Parity of n-Frames with Application to Non-Procrustean Orthogonalization

Abstract

The space of complete orthonormal frames in Euclidean space is not path connected. In fact it has exactly two path components, containing respectively the coordinate frame of n standard coordinates and the frame with two coordinates reversed. The matrices corresponding to these frames have determinant one and minus one respectively. We wish to avoid the implement of the determinant function in many variables, and rather work with these compact spaces by means of fibrations and covering maps. The important fibrations are deletion of the final vector from the frame, and selection of the first vector. The long homotopy sequence for this fibration implies that the equivalence classes of loops form the group of two elements. One generator of this group is a circle of Givens rotations. Dimension three is critical for the proof. It is handled by the Rodrigues formula, together with Invariance of Domain. Another explicit proof that the mapping from unit quaternions is surjective to the main path-component is due to Itzhack. Thus the Orthogonal Group is not path connected, using the Homotopy Lifting Theorem for coverings. But the Givens loop performs an identity lifting, so it cannot be the group generator. As an application we review how quoted algorithms from the astronautical literature, related to the Wahba guidance problem, give rise to an orthogonalization of any matrix that is metrically close to some rotation. This method has a bias in the Frobenius norm, but under realistic conditions serves as well as a Procrustean method.