Swapping modulus and argument in polar coordinates Let $y=f(x)$ in Cartesian coordinates. Swapping the $x$ coordinate with the $y$ coordinate has the effect of reflection about the axis $y=x$.
But if $r=f(\theta )$ in polar coordinates (where $r$ is the modulus and $\theta$ is the argument), what happens when we swap $r$ with $\theta$?
For example, what's the difference between the polar graphs of $r=\theta ^2$ and $\theta =r^2$? Do they look the same? Or is there some kind of reflection as well?


The polar plots of $r= \theta^2, \sqrt \theta, \theta$ are shown in blue,red and grey respectively.
The gray line does not reflect red/blue profiles in a way presently imaginable by me.
This is perhaps because $(r,\theta)$ do not have the same physical dimension and belong to a common type or genre.
$Z=r e^{i \theta}$ is a complex number. I do not know what $U = \theta e^{ir}$ really is.
A very good question, but my answer may be is no good. That is indirectly suggested geometric reciprocal requirement is not answered.
