Let $h\in \mathbb{R}[x,y]$ be a nonzero polynomial and define a plane curve in polar coordinates as $r(\theta) = h(\cos\theta,\sin\theta)$. For all the examples I've looked at, it seems like we can describe this curve as a zero set of a polynomial $f\in \mathbb{R}[x,y]$.
If we use the standard parametrization for the unit circle, we see that the curve above is essentially defined by
$t\mapsto \left(h\left(\frac{1-t^2}{1+t^2},\frac{2t}{1+t^2}\right)\frac{1-t^2}{1+t^2},h\left(\frac{1-t^2}{1+t^2},\frac{2t}{1+t^2}\right)\frac{2t}{1+t^2}\right).$
If the components were both polynomials, then it would be easy to show that we have a zero set of a polynomial. However, the components are rational expressions in terms of $t$. Is there a way to show that these points are the roots of some polynomial?