--- a/sandbox.tex	Sun Dec 11 22:26:09 2011 -0800
+++ b/sandbox.tex	Sun Dec 11 23:03:37 2011 -0800
@@ -9,5 +9,102 @@
 \title{Sandbox}
 \begin{document}
 
+\begin{align*}
+\begin{tikzpicture}[baseline]
+\node[draw] (c) at (0,0) [circle through = {(1,0)}] {$f$};
+\node (d) at (c.east) [circle through = {(0.25,0)}] {};
+\foreach \n in {1,2} {
+	\node (p\n) at (intersection \n of c and d) {};
+	\fill (p\n) circle (2pt);
+}
+\begin{scope}[decoration={brace,amplitude=10,aspect=0.5}]
+	\draw[decorate] (p2.east) -- node[right=2ex] {$a$} (p1.east);
+\end{scope}
+\end{tikzpicture} & = 
+\begin{tikzpicture}[baseline]
+\node[draw] (c) at (0,0) [circle through = {(1,0)}] {};
+\begin{scope}
+\path[clip] (c) circle (1);
+\node[draw,dashed] (d) at (c.east) [circle through = {(0.25,0)}] {};
+\foreach \n in {1,2} {
+	\node (p\n) at (intersection \n of c and d) {};
+}
+\node[left] at (c) {$f$};
+\path[clip] (d) circle (0.75);
+\foreach \y in {1,0.86,...,-1} {
+	\draw[green!50!brown] (0,\y)--(1,\y);
+}
+\end{scope}
+\draw[->,blue] (1.5,-1) node[below] {$a \times I$} -- (0.75,0);
+\end{tikzpicture} \\
+\begin{tikzpicture}[baseline]
+\node[draw] (c) at (0,0) [ellipse, minimum height=2cm,minimum width=2.5cm] {};
+\draw[dashed] (c.north) -- (c.south);
+\node[right=6] at (c) {$g$};
+\node[left=6] at (c) {$f$};
+\end{tikzpicture} & =
+\begin{tikzpicture}[baseline]
+\node[draw] (c) at (0,0) [ellipse, minimum height=2cm,minimum width=2.5cm] {};
+\node[right=9] at (c) {$g$};
+\node[left=9] at (c) {$f$};
+\draw[dashed] (c.north) to[out=-115,in=115] (c.south) to[out=65,in=-65] (c.north);
+\begin{scope}
+\path[clip] (c.north) to[out=-115,in=115] (c.south) to[out=65,in=-65] (c.north);
+\foreach \y in {1,0.86,...,-1} {
+	\draw[green!50!brown] (-1,\y)--(1,\y);
+}
+\end{scope}
+\draw[->,blue] (.75,-1.25) node[below] {$a \times I$} -- (0,-0.25);
+\end{tikzpicture} \\
+\begin{tikzpicture}[baseline]
+\node[draw] (c) at (0,0) [ellipse, minimum height=2cm,minimum width=2.5cm] {};
+\draw[dashed] (c.north) -- (c.south);
+\node[right=18] at (c) {$g$};
+\node[left=10] at (c) {$f$};
+\fill (0,0.4) node (p1) {} circle (2pt);
+\fill (0,-0.4) node (p2) {} circle (2pt);
+\begin{scope}[decoration={brace,amplitude=5,aspect=0.5}]
+	\draw[decorate] (p1.east) -- node[right=0.5ex] {\scriptsize $a$} (p2.east);
+\end{scope}
+\end{tikzpicture} & =
+\begin{tikzpicture}[baseline]
+\node[draw] (c) at (0,0) [ellipse, minimum height=2cm,minimum width=2.5cm] {};
+\node[draw,dashed] (d) at (0,0) [circle, minimum height=1cm,minimum width=1cm] {};
+\draw[dashed] (c.north) -- (d.north) (d.south) -- (c.south);
+\node[right=18] at (c) {$g$};
+\node[left=18] at (c) {$f$};
+\clip (0,0) circle (0.5cm);
+\foreach \y in {1,0.86,...,-1} {
+	\draw[green!50!brown] (-1,\y)--(1,\y);
+}
+\end{tikzpicture} \\
+\end{align*}
 
+\begin{tikzpicture}
+\node[circle,fill=black,inner sep=1pt] (A) at (1.73,0) {};
+\node[circle,fill=black,inner sep=1pt] (B) at (-1.73,0) {};
+\draw[dashed] (A) -- (B);
+\node[circle,fill=black,inner sep=1pt] (C) at (0,0) {};
+\node[circle,fill=black,inner sep=1pt] (D) at (0.8,0) {};
+\begin{scope}[yshift=-1cm]
+\path[clip] (0,0) circle (2);
+\begin{scope}[yshift=2cm]
+\draw (0,0) circle (2);
+\node[circle,fill=black,inner sep=1pt] (L2) at (-90:2) {};
+\node[circle,fill=black,inner sep=1pt] (L1) at (-120:2) {};
+\end{scope}
+\end{scope}
+\begin{scope}[yshift=1cm]
+\path[clip] (0,0) circle (2);
+\begin{scope}[yshift=-2cm]
+\draw (0,0) circle (2);
+\node[circle,fill=black,inner sep=1pt] (U) at (90:2) {};
+\end{scope}
+\end{scope}
+\begin{scope}
+\path[clip] (0,1) circle (2);
+\path[clip] (0,-1) circle (2);
+
+\end{scope}
+\end{tikzpicture}
 \end{document}