  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.
