% jobs
% different p1, p2 shapes
% different Smith function shapes
% set defined on sphere or on projection?

%**************************************************************************
%
% # $Id: AWFE.Rnw 30 2009-08-01 22:00:56Z js229 $
%\VignetteIndexEntry{Details of the Edwards construction}

<<defmakeme,echo=FALSE,eval=FALSE>>=
makeme <- function() {
	if ("package:Vennerable" %in% search()) detach("package:Vennerable")
	library(weaver)
	setwd("C:/JonathanSwinton/Vennerable/pkg/Vennerable/inst/doc")
	Sweave(driver="weaver","AWFE.Rnw",stylepath=FALSE,use.cache=FALSE)
}
makeme()
@


\documentclass[a4paper]{article}


\title{
Edwards-Venn diagrams
}
\author{Jonathan Swinton}

\SweaveOpts{prefix.string=AWFEfig,debug=TRUE,eps=FALSE,echo=FALSE,pdf.version=1.4}
\usepackage{float}
%\usepackage{natbib}
\usepackage{mathptmx}
\usepackage{rotating} 
\usepackage[nodayofweek]{datetime}\longdate
\usepackage{hyperref}
\begin{document}


\maketitle
This vignette is mainly for the use of the package developer.

<<doremove,echo=FALSE>>=
remove(list=setdiff(ls(),"makeme"));
library(Vennerable)
library(grid)
library(RColorBrewer)
 
@

Edwards \cite{edwards:2004} starts by constructing Venn sets on a sphere and then projecting onto the plane.
On the sphere we use polar coordinates with longitude $\theta$ and latitude $\phi$. Arc distance
from the equator is $s$ and height above the equatorial plane is $h$:
\begin{eqnarray*}
s &=& r \frac{\phi}{2\pi}
\\
h &=& r \sin \phi
\end{eqnarray*}
Project down onto the equatorial plane (a polar stereographic projection).
\begin{eqnarray*}
\rho &=& \frac{\cos\phi}{1-\sin\phi} 2r
\\
x &=& \rho \cos \theta
\\
y &=& \rho \sin \theta
\end{eqnarray*}
For the purposes of visualisation in this document, we also use a Mercator projection onto the equatorial cylinder
\begin{eqnarray*}
x &=& r \cos \phi \cos \theta
\\
y &=& r \cos \phi \sin \theta
\end{eqnarray*}
\begin{eqnarray*}
x &=&  r \theta
\\
y &=& h
\end{eqnarray*}

Edwards suggests using the Smith functions. In a Mercator projection the Smith functions are
\begin{eqnarray*}
h_n(\theta) &=& \frac{ \cos(2^{n-2} \theta)}{2^{n-2}}
\end{eqnarray*}

So $h_n(\theta)=T_{n-2}\theta$ where 
\begin{eqnarray*}
T_n(x) &=& \frac{1}{2^n}\cos{2^n x}
\\
&=&  \frac{1}{2^n}\cos\frac{1}{2}{2^{n+1} x}
\\
2^{2n} T_n^2 &=& \frac{1}{2}\left( 1+ 2^{n+1}T_{n+1}\right)
\\
T_{n+1} &=& 2^{n} T_n^2 - \frac{1}{2^{n+1}}
\end{eqnarray*}
So $T_{n+1}=0$ when $T_n=\pm 2^{-n} 1/\sqrt(2)$; $2^nx=\pi/4+(p/2)2\pi$.

<<defsmith>>=

@

<<Sbuilder,cache=TRUE>>=
SetBoundaries <- list()
SetBoundaries[["AWFE"]] <- Vennerable:::makeAWFESets(9,type="AWFE",hmax=.58) # only works up to n=8
SetBoundaries[["AWFEscale"]] <- Vennerable:::makeAWFESets(7,type="AWFEscale",hmax=.6) # can't make it work out to n=9
SetBoundaries[["cog"]] <- Vennerable:::makeAWFESets(9,hmax=.6,type="cog")
SetBoundaries[["battle"]] <- Vennerable:::makeAWFESets(9,type="battle")
@



\begin{figure}[H]\begin{center}
<<shoVD4,fig=TRUE,eval=TRUE>>=
plotSsets <- function(Slist,nsets=length(Slist),scale=4) {
	grid.newpage();pushViewport(plotViewport(c(1,1,1,1)))
	makevp.eqsc(scale*c(-1,1),scale*c(-1,1))
	grid.xaxis();grid.yaxis()
	colix <- rep(brewer.pal(9,"Set1"),2)[1:nsets]

	for (ix in 1:nsets) {
		gp <- list(gpar(col=colix[ix])); names(gp)<- names(Slist[[ix]]@setList)
		PlotSetBoundaries(Slist[[ix]],gp=gp)
	}
}
plotSsets(SetBoundaries[["AWFE"]])
@
\caption{AWFE sets based on Smith functions projected from a sphere}
\label{fig:AWFE}
 \end{center}\end{figure}
	
Edwards (cite) suggests using Smith functions as the set boundaries, as shown in
Figure \ref{fig:AWFE}.



\begin{figure}[H]\begin{center}
<<shoVD4v,fig=TRUE,eval=TRUE>>=
grid.newpage();
pushViewport(plotViewport(c(1,1,1,1)));
makevp.eqsc(4*c(-1,1),4*c(-1,1));
grid.xaxis();grid.yaxis();
V8 <- Venn(numberOfSets=8);
plot(V8,type="AWFE",show=list(SetLabels=FALSE,FaceText="",Faces=TRUE),add=TRUE)
@
\caption{8th order AWFE diagram X}
 \end{center}\end{figure}


\subsection{Rescaled Smith functions}

The amplitude of those functions tends geometrically to zero. If we know in advance 
how many sets we need, we can try rescaling so the amplitudes decreases linearly.
These sets are shown in Figure~\ref{fig:AWFEscale}.

\begin{figure}[H]\begin{center}
<<s2builder,fig=TRUE,cache=TRUE>>=
plotSsets(SetBoundaries[["AWFEscale"]])
@
\caption{AWFE sets based on rescaled Smith functions}
\label{fig:AWFEscale}
 \end{center}\end{figure}
	

@
\subsection{Cog sets}
Alternatively we could replace each sine curve by a toothed cog
\begin{figure}[H]\begin{center}
<<scogbuilder,fig=TRUE>>=
plotSsets(SetBoundaries[["cog"]])

@
\caption{AWFE sets based on cogs}
 \end{center}\end{figure}
We don't offer these as TissueDiagrams


\subsection{Battlement sets}
Alternatively we can square the circle
@

\begin{figure}[H]\begin{center}
<<shoVD8battle,fig=TRUE>>=
plotSsets(SetBoundaries[["battle"]],scale=4.5)
@
\caption{AWFE sets based on battlements}
 \end{center}\end{figure}



\begin{figure}[H]\begin{center}
<<shoVD4c,fig=TRUE,cache=TRUE,eval=TRUE>>=
grid.newpage();pushViewport(plotViewport(c(1,1,1,1)))
makevp.eqsc(4*c(-1,1),4*c(-1,1))
grid.xaxis();grid.yaxis()
V8 <- Venn(numberOfSets=8)
plot(V8,type="battle",show=list(SetLabels=FALSE,FaceText="",Faces=TRUE),add=TRUE)
@
\caption{8th order AWFE diagram based on battlements}
 \end{center}\end{figure}




@
\bibliographystyle{plain}
\bibliography{./Venn}

\end{document}