[h] \begin{figure}[h]
\centering \centering
\includegraphics[width=200pt,bb=0 0 330 162]{./eulerg/eulerg_pic_g_a_unc.jpg} \includegraphics[width=200pt,bb=0 0 330 162]{./eulerg/eulerg_pic_g_a_unc.jpg}
@ -308,7 +317,7 @@ The number of available zones within a {\pic} $P$ does not change
when other contours are added or removed from the diagram when other contours are added or removed from the diagram
that are not, or would not become members of the {\pic} $P$. that are not, or would not become members of the {\pic} $P$.
\end{definition} \end{definition}
his is to say, the the number of zones within a {\pic} is not affected by changes in the diagram That is to say, the the number of zones within a {\pic} is not affected by changes in the diagram
that do not alter the {\pic}. that do not alter the {\pic}.
This allows us to analyses {\pic}s separately, thus reducing the $2^N$ overhead of analysing an Euler diagram for available zones. This allows us to analyses {\pic}s separately, thus reducing the $2^N$ overhead of analysing an Euler diagram for available zones.