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Robin Clark 2010-08-21 14:34:44 +01:00
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eulerg/eulerg.tex
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@ -107,8 +107,8 @@ 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 $.
Another way of expressing this is that $A \cap B \neq \emptyset$ and
$ D \subset E$.
Another way of expressing this is that $D \cap E \neq \emptyset$ and
$ A \subset B$.
\paragraph{Enclosure}
To distinguish between these we can term the $A$, $B$ case to be
@ -158,7 +158,7 @@ $$ B {\enc} A \wedge A {\enc} C \implies B {\enc} C $$
Enlcosure relationships are transitive
\end{definition}
\section{Representing Euler Diagrams as sets of relationships}
\section{Representing Euler Diagrams \\ as sets of relationships}
The diagram in figure \ref{fig:eulerg1} can be represented by the following relationships.
@ -264,7 +264,7 @@ 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}
\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}
@ -289,7 +289,7 @@ see figure \ref{fig:eulerg_pic_g_a_unc}.
\pagebreak[0]
\section{Reduction of searches for available zones}
\section{Reduction of searches \\ for available zones}
Another property of any {\pic} $P$, is that
the maximum number of euler zones within it is
@ -322,7 +322,7 @@ by any other contours. A brute force search for available zones using area opera
is therefore of the order $N.2^N$ (where N is the number of contours in the diagram).
By using the result in definition \ref{picreduction}, we can break the diagram into small segments
(the {\pic}s) which have an order $|P|.2^{|P|}$.
The order of area operations is generally\footnote{In the case where the diagram is not comprised of just one {\pic}, which has no enclosing contours}
The order of area operations is generally\footnote{In the case where the diagram is not comprised of just one {\pic}, which has no enclosing contours.}
reduced by requiring several $|P|.2^{|P|}$
instead of $N.2^N$ as $P \leq N$.