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This paper solves the continuous classification problem for finite clouds of
unlabelled points under Euclidean isometry. The Lipschitz continuity of
required invariants in a suitable metric under perturbations of points is
motivated by the inevitable noise in measurements of real objects.

The best solved case of this isometry classification is known as the SSS
theorem in school geometry saying that any triangle up to congruence (isometry
in the plane) has a continuous complete invariant of three side lengths.

However, there is no easy extension of the SSS theorem even to four points in
the plane partially due to a 4-parameter family of 4-point clouds that have the
same six pairwise distances. The computational time of most past metrics that
are invariant under isometry was exponential in the size of the input. The
final obstacle was the discontinuity of previous invariants at singular
configurations, for example, when a triangle degenerates to a straight line.

All the challenges above are now resolved by the Simplexwise Centred
Distributions that combine inter-point distances of a given cloud with the new
strength of a simplex that finally guarantees the Lipschitz continuity. The
computational times of new invariants and metrics are polynomial in the number
of points for a fixed Euclidean dimension.