Curving an arrow in PGF/TikZ

Question

Consider the following code:

\documentclass[tikz,border=2mm]{standalone}
\usetikzlibrary{arrows.meta,shapes.misc,decorations.pathmorphing,calc}

% http://tex.stackexchange.com/questions/123760/draw-crosses-in-tikz
\tikzset{
  branch point/.style={cross out,draw=black,fill=none,minimum size=2*(#1-\pgflinewidth),inner sep=0pt,outer sep=0pt}, 
  branch point/.default=5
}
\tikzset{
  branch cut/.style={
    decorate,decoration=snake,
    to path={
      (\tikztostart) -- (\tikztotarget) \tikztonodes
    },
    execute at begin to={{
      \draw[thick,green!60!black,-{Stealth[]}] ($(\tikztostart)!.8!-10:(\tikztotarget)$) -- ($(\tikztostart)!.8!10:(\tikztotarget)$) node[scale=.8,pos=.7,above left] {$\times (-1)$};
    }}
  }
}

\begin{document}
\begin{tikzpicture}[x=90pt,y=90pt]
\begin{scope}[shift={(2,0)}]
  % Axes
  \draw[thin,gray,->] (-1,0) -- (1,0) node[right] {$x$};
  \draw[thin,gray,->] (0,-1) -- (0,1) node[above] {$y$};
  % Branch point
  \draw[thick] (0,0) node[branch point,draw=red,thick] {};
  % Branch cut
  \draw[thick,red,branch cut] (0,0) to (140:1.2);
\end{scope}
\end{tikzpicture}
\end{document}

Here is its output.

I want to bend the green arrow like this:

I have attempted the following, without success:

• Replacing the -- in the execute at begin to block with edge [bend right] results in a TeX capacity exceeded, sorry [grouping levels=255] error.
• to [bend right] results in the same error.
• Adding the bend right option to the \draw command does nothing.

How can I do this?

Answer 1

You could use control points:

\documentclass[tikz,border=2mm]{standalone}
\usetikzlibrary{arrows.meta,shapes.misc,decorations.pathmorphing,calc,bending}

% http://tex.stackexchange.com/questions/123760/draw-crosses-in-tikz
\tikzset{
  branch point/.style={cross out,draw=black,fill=none,minimum size=2*(#1-\pgflinewidth),inner sep=0pt,outer sep=0pt}, 
  branch point/.default=5
}
\tikzset{
  branch cut/.style={
    decorate,decoration=snake,
    to path={
      (\tikztostart) -- (\tikztotarget) \tikztonodes
    },
    execute at begin to={{\draw[thick,green!60!black,-{Stealth[]}] ($(\tikztostart)!.8!-10:(\tikztotarget)$) .. controls +(-20pt,5pt) and +(-5pt,2pt) .. ($(\tikztostart)!.8!10:(\tikztotarget)$) node[scale=.8,pos=.7,above left] {$\times (-1)$};
    }}
  }
}

\begin{document}
\begin{tikzpicture}[x=90pt,y=90pt]
\begin{scope}[shift={(2,0)}]
  % Axes
  \draw[thin,gray,->] (-1,0) -- (1,0) node[right] {$x$};
  \draw[thin,gray,->] (0,-1) -- (0,1) node[above] {$y$};
  % Branch point
  \draw[thick] (0,0) node[branch point,draw=red,thick] {};
  % Branch cut
  \draw[thick,red,branch cut] (0,0) to (140:1.2);
\end{scope}
\end{tikzpicture}
\end{document}

Answer 2

Following Gonzalo Medina's suggestion of using a Bézier curve, I modified my code to get the arrow in the shape of a parabola and with an angle of 30 degrees between the arrow and the line joining the two endpoints (I don't know about the parabola, but that 30 degree angle is what bend right is supposed to do).

The advantage is that I can now draw branch cuts (the red snake lines) in other angles — the hard-coding of the control points in his answer does not give the expected effect at those other angles.

\documentclass[tikz,border=2mm]{standalone}
\usetikzlibrary{arrows.meta,shapes.misc,decorations.pathmorphing,calc,bending}

\tikzset{
  branch point/.style={cross out,draw=black,fill=none,minimum size=2*(#1-\pgflinewidth),inner sep=0pt,outer sep=0pt}, 
  branch point/.default=5
}
\tikzset{
  branch cut/.style={
    decorate,decoration=snake,
    to path={
      (\tikztostart) -- (\tikztotarget) \tikztonodes
    },
    execute at begin to={{
      \coordinate (A) at ($(\tikztostart)!.8!-10:(\tikztotarget)$);
      \coordinate (B) at ($(\tikztostart)!.8!10:(\tikztotarget)$);
      \coordinate (AB/3) at ($(A)!1/3!(B)$);
      \coordinate (2AB/3) at ($(A)!2/3!(B)$);
      \coordinate (C) at ($(AB/3)!2/(3*sqrt(3))!-90:(B)$);
      \coordinate (D) at ($(2AB/3)!4/(3*sqrt(3))!-90:(B)$);
      \draw[thick,green!60!black,-{Stealth[]}] (A) .. controls (C) and (D) .. (B) node[scale=.8,pos=.9,above left] {$\times (-1)$};
    }}
  }
}

\begin{document}
\begin{tikzpicture}[x=90pt,y=90pt]
\begin{scope}[shift={(2,0)}]
  % Axes
  \draw[thin,gray,->] (-1,0) -- (1,0) node[right] {$x$};
  \draw[thin,gray,->] (0,-1) -- (0,1) node[above] {$y$};
  % Branch point
  \draw[thick] (0,0) node[branch point,draw=red,thick] {};
  % Branch cut
  \draw[thick,red,branch cut] (0,0) to (20:1.2);
  \draw[thick,red,branch cut] (0,0) to (140:1.2);
  \draw[thick,red,branch cut] (0,0) to (-90:1);
\end{scope}
\end{tikzpicture}
\end{document}

Now I'll play with pgfkeys to be able to adjust some parameters (like the position of the label).

Answer 3

I would draw this picture far more simply. Probably because I don't know or not be able to figured out story behind it ... Of course, it is hard coded:

\documentclass[tikz,border=2mm]{standalone}
\usetikzlibrary{arrows.meta,bending,calc,decorations.pathmorphing,shapes.misc}

\begin{document}
    \begin{tikzpicture}[
    x=90pt, y=90pt,
    decoration = snake
                        ]
% Axes
\path[draw=gray,->]    
    (-1,0) edge node[pos=1,right] {$x$} (1,0)
    (0,-1)  --  (0,1) node[above] {$y$};
% first wave + cross
\draw[red,thick,decorate] (140:1.2) -- (0,0) 
        node[draw,minimum size=1pt,cross out] {};
% arc
\draw[draw=teal,thick,-{Stealth[flex]}] (130:0.9) arc (130:150:0.9) 
        node[above left,inner sep=1pt] {$x(-1)$};
% second wave
\draw[red,thick,decorate] (30:1.2) -- (0,0);
% arc
\draw[draw=teal,thick,-{Stealth[flex]}] (20:0.9) arc (20:40:0.9)
        node[above right,inner sep=1pt] {$x(-1)$};
    \end{tikzpicture}
\end{document}

which gives: