 Unitary transitions of the tautological line over the projective line

For $\zeta =z_2/z_1$ and $\eta =1/\zeta$, the tautological frames $(1,\zeta )$ and $(\eta ,1)$ differ by $1/\zeta$. Normalize their ambient Hermitian form norms; the resulting frames differ by the displayed phase. The unit circle total space is $S^3\subset {\mathbb{C}}^2$ and its projection to the complex projective line is the Hopf fibration. The transition of the dual line is the reciprocal phase.