From c5b688ac71354edeb1f9dd6e55378e95b8bda75d Mon Sep 17 00:00:00 2001 From: Robin Clark Date: Tue, 31 Aug 2010 07:25:12 +0100 Subject: [PATCH] eulerg --- eulerg/eulerg.tex | 10 +++++----- 1 file changed, 5 insertions(+), 5 deletions(-) diff --git a/eulerg/eulerg.tex b/eulerg/eulerg.tex index 3b6deba..1f144d1 100644 --- a/eulerg/eulerg.tex +++ b/eulerg/eulerg.tex @@ -94,7 +94,7 @@ $A$ `enclosed' by $B$. We can express this as a directed relationship. $$ B {\enc} A $$ - +%\clearpage \paragraph{Pair-wise Intersection} In the $D$, $E$ case we have @@ -117,7 +117,7 @@ then B encloses C, see figure \ref{fig:eulerg_enc}. No contour can be both pair-wisely intersected and enclosed with the same contour. \end{definition} - +\clearpage \begin{figure}[h] \centering @@ -164,9 +164,9 @@ 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 on the same graph. -Figures \ref{fig:eulerg1} and \ref{fig:eulergenc} show euler diagrams with corresponding +Figures \ref{fig:eulerg1} and \ref{fig:eulerg_enc} show euler diagrams with corresponding graphs. The next section will introduce the concept of a {\pic} -and will describe graphs where both enclosure and pair-wise +and will present graphs where both enclosure and pair-wise intersection are represented on the same graph. \pagebreak[1] @@ -321,7 +321,7 @@ Searching for an available zone involves finding out if the intersection exists, by any other contours. A brute force search for available zones using area operations is therefore of the order $N.2^N$ (where N is the number of contours in the diagram). -Using $|P|$ to represent the number of conoutrs within a {\pic} +Using $|P|$ to represent the number of contours within a {\pic} and $K$ to represent the number of {\pic}s in a diagram, using the result in definition \ref{picreduction}, we can break the diagram into small segments (the {\pic}s) which have an order $K.2^{|P|}$.