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From: Walter Roberson <roberson@hushmail.com>
Newsgroups: comp.soft-sys.matlab
Subject: Re: Quadric approximation and
Date: Thu, 22 Jul 2010 11:34:49 -0500
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Xref: news.mathworks.com comp.soft-sys.matlab:655302

Alexander Petrov wrote:

> Application of Matlab fit function under 2d-matrix give me an quadratic 
> function: sf(x,y) = p00 + p10*x + p01*y + p20*x^2 + p11*x*y + p02*y^2
> 
> But what I want is an general equation of quadric surface, like on that 
> webpage - http://mathworld.wolfram.com/QuadraticSurface.html
> 
> I've know, that this is a stupid question, but could I really rewrite 
> matlab result as:
> 
> let z= sf(x,y);
> p00 + p10*x + p01*y + p20*x^2 + p11*x*y + p02*y^2 - z  = 0
> 
> Does such a "trick" give me an general equation of quadric surface ?

No. If z = sf(x,y) and the overall surface is to remain quadradic, then 
sf(x,y) would have to be a multinomial in x and y in which the "total power" 
of each term did not exceed 2. Such a multinomial, when substituted into the 
proposed expression, would be of identical form to the p00 <etc> expression 
except with different coefficients. This would not be sufficient to express 
the general quadratic surfaces as described on Wolfram's page, which require 
polynomials in x, y, and z.