Tikz find coordinates of beginning and end of a shortened curved line

Question

I have a shortened curved line drawn between 2 points.
I would like to obtain the coordinates of the beginning and end of that shortened line.

I tried with the following code below
A and B are the coordinates used to define the bended line (red dots)
I would like C and D (black dots) to be the coordinates at the beginning and ending of the shortened line but currently A and C are the same coordinates. Same for B and D as shown in the screenshot

\documentclass[border=10pt]{standalone}
\usepackage{tikz}
\usetikzlibrary{
  arrows.meta,calc,decorations.markings,math,arrows.meta,positioning,automata}
\begin{document}
\begin{tikzpicture}
  \coordinate (A) at (3.5,3.5); 
  \coordinate (B) at (5.1,2.);
\draw[color=red,fill=red] (A) circle (0.15cm);
  \draw[color=red,fill=red] (B) circle (0.15cm);
\draw (A) edge[out=-20,in=140,shorten >=15pt, shorten <=15pt] coordinate[pos=0.]  (C)  coordinate[pos=1] (D)  (B);
\draw[black,fill=black] (C) circle (0.05cm);
  \draw[black,fill=black] (D) circle (0.05cm);
\end{tikzpicture}
\end{document}

More generally I would like

• pos = 0 to correspond to the beginning of the shortened line
• pos = 1 to correspond to the end of the shortened line

So that I can position points on the shortened line wherever I want (like 25% of the shortened line for instance).

Would someone know a solution? Many thanks in advance

Answer 1

Right now I can only think of accessing the points using markings like this:

\documentclass[tikz, border=1cm]{standalone}
\usetikzlibrary{decorations.markings}
\begin{document}
\begin{tikzpicture}
\coordinate (A) at (3.5,3.5); 
\coordinate (B) at (5.1,2.); 
\fill[red] (A) circle[radius=0.15cm];
\fill[red] (B) circle[radius=0.15cm];
\draw[
shorten >=15pt, shorten <=15pt,
postaction=decorate, 
decoration={markings, 
  mark=at position 0 with {\coordinate (C) at (15pt,0);},
  mark=at position 1 with {\coordinate (D) at (-15pt,0);} 
},
] (A) to[out=-20,in=140]  (B); 
\fill (C) circle[radius=0.05cm];
\fill (D) circle[radius=0.05cm];
\end{tikzpicture}
\end{document}

Answer 2

The issue here is that shortening paths is done very late in the path's construction, so things like positioning nodes doesn't notice the shortening. I suspect this is because the shortening of shorten >=15pt is done by the same mechanism as when an arrow tip is added and generally this shouldn't affect the positioning of things along the path.

So one way to achieve this is to force the shortening to happen earlier. Ideally, one would do this before the positioning stuff was calculated, but I don't know where that is done and I happen to know an alternative way to specify points on a path after the path has been defined, so I'm using that instead.

Life is made slightly more complicated by your use of edge, since that creates a separate path that is drawn inside its own scope so we need to use a few globals to get things out of those scopes.

Here's two ways to achieve your goal. In the first, we save the shortened the path and then use the spath3 library to place the circles at its ends. In the second, we save the original path and then use the spath3 library to shorten it - this has the advantage that the shortening is genuinely along its path (the green lines in each picture are the original unshortened paths).

In both cases, we use the spath3 TikZ library to then access coordinates at positions along the shortened path, using the syntax (spath cs:<path name> <position>).

\documentclass{article}
%\url{https://tex.stackexchange.com/q/652865/86}
\usepackage{tikz}
\usetikzlibrary{spath3,intersections}
\makeatletter
\tikzset{
  shorten path early/.code={
    \tikz@addmode{%
      \pgf@prepare@end@of@path
      \pgf@prepare@start@of@path
      \pgfsetshortenstart{0pt}%
      \pgfsetshortenend{0pt}%
    }%
  },
  shorten then name path/.style={
    shorten path early,
    spath/save global=#1
  }
}
\makeatother
\begin{document}
\begin{tikzpicture}
% Version 1: save the shortened path
\begin{scope}
  \coordinate (A) at (3.5,3.5); 
  \coordinate (B) at (5.1,2.);
\draw[color=red,fill=red] (A) circle[radius=0.15cm];
  \draw[color=red,fill=red] (B) circle[radius=0.15cm];
\draw[ultra thick, green] (A) edge[out=-20,in=140] (B);
\draw (A) edge[out=-20,in=140,shorten >=15pt, shorten <=15pt, shorten then name path=short] coordinate[pos=0] (C) coordinate[pos=1] (D) (B);
\draw[black,fill=black] (C) circle[radius=0.05cm];
  \draw[black,fill=black] (D) circle[radius=0.05cm];
\fill[magenta] (spath cs:short 0) circle[radius=0.05cm];
  \fill[magenta] (spath cs:short 1) circle[radius=0.05cm];
\end{scope}
% Version 2: save the path and shorten it afterwards
\begin{scope}[xshift=3cm]
  \coordinate (A) at (3.5,3.5); 
  \coordinate (B) at (5.1,2.);
\draw[color=red,fill=red] (A) circle[radius=0.15cm];
  \draw[color=red,fill=red] (B) circle[radius=0.15cm];
\draw (A) edge[ultra thick, green, out=-20,in=140, spath/save global=short] coordinate[pos=0] (C) coordinate[pos=1] (D) (B);
\tikzset{
  spath/shorten at both ends={short}{15pt}
}
\draw[spath/use=short];
\draw[black,fill=black] (C) circle[radius=0.05cm];
  \draw[black,fill=black] (D) circle[radius=0.05cm];
\fill[magenta] (spath cs:short 0) circle[radius=0.05cm];
  \fill[magenta] (spath cs:short 1) circle[radius=0.05cm];
\end{scope}
\end{tikzpicture}
\end{document}

---

On reflection, I think that the spath3 library should apply the shortening before it saves the path. I've implemented it in the development version (on github). With that version, the following works:

\begin{tikzpicture}
  \coordinate (A) at (3.5,3.5); 
  \coordinate (B) at (5.1,2.);
\draw[color=red,fill=red] (A) circle[radius=0.15cm];
  \draw[color=red,fill=red] (B) circle[radius=0.15cm];
\draw (A) edge[ultra thick, green, out=-20,in=140,shorten >=15pt, shorten <=15pt, spath/save global=short] coordinate[pos=0] (C) coordinate[pos=1] (D) (B);
\draw[black,fill=black] (C) circle[radius=0.05cm];
  \draw[black,fill=black] (D) circle[radius=0.05cm];
\fill[magenta] (spath cs:short 0) circle[radius=0.05cm];
  \fill[magenta] (spath cs:short 1) circle[radius=0.05cm];
\end{tikzpicture}

Answer 3

This is an interesting problem (and 15pt is a lot of shortening)!

It might be better to create a custom to path that just moves the start and target/end coordinate the determined amount and do not use the actual shortening. There is probably also a way to do this with the decorations.markings library.

Basically, all the node positioning along a path is done by TikZ. Every time you use a path operator like   (a move to), -- (a line to) (also -| and |-) or .. controls … .., TikZ saves the start coordinate, the end coordinate and, in the latter case, the control points.

Yes, all those bends and outs and ins are at the end of the day just a .. controls (<p1>) and (<p2>) .. path.

We can change this timer, though. However: the timer doesn't know whether its path is at the start of a path (or after a move) and/or at the end of a path (before a move).

This line

\draw[shorten <=15pt] (0,0) node{S} -- (1,0)
   to[out=90, in=30] node[at start]{x} (2,0);

will start at (15pt,0) and the node x should still be placed at (1,0). This is why I'll provide the key shortening position that actually activates the new timer.

If you're only using paths with one segment, this key can be given to a scope, though, of course.

Code

\documentclass[border=10pt]{standalone}
\usepackage{tikz}
\tikzset{en/.style={edge node={coordinate[pos=#1/10] (c-#1)}}}
\makeatletter
\def\tikz@timer@curve@shorten{% tikz.code.tex, l 4947
  \pgftransformcurveattime{\tikz@time}
    {\pgfpointlineatdistance{\pgf@shorten@start@additional}{\tikz@timer@start}{\tikz@timer@cont@one}}
    {\tikz@timer@cont@one}{\tikz@timer@cont@two}
    {\pgfpointlineatdistance{\pgf@shorten@end@additional}{\tikz@timer@end}{\tikz@timer@cont@two}}}
\tikzset{shortening position/.code=\let\tikz@timer@curve\tikz@timer@curve@shorten}
\makeatother
\begin{document}
\begin{tikzpicture}[radius=.15cm]
  \filldraw[red] (3.5,3.5) coordinate (A) circle[]
                 (5.1,2.0) coordinate (B) circle[];
  \draw (A) edge[out=-20,in=140,shorten >=15pt, shorten <=15pt, en/.list={0,...,10}, shortening position] (B);
\foreach \pos in {0,...,10}
    \draw[gray, shorten <=2pt] (c-\pos) --
      node[right,at end,inner sep=0pt,sloped,font=\tiny]
        {\pgfmathprint{!mod(\pos,2)?\pos:""}} +(45:.2);
% the shortening is done at (0,0), the curve timer doesn't know about this
  \draw[shorten <=15pt] (0,0) node{S} -- (1,0) to[out=90, in=30] node[at start]{x} (2,0);
\end{tikzpicture}
\end{document}

Code

Answer 4

That is an interesting question on path operations. I think TikZ can do that with some efforts, see the above answers.

Here I show that Asymptote can do that easily with the built-in routines: arctime, arcpoint, subpath (see the Asymptote documentation, Paths and Guides). I use the cm unit, so L=15pt/cm; is to change 15pt to cm.

// Determine a point on a given curve 
// via its length on the curve
// http://asymptote.ualberta.ca/
unitsize(1.5cm);
pair A=(3.5,3.5), B=(5.1,2);
path pAB=A..controls A+2dir(-20) and B+2dir(140)..B;
draw(pAB,green+1.5pt);
dot("$A$",align=W,A,red);
dot("$B$",align=E,B,red);
real L=15pt/cm;
real tC=arctime(pAB,L);
pair C=arcpoint(pAB,L);
real tD=arctime(pAB,arclength(pAB)-L);
pair D=arcpoint(pAB,arclength(pAB)-L);
path pCD=subpath(pAB,tC,tD); // a subpath of pAB
draw(pCD);
dot("$C$",align=NE,C);
dot("$D$",align=SW,D);
write("L is ",L);
write("The length of pAB is ",arclength(pAB));
write("The length of pCD is ",arclength(pCD));
write("Checking that 2L+arclength(pCD) ",2L+arclength(pCD));
write("that is arclength(pAB) with error 10^{-12}");
shipout(bbox(5mm,invisible)); 

Asymptote can print out its calculations in the interactive mode, as below for checking that C and D is 15pt from 2 endpoints of the initial curve pAB.

---

Appendix It is even easier to determine a point on a given curve via its relative time (the time on a path at the relative fraction l of its arclength) with point and subpath.

// Determine a point on a given curve 
// via its relative time
// http://asymptote.ualberta.ca/
unitsize(1.5cm);
pair A=(3.5,3.5), B=(5.1,2);
path pAB=A..controls A+2dir(-20) and B+2dir(140)..B;
draw(pAB,green+1.5pt);
dot("$A$",align=W,A,red);
dot("$B$",align=E,B,red);
real tC=.05, tD=.88;         // relative times in [0,1]
pair C=point(pAB,tC);
pair D=point(pAB,tD);
path pCD=subpath(pAB,tC,tD); // a subpath of pAB
draw(pCD);
dot("$C$",align=NE,C);
dot("$D$",align=SW,D);
shipout(bbox(5mm,invisible));