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I have a set of geographic (longitude,latitude,value) data to which I would like to fit surface functions, specifically, the set of quadratic surfaces:

$f(x,y)=Ax^2+Bx^2+Cxy+Dx+Ey+F$

At the moment, I am converting the spherical (or rather, WGS84 geodesic) data to planar data using the Lambert Conformal Conic projection. I then fit my surface functions to the planar data, do whatever other calculations I need, and then invert the projection.

But the planar projection incorporates distortions into the data, and thus the fitting process. It would be nice if it were possible to fit the quadratic surface around the sphere/ellipsoid, but I'm not sure how to go about doing this.

Any thoughts?

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    The form you have will have you fitting paraboloids to your data; what you need is an equation with terms like $z^2$, $yz$, and $xz$...2011-08-11
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    How big is the region you are fitting data over? You may want to look into using spherical harmonics as a basis instead of quadratic polynomials.2011-08-11
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    @Rahul, I'll be finding the intersections of these surfaces later using the Levinberg-Marquadt algorithm - I'm not sure if that would be possible with spherical harmonics, but I'll look into it.2011-08-11
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    @J.M., I'm afraid I don't understand your comment. Since the variables A-F are free to range over all values, including zero, I'd assume my fitting routine is choosing from a variety of possible quadratic surfaces. Paraboloids are just one possibility.2011-08-11
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    $z=Ax^2+Bx^2+Cxy+Dx+Ey+F$ implies that you have ellipses/hyperbolas as your $x-y$ slices, and parabolas as your $x-z$ and $y-z$ slices. To get for instance an ellipsoid, you need your $z$ to be in a quadratic term like $z^2$ or $yz$. LM works with any nonlinear regression problem.2011-08-11
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    A good point, @J.M., and one that should have been obvious. Thank you for that. I'd mentioned ellipsoids because the data set is geographic and therefore being converted from a spherical/ellipsoidal system to the plane.2011-08-11
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    I think @J.M. and I are talking about two different problems. I was assuming you wanted $f$ to be a scalar field defined on a sphere, which assigns a real value to each point. I think J.M. on the other hand is talking about the equation for a [quadric surface](http://en.wikipedia.org/wiki/Quadric). These are two different things, and in retrospect, I'm not sure which one you want. Is the "value" in (longitude, latitude, value) and which $f(x,y)$ produces supposed to be a height/altitude from the base sphere/ellipsoid?2011-08-11

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