Added graphs

eulerg/eulerg.tex

@@ -8,7 +8,11 @@ This paper discusses representing Euler Diagrams as graphs, or sets of relations
 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. 
+to extra unecessary CPU expensive area operations on the concrete diagrams. 
+
+The graph representations presented here form the basis for several algorithms
+and time saving procedures, implemented in the FMMD analysis tool.
+
 }
 }
 { %% Introduction
@@ -17,7 +21,10 @@ This paper discusses representing Euler Diagrams as graphs, or sets of relations
 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. 
+to extra unecessary CPU expensive area operations on the concrete diagrams. 
+
+The graph representations presented here form the basis for several algorithms
+and time saving procedures, implemented in the FMMD analysis tool.
 }
 
 
@@ -46,15 +53,15 @@ processing and parsing of the diagram without resorting to area operations in th
 \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.
+Intersection in a concrete diagram can mean two curves bisecting.
 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.
+$A$ and $B$ exists, even though the curves do not bisect in the concrete plane.
 
-$$ A \cup B \neq \emptyset $$
+$$ A \cap B \neq \emptyset $$
 
 as does the intersection $D$ and $E$
 
-$$ D \cup E \neq \emptyset $$
+$$ D \cap E \neq \emptyset $$
 
 Clearly though these intersections are different, because
 in the $A$, $B$ case 
@@ -102,7 +109,7 @@ Enlcosure relationships are transitive
 
 \section{Representing Euler Diagrams as sets of relationships}
 
-The diagram in figure \ref{fig:eulerg1} can be represented by the foillowing relationships.
+The diagram in figure \ref{fig:eulerg1} can be represented by the following relationships.
 
 $$ B {\enc} A $$
 $$ D {\pin} E $$
@@ -113,7 +120,34 @@ The diagram in figure \ref{fig:eulerg_enc} can be represented by the following r
 $$ B {\enc} A $$
 $$ A {\enc} C $$
 
+\section{Represeting Euler diagrams as graphs}
 
+As the relationships {\em enclosure} and {\pic} are mutually exclusive
+and {\em enclosure} is transitive and {\pic} is not, we can represent
+an {\em enclosure} relationship as a directed vertice and 
+{\pic} as non-directed.
+
+The diagram in figure \ref{fig:eulerg1} can now be represented as a graph thus:
+\begin{figure}[h]
+ \centering
+ \includegraphics[width=70pt,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
+ \caption{Graph Representaion of figure \ref{fig:eulerg1}}
+ \label{fig:eulerg1_g}
+\end{figure}
+
+
+The diagram in figure \ref{fig:eulerg_enc} can now be represented as a graph thus:
+\begin{figure}[h]
+ \centering
+ \includegraphics[width=100pt,bb=0 0 240 43,keepaspectratio=true]{./eulerg/eulerg_enc_g.jpg}
+ % eulerg_enc_g.jpg: 240x43 pixel, 72dpi, 8.47x1.52 cm, bb=0 0 240 43
+ \caption{Graph representation of figure \ref{fig:eulerg_enc}}
+ \label{fig:eulerg_enc_g}
+\end{figure}
+
+
+\pagebreak[0]
 \section{The {\pic}}
 
 Contours may be connected via `pure intersection' relationships to form 
@@ -173,6 +207,54 @@ chain $PIC1$, does not belong to  $PIC1$. Because it encloses one of the contour
 encloses all contours in the chain. 
 Knowing this can save on unecessary area operations on the concrete diagram.
 
+% \subsection{The Pure intersection chain PIC1}
+% \begin{figure}[h]
+%  \centering
+%  \includegraphics[width=200pt,bb=0 0 955 286,keepaspectratio=true]{./eulerg_pic_g.jpg}
+%  % eulerg_pic.jpg: 955x286 pixel, 72dpi, 33.69x10.09 cm, bb=0 0 955 286
+%  \caption{The pure Intersection PIC1 as a graph}
+%  \label{fig:eulerg_pic1}
+% \end{figure}
+
+
+\begin{figure}[h]
+ \centering
+ \includegraphics[width=200pt,bb=0 0 330 158,keepaspectratio=true]{./eulerg/eulerg_pic_g.jpg}
+ % eulerg_pic_g.jpg: 330x158 pixel, 72dpi, 11.64x5.57 cm, bb=0 0 330 158
+ \caption{The Pure Intersection Chain PIC1 as a graph}
+ \label{fig:eulerg_pic_g}
+\end{figure}
+
+Figure \ref{fig:eulerg_pic_g} only shows the {\pic}, but does not show the contour ($A$)
+enclosing $PIC1$. Figure \ref{fig:eulerg_pic_g_a}
+shows contour A enclosing all elements in $PIC1$
+
+\pagebreak[0]
+\subsection{Enclosure and pure intersection in the graph}
+\begin{figure}[h]
+ \centering
+ \includegraphics[width=200pt,bb=0 0 330 162,keepaspectratio=true]{./eulerg/eulerg_pic_g_a.jpg}
+ % eulerg_pic_g_a.jpg: 330x162 pixel, 72dpi, 11.64x5.72 cm, bb=0 0 330 162
+ \caption{PIC1 including enclosing contour A as a graph}
+ \label{fig:eulerg_pic_g_a}
+\end{figure}
+
+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
+see figure \ref{fig:eulerg_pic_g_a_unc}.
+
+\pagebreak[0]
+\subsection{Reducing clutter in the graph}
+\begin{figure}[h]
+ \centering
+ \includegraphics[width=200pt,bb=0 0 330 162]{./eulerg/eulerg_pic_g_a_unc.jpg}
+ % eulerg_pic_g_a_unc.jpg: 330x162 pixel, 72dpi, 11.64x5.72 cm, bb=0 0 330 162
+ \caption{Uncluttered graph of euler diagram in figure \ref{fig:eulerg_pic}}
+ \label{fig:eulerg_pic_g_a_unc}
+\end{figure}
+
+
+\pagebreak[4]
 \section{reduction of searches for available zones}