A matroid on hypergraphs, with applications in scene analysis and geometry

Abstract

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′|≤|A′|+k |B′| -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 |E′|≤|V′| -k for all nonempty subsets. © 1989 Springer-Verlag New York Inc.