How can I set up a macro using TikZ to draw "identify braces"?

Question

An identify brace on a graph tells the reader that the two points "identified" are topologically equivalent. That is, the correct way to understand the graph is to cut it out, roll it into a cylinder, and glue it together at the points being identified. Here's an MS Paint example:

How can I make a TikZ macro for such identify braces? One would typically need horizontal and vertical braces, in both "equals sign up", "equals sign down" (horizontal) or "equals sign left", "equals sign right" (vertical) flavours.

Answer 1

You can start with this. A command \myequivdown which draws something similar to what you want between to nodes. It's easy to replicate for equals going up, right or left. Change command name, angles and distances to your taste.

\documentclass[tikz,border=2mm]{standalone}
\usetikzlibrary{positioning,calc}
\begin{document}
\newcommand{\myequivdown}[2]{
    \path (#1.south)--coordinate[midway] (aux) (#2.south);
    \draw (#1.south) to [out=-89,in=181] ([shift={(-.5mm,-3mm)}]aux)--++(-90:3mm);
    \draw (#2.south) to [out=-91,in=-1] ([shift={(.5mm,-3mm)}]aux)--++(-90:3mm);
}
\begin{tikzpicture}
    \draw (0,0)--(4,0);
    \draw (0.5,.3)--(0.5,-.3) node[below] (zero) {\strut 0};
    \draw (3.5,.3)--(3.5,-.3) node[below] (two) {\strut 2 pi};
    \myequivdown{zero}{two}
\end{tikzpicture}
\end{document}