The visual tip of a pgf arrow tip

Question

In the 'Defining New Arrow Tip Kinds' chapter in the PGF part of the TikZ & PGF manual for version 3.0.1a, the visual tip end of an arrow tip is defined as (p. 1016)

the counterpart of the visual back end for the front.

The visual back end, in turn, is defined as follows. (ibid.)

The visual back end is the position where the path and the the arrow head “meet last” on the path.

To illustrate these and other concepts, the following picture is given.

In terms of this illustration, the visual back end is said to be at (-2,0), and the visual tip end is said to be at (1,0).

I understand the definition of visual back end, but not that of visual tip end. Going by the definitions quoted above ('the visual tip end is the counterpart of the visual back end'), the visual tip end would be the position where the path and the arrow head "meet first" on the path, but this would be somewhere on the x-axis between -3 and -2, not at (1,0).

I'll appreciate it if someone can give a straightforward definition of visual tip end, without referring to the definition of visual back end, and explain, based on this definition, why the visual tip end in the above illustration is at (1,0).

Answer 1

The term visual back end does not necessarily refer to any geometric point on the arrowhead. It is merely a parameter provided to arrow designers.

The motivation to define visual back end in addition to back end is clear only if we consider curved path

In this case, the arrow looks ugly because it does not lie on the path. We expect TikZ to slightly rotate the arrow. For example:

This looks a lot better as the visual back end lies on the solid red line. The bending library will do this job for you with the option flex.

One may also try

This looks good too as the back end(green dot) lies on the solid red line. The bending library will do this job for you with the option flex'.

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The motivation to define visual tip end is the same: when one writes -{Stealth[reversed]}, the library needs the coordinate of the visual back end for this reversed arrow. In this case, the visual tip end of Stealth will be used.

So far I told only half of the story. knowing that the arrow's visual tip end/ back end lies on the original path does not determine the placement of the arrowhead. The following is what TikZ does when flex=1 is used:

• shift the arrowhead so that visual tip end coincides the endpoint of the path;
• rotate the arrowhead so that the visual back end lies on the path.

Therefore, similar to what I said at the beginning of the answer, visual tip end does not necessarily represent any geometric point. It is the arrowhead-designer's choice to assign a point as the visual tip end, which will later be used by TikZ to position the arrowhead.

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Appendix

Figures above are generated by acquiring the source code of the manual. In particular pgfmanual-en-base-arrows.tex line 108-117. As one can easily see, the code draws the path and arrowhead separately.

\documentclass[border=9,tikz]{standalone}
\usetikzlibrary{arrows.meta,bending,calc}
\begin{document}

\begin{tikzpicture}
  \draw [red!50, ,line width=1cm] (0,0) -- (4,0);
  \path [tips, opacity=.25,line width=1cm, -{Stealth[black,line width=0pt,length=4cm, width=4cm, inset=1cm]}] (0,0) -- (6,0);

  \draw [->,thick] (1,0) -- (8,0) node [right] {$x$-axis};
  \draw [->,thick] (5,-2.25) -- (5,2.25) node [above] {$y$-axis};

  \foreach \i in {-3,-2,-1,1,2} \draw (\i+5,-1mm) -- (\i+5,1mm) node [above] {\small$\i$};
  \foreach \i in {-2,-1,1,2} \draw (49mm,\i) -- (51mm,\i) node [right] {\small$\i$};;
\end{tikzpicture}

\begin{tikzpicture}
  \draw [red!50, ,line width=1cm] (0,-4) to[bend left] (4,0);
  \draw [red] (0,-4) to[bend left] (4,0);
  \path [tips, opacity=.25,line width=1cm, -{Stealth[black,line width=0pt,length=4cm, width=4cm, inset=1cm]}] (0,0) -- (6,0);

  \draw [->,thick] (1,0) -- (8,0) node [right] {$x$-axis};
  \draw [->,thick] (5,-2.25) -- (5,2.25) node [above] {$y$-axis};

  \foreach \i in {-3,-2,-1,1,2} \draw (\i+5,-1mm) -- (\i+5,1mm) node [above] {\small$\i$};
  \foreach \i in {-2,-1,1,2} \draw (49mm,\i) -- (51mm,\i) node [right] {\small$\i$};;
\end{tikzpicture}

\begin{tikzpicture}
  \draw [red!50, ,line width=1cm] (0,-4) to[bend left] (4,0);
  \draw [red] (0,-4) to[bend left] (4,0);
  \path [tips, opacity=.25,line width=1cm, -{Stealth[black,line width=0pt,length=4cm, width=4cm, inset=1cm]}] (0,-.6) -- (6,0);

  \draw [->,thick] (1,0) -- (8,0) node [right] {$x$-axis};
  \draw [->,thick] (5,-2.25) -- (5,2.25) node [above] {$y$-axis};

  \foreach \i in {-3,-2,-1,1,2} \draw (\i+5,-1mm) -- (\i+5,1mm) node [above] {\small$\i$};
  \foreach \i in {-2,-1,1,2} \draw (49mm,\i) -- (51mm,\i) node [right] {\small$\i$};;
\end{tikzpicture}

\begin{tikzpicture}
  \draw [red!50, ,line width=1cm] (0,-4) to[bend left] (4,0);
  \draw [red] (0,-4) to[bend left] (4,0);
  \path [tips, opacity=.25,line width=1cm, -{Stealth[black,line width=0pt,length=4cm, width=4cm, inset=1cm]}] (0,-1.3) -- (6,0);
  \fill[green]($(6,0)!4cm!(0,-1.3)$)circle(.1);

  \draw [->,thick] (1,0) -- (8,0) node [right] {$x$-axis};
  \draw [->,thick] (5,-2.25) -- (5,2.25) node [above] {$y$-axis};

  \foreach \i in {-3,-2,-1,1,2} \draw (\i+5,-1mm) -- (\i+5,1mm) node [above] {\small$\i$};
  \foreach \i in {-2,-1,1,2} \draw (49mm,\i) -- (51mm,\i) node [right] {\small$\i$};;
\end{tikzpicture}


\begin{tikzpicture}
  \draw [red!50, ,line width=1cm] (0,0) -- (4,0);
  \path [tips, opacity=.25,line width=1cm, -{Stealth[black,line width=0pt,length=4cm, width=4cm, inset=1cm,reversed]}] (0,0) -- (6,0);

  \draw [->,thick] (1,0) -- (8,0) node [right] {$x$-axis};
  \draw [->,thick] (5,-2.25) -- (5,2.25) node [above] {$y$-axis};

  \foreach \i in {-3,-2,-1,1,2} \draw (\i+5,-1mm) -- (\i+5,1mm) node [above] {\small$\i$};
  \foreach \i in {-2,-1,1,2} \draw (49mm,\i) -- (51mm,\i) node [right] {\small$\i$};;
\end{tikzpicture}

\begin{tikzpicture}[opacity=.25,scale=5]
  \draw [red,line width=1cm,-{Stealth[black,line width=0pt,length=4cm, width=4cm,inset=1cm,flex=0]},preaction={draw,-}]
  (-1,-.5) .. controls (0,-.5) and (0,0) .. (1,0);
\end{tikzpicture}
\begin{tikzpicture}[opacity=.25,scale=5]
  \draw [red,line width=1cm,-{Stealth[black,line width=0pt,length=4cm, width=4cm,inset=1cm,flex=1]},preaction={draw,-}]
  (-1,-.5) .. controls (0,-.5) and (0,0) .. (1,0);
\end{tikzpicture}
\begin{tikzpicture}[opacity=.25,scale=5]
  \draw [red,line width=1cm,-{Stealth[black,line width=0pt,length=4cm, width=4cm,inset=1cm,flex'=1]},preaction={draw,-}]
  (-1,-.5) .. controls (0,-.5) and (0,0) .. (1,0);
\end{tikzpicture}

\end{document}