


\ifthenelse {\boolean{paper}}
{
\abstract{
This paper discusses representing Euler Diagrams as graphs, or sets of relationships.
By representing Euler diagrams in this way,
algorithms to invesigate properties of the diagrams, are possible, without
having to resort
to unecessary CPU expensive area operations on the concrete diagrams. 
}
}
{ %% Introduction
\section{Introduction}
This paper discusses representing Euler Diagrams as graphs, or sets of relationships.
By representing Euler diagrams in this way,
algorithms to invesigate properties of the diagrams, are possible, without
having to resort
to unecessary CPU expensive area operations on the concrete diagrams. 
}


\section{Introduction : Euler Diagram }

Classical Euler diagrams consist of closed curves in the plane which are used to represent sets.
The spaitial relationship between the curves defines the set theoretic relationships, as defined below.
\begin{itemize}
\item Intersection - if the curves defining the area within curves overlap
\item Sub-set      - if a curve is enclosed by another
\item disjoint     - if the curves are separate
\end{itemize}

The definitions above allow us to read an Euler diagram
and write down set theory equations.
The interest here, is to define relationships between the contours, that allow
processing and parsing of the diagram without resorting to area operations in the concerete plane.

\section{Defining `pure intersection' and `enclosure'}
\begin{figure}[h]
 \centering
 \includegraphics[width=200pt,keepaspectratio=true]{./eulerg/eulerg1.jpg}
 % eulerg1.jpg: 513x215 pixel, 72dpi, 18.10x7.58 cm, bb=0 0 513 215
 \caption{An Euler Diagram showing enclosure and Pure Intersection}
 \label{fig:eulerg1}
\end{figure}

The set theory term `intersection' can apply to both the curves overlapping and to the sub-set case.
In conceret diagram terms two curves crossing, can be termed bi-secting.
For instance in diagram \ref{fig:eulerg1} the set theoretic intersection between
$A$ and $B$ exists, even though the curves do no bi-sect in the concrete plane.

$$ A \cup B \neq \emptyset $$

as does the intersection $D$ and $E$

$$ D \cup E \neq \emptyset $$

Clearly though these intersections are different, because
in the $A$, $B$ case 
$$ A \backslash B = \emptyset \wedge B \backslash A \neq \emptyset $$
This is not the case for $D$, $E$ where:
$$ D \backslash E \neq \emptyset \wedge E \backslash D \neq \emptyset $$

\paragraph{Enclosure}
To distinguish between these we can term the $A$, $B$ case to be 
$A$ `enclosed' by $B$. We can express this as a directed relationship.

$$ B {\enc} A $$


\paragraph{Pure Intersection}
In the $D$, $E$ case we have 

We can say that where the areas defined by the curves intersect but no one curve encloses the 
other, we can term this `pure intersection'.
We can express this as a non directed relationship.

$$ D \pin E $$


\paragraph{Mutual exclusivity of `pure intersection' and `enclosure'}

Clearly these two properties are mutually exclusive. No
contour can be both purely intersected and enclosed with the same contour.
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}.

\begin{figure}[h]
 \centering
 \includegraphics[width=200pt,keepaspectratio=true]{./eulerg/eulerg_enc.jpg}
 % eulerg_enc.jpg: 315x269 pixel, 72dpi, 11.11x9.49 cm, bb=0 0 315 269
 \caption{Enclosure, a transitive relationship}
 \label{fig:eulerg_enc}
\end{figure}

$$ B {\enc} A \wedge A {\enc} C \implies B {\enc} C $$

\begin{definition}
Enlcosure relationships are transitive
\end{definition}

\section{Representing Euler Diagrams as sets of relationships}

The diagram in figure \ref{fig:eulerg1} can be represented by the foillowing relationships.

$$ B {\enc} A $$
$$ D {\pin} E $$


The diagram in figure \ref{fig:eulerg_enc} can be represented by the following relationships.

$$ B {\enc} A $$
$$ A {\enc} C $$


\section{The {\pic}}

Contours may be connected via `pure intersection' relationships to form 
`chains' of contours reachable by pure intersection.

Figure \ref{fig:eulerg_pic} shows a {\pic} consisting of contours $M,N,O,P$ and $Q$.

\begin{figure}[h]
 \centering
 \includegraphics[width=300pt,keepaspectratio=true]{./eulerg/eulerg_pic.jpg}
 % eulerg_pic.jpg: 955x286 pixel, 72dpi, 33.69x10.09 cm, bb=0 0 955 286
 \caption{{\pic} with Enclosure}
 \label{fig:eulerg_pic}
\end{figure}

\textbf{rule:}
\begin{definition}
If any contour in a {\pic} is enclosed by any contour not belonging to the chain, 
all the countours within the 
{\pic} will be enclosed by it. This is because a contour 
enclosing which bisects(????) another contour in a {\pic}
becomes part of the pure~intersection~chain. Hmmmm thats true but a better way to say it ????
\end{definition}

%The diagram in figure \ref{fig:eulerg_enc} can be represented by the following relationships.


The diagram in figure \ref{fig:eulerg_pic} can be represented by the following relationships.


$$ M {\pin} N $$
$$ N {\pin} O $$
$$ O {\pin} P $$
$$ O {\pin} Q $$
$$ Q {\enc} P $$
$$ A {\enc} M $$
$$ A {\enc} N $$
$$ A {\enc} O $$
$$ A {\enc} P $$
$$ A {\enc} Q $$


To form the {\pic} we can follow
reachable pure intersection relationships.

$ M {\pin} N {\pin} O {\pin} P $ are part of the same chain.
following from $O$, $O {\pin} Q$.
Thus by the definition of being reachable by pure instersection relationships,$M,N,O,P,Q$
are in the same {\pic}, even though $Q$ encloses $P$.

We can define this {\pic} as $PIC1$ as a set of contours.

$$ PIC1 = \{ M,N,O,P,Q \} $$

Contour $A$, by virtue of not bisecting any contour in the pure instersection
chain $PIC1$, does not belong to  $PIC1$. Because it encloses one of the contours, it
encloses all contours in the chain. 
Knowing this can save on unecessary area operations on the concrete diagram.

\section{reduction of searches for available zones}


Another property of any {\pic} $PIC$, is that
the maximum number of euler zones within it is 

$$ MaxZones =  2^{|PIC|}  $$

Because no contours external to the {\pic} 
bi-sect any in it, no extra zones can be formed.
By enclosing a {\pic} with 
a contour, we change the nature of the zones within
the {\pic}, but the number of zones contributed by the {\pic}
stays the same.
\begin{definition}
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.
\end{definition}


\vspace{40pt}

