  We consider Gaussian quantum circuits that alternate unitary gates and
post-selected weak measurements, with spatial translation symmetry and time
periodicity. We show analytically that such models can host different kinds of
measurement-induced phase transitions detected by entanglement entropy, by
mapping the time evolution and weak measurements to M\"obius transformations.
We demonstrate the existence of a log-law to area-law transition, as well as a
volume-law to area-law transition. For the latter, we compute the critical
exponent $\nu$ for the Hartley, von Neumann and R\'enyi entropies exactly.
