Centering a slash (not) over a (mathematical) glyph?

Question

My aim is to construct a relational symbol which is graphically equivalent the standard infinity sign (coded by\infty) crossed exactly in the middle of it. Yet I am not able to overcome something. To be more precise:

I would like to have slash sign moved exactly to the middle position of the infinity sign. The above was defined in the following way:

\documentclass{standalone}
\begin{document}
\newlength{\length}
\settowidth{\length}{$\infty$}
\newcommand\ninfty{\mathrel{\infty\hspace{-0.5\length/}}} 

$\ninfty$ is not what I am aiming at.

\end{document}

So as you can see my idea was to define the length equal to the length of the infinity sign and then place slash sigh it halfway through it. As it can be seen above it was rather please quarter-way (backwards).

Could you please explain what I did wrong and how can I exactly determine the middle point (in terms of width) of a given mathematical glyph? Or a glyph in general?

Answer 1

Don't reinvent the wheel. ;-)

\documentclass{article}
\usepackage{centernot}
\begin{document}

$\centernot{\infty}$

\end{document}

Some manual adjusting can help:

\documentclass{article}
\usepackage{centernot}
\begin{document}

$\centernot{\mkern-0.35mu\infty}\mkern-0.35mu$

\end{document}

Finding the exact geometric center is not really easy, because it depends on the side bearings of the symbol. Here's a picture showing them

Also the slash has side bearings, so it's a question of balancing them.

Of course, when you're satisfied with the positioning, you can define

\newcommand{\cinfty}{\mathrel{\centernot{\mkern-0.35mu\infty}\mkern-0.35mu}}

A different version that uses the slash rather than \not (and my favorite tool \ooalign).

\documentclass{article}

\makeatletter
\newcommand{\cinfty}{\mathrel{\mathpalette\do@cinfty\relax}}
\newcommand{\do@cinfty}[2]{%
  \vphantom{/}%
  \ooalign{\hidewidth$\m@th#1/$\hidewidth\cr$\m@th#1\infty$}%
}
\begin{document}

$A\cinfty B_{\cinfty}$

\end{document}

Answer 2

\documentclass{article}
\usepackage{stackengine}
\renewcommand\stacktype{L}
\stackMath
\begin{document}
$\stackon[0pt]{\infty}{/}$
\end{document}

If you want the version that is a \mathrel and scales with the math style,

\documentclass{article}
\usepackage{scalerel,stackengine}
\def\notinfty{%
  \renewcommand\stacktype{L}\mathrel{\ensurestackMath{%
  \ThisStyle{\stackon[0pt]{\SavedStyle\infty}{\SavedStyle/}}}}%
}
\begin{document}
$A \notinfty B ~\scriptstyle A\notinfty B ~\scriptscriptstyle A \notinfty B$
\end{document}

Answer 3

\not is usually optimized for the equals sign. Package centernot takes care of the horizontal centering. \infty is wider than the equals sign, see egreg's answer.

The equals sign is vertically centered around the math axis and \not usually follows this symmetry. The symbol \infty, however, just sits on the base line (Computer Modern fonts, cmsy10). Since it seems used with a different meaning as relational symbol, it might be allowed to vertically center it.

The following example centers \infty as \vinfty vertically around the math axis and uses package centernot for the horizontal centering.

\documentclass{article}
\usepackage[T1]{fontenc}
\usepackage[variablett]{lmodern}
\usepackage{amsmath}% for environment "gather*"

\usepackage{centernot}

\makeatletter
\newcommand*{\vinfty}{%
  \mathrel{%
    \mathpalette\@vinfty{}%
  }%
}
\newcommand*{\@vinfty}[2]{%
  % #1: math style
  % #2: unused
  \vcenter{%
    \hbox{$#1\infty\m@th$}%
  }%
}
\makeatother

\newcommand*{\notvinfty}{%
  \centernot\vinfty
}

\begin{document}
\centering

{\scriptsize Vertically centered (\texttt{\textbackslash vinfty})}
\begin{gather*}
  a \vinfty b \notvinfty c
\\
  \scriptstyle a \vinfty b \notvinfty c
\\
  \scriptscriptstyle a \vinfty b \notvinfty c
\end{gather*}
%
{\scriptsize Not vertically centered (\texttt{\textbackslash infty})}
\begin{gather*}
  a \mathrel{\infty} b \centernot\infty c
\\
  \scriptstyle a \mathrel{\infty} b \centernot\infty c
\\
  \scriptscriptstyle a \mathrel{\infty} b \centernot\infty c
\end{gather*}

\end{document}