A swarm of n agents in space defines a point configuration. But two configurations that differ only by a rotation, translation, or relabeling of agents are the same formation. To compare formations meaningfully, you need a metric on the quotient space — the space of configurations modulo symmetries and permutations.
This paper constructs that metric. The formation matching metric optimizes a worst-case assignment error over all ambient symmetries and all relabelings simultaneously. The result is a structured relaxation of Gromov-Hausdorff distance: it inherits the physical interpretability of geometric distance while respecting the combinatorial structure of agent interchangeability.
The key theorem is stability. Persistent homology — the topological signature computed from inter-agent distances — is Lipschitz-continuous with respect to the formation metric. Small changes in formation produce small changes in the persistence diagram. This means the topological fingerprint of a swarm configuration is a reliable monitoring tool: if the formation changes enough to matter, the persistence diagram detects it; if the persistence diagram changes, the formation genuinely changed.
The quotient space has rich geometry. Under natural conditions, it is geodesic (shortest paths exist) but has stratified singularities — lower-dimensional strata where multiple equivalence classes collide. These singularities correspond to formations with enhanced symmetry, where the quotient map fails to be smooth. The geometry of the monitoring space reflects the geometry of the formations it tracks.