Fit points to a plane algorithms, how to iterpret results?

Question

Update: I have modified the Optimize and Eigen and Solve methods to reflect changes. All now return the \"same\" vector allowing for machine precision.

Answer 1

The normal vector of the plane in Eigen solution is the eigenvector for smallest eigenvalue. Some Eigen implementations sort the eigenvalues and eigenvectors some others don't. So in some implementations it's sufficient to take first (or last) eigenvector for normal. In other implementations you have to sort them first. On the other hand the majority of SVD implementations provide sorted values so it's simple first (or last) vector.