  We investigate connections between the geometry of linear subspaces and the
convergence of the alternating projection method for linear projections. The
aim of this article is twofold: in the first part, we show that even in
Euclidean spaces the convergence of the alternating method is not determined by
the principal angles between the subspaces involved. In the second part, we
investigate the properties of the Oppenheim angle between two linear
projections. We discuss, in particular, the question of existence and
uniqueness of "consistency projections" in this context.
