312 lines
10 KiB
TeX
312 lines
10 KiB
TeX
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\ifthenelse {\boolean{paper}}
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{
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\abstract{
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This paper discusses representing Euler Diagrams as graphs, or sets of relationships.
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By representing Euler diagrams in this way,
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algorithms to invesigate properties of the diagrams, are possible, without
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having to resort
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to extra unecessary CPU expensive area operations on the concrete diagrams.
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The graph representations presented here form the basis for several algorithms
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and time saving procedures, implemented in the FMMD analysis tool.
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}
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}
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{ %% Introduction
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\section{Introduction}
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This paper discusses representing Euler Diagrams as graphs, or sets of relationships.
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By representing Euler diagrams in this way,
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algorithms to invesigate properties of the diagrams, are possible, without
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having to resort
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to extra unecessary CPU expensive area operations on the concrete diagrams.
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The graph representations presented here form the basis for several algorithms
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and time saving procedures, implemented in the FMMD analysis tool.
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}
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\section{Introduction : Euler Diagram }
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Classical Euler diagrams consist of closed curves in the plane which are used to represent sets.
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The spaitial relationship between the curves defines the set theoretic relationships, as defined below.
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\begin{itemize}
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\item Intersection - if the curves defining the area within curves overlap
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\item Sub-set - if a curve is enclosed by another
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\item disjoint - if the curves are separate
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\end{itemize}
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The definitions above allow us to read an Euler diagram
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and write down set theory equations.
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The interest here though, is to define relationships between the contours, that allow
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processing and parsing of the diagram without resorting to extra area operations in the concerete plane.
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\section{Defining `pure intersection' and `enclosure'}
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%\begin{figure}[htp]
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% \begin{center}
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% \subfigure[Original image]{\label{fig:edge-a}\includegraphics[scale=0.75]{toucan.eps}}
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% \subfigure[After Laplace edge detection]{\label{fig:edge-b}\includegraphics[scale=0.75]{laplace_toucan.eps}} \\
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% \subfigure[After Sobel edge detection]{\label{fig:edge-c}\includegraphics[scale=0.75]{sobel_toucan.eps}}
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% \end{center}
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% \caption{Various edge detection algorithms}
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% \label{fig:edge}
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%\end{figure}
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%
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%\begin{figure}[h]
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% \centering
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% \includegraphics[width=200pt,keepaspectratio=true]{./eulerg/eulerg1.jpg}
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% % eulerg1.jpg: 513x215 pixel, 72dpi, 18.10x7.58 cm, bb=0 0 513 215
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% \caption{An Euler Diagram showing enclosure and Pure Intersection}
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% \label{fig:eulerg1}
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%\end{figure}
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%
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%
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%
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\begin{figure}[ht]
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\centering
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\subfigure[Euler Diagram]{
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\includegraphics[width=200pt,keepaspectratio=true]{./eulerg/eulerg1.jpg}
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\label{fig:subfig1}
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}
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\subfigure[Graph Representation]{
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\includegraphics[width=50pt,keepaspectratio=true]{./eulerg/eulerg_g.jpg}
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\label{fig:subfig2}
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}
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% \subfigure[Subfigure 3 caption]{
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% \includegraphics[width=100pt,keepaspectratio=true]{./eulerg/eulerg1.jpg}
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% \label{fig:subfig3}
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% }
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\caption{An Euler Diagram showing enclosure and Pure Intersection}
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\label{fig:eulerg1}
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\end{figure}
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The set theory term `intersection' can apply to both the curves overlapping and to the sub-set case.
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Intersection in a concrete diagram can mean two curves bisecting.
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For instance in diagram \ref{fig:eulerg1} the set theoretic intersection between
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$A$ and $B$ exists, even though the curves do not bisect in the concrete plane.
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$$ A \cap B \neq \emptyset $$
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as does the intersection $D$ and $E$
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$$ D \cap E \neq \emptyset $$
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Clearly though these intersections are different, because
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in the $A$, $B$ case
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$$ A \backslash B = \emptyset \wedge B \backslash A \neq \emptyset $$
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This is not the case for $D$, $E$ where:
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$$ D \backslash E \neq \emptyset \wedge E \backslash D \neq \emptyset $$
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\paragraph{Enclosure}
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To distinguish between these we can term the $A$, $B$ case to be
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$A$ `enclosed' by $B$. We can express this as a directed relationship.
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$$ B {\enc} A $$
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\paragraph{Pure Intersection}
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In the $D$, $E$ case we have
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We can say that where the areas defined by the curves intersect but no one curve encloses the
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other, we can term this `pure intersection'.
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We can express this as a non directed relationship.
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$$ D \pin E $$
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\paragraph{Mutual exclusivity of `pure intersection' and `enclosure'}
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Clearly these two properties are mutually exclusive. No
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contour can be both purely intersected and enclosed with the same contour.
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Also enclosure, is transitive. That is to say if B encloses A, and A encloses C
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then B encloses C, see figure \ref{fig:eulerg_enc}.
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\begin{figure}[h]
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\centering
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\subfigure[Euler Diagram]{
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\includegraphics[width=200pt,keepaspectratio=true]{./eulerg/eulerg_enc.jpg}
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% eulerg_enc.jpg: 315x269 pixel, 72dpi, 11.11x9.49 cm, bb=0 0 315 269
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\label{eulerenc}
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}
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\subfigure[Graph Representation]{
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\includegraphics[width=100pt,bb=0 0 240 43,keepaspectratio=true]{./eulerg/eulerg_enc_g.jpg}
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% eulerg_enc_g.jpg: 240x43 pixel, 72dpi, 8.47x1.52 cm, bb=0 0 240 43
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\label{graphenc}
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}
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\caption{Enclosure, a transitive relationship}
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\label{fig:eulerg_enc}
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\end{figure}
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$$ B {\enc} A \wedge A {\enc} C \implies B {\enc} C $$
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\begin{definition}
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Enlcosure relationships are transitive
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\end{definition}
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\section{Representing Euler Diagrams as sets of relationships}
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The diagram in figure \ref{fig:eulerg1} can be represented by the following relationships.
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$$ B {\enc} A $$
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$$ D {\pin} E $$
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The diagram in figure \ref{fig:eulerg_enc} can be represented by the following relationships.
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$$ B {\enc} A $$
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$$ A {\enc} C $$
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\section{Represeting Euler diagrams as graphs}
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As the relationships {\em enclosure} and {\pic} are mutually exclusive
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and {\em enclosure} is transitive and {\pic} is not, we can represent
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an {\em enclosure} relationship as a directed vertice and
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{\pic} as non-directed.
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\pagebreak[0]
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\section{The {\pic}}
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In graph theory a node is said to be reachable from another node
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if you can start at the one node, travel via the edges
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and arrive at the other.
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Contours may be connected via `pure intersection' relationships to form
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`chains' of contours reachable by pure intersection.
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These are termed {\pic}s.
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Figure \ref{fig:eulerg_pic} shows a {\pic} consisting of contours $M,N,O,P$ and $Q$.
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\begin{figure}[h]
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\centering
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\includegraphics[width=300pt,keepaspectratio=true]{./eulerg/eulerg_pic.jpg}
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% eulerg_pic.jpg: 955x286 pixel, 72dpi, 33.69x10.09 cm, bb=0 0 955 286
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\caption{{\pic} with Enclosure}
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\label{fig:eulerg_pic}
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\end{figure}
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\textbf{rule:}
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\begin{definition}
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If any contour in a {\pic} is enclosed by any contour not belonging to the chain,
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all the countours within the
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{\pic} will be enclosed by it. This is because a contour
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enclosing which bisects(????) another contour in a {\pic}
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becomes part of the pure~intersection~chain. Hmmmm thats true but a better way to say it ????
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\end{definition}
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%The diagram in figure \ref{fig:eulerg_enc} can be represented by the following relationships.
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\vbox{
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The diagram in figure \ref{fig:eulerg_pic} can be represented by the following relationships.
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$$ M {\pin} N $$
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$$ N {\pin} O $$
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$$ O {\pin} P $$
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$$ O {\pin} Q $$
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$$ Q {\enc} P $$
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$$ A {\enc} M $$
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$$ A {\enc} N $$
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$$ A {\enc} O $$
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$$ A {\enc} P $$
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$$ A {\enc} Q $$
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}
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To form the {\pic} we can follow
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reachable pure intersection relationships.
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$ M {\pin} N {\pin} O {\pin} P $ are part of the same chain.
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following from $O$, $O {\pin} Q$.
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Thus by the definition of being reachable by pure instersection relationships,$M,N,O,P,Q$
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are in the same {\pic}, even though $Q$ encloses $P$.
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We can define this {\pic} as $PIC1$ as a set of contours.
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$$ PIC1 = \{ M,N,O,P,Q \} $$
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Contour $A$, by virtue of not bisecting any contour in the pure instersection
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chain $PIC1$, does not belong to $PIC1$. Because it encloses one of the contours, it
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encloses all contours in the chain.
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Knowing this can save on unecessary area operations on the concrete diagram.
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\begin{figure}[h]
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\centering
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\includegraphics[width=200pt,bb=0 0 330 158,keepaspectratio=true]{./eulerg/eulerg_pic_g.jpg}
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% eulerg_pic_g.jpg: 330x158 pixel, 72dpi, 11.64x5.57 cm, bb=0 0 330 158
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\caption{Pure Intersection chain PIC1 as a graph}
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\label{fig:eulerg_pic_g}
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\end{figure}
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% \subsection{The Pure intersection chain PIC1}
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% \begin{figure}[h]
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% \centering
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% \includegraphics[width=200pt,bb=0 0 955 286,keepaspectratio=true]{./eulerg_pic_g.jpg}
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% % eulerg_pic.jpg: 955x286 pixel, 72dpi, 33.69x10.09 cm, bb=0 0 955 286
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% \caption{The pure Intersection PIC1 as a graph}
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% \label{fig:eulerg_pic1}
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% \end{figure}
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Figure \ref{fig:eulerg_pic_g} only shows the {\pic}, but does not show the contour ($A$)
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enclosing $PIC1$. Figure \ref{fig:eulerg_pic_g_a}
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shows contour A enclosing all elements in $PIC1$
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\pagebreak[0]
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\subsection{Enclosure and pure intersection in the graph}
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\begin{figure}[h]
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\centering
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\includegraphics[width=200pt,bb=0 0 330 162,keepaspectratio=true]{./eulerg/eulerg_pic_g_a.jpg}
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% eulerg_pic_g_a.jpg: 330x162 pixel, 72dpi, 11.64x5.72 cm, bb=0 0 330 162
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\caption{PIC1 including enclosing contour A as a graph}
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\label{fig:eulerg_pic_g_a}
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\end{figure}
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Because we know that a contour enclosing a contour within a {\pic} but not belonging
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to it, encloses all elements of the {\pic}, we can draw this in a less cluttered way
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see figure \ref{fig:eulerg_pic_g_a_unc}.
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\pagebreak[0]
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\subsection{Reducing clutter in the graph}
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\begin{figure}[h]
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\centering
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\includegraphics[width=200pt,bb=0 0 330 162]{./eulerg/eulerg_pic_g_a_unc.jpg}
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% eulerg_pic_g_a_unc.jpg: 330x162 pixel, 72dpi, 11.64x5.72 cm, bb=0 0 330 162
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\caption{Uncluttered graph of euler diagram in figure \ref{fig:eulerg_pic}}
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\label{fig:eulerg_pic_g_a_unc}
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\end{figure}
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\pagebreak[4]
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\section{reduction of searches for available zones}
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Another property of any {\pic} $P$, is that
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the maximum number of euler zones within it is
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$$ MaxZones = 2^{|P|} $$
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Because no contours external to the {\pic}
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bisect any in it, no extra zones can be formed.
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By enclosing a {\pic} with
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a contour, we change the nature of the zones within
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the {\pic}, but the number of zones contributed by the {\pic}
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stays the same.
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\begin{definition}
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A pure intersection chain has a maximum number of possible Euler zones, and exists as independent entities in the diagram. This
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allows us to analyses {\pic}s separately, thus reducing the $2^N$ overhead of analysing an Euler diagram for available zones.
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\end{definition}
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This is to say, the the number of zones within a {\pic} is not affected by changes in the diagram
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that do not alter the {\pic}.
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\vspace{40pt}
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