morning edit

eulerg/eulerg.tex

@@ -8,7 +8,7 @@ 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 CPU expensive area operations on the concrete diagrams. 
+to unecessary CPU expensive area operations on the concrete diagrams. 
 }
 }
 { %% Introduction
@@ -17,7 +17,7 @@ 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 CPU expensive area operations on the concrete diagrams. 
+to unecessary CPU expensive area operations on the concrete diagrams. 
 }
 
 
@@ -31,6 +31,10 @@ The spaitial relationship between the curves defines the set theoretic relations
 \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]
@@ -42,8 +46,9 @@ The spaitial relationship between the curves defines the set theoretic relations
 \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.
+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.
 
 $$ A \cup B \neq \emptyset $$
 
@@ -91,6 +96,10 @@ then B encloses C, see figure \ref{fig:eulerg_enc}.
 
 $$ B {\enc} A \wedge A {\enc} C \implies B {\enc} C $$
 
+\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.
@@ -105,28 +114,29 @@ $$ B {\enc} A $$
 $$ A {\enc} C $$
 
 
-\section{The Pure Intersection chain}
+\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 pure intersection chain consisting of contours $M,N,O,P$ and $Q$.
+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_pic.jpg}
  % eulerg_pic.jpg: 955x286 pixel, 72dpi, 33.69x10.09 cm, bb=0 0 955 286
- \caption{Pure Intersection Chain with Enclosure}
+ \caption{{\pic} 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, 
+\begin{definition}
+If any contour in a {\pic} 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
+{\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.
 
@@ -146,16 +156,41 @@ $$ A {\enc} P $$
 $$ A {\enc} Q $$
 
 
-To form the pure intersection chain we can follow
+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 pure intersection chain, even though $Q$ encloses $P$.
+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, 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.
+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}

style.tex

@@ -76,6 +76,8 @@
 \newcommand{\bcs}{\em base~components}
 \newcommand{\enc}{\ensuremath{\stackrel{enc}{\longrightarrow}}}
 \newcommand{\pin}{\ensuremath{\stackrel{pi}{\longleftrightarrow}}}
+\newcommand{\pic}{\em pure~intersection~chain}
+
 %----- Display example text (#1) in typewriter font
  
 %\newcommand{\example}[1]{\\ \smallskip\hspace{1in}{\tt #1}\hfil\\