parallel shifting of arbitrary path with to[bend=..] but only between two given coordinates

Question

I am trying to change TikZ: draw only a certain central length of a given path to fit my needs. I have a path that is built with \draw (A) to [bend right=30] (B); I want to now shift this path in a parallel way but only show a segment of it. I am hoping this segment can be arbitrary chosen with two different coordinates. Here is my code:

\documentclass[12pt, thmsa]{article}
\usepackage{tikz,pgf,  pgfplots}
\pgfdeclaredecoration{ignore}{final}
{
\state{final}{}
}

% Declare the actual decoration.
\pgfdeclaremetadecoration{myshortershift}{initial}{
  \state{initial}[
  width=1pt,
        next state=myshortershift
    ]
    {\decoration{moveto}}

    \state{myshortershift}[
        width={\the\pgfdecorationsegmentlength},
        next state=final
    ]
    {\decoration{curveto}}

    \state{final}
    {\decoration{ignore}}
  }
  \tikzset{myshortershift segment/.style={decoration={myshortershift},decorate, segment length=#1}}

\begin{document}
 \begin{tikzpicture}[>=latex]
  \begin{axis}[
    axis x line=bottom,
    axis y line=left,
    xmin=0, xmax=10, 
    ymin=0, ymax=10,
     x label style={at={(axis description cs:1,-.01)},anchor=south},
    ylabel style={at={(axis description cs:0.1,1)},anchor=west, rotate=270}, 
    xlabel={$X$},
    ylabel={$Y$},
    ytick=\empty,
    xtick=\empty,
    ]
    \coordinate (A) at (axis cs:2.2, 9.5);
    \coordinate (B) at (axis cs:8, .5);
    \draw (A) to [bend right=30] (B);
 \draw[blue] (axis cs:2.2+1, 9.5+1) to [bend right=30]  (axis cs:8+1,.5+1);
    \draw [blue,->, dotted, myshortershift segment=1.75cm] (axis cs:2.2+.5,9.5+.5) to [bend right=30]  (axis cs:8+.5,.5+.5);
    \draw [blue,->, dotted, myshortershift segment=1.75cm] (axis cs:8+.5,.5+.5) to [bend left=30]  (axis cs:2.2+.5,9.5+.5);
  \end{axis}
  \end{tikzpicture} 
\end{document}

I am trying to get the dotted arrows to start and stop at any two given coordinate (these will be derived from some intersections commands)

Thank you

Answer 1

Welcome to TeX-SE! This defines a shifted curve, and draws the relevant intersection segments with two lines (thin red to guide the eye). It also draws the segment between the two intersections. To this end, one has to first define an auxiliary path (called rest1) which starts from the first intersection and goes to the end of the curve. And then draw the first intersection segment of this path with the second line. Please note also that in recently recent version of pgfplots, which you can access with \pgfplotsset{compat=...}, axis cs: is no longer needed.)

\documentclass[12pt, thmsa]{article}
\usepackage{pgfplots}
\pgfplotsset{compat=1.16}
\usepgfplotslibrary{fillbetween}
\pgfdeclaredecoration{ignore}{final}
{
\state{final}{}
}

% Declare the actual decoration.
\pgfdeclaremetadecoration{myshortershift}{initial}{
  \state{initial}[
  width=1pt,
        next state=myshortershift
    ]
    {\decoration{moveto}}

    \state{myshortershift}[
        width={\the\pgfdecorationsegmentlength},
        next state=final
    ]
    {\decoration{curveto}}

    \state{final}
    {\decoration{ignore}}
  }
  \tikzset{myshortershift segment/.style={decoration={myshortershift},decorate, segment length=#1}}

\begin{document}


 \begin{tikzpicture}[>=latex]
  \begin{axis}[
    axis x line=bottom,
    axis y line=left,
    xmin=0, xmax=10, 
    ymin=0, ymax=10,
     x label style={at={(axis description cs:1,-.01)},anchor=south},
    ylabel style={at={(axis description cs:0.1,1)},anchor=west, rotate=270}, 
    xlabel={$X$},
    ylabel={$Y$},
    ytick=\empty,
    xtick=\empty,
    ]
    \coordinate (A) at (2.2, 9.5);
    \coordinate (B) at (8, .5);
    \draw (A) to [bend right=30] (B);
    \draw[blue] (2.2+1, 9.5+1) to [bend right=30]  (8+1,.5+1);
    \path[%draw=blue!30,ultra thin,
     name path=shifted curve] (2.2+2, 9.5+2) to [bend right=30]  (8+2,.5+2);
    \path[draw=red!30,ultra thin,name path=line1] (2.2,8) -- (10,8);
    \path[draw=red!30,ultra thin,name path=line2] (2.2,4) -- (10,4);
    \draw[-latex,blue,very thick,
        intersection segments={of=shifted curve and line1,sequence={A0}}];
    \draw[-latex,blue,very thick,
        intersection segments={of=shifted curve and line2,sequence={A1[reverse]}}];
    \path[name path=rest1,intersection segments={of=shifted curve and line1,sequence={A1}}];    
    \draw[latex-latex,red,very thick,
        intersection segments={of=rest1 and line2,sequence={A0}}];
%     \draw [blue,->, dotted, myshortershift segment=1.75cm] (axis cs:2.2+.5,9.5+.5) to [bend right=30]  (axis cs:8+.5,.5+.5);
%     \draw [blue,->, dotted, myshortershift segment=1.75cm] (axis cs:8+.5,.5+.5) to [bend left=30]  (axis cs:2.2+.5,9.5+.5);
  \end{axis}
  \end{tikzpicture}
\end{document}