Shorten curved arrow proportionally to length

Question

I am drawing curved arrows in tikz-cd, using the method devised by @AndréC in this answer. There are situations where I want to shorten these arrows, proportional to their arc length. However, the naïve solution (using shorten) is not suitable, as it creates distorted curves (as discussed, for instance, in this question). As an example (Example 1), the following two curves have the same input path, but the red curve has been shortened. What should happen is the red curve should overlay the black curve.

Note that it is not just the position of the shortened curve that is incorrect: the shape is incorrect too: in the example (Example 2) below, I've positioned the curves so they have the same apex, but the shortened curve does not overlay the original (the amount of distortion depends on the curve width and height).

The other issue with shorten is that it requires an absolute length to shorten by, whereas I want to specify it proportionally with respect to the length of the curve itself.

I attempted to manually draw an arrow head with decorations.markings, but this does not properly calculate the right size for the arrowhead (or take into account the existing style of the head or tail). In the example below (Example 3), the arrow head further up the curve should be the same size as the one at the tip.

My idea was to shorten this arrow using a custom dash pattern, but this seems like a hack that is likely in practice to fail in edge cases. Ideally, the curve path itself should be changed, which would handle the existing style/arrowheads/tails/etc. The option pgfpathcurvebetweentime seemed like a possible solution, but I could not figure out how to integrate it with the custom curve style in the two examples.

How might one add an option to the custom curve style to allow the curve to be shortened (e.g. curve={height=-40pt, shorten=0.2})? In practice, shortening symmetrically from both ends is usually sufficient, but having options to shorten the start and end differently would be even better if it doesn't add much extra complexity.

For a demonstration of what I intend by "shortening a curved arrow", see @Thruston's example.

Example 1

\documentclass{article}
\usepackage{tikz-cd}
\usetikzlibrary{calc}

\begin{document}

{\tikzset{curve/.style={settings={#1},to path={(\tikztostart)
    .. controls ($(\tikztostart)!\pv{pos}!(\tikztotarget)!\pv{height}!270:(\tikztotarget)$)
    and ($(\tikztostart)!1-\pv{pos}!(\tikztotarget)!\pv{height}!270:(\tikztotarget)$)
    .. (\tikztotarget)\tikztonodes}},
    settings/.code={\tikzset{quiver/.cd,#1}
        \def\pv##1{\pgfkeysvalueof{/tikz/quiver/##1}}},
    quiver/.cd,pos/.initial=0.35,height/.initial=0}

\[\begin{tikzcd}
    {\bullet} & {\bullet}
    \arrow[from=1-1, to=1-2, curve={height=-40pt}, shorten <= 30pt, shorten >= 30pt, color=red]
    \arrow[from=1-1, to=1-2, curve={height=-40pt}]
\end{tikzcd}\]}

\end{document}

Example 2

\documentclass{article}
\usepackage{tikz-cd}
\usetikzlibrary{calc}

\begin{document}

{\tikzset{curve/.style={settings={#1},to path={(\tikztostart)
    .. controls ($(\tikztostart)!\pv{pos}!(\tikztotarget)!\pv{height}!270:(\tikztotarget)$)
    and ($(\tikztostart)!1-\pv{pos}!(\tikztotarget)!\pv{height}!270:(\tikztotarget)$)
    .. (\tikztotarget)\tikztonodes}},
    settings/.code={\tikzset{quiver/.cd,#1}
        \def\pv##1{\pgfkeysvalueof{/tikz/quiver/##1}}},
    quiver/.cd,pos/.initial=0.35,height/.initial=0}

\[\begin{tikzcd}
    {\bullet} &&& {\bullet}
    \arrow[from=1-1, to=1-4, curve={height=-60pt}, shorten <= 30pt, shorten >= 30pt, color=red, yshift=-11pt]
    \arrow[from=1-1, to=1-4, curve={height=-60pt}]
\end{tikzcd}\]}

\end{document}

Example 3

\documentclass{article}
\usepackage{tikz-cd}
\usetikzlibrary{calc}
\usetikzlibrary{decorations.markings}

\begin{document}

{\tikzset{curve/.style={settings={#1},to path={(\tikztostart)
    .. controls ($(\tikztostart)!\pv{pos}!(\tikztotarget)!\pv{height}!270:(\tikztotarget)$)
    and ($(\tikztostart)!1-\pv{pos}!(\tikztotarget)!\pv{height}!270:(\tikztotarget)$)
    .. (\tikztotarget)\tikztonodes},
    % Arrow head and tail
    decoration={
      markings,
      mark=at position 0.8 with {\arrow{>}}}
    , postaction={decorate}},
    %
    settings/.code={\tikzset{quiver/.cd,#1}
        \def\pv##1{\pgfkeysvalueof{/tikz/quiver/##1}}},
    quiver/.cd,pos/.initial=0.35,height/.initial=0}

\[\begin{tikzcd}
    {\bullet} & {\bullet}
    \arrow[Rightarrow, from=1-1, to=1-2, curve={height=-40pt}]
\end{tikzcd}\]}

\end{document}

Answer 1

Update: 2023-04-8: This is now in the version of spath3 available on CTAN

Now that this is implemented in the version of spath3 on CTAN, the code below can be considerably simplified.

\documentclass{article}
%\url{https://tex.stackexchange.com/q/568648/86}
\usepackage{tikz-cd}
\usetikzlibrary{
  calc,
  spath3
}



\def\pv#1{%
  \pgfkeysvalueof{/tikz/quiver/#1}%
}


\tikzset{
  curve/.style={
    quiver/.cd,
    #1,
    /tikz/.cd,
    to path={
      (\tikztostart) .. controls ($(\tikztostart)!\pv{pos}!(\tikztotarget)!\pv{height}!270:(\tikztotarget)$)
    and ($(\tikztostart)!1-\pv{pos}!(\tikztotarget)!\pv{height}!270:(\tikztotarget)$)
    .. (\tikztotarget)\tikztonodes}
  },
  quiver/.cd,
  pos/.initial=0.35,
  height/.initial=0
}

\begin{document}

\begin{tikzpicture}
\draw[spath/save=curve, ultra thick] (0,0) .. controls +(1,2) and +(-1,2) .. (3,0);
\tikzset{
%  spath/split at keep start={curve}{.9},
%  spath/split at keep end={curve}{.1/.9},
  spath/split at keep middle={curve}{.1}{.9},
}
\draw[spath/use={curve},red];
\end{tikzpicture}

\begin{tikzcd}
{\bullet} & {\bullet} & {\bullet}
\arrow[
  from=1-1,
  to=1-2,
  curve={
    height=-40pt
  }
]
\arrow[
  from=1-2,
  to=1-3,
  curve={
    height=-40pt
  },
  /tikz/spath/at end path construction={
    \tikzset{spath/split at keep middle={current}{.1}{.9}}
  },
  color=red,
]
\end{tikzcd}

\end{document}

This produces:

---

The capability to do this is essentially in my spath3 package. The existing user-level code shortens paths by a dimension so I needed to do a bit of coding to add the required functionality to the user interface, this will probably end up in the library in some form or other as it feels quite useful (I just need to think about consistency between the various commands that I have). So don't be alarmed by the stuff in the preamble - that'll end up in the package eventually.

\documentclass{article}
%\url{https://tex.stackexchange.com/q/568648/86}
\usepackage{tikz-cd}
\usetikzlibrary{
  calc,
  spath3
}

\makeatletter
\ExplSyntaxOn

\cs_new_protected_nopar:Npn \spath_discard_after:Nn #1#2
{
  \spath_split_at:NNVn \l_tmpa_tl \l_tmpb_tl #1 {#2}
  \tl_set_eq:NN #1 \l_tmpa_tl
}

\cs_new_protected_nopar:Npn \spath_discard_before:Nn #1#2
{
  \spath_split_at:NNVn \l_tmpa_tl \l_tmpb_tl #1 {#2}
  \tl_set_eq:NN #1 \l_tmpb_tl
}

\cs_new_protected_nopar:Npn \spath_discard_outside:Nnn #1#2#3
{
  \spath_split_at:NNVn \l_tmpa_tl \l_tmpb_tl #1 {#3}
  \spath_split_at:NNVn \l_tmpa_tl \l_tmpb_tl \l_tmpa_tl {(#2)/(#3)}
  \tl_set_eq:NN #1 \l_tmpb_tl
}

\cs_generate_variant:Nn \spath_discard_after:Nn {cn}
\cs_generate_variant:Nn \spath_discard_before:Nn {cn}
\cs_generate_variant:Nn \spath_discard_outside:Nnn {cnn}

\tl_new:N \l__tikzspath_tikzpath_finish_tl

\cs_new_protected_nopar:Npn \spath_at_end_of_path_construction:
{
  \tl_use:N \l__tikzspath_tikzpath_finish_tl
}
\tl_put_left:Nn \tikz@finish {\spath_at_end_of_path_construction:}

\tikzset{
  spath/.cd,
  at~ end~ path~ construction/.code={%
    \tl_put_right:Nn \l__tikzspath_tikzpath_finish_tl {#1}
  },
  discard~ after~ point/.code~ 2~ args={
    \__tikzspath_maybe_current_path_reuse:nnn
    {
      \__tikzspath_check_path:nnn
      {
        \spath_discard_after:cn
      }
    } {#1} { {} {#2} }
  },
  discard~ before~ point/.code~ 2~ args={
    \__tikzspath_maybe_current_path_reuse:nnn
    {
      \__tikzspath_check_path:nnn
      {
        \spath_discard_after:cn
      }
    } {#1} { {} {#2} }
  },
  discard~ outside~ points/.code~ n~ args={3}{
    \__tikzspath_maybe_current_path_reuse:nnn
    {
      \__tikzspath_check_path:nnn
      {
        \spath_discard_outside:cnn
      }
    } {#1} { {} {#2} {#3} }
  },
}

\ExplSyntaxOff
\makeatother

\def\pv#1{%
  \pgfkeysvalueof{/tikz/quiver/#1}%
}


\tikzset{
  curve/.style={
    quiver/.cd,
    #1,
    /tikz/.cd,
    to path={
      (\tikztostart) .. controls ($(\tikztostart)!\pv{pos}!(\tikztotarget)!\pv{height}!270:(\tikztotarget)$)
    and ($(\tikztostart)!1-\pv{pos}!(\tikztotarget)!\pv{height}!270:(\tikztotarget)$)
    .. (\tikztotarget)\tikztonodes}
  },
  quiver/.cd,
  pos/.initial=0.35,
  height/.initial=0
}

\begin{document}

\begin{tikzcd}
{\bullet} & {\bullet} & {\bullet}
\arrow[
  from=1-1,
  to=1-2,
  curve={
    height=-40pt
  }
]
\arrow[
  from=1-2,
  to=1-3,
  curve={
    height=-40pt
  },
  /tikz/spath/at end path construction={
    \tikzset{spath/discard outside points={current}{.1}{.9}}
  },
  color=red,
]
\end{tikzcd}

\end{document}

NB The link above is to the github repository as the development version is a little ahead of the ctan at the moment.

Answer 2

Is this what you mean?

This is done in Metapost, purely for comparison, and in the hope that some one else here can show us how to do the equivalent in TikZ.

\documentclass[border=5mm]{standalone}
\usepackage{luamplib}
\begin{document}
\mplibtextextlabel{enable}
\begin{mplibcode}
input colorbrewer-rgb
beginfig(1);
    z0 = -z1 = 34 left;
    path a; a = z0 .. controls (-13, 55) and (13, 55) .. z1;
    ahangle := 30;
    ahlength := 2;
    for s=1 upto 8:
        drawarrow a 
            cutbefore fullcircle scaled 10s shifted z0
            cutafter  fullcircle scaled 10s shifted z1
            withcolor Blues[9][s];
    endfor
    drawdot z0 withpen pencircle scaled dotlabeldiam;
    drawdot z1 withpen pencircle scaled dotlabeldiam;
endfig;
\end{mplibcode}
\end{document}

(Compile with lualatex).

What I have done each is draw the same curve each time, but with more of it "cut off" at each end.

Answer 3

This is a topic I also struggle with for some time. I found something that could be useful on this topic: Tikz Graphics: Curved arrow drawn parallel to curved line

I adapted it to the current question and am almost happy with it:

\documentclass{article}
\usepackage{tikz}
\begin{document}
\thispagestyle{empty}

\begin{tikzpicture}

    % Nodes for start and end
    \node at (0,0) (A) {};  
    \node at (5,0) (B) {};  

  
    % Path from start to end, using Bezier definition (no to[bend] allowed)
    % The path is cut into 20 parts, using 'pos'
    \path  (A) .. controls +(2,4) and +(-2,4).. (B)
        {\foreach \t [count=\i] in {0.05,0.10,0.15,...,0.85,0.90,0.95,1}
            {
            coordinate[pos=\t] (p\i)
            }
        };
  
    % naming p0 and p20 to allow the loop to draw the complete first curve
    \coordinate (p0) at (A);
    \coordinate (p20) at (B);
    
    % Drawing the 9 other curves
    \foreach \k in {0,...,9}
        {
        \pgfmathtruncatemacro\l{20-\k}      % p\k is the start point, p\l is the end point
        \pgfmathtruncatemacro\clr{10*\k}    % fading the colour of the curves
        
        % Now draw (segments) between all p\i points from p\k to p\l
        \draw[cap=round,line width=1.5pt,-stealth,blue!\clr] (p\k)
            {\foreach \i in {\k,...,\l} {-- (p\i) }};
        }
    
\end{tikzpicture}
\end{document}

Now, the main issue of this solution is that it doesn't draw curves but broken lines. This could easily be fixed by computing more points on the original path. You may want to play with it, while finding a better solution.

Answer 4

\documentclass[tikz, border=1 cm]{standalone}
\usetikzlibrary{decorations.markings}
\begin{document}
\begin{tikzpicture}
\newcommand{\mypath}{(0,0) to[out=80, in=100, min distance=1cm] (1,0)}
\newcommand{\mystart}{0.2} %start path at 20%
\newcommand{\myend}{0.8}   %end path at 80%
\draw[ultra thick] \mypath;
\draw[
draw=none,
preaction={decoration={markings, mark=at position \myend with {\arrow{>[green]}}}, decorate},
postaction={draw, green, decoration={curveto, 
pre=moveto, pre length=\mystart*\pgfmetadecoratedpathlength,
post=moveto, post length=(1-\myend)*\pgfmetadecoratedpathlength, 
}, decorate}
] \mypath;
\end{tikzpicture}
\end{document}

---

Edit: Example 1 with the shortened arrow:

\documentclass[tikz, border=1 cm]{standalone}
\usepackage{tikz-cd}
\usetikzlibrary{calc, decorations.markings}
\begin{document}
\tikzset{curve/.style={settings={#1},to path={(\tikztostart)
    .. controls ($(\tikztostart)!\pv{pos}!(\tikztotarget)!\pv{height}!270:(\tikztotarget)$)
    and ($(\tikztostart)!1-\pv{pos}!(\tikztotarget)!\pv{height}!270:(\tikztotarget)$)
    .. (\tikztotarget)\tikztonodes}},
    settings/.code={\tikzset{quiver/.cd,#1}
        \def\pv##1{\pgfkeysvalueof{/tikz/quiver/##1}}},
    quiver/.cd,pos/.initial=0.35,height/.initial=0}

\begin{tikzcd}
    {\bullet} & {\bullet}
    \arrow[from=1-1, to=1-2, curve={height=-40pt}]
    \arrow[from=1-1, to=1-2, curve={height=-40pt},
    -_,
    draw=none,
    preaction={decoration={markings, mark=at position 0.8 with {\arrow{>[green]}}}, decorate},
    postaction={draw, green, decoration={curveto, 
    pre=moveto, pre length=0.2*\pgfmetadecoratedpathlength,
    post=moveto, post length=(1-0.8)*\pgfmetadecoratedpathlength, 
    }, decorate}]
\end{tikzcd}
\end{document}

Answer 5

Here is my proposal:

On the previous style, I added the shift parameter which allows me to shift the curve perpendicular to the path. To do this, I applied the idea I developed in my answer to the question Annotating a table with arrows.

The new style is called curve 1, I put as default height=-40 pt as in your examples.

\documentclass{article}
\usepackage{tikz-cd}
\usetikzlibrary{calc}

\begin{document}

\tikzset{curve 1/.style={settings={#1},
                        to path={($(\tikztostart)!\pv{shift}!90:(\tikztotarget)$) .. controls ($(\tikztostart)!\pv{pos}!(\tikztotarget)!\pv{height}!270:(\tikztotarget)$)
    and ($(\tikztostart)!1-\pv{pos}!(\tikztotarget)!\pv{height}!270:(\tikztotarget)$)
    .. ($(\tikztotarget)!\pv{shift}!270:(\tikztostart)$)\tikztonodes}},
    settings/.code={\tikzset{quiver/.cd,#1}
        \def\pv##1{\pgfkeysvalueof{/tikz/quiver/##1}}},
    quiver/.cd,pos/.initial=0.35,height/.initial=-40pt,shift/.initial=0pt}
    
\[\begin{tikzcd}
    {\bullet} & {\bullet}
    \arrow[from=1-1, to=1-2, curve 1={shift=2mm},color=blue,thick]
    \arrow[from=1-1, to=1-2, curve 1={shift=4mm},color=cyan,thick]
    \arrow[from=1-1, to=1-2, curve 1={shift=6mm},color=green,thick]
    \arrow[from=1-1, to=1-2, curve 1={shift=8mm},color=yellow,thick]
    \arrow[from=1-1, to=1-2, curve 1={shift=10mm},color=orange,thick]
    \arrow[from=1-1, to=1-2, curve 1={shift=12mm},color=red,thick]
\end{tikzcd}\]

\end{document}

Answer 6

Since I don't really understand what still refrains you to use this method, I made some adjustments, in order to make it easier to use. In this new version, you can decide the position of both starting and ending points of your arrow, and the number of points to be computed, in order to avoid the uggly broken line effect on strongly curved lines.

\documentclass{standalone}
\usepackage{tikz}
\begin{document}
\thispagestyle{empty}

\begin{tikzpicture}

    % First, the starting and ending nodes
    \draw (0,0) node[circle, inner sep=0.8pt, fill=black, label={below:{$A$}}] (A) {};  
    \draw (5,1) node[circle, inner sep=0.8pt, fill=black, label={above:{$B$}}] (B) {};
    
    % Define the path between the two nodes (Bezier curve)
    \def\route{(A) .. controls +(2,5) and +(-2,-3).. (B)}
    
    % Drawing the complete path
    \draw[orange] \route;
    
    %%%%%%%%%%%%%%
    % Blue arrow %
    %%%%%%%%%%%%%%
    
    \def\start{0.5}     % Starting point position
    \def\len{0.45}      % Arrow length (percentage of total line length)
    \def\numpoints{30}  % Number of points to draw segments between them
    
    % Computing the 30 points from the arrow start and the arrow end
    \path  \route 
        {\foreach \i in {1,...,\numpoints} { coordinate[pos=\start+\len*\i/\numpoints] (p\i) } };
        
    % Drawing a broken line linking all 30 points, with the arrow decoration at the end
    \draw[blue, cap=round, opacity=0.5, line width=1.5pt,->>] (p1)
        { \foreach \j in {2,...,\numpoints} {-- (p\j) } };
    
    %%%%%%%%%%%%%
    % Red arrow %
    %%%%%%%%%%%%%
    \def\start{0.4} \def\len{-0.3} \def\numpoints{30}
    \path  \route 
        {\foreach \i in {1,...,\numpoints} { coordinate[pos=\start+\len*\i/\numpoints] (p\i) } };
    \draw[red, cap=round, opacity=0.5, line width=1.5pt,->>] (p1)
        { \foreach \j in {2,...,\numpoints} {-- (p\j) } };
    
\end{tikzpicture}


\end{document}

Hope this will give you a better insight on this.

edit: I realize now that you would want to do this, using @AndréC 's method of creating curves between points. If you don't find how to implement this, let us know, sure someone will find a way.