[Tex/LaTex] Fill the area between several paths

fillbetweentikz-pgf

How can the PGF library fillbetween be used to fill the area marked "A" in this figure? Here is the preliminary code for the figure:

\documentclass[tikz,border=10pt]{standalone}
\usetikzlibrary{arrows,intersections}
\begin{document}
\begin{tikzpicture}[
    thick,
    >=stealth'
  ]

  % Setup
% \draw [help lines] grid (8,5);
  \coordinate (O) at (0,0);

  % Axes
  \draw[name path=x-axis,->] (-0.3,0) -- (8,0) coordinate[label = {below:$\tau$}] (xmax);
  \draw[name path=y-axis,->] (0,-0.3) -- (0,5) coordinate[label = {right:$v(\tau)$}] (ymax);

  % Value function
  \draw [blue, name path=value] (.5,4.5) .. controls (2,1)  .. (7.5,.5);

  \path [name path=level] (0,1) -- (8,1);
  \path [name intersections={of=level and value}];
    \draw[thin,gray] (0,1) node[black,left] {$\sigma$} -- (intersection-1);
    \fill[red] (intersection-1) circle (1pt);
    \draw[thin,gray,name path=taubp] (intersection-1) -- ++(0,-1) node[black,below] {$\tau^*$};

  \path [name path=levelup] (0,1.5) -- (8,1.5);
  \path [name intersections={of=levelup and value}];
    \draw[thin,gray] (0,1.5) node[black,left] {$\sigma+\epsilon$} -- (intersection-1);
    \fill[red] (intersection-1) circle (1pt);
    \draw[thin,gray,name path=taubp] (intersection-1) -- ++(0,-1.5) node[black,below] {$\tau^k$};

  \draw (2.6,1.15) node {A}; % temporary marker

\end{tikzpicture}
\end{document}

Best Answer

As I say in a comment, I'm not sure if the fillbetween library can be used here, but there are other options. Here is one option that uses \clip. Note I gave names to the intersections you found with by={..}.

\documentclass[tikz,border=10pt]{standalone}
\usetikzlibrary{arrows,intersections,backgrounds} % added backgrounds
\begin{document}
\begin{tikzpicture}[
    thick,
    >=stealth'
  ]

  % Setup
% \draw [help lines] grid (8,5);
  \coordinate (O) at (0,0);

  % Axes
  \draw[name path=x-axis,->] (-0.3,0) -- (8,0) coordinate[label = {below:$\tau$}] (xmax);
  \draw[name path=y-axis,->] (0,-0.3) -- (0,5) coordinate[label = {right:$v(\tau)$}] (ymax);

  % Value function
  \draw [blue, name path=value] (.5,4.5) .. controls (2,1)  .. (7.5,.5);

  \path [name path=level] (0,1) -- (8,1);
  \path [name intersections={of=level and value,by={lev}}];
    \draw[thin,gray] (0,1) node[black,left] {$\sigma$} -- (lev);
    \fill[red] (lev) circle (1pt);
    \draw[thin,gray,name path=taubp] (lev) -- ++(0,-1) node[black,below] {$\tau^*$};

  \path [name path=levelup] (0,1.5) -- (8,1.5);
  \path [name intersections={of=levelup and value,by={levup}}];
    \draw[thin,gray] (0,1.5) node[black,left] {$\sigma+\epsilon$} -- (levup);
    \fill[red] (levup) circle (1pt);
    \draw[thin,gray,name path=taubp] (levup) -- ++(0,-1.5) node[black,below] {$\tau^k$};

  \draw (2.6,1.15) node {A}; % temporary marker

\begin{scope}[on background layer]
\clip (lev) rectangle (levup);
\fill [blue!20] (.5,4.5) .. controls (2,1)  .. (7.5,.5) |- (O) -- cycle;
\end{scope}
\end{tikzpicture}
\end{document}

`pgfplots`

For fun, here is one possible method for making a diagram like this with pgfplots and the fillbetween library.

\documentclass[border=10pt]{standalone}
\usepackage{pgfplots}
\pgfplotsset{compat=1.11}
\usepgfplotslibrary{fillbetween}
\begin{document}
\begin{tikzpicture}[
  declare function={f(\x)=exp(-0.5*\x+1.5)+0.5;} % some random function that looked OK
]

% for convenience, save the two x-values we want red dots at in macros
\newcommand\XTickA{2.5}
\newcommand\XTickB{4}
\begin{axis}[
  axis lines=center,
  xmin=-0.3,xmax=8,
  ymin=-0.3,ymax=5,
  % xtick/ytick=data means that ticks are placed at the
  % data points belonging to the first \addplot   
  xtick=data,
  ytick=data,
  typeset ticklabels with strut, % improves vertical alignment
  xticklabels={$\tau^k$,$\tau^*$},
  yticklabels={$\sigma+\epsilon$,$\sigma$},
  xlabel=$\tau$,
  ylabel=$v(\tau)$,
  domain=0.5:7.5,
  smooth
]

% because we have xtick=data,ytick=data, we first plot
% something with samples at our two x-values
% xcomb makes horizontal lines from the y-axis to the point
\addplot [xcomb, gray, samples at={\XTickA,\XTickB}] {f(x)};

% path used for lower boundary of the filled area
% if you have a compat setting lower than 1.11, you need to use
% (axis cs:x,y) for this path instead of (x,y)
\path [name path=level] (\XTickB,{f(\XTickB)}) -- (0,{f(\XTickB)});

% plot the function
\addplot [name path=value] {f(x)};

% ycomb makes vertical lines from x-axis
% add the red dots as markers
\addplot [ycomb,
          mark=*,
          mark options={red,mark size=1pt},
          gray,
          samples at={\XTickA,\XTickB}] {f(x)};

% the filling
% use soft clip to limit the domain of the filling
% I had to use a value slightly higher than 4.0 to get the
% correct filling, possibly due to some roundoff error.
\addplot [blue!10] fill between[of=value and level,
                                soft clip={domain=2.5:4.01}
                               ];

\end{axis}


\end{tikzpicture}
\end{document}

Related Solutions

[Tex/LaTex] Numerical conditional within tikz keys

I am definitely unfamiliar with both beamer and tikz (do not quite get what the \only are supposed to do) but perhaps this could go in the direction you want:

\documentclass{beamer}
\usepackage{tikz}
\usetikzlibrary{chains}
\newcounter{count}

% helper macro:
\long\def\GobToSemiColon #1;{}

\newcommand\myPicture{
\begin{tikzpicture}
  \begin{scope}[start chain = going below]
  \ifnum\value{count}<1 \expandafter\GobToSemiColon\fi
  \ifnum\value{count}>3 \expandafter\GobToSemiColon\fi
  \node[draw, rectangle, on chain] {display only when counter is between
    1 and 3};

  \ifnum\value{count}>-1 \expandafter\GobToSemiColon\fi
  \node[draw, rectangle, on chain] {display only when counter is
    negative}; 

  \ifnum\value{count}<100 \expandafter\GobToSemiColon\fi
  \ifnum\value{count}>200 \expandafter\GobToSemiColon\fi  
  \node[draw, rectangle, on chain] {display only if counter is between
    100 and 200};

  \ifnum\value{count}<3 \expandafter\GobToSemiColon\fi
  \ifnum\value{count}>20 \expandafter\GobToSemiColon\fi
  \node[draw, circle, on chain] {only when counter is in the range 3 to 20};
  \end{scope}
\end{tikzpicture}
}

\begin{document}
\begin{frame}
\only{\setcounter{count}{-3}\myPicture}
\only{\setcounter{count}{105}\myPicture}
\only{\setcounter{count}{39}\myPicture}
\only{\setcounter{count}{2}\myPicture}
\only{\setcounter{count}{5}\myPicture}
\end{frame}
\end{document}

beamer

[Tex/LaTex] TikZ: Drawing an arc from an intersection to an intersection

A solution which allows to draw intersection segments of any two intersections is available as tikz library fillbetween.

This library works as general purpose tikz library, but it is shipped with pgfplots and you need to load pgfplots in order to make it work:

\documentclass{standalone}

\usepackage{tikz}
\usepackage{pgfplots}
\usetikzlibrary{fillbetween}

\begin{document}

\begin{tikzpicture}

\draw [name path=red,red] (120:1.06) circle (1.9);
%\draw [name path=yellow,yellow] (0:1.06) circle (2.12);
\draw [name path=green,green!50!black] (0:0.77) circle (2.41);
\draw [name path=blue,blue] (0:0) circle (1.06);

% substitute this temp path by `\path` to make it invisible:
\draw[name path=temp1, intersection segments={of=red and blue,sequence=L1}];
\draw[red,-stealth,ultra thick, intersection segments={of=temp1 and green,sequence=L3}];

\end{tikzpicture}

\end{document}

enter image description here

The key intersection segments is described in all detail in the pgfplots reference manual section "5.6.6 Intersection Segment Recombination"; the key idea in this case is to

create a temporary path temp1 which is the first intersection segment of red and blue, more precisely, it is the first intersection segment in the Left argument in red and blue : red. This path is drawn as thin black path. Substitute its \draw statement by \path to make it invisible.
Compute the desired intersection segment by intersecting temp1 and green and use the correct intersection segment. By trial and error I figured that it is the third segment of path temp1 which is written as L3 (L = left argument in temp1 and green and 3 means third segment of that path).

The argument involves some trial and error because fillbetween is unaware of the fact that end and startpoint are connected -- and we as end users do not see start and end point.

Note that you can connect these path segments with other paths. If such an intersection segment should be the continuation of another path, use -- as before the first argument in sequence. This allows to fill paths segments:

\documentclass{standalone}

\usepackage{tikz}
\usepackage{pgfplots}
\usetikzlibrary{fillbetween}

\begin{document}

\begin{tikzpicture}

\draw [name path=red,red] (120:1.06) circle (1.9);
%\draw [name path=yellow,yellow] (0:1.06) circle (2.12);
\draw [name path=green,green!50!black] (0:0.77) circle (2.41);
\draw [name path=blue,blue] (0:0) circle (1.06);

% substitute this temp path by `\path` to make it invisible:
\draw[name path=temp1, intersection segments={of=red and blue,sequence=L1}];
\draw[red,fill=blue,-stealth,ultra thick, intersection segments={of=temp1 and green,sequence=L3}]  
    [intersection segments={of=temp1 and green, sequence={--R2}}]
;

\end{tikzpicture}

\end{document}