  Let A be a unital C*-algebra. Denote by P the space of selfadjoint
projections of A. We study the relationship between P and the spaces of
projections P_a determined by the different involutions #_a induced by positive
invertible elements a in A. The maps f_p: P \to P_a sending p to the unique q
in P_a with the same range as p and \Omega_a: P_a \to P sending q to the
unitary part of the polar decomposition of the symmetry 2q-1 are shown to be
diffeomorphisms. We characterize the pairs of idempotents q, r in A with
|q-r|<1 such that there exists a positive element a in A verifying that q, r
are in P_a. In this case q and r can be joined by an unique short geodesic
along the space of idempotents Q of A.
