Added euler diagrams as graphs

eulerg/Makefile

@@ -0,0 +1,24 @@
+
+#
+# Make the propositional logic diagram a paper
+#
+SOURCE = eulerg.tex
+
+paper:	paper.tex eulerg_paper.tex $(SOURCE)
+	#cat introduction.tex | sed s/chapter/paper/ > introduction.tex
+	#latex paper.tex
+	#dvipdf paper     pdflatex cannot use eps ffs
+	pdflatex paper.tex
+	mv paper.pdf eulerg_paper.pdf
+	okular eulerg_paper.pdf
+
+
+# Remove the need for referncing graphics in subdirectories
+#
+eulerg_paper.tex: eulerg.tex paper.tex $(SOURCE)
+	cat eulerg.tex | sed 's/eulerg\///'   > eulerg_paper.tex
+
+
+bib: $(SOURCE)	
+	bibtex paper
+	

eulerg/eulerg.tex

@@ -0,0 +1,162 @@
+
+
+
+\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 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 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}
+
+
+\section{Defining `pure intersection' and `enclosure'}
+\begin{figure}[h]
+ \centering
+ \includegraphics[width=200pt,keepaspectratio=true]{./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.
+For instance in diagram \ref{fig:eulerg1} the intersection between
+$A$ and $B$ exists.
+
+$$ A \cup B \neq \emptyset $$
+
+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_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}
+
+$$ B {\enc} A \wedge A {\enc} C \implies B {\enc} C $$
+
+\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 Pure Intersection chain}
+
+Contours may be connected via `pure intersection' relationships to form 
+`chains' of contours reachable by pure intersection.
+
+Figure \ref{fig:eulerg_pic} shows a pure intersection chain consisting of contours $M,N,O,P$ and $Q$.
+
+\begin{figure}[h]
+ \centering
+ \includegraphics[width=300pt,keepaspectratio=true]{./eulerg_pic.jpg}
+ % eulerg_pic.jpg: 955x286 pixel, 72dpi, 33.69x10.09 cm, bb=0 0 955 286
+ \caption{Pure Intersection Chain with Enclosure}
+ \label{fig:eulerg_pic}
+\end{figure}
+
+\textbf{rule:}
+If any contour in a pure intersection chain is enclosed by any contour not belonging to the chain, 
+all the countours within the 
+pure intersection chain will be enclosed by it. This is because a contour 
+enclosing which bisects(????) another contour in a pure intersection chain
+becomes part of the pure~intersection~chain. 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_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 pure intersection chain 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 pure intersection chain, even though $Q$ encloses $P$.
+Contour $A$, by virtue of not bisecting any contour in the pure instersection
+chain, does not belong to it. 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.
+
+
+\vspace{40pt}
+

eulerg/eulerg.tex.backup

@@ -0,0 +1,119 @@
+
+
+
+\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 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 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}
+
+
+\section{Defining `pure intersection' and `enclosure'}
+\begin{figure}[h]
+ \centering
+ \includegraphics[width=200pt,keepaspectratio=true]{./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.
+For instance in diagram \ref{fig:euler1} the intersection between
+$A$ and $B$ exists.
+
+$$ A \cup B \neq \emptyset $$
+
+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_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}
+
+$$ B {\enc} A \wedge A {\enc} C \implies B {\enc} C $$
+
+\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 Pure Intersection chain}
+
+Contours may be connected via `pure intersection' relationships to form 
+`chains' of contours reachable by pure intersection.
+
+Figure \ref{fig:eulerg_pic} shows a pure intersection chain consisting of contours $M,N,O,P$ and $Q$.
+
+
+\textbf{rule:}
+If any contour in a pure intersection chain is enclosed by any contour, all countours within the 
+pure intersection chain will be enclosed by it.
+

eulerg/paper.tex

@@ -0,0 +1,35 @@
+
+\documentclass[a4paper,10pt]{article}
+\usepackage{graphicx}
+\usepackage{fancyhdr}
+\usepackage{tikz}
+\usepackage{amsfonts,amsmath,amsthm}
+\usepackage{algorithm}
+\usepackage{algorithmic}
+\usepackage{ifthen}
+\newboolean{paper}
+\setboolean{paper}{true} % boolvar=true or false
+
+\input{../style}
+
+%\newtheorem{definition}{Definition:}
+
+\begin{document}
+\pagestyle{fancy}
+
+%\outerhead{{\small\bf Symptom Extraction Process}}         
+%\innerfoot{{\small\bf R.P. Clark } }   
+                                              % numbers at outer edges
+\pagenumbering{arabic}                        % Arabic page numbers hereafter
+\author{R.P.Clark}
+\title{Euler Diagrams as Graphs}
+\maketitle 
+\input{eulerg_paper}
+
+\bibliographystyle{plain}
+\bibliography{../vmgbibliography,../mybib}
+
+\today
+
+
+\end{document}

style.tex

@@ -74,6 +74,8 @@
 \newcommand{\dcs}{\em derived~components}
 \newcommand{\bc}{\em base~component}
 \newcommand{\bcs}{\em base~components}
+\newcommand{\enc}{\ensuremath{\stackrel{enc}{\longrightarrow}}}
+\newcommand{\pin}{\ensuremath{\stackrel{pi}{\longleftrightarrow}}}
 %----- Display example text (#1) in typewriter font
  
 %\newcommand{\example}[1]{\\ \smallskip\hspace{1in}{\tt #1}\hfil\\

thesis.tex

@@ -86,6 +86,7 @@
 \chapter {Software as PLDs}
 \input{sw_as_plds/sw_as_plds}
 
+
 \typeout{ ----------------  Mechanical Sub-systems as PLDs}
 \chapter {Common Mechanical Sub-systems as PLDs}
 %\input{mech_as_plds/mech_as_plds}
@@ -136,8 +137,11 @@ for incorrect temperature.
 reference the MSC document and describe the Java extension classes.
 Software documentation for fmmd tool.
 
+\typeout{ ----------------  Euler Diagrams represented as graphs}
+\chapter {Euler Diagrams Represented as graphs}
+\input{eulerg/eulerg}
 
-\chapter{Algorithms and Mathematical Relationships Discovered}
+\chapter{Fast Zone Discrimination Algorithm}
 \input{fzd/fzd}
 
 \chapter{Milli Volt Amp with Safety Resistor}