3D Least Squares Plane

Question

What\'s the algorithm for computing a least squares plane in (x, y, z) space, given a set of 3D data points? In other words, if I had a bunch of points like (1, 2, 3), (4, 5

Answer 1

See 'Least Squares Fitting of Data' by David Eberly for how I came up with this one to minimize the geometric fit (orthogonal distance from points to the plane).

bool Geom_utils::Fit_plane_direct(const arma::mat& pts_in, Plane& plane_out)
{
    bool success(false);
    int K(pts_in.n_cols);
    if(pts_in.n_rows == 3 && K > 2)  // check for bad sizing and indeterminate case
    {
        plane_out._p_3 = (1.0/static_cast(K))*arma::sum(pts_in,1);
        arma::mat A(pts_in);
        A.each_col() -= plane_out._p_3; //[x1-p, x2-p, ..., xk-p]
        arma::mat33 M(A*A.t());
        arma::vec3 D;
        arma::mat33 V;
        if(arma::eig_sym(D,V,M))
        {
            // diagonalization succeeded
            plane_out._n_3 = V.col(0); // in ascending order by default
            if(plane_out._n_3(2) < 0)
            {
                plane_out._n_3 = -plane_out._n_3; // upward pointing
            }
            success = true;
        }
    }
    return success;
}

Timed at 37 micro seconds fitting a plane to 1000 points (Windows 7, i7, 32bit program)