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.

 

[pat_connell]

Unfortunately, there is no general equation that will accurately represent a simple closed curve using a polynomial. Approximating curves with polynomials is a difficult problem and requires significant mathematical knowledge, depending on the type of curve you are trying to approximate. For example, if you are trying to approximate an ellipse, you may use the polar form of an ellipse, which can be represented by a 4th-degree polynomial. The most common way to approximate a curve is to use a spline function, which is a piecewise function that can be used to fit data points. Splines are often used in computer graphics applications and are commonly used to approximate curves.