I know a circle can be parametrized by $x(t) = cos(t)$, $y(t) = sin(t)$, which is also the solution to the algebraic equation $f(x,y) = x^2 + y^2 = 1$. I wonder about linear combinations of these, such as
$x(t) = 3cos(t) - cos(2t),~ y(t) = 3sin(t) - sin(2t)$.
Is this the solution to an algebraic equation in x and y, and when is the curve defined by
$(x(t), y(t)) = \sum\limits_{i = 1}^{n} [r_icos(s_i\theta),~ r_isin(s_i\theta)]$
algebraic?
I am a college senior doing independent research on a related topic and I've searched a lot but found nothing, yet someone must have considered this before, so maybe I'm not using the right wording; please point me to a reference.