Rigidity matrix

The d-dimensional rigidity matrix of a framework (G,p) is a matrix $R(G,p) \in \mathbb{R}^{|E| \times d|V|}$ defined as follows. Each row represents an edge while the columns are grouped into sets of d columns, each group representing a node. For an edge $uv \in E$, the d entries corresponding to a node u (respectively the node v) are the d different coordinates of the vector p(u)-p(v) (respectively p(v)-p(u)). The rest of the entries are zeroes.

The d-dimensional rigidity matrix of a graph G is the rigidity matrix of a generic realization of G in d dimensions. Equivalently, it is the matrix of indeterminates obtained by replacing every distinct value in the previous matrix by different indeterminates.

Rigidity matroid

The d-dimensional rigidity matroid of a framework is the matroid defined by the rows of the rigidity matrix of the framework. The d-dimensional rigidity matroid of a graph G is the rigidity matroid of a generic realization of G. It can be seen that all generic realizations give the same matroid.

The two-dimensional rigidity matroid of a graph is always a count matroid. However, it is an open problem to characterize the rank function of the 3-dimensional rigidity matroid.