fuse different paths

Question

Suppose that we have four nodes a, b, c, and d. For example :

\coordinate [label=left:a]  (a) at (0,4);
\coordinate [label=right:b] (b) at (4,4);
\coordinate [label=left:c]  (c) at (0,0);
\coordinate [label=right:d] (d) at (4,0);

Let's draw some arbitrary paths between every couple in {(a,b), (a,c), (c,d), (d,b), (b,a)}. Example :

\draw [red]    (a) to [bend left=30]                   (c);
\draw [blue]   (b) to [out=45, in= -50]                (a);
\draw [orange] (c) to [controls=+(45:6) and +(170:6)]  (d);
\draw [green!60!black,decorate,decoration={snake,pre length=1pt}] (d) -- (b);

To my knowledge

\path (a) -- (c) -- (d) -- (b) -- cycle;

is considered as one path, so therefore we can fill the area inside this path; But in the example above we have four paths and therefore the fill area is senseless for tikZ.

My Question : Is it possible to fuse all of paths (in example) to create one fillable path like \fill [blue!10] (a) -- (c) -- (d) -- (b) -- cycle; or something else ? In other way, I want to hatch the surface S.

All code :

\documentclass[tikz, border=1cm]{standalone}

\usepackage{tikz}
\usetikzlibrary{decorations.pathmorphing}

\mathversion{bold}

\begin{document}

\begin{tikzpicture}
\coordinate [label=left:a]  (a) at (0,4);
\coordinate [label=right:b] (b) at (4,4);
\coordinate [label=left:c]  (c) at (0,0);
\coordinate [label=right:d] (d) at (4,0);

\foreach \p in {a,b,c,d}{
\fill[red] (\p) circle (2pt);}

\draw [red]    (a) to [bend left=30]                   (c);
\draw [blue]   (b) to [out=45, in= -50]                (a);
\draw [orange] (c) to [controls=+(45:6) and +(170:6)]  (d);
\draw [green!60!black,decorate,decoration={snake,pre length=1pt}] (d) -- (b);
\node at (2,2.5) {$S$};

\end{tikzpicture}

\end{document}

To be more clear, I encountered this problem when creating a (breakable) tcolorbox

I want to fill the empty surface S, but with four paths I cannot do it.

Answer 1

Yes, of course. You cannot draw the path in different colors, but certainly you can combine the stretches to be filled, and draw the colored stretches either explicitly separately, or via edges.

\documentclass[tikz, border=1cm]{standalone}

\usepackage{tikz}
\usetikzlibrary{decorations.pathmorphing}

\mathversion{bold}

\begin{document}

\begin{tikzpicture}
\coordinate [label=left:a]  (a) at (0,4);
\coordinate [label=right:b] (b) at (4,4);
\coordinate [label=left:c]  (c) at (0,0);
\coordinate [label=right:d] (d) at (4,0);

\foreach \p in {a,b,c,d}{
\fill[red] (\p) circle (2pt);}
\path[decoration={snake,pre length=1pt},fill=blue!20]
 (a) to [bend left=30]                   (c)
 to [controls=+(45:6) and +(170:6)](d)
 decorate {  -- (b)}  
 to [out=45, in= -50]   cycle;
\draw [red]    (a) to [bend left=30]                   (c);
\draw [blue]   (b) to [out=45, in= -50]                (a);
\draw [orange] (c) to [controls=+(45:6) and +(170:6)]  (d);
\draw [green!60!black,decorate,decoration={snake,pre length=1pt}] (d) -- (b);
\node at (2,2.5) {$S$};
\end{tikzpicture}

\end{document}

and adding even odd rule, i.e.

\path[decoration={snake,pre length=1pt},fill=blue!20,even odd rule]
 (a) to [bend left=30]                   (c)
 to [controls=+(45:6) and +(170:6)](d)
 decorate {  -- (b)}  
 to [out=45, in= -50]   cycle;

yields