Nearest plane to non-coplanar points?

Question

I have a number of non-coplanar 3D points and I want to calculate the nearest plane to them (They will always form a rough plane but with some small level of variation). This can be done by solving simultaneous linear equations, one for each point, of the form:

"Ax + By + Cz + D = 0"

The problem I'm having at the moment is twofold.

Firstly since the points are 3D floats they can't be relied on to be precise due to rounding errors.

Secondly all of the methods to solving linear equations programatically that I have found thus far involve using NXN matrices which severely limits what I would be able to do given that I have 4 unknowns and any number of linear equations (due to the variation in the number of 3D points).

Does anyone have a decent way to either solve the simultaneous linear equations without these constraints or, alternatively, a better way to calculate the nearest plane to non-coplanar points? (The precision of the plane calculation is not too much of a concern)

Thanks! :)

Answer 1

If your points are all close to the plane, you have a choice between ordinary least squares (where you see Z as a function of two independent variables X and Y and you minimize the sum of squared vertical distances to the plane), or total least squares (all variables independent, minimize the sum of normal distances). The latter requires a 3x3 SVD. (See http://en.wikipedia.org/wiki/Total_least_squares, unfortunately not the easiest presentation.)

If some of the points are outliers, you will need to resort to robust fitting methods. One of them is RANSAC: choose three points are random, build their plane and compute the sum of distances of all points to the plane, as a measure of fitness. Keep the best result after N drawings.

Answer 2

There are numerical methods for linear regression, which calculates the nearest line y=mx+c to a set of points. Your solution will be similar, only it has one more dimension and is thus a "planar regression".

If you don't care the mathematical accuracy of the algorithm and just want to get a rough result, then perhaps you'd randomly 3 points to construct a plane vector, then adjust it incrementally as you go through the rest of the points. Just some thoughts...