  Rigid structures such as cars or any other solid objects are often
represented by finite clouds of unlabeled points. The most natural equivalence
on these point clouds is rigid motion or isometry maintaining all inter-point
distances. Rigid patterns of point clouds can be reliably compared only by
complete isometry invariants that can also be called equivariant descriptors
without false negatives (isometric clouds having different descriptions) and
without false positives (non-isometric clouds with the same description). Noise
and motion in data motivate a search for invariants that are continuous under
perturbations of points in a suitable metric. We propose the first continuous
and complete invariant of unlabeled clouds in any Euclidean space. For a fixed
dimension, the new metric for this invariant is computable in a polynomial time
in the number of points.
