ABSTRACT
Frameworks which have been most successfully studied are those which lie in the plane, such as the mechanisms studied by A. B. Kempe. In the 1970s, Ethan D. Bolker and Henry Crapo developed a criterion for how to rigidify a grid of squares in the plane using diagonal braces along a selection of the square diagonals. A more theoretically tractable notion is infinitesimal rigidity, which forbids not actual motions of the vertices, but infinitesimal ones. Every placement of a complete graph is globally rigid, so every rigid framework can be made globally rigid by adding a sufficient number of braces. The abstract rigidity matroid generalizes the infinitesimal rigidity matroid, which encodes the independence structure of the rigidity transformation. Matroid on the edges of a framework in which the independent sets are those edge sets whose corresponding rows in the rigidity matrix are linearly independent.
