How to do a non-linear transformation with tikz-pgf?

Question

Consider the following plot of the (-2, -1) -- (3, 9) line segment:

I want to apply a transformation so that the horizontal axis is the polar angle of a point, and the vertical axis is the distance between that point and the origin.

It should look like this:

I've taken a look at this question so I believe this should be possible, but I don't really understand the code inside \def\polartransformation.

Any hint about how to do this or links to documentation about those pgfmath functions and pgftransformnonlinear would be appreciated.

Answer 1

First of all, I think that the otherwise great pgfmanual is not entirely correct. The manual section 107.4.2 states

% \pgf@x will contain the radius angle
% \pgf@y will contain the distance
\pgfmathsincos@{\pgf@sys@tonumber\pgf@x}%
% pgfmathresultx is now the cosine of radius angle and 
% pgfmathresulty is the sine of radius angle 
\pgf@x=\pgfmathresultx\pgf@y% 
\pgf@y=\pgfmathresulty\pgf@y%

What the code in the pgfmanual is probably doing is to express the x coordinate in pt, then take the cos and sin of x/pt (i.e. if x=50pt then it will return cos(50)), and to multiply the outcome by the y coordinate, i.e.

(x_new,y_new) = (y_old cos(x_old/pt), y_old sin(x_old/pt))

This leads to the code:

\documentclass{article}
\usepackage{tikz}
\usepgfmodule{nonlineartransformations}
\tikzset{declare function={mymod(\x)=\x-int(\x);}}
\makeatletter
\def\slowtransformation{% modified version of the manual 103.4.2 Installing Nonlinear Transformation
\typeout{before:\space\the\pgf@x\space\the\pgf@y}%
\edef\oriX{\the\pgf@x}%
\edef\oriY{\the\pgf@y}%
\pgfmathsetmacro{\myAngle}{mod(360+atan2(\oriY,\oriX),360)}
\pgfmathsetmacro{\myRadius}{veclen(\oriX,\oriY)}
\typeout{original\space x=\oriX\space y=\oriY}
\typeout{radius=\myRadius\space angle=\myAngle}
\setlength{\pgf@x}{\myAngle pt}
\setlength{\pgf@y}{\myRadius pt}
} 
\def\fastertransformation{% modified version of the manual 103.4.2 Installing Nonlinear Transformation
\pgfmathsetmacro{\myAngle}{mod(720+atan2(\pgf@y,\pgf@x),360)}
\pgfmathsetmacro{\myRadius}{veclen(\pgf@x,\pgf@y)}
\setlength{\pgf@x}{\myAngle pt}
\setlength{\pgf@y}{\myRadius pt}
} 
\makeatother
\begin{document}
\begin{tikzpicture}
\draw[-latex] (-2.3,-2) -- (3.5,-2) node[below]{$x$};
\draw[-latex] (0,-2.3) -- (0,4.5) node[left]{$y$};
\draw[blue]  (-2, -1) -- (3, 4);
\end{tikzpicture}\\
\begin{tikzpicture}
\draw[-latex] (-0.3,0) -- (7.5,0) node[below]{$\varphi$};
\draw[-latex] (0,-0.3) -- (0,4.5) node[left]{$r$};
\pgftransformnonlinear{\slowtransformation}
\draw[blue]  (-2, -1) -- (3, 4);
\end{tikzpicture}
\begin{tikzpicture}
\draw[-latex] (-0.3,0) -- (7.5,0) node[below]{$\varphi$};
\draw[-latex] (0,-0.3) -- (0,4.5) node[left]{$r$};
\pgftransformnonlinear{\fastertransformation}
\draw[blue]  (-2, -1) -- (3, 4);
\end{tikzpicture}
\end{document}

There are two identical transformations, the first one (\slowtransformation) is more explicit and issues \typeouts to understand what's going on, whereas the second one (\fastertransformation) is a bit faster.

NOTE:

• One has to be careful with options like scale=... and the like. This transformation will take the very coordinates after all scale and so on transformations, and then map them to polar coordinates. Therefore, if one has, say, [yscale=0.5] and draws a circle around the origin, the coordinates are those of an ellipse, and the transformation does then not map them to a horizontal line (which it does without any such additional transformations).

• The current code will produce angles between 0 and 360 degrees, and this is controlled by mod(360+atan2(\oriY,\oriX),360). If you use different conventions, you need to adjust this bit of code accordingly.

UPDATES: Fixed two a stupid mistake in the computation of the angle (atan2(y,x) vs atan2(x,y)) and made the axes labels more appropriate. I also removed all scale directives and added more explanations.