223 \begin{align*}

   224 \begin{tikzpicture}[baseline]

   225 \node[draw] (c) at (0,0) [circle through = {(1,0)}] {$f$};

   226 \node (d) at (c.east) [circle through = {(0.25,0)}] {};

   227 \foreach \n in {1,2} {

   228 	\node (p\n) at (intersection \n of c and d) {};

   229 	\fill (p\n) circle (2pt);

   230 }

   231 \begin{scope}[decoration={brace,amplitude=10,aspect=0.5}]

   232 	\draw[decorate] (p2.east) -- node[right=2ex] {$a$} (p1.east);

   233 \end{scope}

   234 \end{tikzpicture} & = 

   235 \begin{tikzpicture}[baseline]

   236 \node[draw] (c) at (0,0) [circle through = {(1,0)}] {};

   237 \begin{scope}

   238 \path[clip] (c) circle (1);

   239 \node[draw,dashed] (d) at (c.east) [circle through = {(0.25,0)}] {};

   240 \foreach \n in {1,2} {

   241 	\node (p\n) at (intersection \n of c and d) {};

   242 }

   243 \node[left] at (c) {$f$};

   244 \path[clip] (d) circle (0.75);

   245 \foreach \y in {1,0.86,...,-1} {

   246 	\draw[green!50!brown] (0,\y)--(1,\y);

   247 }

   248 \end{scope}

   249 \draw[->,blue] (1.5,-1) node[below] {$a \times I$} -- (0.75,0);

   250 \end{tikzpicture} \\

   251 \begin{tikzpicture}[baseline]

   252 \node[draw] (c) at (0,0) [ellipse, minimum height=2cm,minimum width=2.5cm] {};

   253 \draw[dashed] (c.north) -- (c.south);

   254 \node[right=6] at (c) {$g$};

   255 \node[left=6] at (c) {$f$};

   256 \end{tikzpicture} & =

   257 \begin{tikzpicture}[baseline]

   258 \node[draw] (c) at (0,0) [ellipse, minimum height=2cm,minimum width=2.5cm] {};

   259 \node[right=9] at (c) {$g$};

   260 \node[left=9] at (c) {$f$};

   261 \draw[dashed] (c.north) to[out=-115,in=115] (c.south) to[out=65,in=-65] (c.north);

   262 \begin{scope}

   263 \path[clip] (c.north) to[out=-115,in=115] (c.south) to[out=65,in=-65] (c.north);

   264 \foreach \y in {1,0.86,...,-1} {

   265 	\draw[green!50!brown] (-1,\y)--(1,\y);

   266 }

   267 \end{scope}

   268 \draw[->,blue] (.75,-1.25) node[below] {$a \times I$} -- (0,-0.25);

   269 \end{tikzpicture} \\

   270 \begin{tikzpicture}[baseline]

   271 \node[draw] (c) at (0,0) [ellipse, minimum height=2cm,minimum width=2.5cm] {};

   272 \draw[dashed] (c.north) -- (c.south);

   273 \node[right=18] at (c) {$g$};

   274 \node[left=10] at (c) {$f$};

   275 \fill (0,0.4) node (p1) {} circle (2pt);

   276 \fill (0,-0.4) node (p2) {} circle (2pt);

   277 \begin{scope}[decoration={brace,amplitude=5,aspect=0.5}]

   278 	\draw[decorate] (p1.east) -- node[right=0.5ex] {\scriptsize $a$} (p2.east);

   279 \end{scope}

   280 \end{tikzpicture} & =

   281 \begin{tikzpicture}[baseline]

   282 \node[draw] (c) at (0,0) [ellipse, minimum height=2cm,minimum width=2.5cm] {};

   283 \node[draw,dashed] (d) at (0,0) [circle, minimum height=1cm,minimum width=1cm] {};

   284 \draw[dashed] (c.north) -- (d.north) (d.south) -- (c.south);

   285 \node[right=18] at (c) {$g$};

   286 \node[left=18] at (c) {$f$};

   287 \clip (0,0) circle (0.5cm);

   288 \foreach \y in {1,0.86,...,-1} {

   289 	\draw[green!50!brown] (-1,\y)--(1,\y);

   290 }

   291 \end{tikzpicture} 

   292 \end{align*}

   293 \todo{fourth case}