Rigidity theory is the study of geometric configurations determined, locally or globally, by specified constraints. A central model is a bar-and-joint framework 
, consisting of a graph
and a placement of its vertices in Euclidean
space, with fixed lengths on its edges (Schulze
and Whiteley 2017, pp. 1593-1594).
Rigidity of a placement means that every sufficiently nearby placement satisfying the same edge-length constraints is congruent to it. Infinitesimal rigidity instead tests the linearized constraints through the rigidity matrix. A generically rigid graph is rigid at generic placements in a fixed dimension, so rigidity of a particular placement must be distinguished from this property of the underlying graph (Asimow and Roth 1978).
Global rigidity requires every equivalent placement in the same dimension, not just nearby placements, to be congruent. Universal rigidity allows equivalent placements in arbitrary higher dimensions as well. An equivalent placement preserves the specified edge lengths, whereas a congruent placement preserves all pairwise distances (Jordán and Whiteley 2017, pp. 1661-1663).
Equilibrium stress supplies a related static formulation. Combinatorial constructions such as vertex splitting are studied for their ability to preserve rigidity under suitable hypotheses. The field also studies broader constraints in structures such as tensegrities (Schulze and Whiteley 2017).
