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Robin Clark 2010-08-17 14:29:07 +01:00
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eulerg/eulerg.tex
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@ -40,8 +40,8 @@ The spaitial relationship between the curves defines the set theoretic relations
The definitions above allow us to read an Euler diagram The definitions above allow us to read an Euler diagram
and write down set theory equations. and write down set theory equations.
The interest here, is to define relationships between the contours, that allow The interest here though, is to define relationships between the contours, that allow
processing and parsing of the diagram without resorting to area operations in the concerete plane. processing and parsing of the diagram without resorting to extra area operations in the concerete plane.
\section{Defining `pure intersection' and `enclosure'} \section{Defining `pure intersection' and `enclosure'}
\begin{figure}[h] \begin{figure}[h]
@ -130,7 +130,7 @@ an {\em enclosure} relationship as a directed vertice and
The diagram in figure \ref{fig:eulerg1} can now be represented as a graph thus: The diagram in figure \ref{fig:eulerg1} can now be represented as a graph thus:
\begin{figure}[h] \begin{figure}[h]
\centering \centering
\includegraphics[width=70pt,bb=0 0 128 108,keepaspectratio=true]{./eulerg/eulerg_g.jpg} \includegraphics[width=50pt,bb=0 0 128 108,keepaspectratio=true]{./eulerg/eulerg_g.jpg}
% eulerg_g.jpg: 128x108 pixel, 72dpi, 4.52x3.81 cm, bb=0 0 128 108 % eulerg_g.jpg: 128x108 pixel, 72dpi, 4.52x3.81 cm, bb=0 0 128 108
\caption{Graph Representaion of figure \ref{fig:eulerg1}} \caption{Graph Representaion of figure \ref{fig:eulerg1}}
\label{fig:eulerg1_g} \label{fig:eulerg1_g}
@ -175,9 +175,9 @@ becomes part of the pure~intersection~chain. Hmmmm thats true but a better way t
%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.
\vbox{
The diagram in figure \ref{fig:eulerg_pic} can be represented by the following relationships. The diagram in figure \ref{fig:eulerg_pic} can be represented by the following relationships.
$$ M {\pin} N $$ $$ M {\pin} N $$
$$ N {\pin} O $$ $$ N {\pin} O $$
$$ O {\pin} P $$ $$ O {\pin} P $$
@ -188,7 +188,7 @@ $$ A {\enc} N $$
$$ A {\enc} O $$ $$ A {\enc} O $$
$$ A {\enc} P $$ $$ A {\enc} P $$
$$ A {\enc} Q $$ $$ A {\enc} Q $$
}
To form the {\pic} we can follow To form the {\pic} we can follow
reachable pure intersection relationships. reachable pure intersection relationships.
@ -258,13 +258,13 @@ see figure \ref{fig:eulerg_pic_g_a_unc}.
\section{reduction of searches for available zones} \section{reduction of searches for available zones}
Another property of any {\pic} $PIC$, is that Another property of any {\pic} $P$, is that
the maximum number of euler zones within it is the maximum number of euler zones within it is
$$ MaxZones = 2^{|PIC|} $$ $$ MaxZones = 2^{|P|} $$
Because no contours external to the {\pic} Because no contours external to the {\pic}
bi-sect any in it, no extra zones can be formed. bisect any in it, no extra zones can be formed.
By enclosing a {\pic} with By enclosing a {\pic} with
a contour, we change the nature of the zones within a contour, we change the nature of the zones within
the {\pic}, but the number of zones contributed by the {\pic} the {\pic}, but the number of zones contributed by the {\pic}
@ -273,6 +273,8 @@ stays the same.
A pure intersection chain has a maximum number of possible Euler zones, and exists as independent entities in the diagram. This A pure intersection chain has a maximum number of possible Euler zones, and exists as independent entities in the diagram. This
allows us to analyses {\pic}s separately, thus reducing the $2^N$ overhead of analysing an Euler diagram for available zones. allows us to analyses {\pic}s separately, thus reducing the $2^N$ overhead of analysing an Euler diagram for available zones.
\end{definition} \end{definition}
This 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}.
\vspace{40pt} \vspace{40pt}