Carto-polar curve

Question

Is there any plane curve that has the same equation in cartesian as in polar coordinates?

To be more specific, is there a function such that $f(f(x)\cos x)=f(x)\sin x, \forall x$?

Answer 1

I don't think so. Start with a plane curve in cartesian coordinates, $y = f(x)$. The same curve in polar coordinates would have the equation $r = g(\theta\,)$.

If I understand your question correctly, you're asking whether there exists a pair of functions $f(\cdot)$ and $g(\cdot)$ with identical functional forms. If so, that would imply $y=r$ and $f(x) = g(\theta\,)$ but you can't have $y=r$ and $y=r\sin\theta$ for all values of $\theta$.