Today I read a paper titled “On Linear Spaces of Polyhedral Meshes”
The abstract is:
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