robin/Robin_PHD
1
0
Fork 0
Code Issues Pull Requests Actions Packages Projects Releases Wiki Activity

Spell checked

Browse Source
This commit is contained in:
Robin Clark 2010-08-23 08:48:45 +01:00
parent 63d36851e8
commit c11490049b
1 changed files with 16 additions and 16 deletions
Show Stats Download Patch File Download Diff File Expand all files Collapse all files

32
eulerg/eulerg.tex
View File

@ -6,9 +6,9 @@
\abstract{ \abstract{
This paper discusses representing Euler Diagrams as graphs, or sets of relationships. This paper discusses representing Euler Diagrams as graphs, or sets of relationships.
By representing Euler diagrams in this way, By representing Euler diagrams in this way,
algorithms to invesigate properties of the diagrams, are possible, without algorithms to investigate properties of the diagrams, are possible, without
having to resort having to resort
to extra unecessary CPU expensive area operations on the concrete diagrams. to extra unnecessary CPU expensive area operations on the concrete diagrams.
The graph representations presented here form the basis for several algorithms The graph representations presented here form the basis for several algorithms
and time saving procedures, implemented in the FMMD analysis tool. and time saving procedures, implemented in the FMMD analysis tool.
@ -19,9 +19,9 @@ and time saving procedures, implemented in the FMMD analysis tool.
\section{Introduction} \section{Introduction}
This chapter discusses representing Euler Diagrams as graphs, or sets of relationships. This chapter discusses representing Euler Diagrams as graphs, or sets of relationships.
By representing Euler diagrams in this way, By representing Euler diagrams in this way,
algorithms to invesigate properties of the diagrams, are possible, without algorithms to investigate properties of the diagrams, are possible, without
having to resort having to resort
to extra unecessary CPU expensive area operations on the concrete diagrams. to extra unnecessary CPU expensive area operations on the concrete diagrams.
The graph representations presented here form the basis for several algorithms The graph representations presented here form the basis for several algorithms
and time saving procedures, implemented in the FMMD analysis tool. and time saving procedures, implemented in the FMMD analysis tool.
@ -31,7 +31,7 @@ and time saving procedures, implemented in the FMMD analysis tool.
\section{Introduction : Euler Diagram } \section{Introduction : Euler Diagram }
Classical Euler diagrams consist of closed curves in the plane which are used to represent sets. 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. The spatial relationship between the curves defines the set theoretic relationships, as defined below.
\begin{itemize} \begin{itemize}
\item Intersection - if the curves defining the area within curves overlap \item Intersection - if the curves defining the area within curves overlap
\item Sub-set - if a curve is enclosed by another \item Sub-set - if a curve is enclosed by another
@ -41,7 +41,7 @@ The spaitial relationship between the curves defines the set theoretic relations
The definitions above allow us to read an Euler diagram The definitions above allow us to read an Euler diagram
and write down set theory equations. and write down set theory equations.
The interest here though, is to define relationships between the contours, that allow The interest here though, is to define relationships between the contours, that allow
processing and parsing of the diagram without resorting to extra area operations in the concerete plane. processing and parsing of the diagram without resorting to extra area operations in the concrete plane.
\section{Defining `pair-wise intersection' \\ and `enclosure'} \section{Defining `pair-wise intersection' \\ and `enclosure'}
@ -140,7 +140,7 @@ $$ B {\enc} A \wedge A {\enc} C \implies B {\enc} C $$
\begin{definition} \begin{definition}
\label{def:enctrans} \label{def:enctrans}
Enlcosure relationships are transitive. Enclosure relationships are transitive.
\end{definition} \end{definition}
@ -158,7 +158,7 @@ The diagram in figure \ref{fig:eulerg_enc} can be represented by the following r
$$ B {\enc} A $$ $$ B {\enc} A $$
$$ A {\enc} C $$ $$ A {\enc} C $$
\section{Represeting Euler diagrams as graphs} \section{Representing Euler diagrams as graphs}
As the relationships {\em enclosure} and {\pic} are mutually exclusive As the relationships {\em enclosure} and {\pic} are mutually exclusive
and {\em enclosure} is transitive and {\pic} is not, we can represent and {\em enclosure} is transitive and {\pic} is not, we can represent
@ -187,7 +187,7 @@ Figure \ref{fig:eulerg_pic} shows a {\pic} consisting of contours $M,N,O,P$ and
\begin{definition} \begin{definition}
\label{def:encpic} \label{def:encpic}
If any contour in a {\pic} is enclosed by any contour not belonging to the chain, If any contour in a {\pic} is enclosed by any contour not belonging to the chain,
all the countours within the all the contours within the
{\pic} will be enclosed by it. {\pic} will be enclosed by it.
\end{definition} \end{definition}
@ -218,17 +218,17 @@ reachable pair-wise intersection relationships.
$ M {\pin} N {\pin} O {\pin} P $ are part of the same chain. $ M {\pin} N {\pin} O {\pin} P $ are part of the same chain.
following from $O$, $O {\pin} Q$. following from $O$, $O {\pin} Q$.
Thus by the definition of being reachable by pair-wise instersection relationships,$M,N,O,P,Q$ Thus by the definition of being reachable by pair-wise intersection relationships,$M,N,O,P,Q$
are in the same {\pic}, 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. We can define this {\pic} as $PIC1$ as a set of contours.
$$ PIC1 = \{ M,N,O,P,Q \} $$ $$ PIC1 = \{ M,N,O,P,Q \} $$
Contour $A$, by virtue of not bisecting any contour in the pair-wise instersection Contour $A$, by virtue of not bisecting any contour in the pair-wise intersection
chain $PIC1$, does not belong to $PIC1$. Because it encloses one of the contours, it chain $PIC1$, does not belong to $PIC1$. Because it encloses one of the contours, it
encloses all contours in the chain. encloses all contours in the chain.
Knowing this can save on unecessary area operations on the concrete diagram. Knowing this can save on unnecessary area operations on the concrete diagram.
\begin{figure}[h] \begin{figure}[h]
\centering \centering
@ -259,7 +259,7 @@ we can represent them both on the same graph, see figure \ref{fig:eulerg_pic_g_a
\centering \centering
\includegraphics[width=200pt,bb=0 0 330 162,keepaspectratio=true]{./eulerg/eulerg_pic_g_a.jpg} \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 % eulerg_pic_g_a.jpg: 330x162 pixel, 72dpi, 11.64x5.72 cm, bb=0 0 330 162
\caption{Graph of euler diagram in figure \ref{fig:eulerg_pic}.} \caption{Graph of Euler diagram in figure \ref{fig:eulerg_pic}.}
\label{fig:eulerg_pic_g_a} \label{fig:eulerg_pic_g_a}
\end{figure} \end{figure}
@ -267,7 +267,7 @@ we can represent them both on the same graph, see figure \ref{fig:eulerg_pic_g_a
\subsection{Reducing clutter in the graph} \subsection{Reducing clutter in the graph}
Contour A encloses the pure intersection chain $PIC1$; Contour A encloses the pure intersection chain $PIC1$;
using definition \ref{def:encpic}, we can draw this in a less cluttered way, using definition \ref{def:encpic}, we can draw this in a less cluttered way,
by only drawing one enclosure ralationship to the {\pic} by only drawing one enclosure relationship to the {\pic}
(see figure \ref{fig:eulerg_pic_g_a_unc}). (see figure \ref{fig:eulerg_pic_g_a_unc}).
We only need to show contour A enclosing one member of the {\pic} $PIC1$ We only need to show contour A enclosing one member of the {\pic} $PIC1$
in order to show that contour A encloses all contours in $PIC1$. in order to show that contour A encloses all contours in $PIC1$.
@ -278,7 +278,7 @@ in order to show that contour A encloses all contours in $PIC1$.
\centering \centering
\includegraphics[width=200pt,bb=0 0 330 162]{./eulerg/eulerg_pic_g_a_unc.jpg} \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 % 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}.} \caption{Uncluttered graph of Euler diagram in figure \ref{fig:eulerg_pic}.}
\label{fig:eulerg_pic_g_a_unc} \label{fig:eulerg_pic_g_a_unc}
\end{figure} \end{figure}
@ -287,7 +287,7 @@ in order to show that contour A encloses all contours in $PIC1$.
\section{Reduction of searches \\ for available zones} \section{Reduction of searches \\ for available zones}
Another property of any {\pic} $P$, is that Another property of any {\pic} $P$, is that
the maximum number of euler zones within it is the maximum number of Euler zones within it is
$$ MaxZones = 2^{|P|} $$ $$ MaxZones = 2^{|P|} $$