TikZ/PGF: cut, intersect, unify paths getting a new path

Question

Sorry, I am fairly new to TikZ.

Is there a set-theory way of cutting, intersecting, unifying paths like in (proprietary) GUI applications? Or better asked, what is the tikz approach to to deal with that problem? Let's say I've built several paths, in GUI-land I would select two of them and do an operation resulting in a new path. But how do I define a reusable tikz-path in the first place without drawing it right away? (A link to a systematic macro-tutorial would be very much appreciated.)

There are a lot of snippets out there, but for a beginner it`s all quite confusing. There are pics and shapes and macros and so on, but in the end there are just paths. Do I really have to start studying the underlying layers?

Here is a real world problem:

\documentclass[tikz,border=0.2cm]{standalone}
\usetikzlibrary{scopes}
\begin{document}
\begin{tikzpicture}[scale=4]
    %\draw [help lines,step=0.1] (-0.2,-0.8) grid (1,0.8);
    \path [fill=gray!50] (-0.2,-0.8) rectangle (1,0.8);
    \begin{scope} %[every path/.style={fill=white}]
        \draw (0,-0.65) --
            ++(0  , 0.3) .. controls +( 0.25,0) and +(-0.5,0) ..
            ++(0.8, 0.7) --
            ++(0  ,-0.3) .. controls +(-0.25,0) and +( 0.5,0) ..
            cycle;
        \draw (0,-0.15) rectangle +(0.8,0.3);
        \draw (0, 0.65)  -- %%% Yes is is the path from above flipped over. I just couldn't figure out how to do it properly
            ++(0  ,-0.3) .. controls +( 0.25,0) and +(-0.5,0) ..
            ++(0.8,-0.7) -- 
            ++(0  , 0.3) .. controls +(-0.25,0) and +( 0.5,0) ..
            cycle;
    \end{scope}
\end{tikzpicture}
\end{document}

Step by step procedure:

0. Create 3 paths: "rising bent rectange", "bar", "falling bent rectangle"

1. Select "bar" and "rising" paths.

2. Cut "bar" from "rising" path.

3. Select resulting paths and "falling" path.

4. Cut "falling" from other paths.

4. Select "bar" and "falling" path.

5. Cut "falling" from "bar".

6. Select resulting paths and "falling". ("rising" and "bar" can be discarded now)

7. Cut through the background (using the 3 original paths maybe)

8. Final result:

and without the grid in the back

Answer 1

Here's an example using my spath3 library that can be used to split and recombine paths. I'm not very clear what your desired outcome is, so this is intended just as a start to help figure that out.

Firstly, I take your primary path (the curve) and use it to construct the curved region, resaving that as a path. Then I intersect that against itself (but flipped) to split the two paths into segments. Finally, I draw certain of those segments to create the outer region.

\documentclass[tikz,border=0.2cm]{standalone}
%\url{https://tex.stackexchange.com/q/585037/86}
\usetikzlibrary{scopes,spath3,intersections}
\begin{document}
\begin{tikzpicture}[scale=4]
    %\draw [help lines,step=0.2] (-0.2,-0.8) grid (1,0.8);
    \path [fill=gray!50] (-0.2,-0.8) rectangle (1,0.8);
\begin{scope} %[every path/.style={fill=white}]
\path[spath/save=curve] (0,-0.35) .. controls +( 0.25,0) and +(-0.5,0) .. ++(0.8, 0.7);
\path
[
  spath/save=lower path,
  spath/use=curve
]
-- ++(0,-.3)
[
  spath/use={curve,move,weld,transform={scale=-1}}
]
-- cycle;
\path[spath/save=rectangle] (0,-0.15) rectangle +(0.8,0.3);
\tikzset{
  spath/clone={upper path}{lower path},
  spath/transform={upper path}{yscale=-1},
  spath/split at intersections={upper path}{lower path},
  spath/get components of={upper path}\Ucpts,
  spath/get components of={lower path}\Lcpts,
}
\draw[
  spath/use=\getComponentOf\Ucpts{3},
  spath/use=\getComponentOf\Ucpts{4},
  spath/use=\getComponentOf\Lcpts{3},
  spath/use=\getComponentOf\Lcpts{4},
  spath/use=\getComponentOf\Ucpts{6},
  spath/use=\getComponentOf\Lcpts{6},
];
    \end{scope}
\end{tikzpicture}
\end{document}

Result:

Answer 2

Here's an answer to show what is possible using just clipping. I do use my spath3 library to construct the paths and to re-use them, but you could do away with that by just cutting-and-pasting.

\documentclass[tikz,border=0.2cm]{standalone}
%\url{https://tex.stackexchange.com/q/585037/86}
\usetikzlibrary{scopes,spath3,intersections,patterns}
\begin{document}
\begin{tikzpicture}[scale=4]
    %\draw [help lines,step=0.2] (-0.2,-0.8) grid (1,0.8);
\begin{scope} %[every path/.style={fill=white}]
\path[spath/save=curve] (0,-0.35) .. controls +( 0.25,0) and +(-0.5,0) .. ++(0.8, 0.7);
\path
[
  spath/save=lower path,
  spath/use=curve,
]
-- ++(0,-.3)
[
  spath/use={curve,move,weld,transform={scale=-1}},
]
-- cycle;
\path[spath/save=rectangle] (0,-0.15) rectangle +(0.8,0.3);
\tikzset{
  spath/clone={upper path}{lower path},
  spath/transform={upper path}{yscale=-1},
}
\path[preaction={fill=red!50!white},pattern=bricks, pattern color=white] (-0.2,-0.8) rectangle (1,0.8);
\begin{scope}[even odd rule]
\clip (-0.2,-0.8) rectangle (1,0.8) [spath/use=upper path];
\begin{scope}[even odd rule]
\clip (-0.2,-0.8) rectangle (1,0.8) [spath/use=rectangle];
\begin{scope}[even odd rule]
\clip (-0.2,-0.8) rectangle (1,0.8) [spath/use=lower path];
\path [fill=gray!50] (-0.2,-0.8) rectangle (1,0.8);
\end{scope}
\draw[spath/use=lower path];
\end{scope}
\draw[spath/use=rectangle];
\end{scope}
\draw[spath/use=upper path];
\end{scope}
\end{tikzpicture}
\end{document}

I've put the pattern in to show that you really are seeing through the paths and not filling them in white.