Dimensioning the radius of a circle: `rotate around` doesn't work with polar coordinates?

Question

This question partly extends on this post.

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I am using the following command line to rotate a line that I use to dimension the radius of a circle, as mentioned in an answer to the above mentioned post:

\draw[-latex, rotate around={45:(0,0)}] (0,0) -- (6,0)  node [midway,fill=white] {$\text{Text}$};

The problem is, I am not using (6,0) but a point (p) that is calculated with polar coordinates. As soon as I replace (6,0) by (p), the rotation doesn't work.

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How do I workaround the fact that rotate around apparently doesn't work for points in polar coordinates?

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Edit: I added the exact code I used to calculate the point with polar coordinates (it's called last1):

\documentclass{standalone}
\newcommand{\ts}[1]{\textnormal{#1}}
\usepackage{tikz} 
\usetikzlibrary{positioning,arrows,calc,math,angles,quotes,trees,mindmap,through}
%include other needed packages here   
\begin{document}
\begin{tikzpicture}
\draw[-latex] (-4,0) -- (4,0) node[right]{$\ts{Re}(r)$};
\draw-latex -- (0,4) node[above] {$\ts{Im}(r)$};
\def\z{0}
\def\zz{-1.5}
\def\zzz{-5}
\coordinate (last1) at (0:0);
\coordinate (last2) at (0:0);
\coordinate (last3) at (0:0);
\foreach \y in {1,2,3,4,5,6,7,8,9,10,11,12,13,14}
  {
    \pgfmathsetmacro{\o}{0.05random(0,1000)};
\pgfmathsetmacro{\t}{0.005random(0,100)};
    \draw[->] (last1) -- ++ (\o\z:\t) coordinate (last1);
    \draw[->] (last2) -- ++ (\o\zz:\t) coordinate (last2);
    \draw[->] (last3) -- ++ (\o*\zzz:\t) coordinate (last3);

  };
\node at (0,0) [draw,dotted, circle through={(last1)}] {};
\draw[-latex,line width=0.7mm] (0:0) -- (last3);
\draw[-latex, rotate around={45:(0,0)}] (0,0) -- (last1)  node [midway,fill=white] {$\sum |b_{z_1z_2\cdots z_N}|^2 =1$};
\end{tikzpicture}
\end{document}

Answer 1

I have to admit that it took me a while to get what you mean, so I try to restate it with my own words:

The problem is that \draw[-latex, rotate around={45:(0,0)}] (0,0) -- (4,0); would work (the resulting line is rotated by 45 degrees). Similary, \draw[-latex, rotate around={45:(0,0)}] (0,0) -- (0:4); would work. But as soon as you replace one of the coordinates of your path with a previously defined coordinate, the transformation does not work anymore. So \draw[-latex, rotate around={45:(0,0)}] (0,0) -- (last1); would actually not result in a rotated line, although the coordinate (last1) is defined as (0:<x>).

This problem seems to be the same as in this other question. The reason for this is that the coordinate was defined outside the scope of the transformation (in fact it is not exactly a scoping issue but a feature that applies an inverse transformation on anchor points and which is quite sensible in most cases*; thanks to Qrrbrbirlbel!). I can actually understand that this behaviour is quite peculiar, at least not directly obvious.

So, what you need to do is to explicitly transform the previously defined coordinate. Since the other coordinate of your path is (0,0), you don't need to transform the rest of the path: You can write \draw[-latex] (0,0) -- ([rotate=45]last1); and will get a line rotated by 45 degrees.

\documentclass[border=10pt]{standalone}
\usepackage{tikz}
\begin{document}
\begin{tikzpicture}
\draw[-latex, red, rotate around={10:(0,0)}] (0,0) -- (4,0);
\draw[-latex, magenta, rotate around={20:(0,0)}] (0,0) -- (0:4);
% behaviour not as expected
\coordinate (last1) at (4,0);
\draw[-latex, blue, thick, rotate around={30:(0,0)}] (0,0) -- (last1);
% behaviour not as expected
\coordinate (last2) at (0:4);
\draw[-latex, cyan, rotate around={40:(0,0)}] (0,0) -- (last2);
\coordinate (last3) at (4,0);
\draw[-latex, green] (0,0) -- ([rotate=50]last3);
\coordinate (last4) at (0:4);
\draw[-latex, yellow] (0,0) -- ([rotate=60]last4);
\end{tikzpicture}
\end{document}

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^{* See section 106.3.1 "Referencing Anchors of Nodes in the Same Picture" in the PGF manual.}