worked on fzd and images

fzd/fzd.tex

@@ -1,67 +1,47 @@
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-
-%\documentclass[reviewversion,strict]{twocolconf}
-
-
-%\conferencename{EULER2005}
-
-
-% \begin{document}
-
-
-% \papertitle { Fast Zone Discrimination \thanks{Version 1.0 2004JUL31 
-% {\tt twocolconf} class.}}
-% {
-% R P Clark\thanks{footnote}\\
-% \em Written as a by product of constraint editor production \\
-% \email{r.clark@energytechnologycontrol.com} \\
-%\and
-%Second author \\
-%\em Affiliation \\
-%\em Affiliation  \\
-%\email{x@any.tld}\\
-%\and
-%Third author \\
-%\em Affiliation \\
-%\em Affiliation  \\
-%\email{y@any.tld}\\
-%}
-
-%\headertitle{Enables an alternate running header, if applicable to current layout}
-
-
-% \begin{abstract}
-
-% This paper concentrates on an algorithmic method for determining the
-% available zones in an Euler Diagram. It introduces the concepts of pure
-%  and enclosing relations between contours and chains of pure intersections.
-% By representing 
-% spider/constraint diagrams as directed graphs, this then develops rules and 
-% a recursive strategy
-%  determining the available zones, and by deduction, eliminating examining contour combinations that
-% will not contain available zones.
-% \end{abstract}
-
-
 
 \section{Introduction : Euler Diagram and Zones Available for use}
 
+Euler diagrams consist of closed curves in the plane which are used to represent sets.
+The spaitial ralationship between the curves defines the set theoretic relationships.
+\begin{itemize}
+\item Intersection - if the curves overlap
+\item Sub-set      - if a curve is enclosed by another
+\item disjoint     - if the curves are separate
+\end{itemize}
 
-\subsection{Defining Available Zones}
+A zone is defined as a region of the plane where a set of curves will enclose it
-
+and another set is disjoint from it.
-An Euler diagram as opposed to a Venn diagram defines a universe of discourse.
+A Venn diagram is an Euler diagram where all possible zones are present.
-In a Venn diagram all possible zones are visible and available
+Thus if we have a Venn diagram with $N$ countours it will have $2^N$
-for placing of existential objects.% \cite{wiki}.
+zones.
-\par
+%
-An Euler diagram, by not having to make all possible zones available
+An Euler diagram, does not have to  make all possible zones available.
-restricts this and can place conjunctive constraints on the number combinations of
+Thus we can make certain combinations of contours unavailable
-attributes that the diagram respesents.
+by drawing the diagrams. Or in other words we can deliberately restrict
-\par
+not only combinations of contours but also the number of available zones.
-When performing  logical reasoning on euler diagrams with added existential objects,
+%
-it is important to note which available zones do not have objects associated with them.
+In Constraint Diagram and PLD's Euler diagrams are used
-These represent cases where the user has left them undefined, or considers them to
+and objects are placed upon the available zones.
+%
+When performing  logical reasoning on euler diagrams,
+it is important to note which available zones do not have objects associated with them in order to flag the unhandled cases in the diagram.
+These could represent cases where the user has left them undefined, or considers them to
 be a general case. Either way they need to be flagged as an ommission error or
-collected. In order to do this a method of finding all available zones in an euler diagram is necessary.
+collected. In order to do this a method of finding all available zones is necessary.
+%
+Also note that in a complicated diagram, a zone may, at first glance, appear available, but could be covered-up or obscured by
+an enclosing contour.
+
+%The sizes or shapes of the curves are not important, the significance of the diagram is in how they overlap.
+%The spatial relationships between the regions bounded by each curve (overlap, containment or neither) corresponds to set-theoretic relationships (intersection, subset and disjointness).
+%
+%Each Euler curve divides the plane into two regions or "zones": the interior, which symbolically represents the elements of the set, and the exterior, which represents all elements which are not members of the set. 
+%Curves which do not intersect represent disjoint sets. Two curves which intersect represent sets that have common elements; the zone inside both curves represents the set of elements common to both sets (the intersection of the sets). A curve which is contained completely within another represents a subset of it.
+%
+%Venn diagrams are a more restrictive form of Euler diagrams. 
+%A Venn diagram must contain all the possible zones of overlap between its curves, representing all combinations of inclusion/exclusion of its constituent sets, but in an Euler diagram some zones might be missing. 
+%Therefore there is only one Venn diagram representing the relationships between n sets, with 2n zones, but there may be many Euler diagrams.
+%(An example is given below in the History section; in the top-right illustration the O and I diagrams are merely rotated; Venn stated that this difficulty in part led him to develop his diagrams).
 
 \par
 
@@ -75,21 +55,38 @@ For a zone to be available for use it must
 
 \par
 
+
+Each zone can be defined by the contours that intersect to define it and the remaining 
+contours in the diagram.
 Thus for a diagram $ D $ consisting of a set of zones $ Z $ where the  zones in the diagram 
 $ Z_{n} $
  are  defined as a sets of intersection contours $ A_{n} $, and exclusion sets 
 $ B_{n} $.
 
-\subsubsection{Testing for Obscuration}
 
-Obscuration is tested for by being able to subtract one shape from another
+\subsubsection{Testing a Zone for Existance}
-with a resultant shape only containg the remainder of the subtraction.
+
-This is as defined in the Java Area classes \cite{javaarea} .
+The Java Area, Shape and Polygon classes, provide functions to
-A zone obscured by other contours is one that forms the area of intersection
+intersect, subtract and `exclusive or' Areas on the plane. They are thus
+very useful in testing objects drawn under a Java environment.
+%
+To determine if a zone exists we can apply the intersection
+functions to all the $A_{N}$ contours. If any Area remains
+the zone exists. To check that the zone is available, we must
+also ensure that it is not covered over by a contour in the $ B_{N}$ set.
+
+\subsubsection{Testing for Obscuration}
+Firstly apply intersection to all the $A_{n}$ contours in the Zone.
+We are then left with the area of intersection only.
+%
+From this we can subtract all the areas from the $B_{n}$ contours.
+If the zone was obscured, or covered up the Area object will register that it has no area on the plane.
+%
+In other words, a zone obscured by other contours is one that forms the area of intersection
 of the set $ A_{n} $ and then having all the areas from set $ B_{n} $ subtracted from it
 no surface area left.
-
+%
-Firstly let us define the meaning of availability in concrete area terms.
+Firstly let us define the meaning of availability in concrete area termsi by means of an example diagram.
 In figure \ref{fig:ex1}, there is an Euler diagram with the following zones available.
 
 
@@ -119,25 +116,39 @@ In figure \ref{fig:ex1}, there is an Euler diagram with the following zones avai
 
  Note $B \cap C$ and $ C $ are not available in this diagram because it is impossible to place objects on them.
  Objects could be placed on $ B \cap C \cap D $ and $ B \cap D $ however.
+\begin{figure}[h]
+ \centering
+ \includegraphics[width=200pt,bb=0 0 315 217,keepaspectratio=true]{fzd/exampleareasubtraction1.jpg}
+ % exampleareasubtraction1.jpg: 438x301 pixel, 100dpi, 11.13x7.65 cm, bb=0 0 315 217
+ \caption{Simple Euler Diagram}
+ \label{fig:ex1}
+\end{figure}
 
-
+% 
- \begin{figure}
+%  \begin{figure}
- \vskip 4.2cm
+%  \vskip 4.2cm
- \special{psfile=fzd/exampleareasubtraction1.eps hoffset=0 voffset=-10 hscale=60 vscale=60}\caption[Area Intersection]{
+%  \special{psfile=fzd/exampleareasubtraction1.eps hoffset=0 voffset=-10 hscale=60 vscale=60}\caption[Area Intersection]{
- Simple Euler Diagram
+%  Simple Euler Diagram
- \label{fig:ex1}}
+%  \label{fig:ex1}}
- \end{figure}
+%  \end{figure}
 
  Examining the intersection $ A \cap B $ the corresponding Area is shown in red 
  in figure \ref{fig:ex2}.
 
+\begin{figure}[h]
+ \centering
+ \includegraphics[width=200pt,bb=0 0 315 217,keepaspectratio=true]{fzd/exampleareasubtraction2.jpg}
+ % exampleareasubtraction2.jpg: 438x301 pixel, 100dpi, 11.13x7.65 cm, bb=0 0 315 217
+ \caption{Area representation of intersection between A and B}
+ \label{fig:ex2}
+\end{figure}
 
- \begin{figure}
+%  \begin{figure}
- \vskip 4.2cm
+%  \vskip 4.2cm
- \special{psfile=fzd/exampleareasubtraction2.eps hoffset=0 voffset=-10 hscale=60 vscale=60}\caption[Area Intersection]{
+%  \special{psfile=fzd/exampleareasubtraction2.eps hoffset=0 voffset=-10 hscale=60 vscale=60}\caption[Area Intersection]{
- Area representation of intersection between set A and B
+%  Area representation of intersection between set A and B
- \label{fig:ex2}}
+%  \label{fig:ex2}}
- \end{figure}
+%  \end{figure}
 
  The area to be subtracted is shown in blue in figure \ref{fig:ex3}.
  Note that here the intersection exists
@@ -148,13 +159,20 @@ In figure \ref{fig:ex1}, there is an Euler diagram with the following zones avai
  areas comprised of the other contours  $ C \cup D $ 
  (in fact in this diagram  only $ D $ is required to prove obscuration
  because $ C \subset D $ ).
-
+\begin{figure}[h]
-\begin{figure}
+ \centering
-\vskip 4cm
+ \includegraphics[width=200pt,bb=0 0 315 217,keepaspectratio=true]{fzd/exampleareasubtraction3.jpg}
-\special{psfile=fzd/exampleareasubtraction3.eps hoffset=0 voffset=-10 hscale=60 vscale=60}\caption[Area Intersection]{
+ % exampleareasubtraction3.jpg: 438x301 pixel, 100dpi, 11.13x7.65 cm, bb=0 0 315 217
-Area representation of exclusion Zone to be subtracted
+ \caption{Area representation of exclusion zone to be subtracted}
-\label{fig:ex3}}
+ \label{fig:ex3}
 \end{figure}
+% 
+% \begin{figure}
+% \vskip 4cm
+% \special{psfile=fzd/exampleareasubtraction3.eps hoffset=0 voffset=-10 hscale=60 vscale=60}\caption[Area Intersection]{
+% Area representation of exclusion Zone to be subtracted
+% \label{fig:ex3}}
+% \end{figure}
 
 %\begin{picture}(200,200)(10,10)
 %\put(20,0){\circle{20}}
@@ -335,13 +353,20 @@ is small in comparison with $2^{N}$ the algorithm becomes far more efficient.
 Examples of complexity savings are shown in section \ref{complexity}.
 
 
-
+\begin{figure}[h]
-\begin{figure}
+ \centering
-\vskip 6cm
+ \includegraphics[width=200pt,bb=0 0 642 482,keepaspectratio=true]{fzd/piee.jpg}
-\special{psfile=fzd/piee.ps hoffset=0 voffset=-10 hscale=40 vscale=40}\caption[Pure Intersection Zone Capture Method]{
+ % piee.jpg: 891x669 pixel, 100dpi, 22.63x16.99 cm, bb=0 0 642 482
-A Pure Intersection  and an Enclosure
+ \caption{A Pure Intersection and an Enclosure}
-\label{fig:piee}}
+ \label{fig:piee}
 \end{figure}
+
+% \begin{figure}
+% \vskip 6cm
+% \special{psfile=fzd/piee.ps hoffset=0 voffset=-10 hscale=40 vscale=40}\caption[Pure Intersection Zone Capture Method]{
+% A Pure Intersection  and an Enclosure
+% \label{fig:piee}}
+% \end{figure}
 \subsection { Relationships between Contours }
 
 The algorithm for fast finding of available zones depends upon defining three new relationships between contours.
@@ -362,12 +387,20 @@ A countour is said to be enclosed if it fits completely within another contour.
 
 \subsubsection { Pure Intersection Chains }
 
-\begin{figure}
+\begin{figure}[h]
-\vskip 6cm
+ \centering
-\special{psfile=fzd/pic.ps hoffset=0 voffset=0 hscale=40 vscale=40}\caption[Pure Intersection Zone Capture Method]{
+ \includegraphics[width=200pt,bb=0 0 642 482]{fzd/pic.jpg}
-Pure Intersection Zone Chain
+ % pic.jpg: 891x669 pixel, 100dpi, 22.63x16.99 cm, bb=0 0 642 482
-\label{fig:pic}}
+ \caption{Pure Intersection Zone Chain}
+ \label{fig:pic}
 \end{figure}
+% 
+% \begin{figure}
+% \vskip 6cm
+% \special{psfile=fzd/pic.ps hoffset=0 voffset=0 hscale=40 vscale=40}\caption[Pure Intersection Zone Capture Method]{
+% Pure Intersection Zone Chain
+% \label{fig:pic}}
+% \end{figure}
 
 Pairs of contours may belong to the same pure intersection chain.
 Pure Intersection chains are a chain of contours that can all 
@@ -427,14 +460,21 @@ This forms a list of relationship pairs from the cross product of all the contou
 \endequation
 
 
-
+\begin{figure}[h]
-\begin{figure}
+ \centering
-\vskip 6cm
+ \includegraphics[width=200pt,bb=0 0 452 290,keepaspectratio=true]{fzd/pice.jpg}
-\special{psfile=fzd/pice.ps hoffset=0 voffset=0 hscale=40 vscale=40}\caption[Pure Intersection Zone Capture Method]{
+ % pice.jpg: 628x403 pixel, 100dpi, 15.95x10.24 cm, bb=0 0 452 290
-Pure Intersection Chain with an implicit Enclosure
+ \caption{Pure Intersection Chain with implicit enclosure}
-\label{fig:picwaie}}
+ \label{fig:picwaie}
 \end{figure}
 
+% \begin{figure}
+% \vskip 6cm
+% \special{psfile=fzd/pice.ps hoffset=0 voffset=0 hscale=40 vscale=40}\caption[Pure Intersection Zone Capture Method]{
+% Pure Intersection Chain with an implicit Enclosure
+% \label{fig:picwaie}}
+% \end{figure}
+
 \subsubsection {Determining  Enclosure }
 
 When a contour completely encloses another contour, it has an enclosing relationship.
@@ -470,21 +510,37 @@ intersections and red for enclosures for the examples that follow.
 
 
 Figure \ref{fig:pig1} Shows a CDG for the diagram in figure \ref{fig:pic}. Note that traversing through this graph reveals all the intersections, and that there are no loops in the traversal, meaning that no multiple intersections exist. 
-\begin{figure}
+
-\vskip 6cm
+\begin{figure}[h]
-\special{psfile=fzd/pig1.ps hoffset=0 voffset=0 hscale=40 vscale=40}\caption[Pure Intersection Zone Capture Method]{
+ \centering
-Coloured Directed Graph of Pure Intersection Chain 
+ \includegraphics[width=200pt,bb=0 0 361 163]{fzd/pig1.jpg}
-\label{fig:pig1}}
+ % pig1.jpg: 502x227 pixel, 100dpi, 12.75x5.77 cm, bb=0 0 361 163
+ \caption{Coloured Directed Graph of Pure Intersection Chain}
+ \label{fig:pig1}
 \end{figure}
+% 
+% \begin{figure}
+% \vskip 6cm
+% \special{psfile=fzd/pig1.ps hoffset=0 voffset=0 hscale=40 vscale=40}\caption[Pure Intersection Zone Capture Method]{
+% Coloured Directed Graph of Pure Intersection Chain 
+% \label{fig:pig1}}
+% \end{figure}
 
 
-
+\begin{figure}[h]
-\begin{figure}
+ \centering
-\vskip 6cm
+ \includegraphics[width=200pt,bb=0 0 352 307,keepaspectratio=true]{fzd/pig2.jpg}
-\special{psfile=fzd/pig2.ps hoffset=0 voffset=0 hscale=40 vscale=40}\caption[Pure Intersection Zone Capture Method]{
+ % pig2.jpg: 489x427 pixel, 100dpi, 12.42x10.85 cm, bb=0 0 352 307
-Coloured Directed Graph of Pure Intersection Chain with an implicit Enclosure
+ \caption{Coloured Directed Graph of Pure Intersection Chain with an implicit enclosure}
-\label{fig:pig2}}
+ \label{fig:pig2}
 \end{figure}
+% 
+% \begin{figure}
+% \vskip 6cm
+% \special{psfile=fzd/pig2.ps hoffset=0 voffset=0 hscale=40 vscale=40}\caption[Pure Intersection Zone Capture Method]{
+% Coloured Directed Graph of Pure Intersection Chain with an implicit Enclosure
+% \label{fig:pig2}}
+% \end{figure}
 
 
 \subsubsection { Graph Traversal }
@@ -538,31 +594,53 @@ Multiple intersections due to enclosure are discovered by traversing the
 enclosure relations.
 \par
 
-\label{fzd}
+\begin{figure}[h]
-\begin{figure}
+ \centering
-\vskip 6cm
+ \includegraphics[width=200pt,bb=0 0 375 377,keepaspectratio=true]{fzd/abc1.jpg}
-\special{psfile=fzd/abc1.eps hoffset=0 voffset=0 hscale=60 vscale=60}\caption[Pure Intersection Zone Capture Method]{
+ % abc1.jpg: 521x524 pixel, 100dpi, 13.23x13.31 cm, bb=0 0 375 377
-Circular reference with multiple zone
+ \caption{circular Reference with multiple zone}
-\label{fig:abc1}}
+ \label{fig:abc1}
 \end{figure}
+% 
+% \label{fzd}
+% \begin{figure}
+% \vskip 6cm
+% \special{psfile=fzd/abc1.eps hoffset=0 voffset=0 hscale=60 vscale=60}\caption[Pure Intersection Zone Capture Method]{
+% Circular reference with multiple zone
+% \label{fig:abc1}}
+% \end{figure}
 
-
+\begin{figure}[h]
-\label{fzd}
+ \centering
-\begin{figure}
+ \includegraphics[width=200pt,bb=0 0 390 371]{fzd/abc2.jpg}
-\vskip 6cm
+ % abc2.jpg: 542x515 pixel, 100dpi, 13.77x13.08 cm, bb=0 0 390 371
-\special{psfile=fzd/abc2.eps hoffset=0 voffset=-20 hscale=60 vscale=60}\caption[Pure Intersection Zone Capture Method]{
+ \caption{Circular reference without multiple zone}
-Circular reference without multiple zone
+ \label{fig:abc2}
-\label{fig:abc2}}
 \end{figure}
+% 
+% \label{fzd}
+% \begin{figure}
+% \vskip 6cm
+% \special{psfile=fzd/abc2.eps hoffset=0 voffset=-20 hscale=60 vscale=60}\caption[Pure Intersection Zone Capture Method]{
+% Circular reference without multiple zone
+% \label{fig:abc2}}
+% \end{figure}
 
-
+\begin{figure}[h]
-\label{fzd}
+ \centering
-\begin{figure}
+ \includegraphics[width=200pt,bb=0 0 642 482,keepaspectratio=true]{fzd/abcgraph.jpg}
-\vskip 6cm
+ % abcgraph.jpg: 891x669 pixel, 100dpi, 22.63x16.99 cm, bb=0 0 642 482
-\special{psfile=fzd/abcgraph.ps hoffset=0 voffset=-10 hscale=30 vscale=30}\caption[Pure Intersection Zone Capture Method]{
+ \caption{Pure Intersection Zone and Enclosure}
-Pure Intersection Zone and Enclosure
+ \label{fig:abdgraph}
-\label{fig:abcgraph}}
 \end{figure}
+% 
+% \label{fzd}
+% \begin{figure}
+% \vskip 6cm
+% \special{psfile=fzd/abcgraph.ps hoffset=0 voffset=-10 hscale=30 vscale=30}\caption[Pure Intersection Zone Capture Method]{
+% Pure Intersection Zone and Enclosure
+% \label{fig:abcgraph}}
+% \end{figure}
 
 In order to examine multiple intersections, the spanning tree from the contour under inspection must be 
 recursively iterated (only within the pure intersection chain it belongs to i.e. within the subset $G$).
@@ -767,12 +845,21 @@ This diagram therefore requires $128 + 2.(9 + 9) \equiv 146  $ area compares.
 2.N^{2}  + \frac{N}{4}.(18)  
 \endequation
 
-\begin{figure}
+
-\vskip 6cm
+\begin{figure}[h]
-\special{psfile=fzd/tripples.ps hoffset=0 voffset=-10 hscale=40 vscale=40}\caption[Two Enclosed Venn 3]{
+ \centering
-Two Enclosed Ven 3
+ \includegraphics[width=200pt,bb=0 0 585 410,keepaspectratio=true]{fzd/tripples.jpg}
-\label{fig:tev3}}
+ % tripples.jpg: 813x569 pixel, 100dpi, 20.65x14.45 cm, bb=0 0 585 410
+ \caption{Two Enclosed Venn 3}
+ \label{fig:tev3}
 \end{figure}
+% 
+% \begin{figure}
+% \vskip 6cm
+% \special{psfile=fzd/tripples.ps hoffset=0 voffset=-10 hscale=40 vscale=40}\caption[Two Enclosed Venn 3]{
+% Two Enclosed Ven 3
+% \label{fig:tev3}}
+% \end{figure}
 
 
 \subsection {Extrapolating for N Contour Diagrams}
@@ -784,21 +871,36 @@ against diagram complexity
 can be drawn. These graphs clearly shows that the fzd method efficiency increases with the 
 number of contours in a diagram. 
 
-
+\begin{figure}[h]
-\begin{figure}
+ \centering
-\vskip 6cm
+ \includegraphics[width=200pt,bb=0 0 414 308,keepaspectratio=true]{fzd/perf1.jpg}
-\special{psfile=fzd/perf1.ps hoffset=0 voffset=-10 hscale=60 vscale=60}\caption[Performance Comparison]{
+ % perf1.jpg: 575x428 pixel, 100dpi, 14.60x10.87 cm, bb=0 0 414 308
-Perfomance from 0 to 8 contours
+ \caption{Performace from 0 to 8 contours}
-\label{fig:perf1}}
+ \label{fig:perf1}
 \end{figure}
 
-\begin{figure}
+% \begin{figure}
-\vskip 6cm
+% \vskip 6cm
-\special{psfile=fzd/perf2.ps hoffset=0 voffset=-10 hscale=60 vscale=60}\caption[Performance Comparison]{
+% \special{psfile=fzd/perf1.ps hoffset=0 voffset=-10 hscale=60 vscale=60}\caption[Performance Comparison]{
-Performance from 8 to 64 contours
+% Perfomance from 0 to 8 contours
-\label{fig:perf2}}
+% \label{fig:perf1}}
+% \end{figure}
+
+\begin{figure}[h]
+ \centering
+ \includegraphics[width=200pt,bb=0 0 414 308,keepaspectratio=true]{fzd/perf2.jpg}
+ % perf2.jpg: 575x428 pixel, 100dpi, 14.60x10.87 cm, bb=0 0 414 308
+ \caption{Performace from 8 to 64 contours}
+ \label{fig:perf2}
 \end{figure}
 
+% \begin{figure}
+% \vskip 6cm
+% \special{psfile=fzd/perf2.ps hoffset=0 voffset=-10 hscale=60 vscale=60}\caption[Performance Comparison]{
+% Performance from 8 to 64 contours
+% \label{fig:perf2}}
+% \end{figure}
+
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 % ALREADY DONE ! Obscuration is now only looked at within pure intersection chains.