Hyperspheres and hyperplanes fitted seamlessly by algebraic constrained total least-squares

For each finite set of points in a Euclidean space of any dimension, the algorithm presented here determines all the algebraically best fitting circles or lines, spheres or planes, or hyperspheres or hyperplanes, in a seamless manner from spherical through affine manifolds. In particular, affine submanifolds of any dimensions are not singularities of the algorithm. To this end, the algorithm combines projective geometry, Coopeʹs and Gander et al.ʹs layouts of the equations, and Golub et al.ʹs generalization of the Schmidt–Mirsky matrix approximation theorem to solve the equations. The resulting best fitting manifolds remain invariant under rigid transformations. Moreover, if the best fitting manifold is affine, then it coincides with Golub and Van Loanʹs affine manifold of Total Least-Squares. Thus the algorithm can also fit hyperspheres in a manner that remains robust with data lying near a hyperplane. Furthermore, an analysis with a theorem of Wedinʹs shows that the fitted hyperspheresʹ sensitivity to perturbations of the data increases as the colinearity of the data increases.