  Polyhedral meshes (PM) - meshes having planar faces - have enjoyed a rise in
popularity in recent years due to their importance in architectural and
industrial design. However, they are also notoriously difficult to generate and
manipulate. Previous methods start with a smooth surface and then apply
elaborate meshing schemes to create polyhedral meshes approximating the
surface. In this paper, we describe a reverse approach: given the topology of a
mesh, we explore the space of possible planar meshes with that topology.
  Our approach is based on a complete characterization of the maximal linear
spaces of polyhedral meshes contained in the curved manifold of polyhedral
meshes with a given topology. We show that these linear spaces can be described
as nullspaces of differential operators, much like harmonic functions are
nullspaces of the Laplacian operator. An analysis of this operator provides
tools for global and local design of a polyhedral mesh, which fully expose the
geometric possibilities and limitations of the given topology.
