Representing Polyhedra: Faces Are Better than Vertices

In this paper, we investigate the reconstruction of planar-faced polyhedra given their spherical dual representation. We prove that the spherical dual representation is unambiguous for all genus 0 polyhedra and that a genus 0 polyhedron can be uniquely reconstructed in polynomial time. We also prove that when the degree of spherical dual representation is at most four, the representation is unambiguous for polyhedra of any genus.