Modifying `*` and `o` style tikz arrows so that they are centered at the end of line

Question

I would like to use tikz * and o arrows to indicate included/excluded endpoints in plots of piecewise functions, like for example the following one:

However, it seems that tikz always aligns the "tip" of the arrow with the end of the line, so that the code

\documentclass{article} 
\usepackage{tikz}
\usetikzlibrary{arrows}
\begin{document}
\begin{tikzpicture}
   \draw[help lines] (-.1,-.1) grid (2.1,2.1);
   \draw[thick,-*] (0,1.5) -- (1,.5);
   \draw[thick,o-] (1,1.5) -- (2,.5);
\end{tikzpicture}
\end{document}

produces

where I would like the circles to be vertically aligned on the grid line.

Is there a parameter or code I can modify to get the desired behavior?

Answer 1

There are the shorten < and shorten > option which allow you to shorten the arrow length from the start and end of the line, respectively. It also accepts negative values which then extend the arrow head forwards:

\documentclass{standalone}
\usepackage{tikz}
\usetikzlibrary{arrows}
\begin{document}
\begin{tikzpicture}[shorten >=-3pt,shorten <=-3pt]
   \draw[help lines] (-.1,-.1) grid[step=.5] (2.1,2.1);
   \draw[thick,-*] (0,1.5) -- (1,.5);
   \draw[thick,o-] (1,1.5) -- (2,.5);
\end{tikzpicture}
\end{document}

The 3pt was just a lucky guess, but seems to be the correct value for this arrow heads.

Result:

Answer 2

If you want to avoid having to guess the value for the shorten < key, you can define new arrow heads that will place the circles precisely at the specified coordinate:

\documentclass[]{standalone}

\usepackage{tikz}
\usepackage{pgfplots}
\usepgflibrary{arrows}

\makeatletter
\pgfarrowsdeclare{center*}{center*}
{
  \pgfarrowsleftextend{+-.5\pgflinewidth}
  \pgfutil@tempdima=0.4pt%
  \advance\pgfutil@tempdima by.2\pgflinewidth%
  \pgfarrowsrightextend{4.5\pgfutil@tempdima}
}
{
  \pgfutil@tempdima=0.4pt%
  \advance\pgfutil@tempdima by.2\pgflinewidth%
  \pgfsetdash{}{+0pt}
  \pgfpathcircle{\pgfqpoint{4.5\pgfutil@tempdima}{0bp}}{4.5\pgfutil@tempdima}
  \pgfusepathqfillstroke
}

\pgfarrowsdeclare{centero}{centero}
{
  \pgfarrowsleftextend{+-.5\pgflinewidth}
  \pgfutil@tempdima=0.4pt%
  \advance\pgfutil@tempdima by.2\pgflinewidth%
  \pgfarrowsrightextend{4.5\pgfutil@tempdima}
}
{
  \pgfutil@tempdima=0.4pt%
  \advance\pgfutil@tempdima by.2\pgflinewidth%
  \pgfsetdash{}{+0pt}
  \pgfpathcircle{\pgfqpoint{4.5\pgfutil@tempdima}{0bp}}{4.5\pgfutil@tempdima}
  \pgfusepathqstroke
}
\makeatother

\begin{document}
\begin{tikzpicture}
   \draw[help lines] (-.1,-.1) grid (2.1,2.1);
   \draw[thick,-center*] (0,1.5) -- (1,.5);
   \draw[thin,centero-] (1,1.5) -- (2,.5);
\end{tikzpicture}
\end{document}

Answer 3

With the newer arrows.meta library, this can be achieved by setting the sep to the negative of half the arrow tip's length. Since the default value for the length of the Circle arrow tip is

length  = +2.39365pt +3.191538

we'll be using

Circle[sep=-1.196825pt -1.595769]

Code

\documentclass[tikz]{standalone}
\usetikzlibrary{arrows.meta}
\begin{document}
\begin{tikzpicture}[
  */.tip={Circle[sep=-1.196825pt -1.595769]},
  o/.tip={Circle[sep=-1.196825pt -1.595769, open]},
]
   \draw[help lines] (-.1,-.1) grid (2.1,2.1);
   \draw[thick, -*] (0,1.5) -- (1,.5);
   \draw[thick, o-] (1,1.5) -- (2,.5);
\end{tikzpicture}
\end{document}

Output