Discrete PolyCube Surface Flows

Discrete flows have been of particular interest to researchers in discrete differential geometry, computational geometry, and computer graphics, due to their connection to the Plateau problem in mathematics, mechanical behavior of various shells and membranes, and their application to shape sciene problems such as surface fairing, parameterization, collision modeling, registration, and interpolation. We study a new flow, formulated as a variational problem on certain bundles of quotient spaces of rotations over the surface, and describe algorithms to discretize and simulate it. The flow has the following property: stationary surfaces of the flow are polycubes, polyhedra whose facets meet only at right angles. Spheres flow to cubes, and more generally, surface features sharpen or flatten. Moreover the flow is intrinsic to the surface itself, and does not require a choice of preferred planes or directions in the ambient space. This flow has potential applications in crystallograpy, as well as simplification, discretization, and hexahedralization of shapes.