Abstract
Swarm and constellation reconfiguration can be viewed as motion of an unordered point configuration in an ambient space. Here, we provide persistence-stable, symmetry-invariant geometric representations for comparing and monitoring multi-agent configuration data. We introduce a quotient formation space \mathcal{S}_n(M,G)=M^n/(G\times S_n) and a formation matching metric d_{M,G} obtained by optimizing a worst-case assignment error over ambient symmetries g\in G and relabelings \sigma\in S_n. This metric is a structured, physically interpretable relaxation of Gromov--Hausdorff distance: the induced inter-agent metric spaces satisfy d_{\mathrm{GH}}(X_x,X_y)\le d_{M,G}([x],[y]). Composing this bound with stability of Vietoris--Rips persistence yields d_B(\Phi_k([x]),\Phi_k([y]))\le d_{M,G}([x],[y]), providing persistence-stable signatures for reconfiguration monitoring. We analyze the metric geometry of (\mathcal{S}_n(M,G),d_{M,G}): under compactness/completeness assumptions on M and compact G it is compact/complete and the metric induces the quotient topology; if M is geodesic then the quotient is geodesic and exhibits stratified singularities along collision and symmetry strata, relating it to classical configuration spaces. We study expressivity of the signatures, identifying symmetry-mismatch and persistence-compression mechanisms for non-injectivity. Finally, in a phase-circle model we prove a conditional inverse theorem: under semicircle support and a gap-labeling margin, the H_0 signature is locally bi-Lipschitz to d_{M,G} up to an explicit factor, yielding two-sided control. Examples on \mathbb{S}^2 and \mathbb{T}^m illustrate satellite-constellation and formation settings.
