A Matroid on Hypergraphs with Applications in Scene Analysis and Geometry

Following a conjecture of Sugihara, we characterize, combinatorially, the plane pictures of vertices and faces which lift to sharp three-dimensional scenes with plane faces. We also prove two generalizations of Laman's theorem on infinitesimally rigid plane frameworks. Both results are special cases of a representation theorem for the k-plane matroid of an incidence graph G = (A, B; I). The independent sets of incidences are characterized by I I'l-I A'[ + k l B ' I-k for all nonempty subsets, and the incidences are represented by rows of a matrix which uses indeterminate points in k-space for the vertices in A. Underlying this result is the simpler depth k matroid of a hypergraph H = (V, E) in which an independent set of edges satisfies I E'[ <-I V ' t-k for all nonempty subsets.