Suppose I have a collection of $N$ points in the plane, $S=\{(x_i,y_i) \}_{i=1}^N$, and I want to describe the points as an approximate zero level-set of a polynomial of degree $d$, where $d$ is given. That is, I want to find a polynomial of the form $$P_d(x,y) :=\sum_{n=0}^d \sum_{k=0}^n a_{k,n-k} x^k y^{n-k}$$ such that all points in $S$ satisfy $$P_d(x_i,y_i) \approx 0.$$ Is this "implicit curve fitting", well known? My intuition from standard curve fitting is to try minimizing $$\sum_{i=1}^N P_d(x_i,y_i)^2 $$ with respect to all of the coefficients $\{a_{k,l}\}$. If there's a better way of doing this I'd like to know it.
Thank you!
See Taubin's papers, especially:
Estimation Of Planar Curves, Surfaces And Nonplanar Space Curves Defined By Implicit Equations, With Applications To Edge And Range Image Segmentation, by G. Taubin. In IEEE Transactions on Pattern Analysis and Machine Intelligence, November 1991.