Darned ellipsoid !

[krtxmrtz]

Does anyone know any free/shareware pack for fitting a cloud of points in space to a surface?

The idea is, I have a number of (say, about half a dozen) experimental points that should be theoretically distributed on the surface of an ellipsoid:

z = Sqr(1 - (x/a)² -(y/b)²)

The problem is determining a and b so the error (or the sum of the squared errors) is a minimum. The error would be the distance from each point to the surface (measured along a direction which is perpendicular to the tangent plane to the surface).

Now, the above equation corresponds to an ellipsoid with its center at the coordinate origin and its main (symmetry) axes coincident with the corrdinate axes x,y & z.

However, the points are not necessarily distributed around my coordinate origin so you've got to make a translation. And secondly, the ellipsoid's main axes could be arbitrarily oriented in space. Therefore you have to somehow determine what the most probable orientation would be and rotate the coordinate axes accordingly.

I kind of solved the problem using Excel but the whole thing was very messy and I wouldn't waste any more time on it. That's why I'd like to use someone else's (better!?) software.

[krtxmrtz]

Originally posted by krtxmrtz

z = Sqr(1 - (x/a)² -(y/b)²)

Btw, maybe it's more "correct" to write the equation in this fashion.

(x/a)² + (y/b)² + (z/c)² = 1

[kedaman]

I think placing the origin in the center of the cloud would usually give best results, but on the other hand the points might as well be originated along one side of the surface with no error. Does the solution certainly have to be minimal?

[krtxmrtz]

The points are not just "any points", i.e. they are not random. They will most likely be distributed throughout (very near) the surface.

Precisely, the main problem is what you justly point out. How best determine whether you should place the new origin at the geometrical center of the cloud or elsewhere, and how to decide this mathematically. Maybe one has to try a more general fit (i.e. a more general / complex model than the equation I've written) without doing any previous transformation (translation + rotations).

The minimal error condition was the requirement I thought would help me best in crunching the algebra but there may be other useful methods. I haven't thought of any other though.