  We show that the geometric phase between any two states, including orthogonal
states, can be computed and measured using the notion of projective
measurement, and we show that a topological number can be extracted in the
geometric phase change in an infinitesimal loop near an orthogonal state. Also,
the Pancharatnam phase change during the passage through an orthogonal state is
shown to be either $\pi$ or zero (mod $2\pi$). All the off-diagonal geometric
phases can be obtained from the projective geometric phase calculated with our
generalized connection.
