Representing polyhedra: faces are better than vertices

This paper investigates the reconstruction of planar-faced polyhedra given their spherical dual representation. The spherical dual representation for any genus 0 polyhedron is shown to be unambiguous and to be uniquely reconstructible in polynomial time. It is also shown that when the degree of the spherical dual representation is at most four, the representation is unambiguous for polyhedra of any genus. The first result extends, in the case of planar-faced polyhedra, the well known result that a vertex or face connectivity graph represents a polyhedron unambiguously when the graph is triconnected and planar. The second result shows that when each face of a polyhedron of arbitrary genus has at most four edges, the polyhedron can be reconstructed uniquely. This extends the previous result that a polyhedron can be uniquely reconstructed when each face of the polyhedron is triangular. As a consequence of this result, faces are a more powerful representation than vertices for polyhedra whose faces have three or four edges. A result of the reconstruction algorithm is that high level features of the polyhedron are naturally extracted. Both results explicitly use the fact that the faces of the polyhedron are planar. It is conjectured that the spherical dual representation is unambiguous for polyhedra of any genus.