  This paper deals with the problem of reconstructing the path of a vehicle in
an unknown environment consisting of planar structures using sound. Many
systems in the literature do this by using a loudspeaker and microphones
mounted on a vehicle. Symmetries in the environment lead to solution
ambiguities for such systems. We propose to resolve this issue by placing the
loudspeaker at a fixed location in the environment rather than on the vehicle.
The question of whether this will remove ambiguities regardless of the
environment geometry leads to a question about breaking symmetries that can be
phrased in purely mathematical terms. We solve this question in the affirmative
if the geometry is in dimension three or bigger, and give counterexamples in
dimension two. Excluding the rare situations where the counterexamples arise,
we also give an affirmative answer in dimension two. Our results lead to a
simple path reconstruction algorithm for a vehicle carrying four microphones
navigating within an environment in which a loudspeaker at a fixed position
emits short bursts of sounds. This algorithm could be combined with other
methods from the literature to construct a path tracking system for vehicles
navigating within a potentially symmetric environment.
