Surface fitting through the least squares' general regression matrix formula.

Question

The last page of this paper attemps to fit a paraboloid to a set of given 3D points through the general linear regression matrix formula , my question here is to what extent is this approach valid? can I surface fit Z=af(x,y)+bg(x,y)+c*h(x,y)+.... using the same approach?

Answer 1

You have $n$ data points$(x_i,y_i,z_i)$ and you want to fit a model $$z=a+b x+c y+d x^2+e x y+ f y^2$$

Let us define $t_i=x^2_i$, $u_i=x_i y_i$ and $v_i=y_i^2$ which makes the model to be $$z=a+bx+c y+d t+e u+f v$$ which is a multilinear regression model. Just write the classical matrix form and replace (if you want) the $(t,u,v)$ by their definition.