Plot along path in TikZ

Question

I'm trying to automate drawing filled plots along (mostly straight) paths. A simple example of what I'm trying to achieve:

\documentclass[border=2cm]{standalone}  
\usepackage{tikz}

\usetikzlibrary{backgrounds}

\begin{document}
    \begin{tikzpicture}[gridded, 
            every path/.style={thick, red ,fill, fill opacity=.5},
            every plot/.style={smooth}]

        \begin{scope}
            \draw (0,1) -- plot[domain=0:5, shift={(0,1)}, rotate={atan(3/4)}] (\x,0.05*\x^2) -- (4,4) -- cycle;
            \draw[black] (0,1) -- (4,4);
        \end{scope}

        \begin{scope}[xshift=5cm]
            \draw (0,3) -- plot[domain=0:sqrt(17), shift={(0,3)}, rotate={atan(-1/4)}] (\x,{0.5*sin(3*\x r)}) -- (4,2) -- cycle;
            \draw[black] (0,3) -- (4,2);
        \end{scope}
    \end{tikzpicture}
\end{document}

But how can I automate the process (maybe even for curved paths?) How can I

• Shift the plot to the last coordinate (I tried it with \pgfextractxy, but was not able to get it to work)
• Rotate it (Works quite ok with the atan(), but what about vertical paths?)
• Set the domain to the length of the path (Tried it with the let-option and the veclen command, but I got the "dimension too large"-error)
• Have it implemented, so that I can repeat it many times (e.g. \draw (A) -- plot[on path] (\x, <function>) -- (B))

Any ideas are appreciated!

Edit

As it seems that my intentions were not very clear: At the moment I would be happy with a solution for straight paths only. (The option for curved paths would be nice to have, though) So my goal is to have a simple method plotting a function between two given points on the canvas without having to recalculate my rotation, shift and domain every time.

Answer 1

This plots along straight lines.

\documentclass[tikz,border=3.14mm]{standalone}

\usetikzlibrary{calc}

\begin{document}
    \begin{tikzpicture}[%gridded, 
            every path/.style={thick, red ,fill, fill opacity=.5},
            every plot/.style={smooth},
            plot along line/.style args={from #1 to #2}{insert
            path={
            let \p1=($#2-#1$),\n1={veclen(\x1,\y1)/1cm},\n2={atan2(\y1,\x1)}
            in   [shift={#1},rotate=\n2,domain=0:\n1]
            }}]

        \begin{scope}
            \draw {[plot along line=from {(0,1)} to {(4,4)}] 
            plot (\x,0.05*\x^2) }-- (4,4) -- cycle;
            \draw[black] (0,1) -- (4,4);
        \end{scope}

        \begin{scope}[xshift=5cm]
            \draw[plot along line=from {(0,3)} to {(4,2)}] plot (\x,{0.5*sin(3*\x r)})
            coordinate (end) --cycle;
            \draw[black] (0,3) -- (end);
        \end{scope}
    \end{tikzpicture}
\end{document}

You can also add styles that plot along curved coordinate systems, using \usepgfmodule{nonlineartransformations} \usepgflibrary{curvilinear}.

\documentclass[tikz,border=3.14mm]{standalone}

\usetikzlibrary{calc}
\usepgfmodule{nonlineartransformations} 
\usepgflibrary{curvilinear}
\begin{document}
\begin{tikzpicture}
  \draw [help lines] (0,0) grid (3,2);
  {
    \pgfsetcurvilinearbeziercurve
      {\pgfpoint{0mm}{20mm}}
      {\pgfpoint{11mm}{20mm}}
      {\pgfpoint{20mm}{11mm}}
      {\pgfpoint{20mm}{0mm}}
\makeatletter     
\pgftransformnonlinear{\pgfpointcurvilinearbezierorthogonal\pgf@x\pgf@y}%
\makeatother
\draw (0,-30pt) grid [step=10pt] (80pt,30pt); 
\draw[blue,thick] plot[domain=0:2.8,samples=51] ({\x},{sin(180*\x)});
}
  \draw[red, very thick]
    (0mm,20mm) .. controls (11mm,20mm) and (20mm,11mm) .. (20mm,0mm);
\end{tikzpicture}
\end{document}

This example is more or less taken from the pgfmanual except that I added a plot of a function. However, I do not know what precisely you expect to be done? How do you specify the coordinate system?