  Assume that a ground-based vehicle moves in a room with walls or other planar
surfaces. Can the vehicle reconstruct the positions of the walls from the
echoes of a single sound event? We assume that the vehicle carries some
microphones and that a loudspeaker is either also mounted on the vehicle or
placed at a fixed location in the room. We prove that the reconstruction is
almost always possible if (1) no echoes are received from floors, ceilings or
sloping walls and the vehicle carries at least three non-collinear microphones,
or if (2) walls of any inclination may occur, the loudspeaker is fixed in the
room and there are four non-coplanar microphones.
  The difficulty lies in the echo-matching problem: how to determine which
echoes come from the same wall. We solve this by using a Cayley-Menger
determinant. Our proofs use methods from computational commutative algebra.

Commutative Algebra;Signal Processing;Metric Geometry