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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 thek-plane matroid of an incidence graphG=(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 ink-space for the vertices inA. Underlying this result is the simpler depthk matroid of a hypergraphH=(V, E) in which an independent set of edges satisfies |E′|≤|V′| −k for all nonempty subsets.

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