  Ultra sensitive ring laser gyroscopes are regarded as potential detectors of
the general relativistic frame-dragging effect due to the rotation of the
Earth: the project name is GINGER (Gyroscopes IN GEneral Relativity), a
ground-based triaxial array of ring lasers aiming at measuring the Earth
rotation rate with an accuracy of 10^-14 rad/s. Such ambitious goal is now
within reach as large area ring lasers are very close to the necessary
sensitivity and stability. However, demanding constraints on the geometrical
stability of the laser optical path inside the ring cavity are required. Thus
we have started a detailed study of the geometry of an optical cavity, in order
to find a control strategy for its geometry which could meet the specifications
of the GINGER project. As the cavity perimeter has a stationary point for the
square configuration, we identify a set of transformations on the mirror
positions which allows us to adjust the laser beam steering to the shape of a
square. We show that the geometrical stability of a square cavity strongly
increases by implementing a suitable system to measure the mirror distances,
and that the geometry stabilization can be achieved by measuring the absolute
lengths of the two diagonals and the perimeter of the ring.
