From c34973f9a885b2b444a7fdc51160a8e004b6a8d7 Mon Sep 17 00:00:00 2001 From: Robin Clark Date: Tue, 17 Aug 2010 13:53:52 +0100 Subject: [PATCH] Added graphs --- eulerg/eulerg.tex | 96 +++++++++++++++++++++++++++++++--- eulerg/eulerg_enc_g.dia | Bin 0 -> 1118 bytes eulerg/eulerg_enc_g.jpg | Bin 0 -> 2411 bytes eulerg/eulerg_g.dia | Bin 0 -> 1236 bytes eulerg/eulerg_g.jpg | Bin 0 -> 2596 bytes eulerg/eulerg_pic_g.dia | Bin 0 -> 1468 bytes eulerg/eulerg_pic_g.jpg | Bin 0 -> 4729 bytes eulerg/eulerg_pic_g_a.dia | Bin 0 -> 1827 bytes eulerg/eulerg_pic_g_a.jpg | Bin 0 -> 9251 bytes eulerg/eulerg_pic_g_a_unc.dia | Bin 0 -> 1634 bytes eulerg/eulerg_pic_g_a_unc.jpg | Bin 0 -> 5794 bytes 11 files changed, 89 insertions(+), 7 deletions(-) create mode 100644 eulerg/eulerg_enc_g.dia create mode 100644 eulerg/eulerg_enc_g.jpg create mode 100644 eulerg/eulerg_g.dia create mode 100644 eulerg/eulerg_g.jpg create mode 100644 eulerg/eulerg_pic_g.dia create mode 100644 eulerg/eulerg_pic_g.jpg create mode 100644 eulerg/eulerg_pic_g_a.dia create mode 100644 eulerg/eulerg_pic_g_a.jpg create mode 100644 eulerg/eulerg_pic_g_a_unc.dia create mode 100644 eulerg/eulerg_pic_g_a_unc.jpg diff --git a/eulerg/eulerg.tex b/eulerg/eulerg.tex index 107a118..9f9a8f8 100644 --- a/eulerg/eulerg.tex +++ b/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} diff --git a/eulerg/eulerg_enc_g.dia b/eulerg/eulerg_enc_g.dia new file mode 100644 index 0000000000000000000000000000000000000000..a76ef9e3b296f384d1a31782bcf1da14c4cb6bad GIT binary patch literal 1118 zcmV-k1flyMiwFP!000001MOK$kE1vczW1*{^mT?{-kzRhG_%^(9#-1J&YmL!ZsHxs zM8-)vm;LQ)1L=fU9uRufqaX#zPPsl;QD0SIK7V;xQf-5IMp8D_fnn+zW>FdwmQ3}( zzyJ0z(7((cKF0)olD~wbr6yOfOpT`cLWtF;$z;3T8g!Q-k#d8QwUOb;zlc&akwO!F z_MmCU3SuOXnk#3K5S)Z-fi;Gfc&dje`jPN-&0@WfD&?Y-rd-<~I@KSa^GlzUY9k+fT*pikI^!<*eB&+|sZdCQmC(h5z@VRl01pCLSIodyqrf&{Kh!8?Y>ru+8*;6AO(|9NJkZ|> zY`u(D^9(#IOSsan12xYnX31&hxI-b%a`Qq2^~hyZ$>`4-)Z71~ zEhJymi;kp+Nb?sI?eMRB&W#{wSqsXwpmXj1O|7<}+c4T0-Y7!L@q8K04s`7iSq){a zb3l-{ZqV{n2?VyUXyij++9NZ&%gpwYX}PwNbRGZB@Slp|U;x9dks1CSzlbA-gOcIy z@KY1p{a}q;u_XjBplO-jlOnY1I1uA9}%83tO7 zzsq8<4^F+6zNnq<1B56^*b=ut%tN}p6s7g7_BuXn@-6kx{I|(P-tcY%jGT%T!afs0 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