12 

    12 \begin{align*}

    13 

    13 \begin{tikzpicture}[baseline]

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

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

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

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

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

    19 }

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

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

    22 \end{scope}

    23 \end{tikzpicture} & = 

    24 \begin{tikzpicture}[baseline]

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

    26 \begin{scope}

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

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

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

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

    31 }

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

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

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

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

    36 }

    37 \end{scope}

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

    39 \end{tikzpicture} \\

    40 \begin{tikzpicture}[baseline]

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

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

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

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

    45 \end{tikzpicture} & =

    46 \begin{tikzpicture}[baseline]

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

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

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

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

    51 \begin{scope}

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

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

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

    55 }

    56 \end{scope}

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

    58 \end{tikzpicture} \\

    59 \begin{tikzpicture}[baseline]

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

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

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

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

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

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

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

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

    68 \end{scope}

    69 \end{tikzpicture} & =

    70 \begin{tikzpicture}[baseline]

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

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

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

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

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

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

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

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

    79 }

    80 \end{tikzpicture} \\

    81 \end{align*}