diff -r dec5ea4f3452 -r fea0cfe78103 text/appendixes/comparing_defs.tex
--- a/text/appendixes/comparing_defs.tex	Sun Dec 11 23:03:37 2011 -0800
+++ b/text/appendixes/comparing_defs.tex	Sun Dec 11 23:18:36 2011 -0800
@@ -220,6 +220,77 @@
 $$
 \mathfig{0.5}{triangle/triangle2}
 $$
+\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*}
+\todo{fourth case}
 \caption{Examples of inserting or deleting product regions.}
 \label{fig:product-regions}
 \end{figure}