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I want to calculate the curvature of a surface defined by a set of $(x,y,z)$ coordinates. So I fitted this formula to the set of points and obtained values of $a, \dots,k$ with $R^2 \sim 0.98$:

$$z = a + bx + cy + dx^2 + fy^2 + gx^3 + hy^3 + ixy + jx^2y + kxy^2.$$

Now I am wondering how to calculate the curvature of the surface from this equation.

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Why did you exclude x.y and x.y.z. terms ?

Starting from first/second fundamental form of surface theory, evaluate coefficients E,F and G, e,f and g.

Start with special case Monge's form z = f(x,y) in second degree formula

$$z = a + bx + cy + A x y +dx^2 + fy^2 $$

before the third degree.

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