General simple closed curve equation

Integratethis

I have a list of points on a 2D simple closed curve and I'd like to approximate that curve using a polynomial such that the approximation will be given by:
Ʃa_i,jxⁱy^j = 0

However, I still need to limit the a_i,j to make sure the approximation is also a simple closed curve, while still keeping the equation as general as possible(so it can be used for all sorts of simple closed curves).

I haven't found a general equation such as this, and I was hoping you guys could point me in the right direction(or just tell me if such a thing doesn't exist).

Thanks.