How can I create a polar plot on a Cartesian grid?

Question

I'm currently using tikzpicture and pgfplots to plot this polar function:

\documentclass{article}
\usepackage{pgfplots}
\usepackage{tikz}
\begin{document}

\begin{tikzpicture}
\begin{polaraxis}
    \addplot[red,domain=0:360,samples=360,smooth] (x,{sqrt(4/(cos(4*x)+3))});
\end{polaraxis}
\end{tikzpicture}

\end{document}

However, this plots it on a polar axis, which doesn't really make sense in context. (It's defined implicitly as x⁴ + y⁴ = x² + y² and I'm just using polar coordinates to easily plot it.)

How can I get the same nice graph with the correct Cartesian grid, as Wolfram|Alpha displays it?

I considered \begin{polaraxis}[hide axis] and an additional \begin{axis} but that seems like it would require manual alignment, etc.

Is there a clean way to do this?

Answer 1

You can tell pgfplots that the input is actually given in polar coordinates using data cs=polar. Pgfplots will automatically transform it to the output coordinate system:

\documentclass{standalone}
\usepackage{pgfplots}
\usepackage{tikz}
\begin{document}

\begin{tikzpicture}
\begin{axis}[
    axis lines=center,
    axis equal image,
    enlargelimits=true,
     ]
    \addplot[data cs=polar,red,domain=0:360,samples=360,smooth] (x,{sqrt(4/(cos(4*x)+3))});
\end{axis}
\end{tikzpicture}

\end{document}

This key can also be used to provide cartesian coordinates in polar axis or other variations.

Answer 2

You can transform the polar coordinates to cartesian using an x filter and a y filter. If you wrap those in a style like this:

\pgfplotsset{
    interpret as polar/.style={
            x filter/.code=\pgfmathparse{cos(rawx)*rawy},
            y filter/.code=\pgfmathparse{sin(rawx)*rawy}
        }
}

you can just add interpret as polar to your \addplot options:

\documentclass{article}
\usepackage{pgfplots}

\pgfplotsset{
    interpret as polar/.style={
            x filter/.code=\pgfmathparse{cos(rawx)*rawy},
            y filter/.code=\pgfmathparse{sin(rawx)*rawy}
        }
}

\begin{document}
\begin{tikzpicture}
\begin{axis}[
            axis lines=center,
            axis equal image,
            enlargelimits=true
        ]
    \addplot[
            red, thick,
            domain=0:360,
            samples=360,
            smooth,
            interpret as polar
        ] (x,{sqrt(4/(cos(4*x)+3))});
\end{axis}
\end{tikzpicture}
\end{document}