Polycubes via Dual Loops

We present a complete characterization of polycubes of any genus based on their dual structure: a collection of oriented loops which run in each of the axis directions and capture polycubes via their intersection patterns. A polycube loop structure uniquely corresponds to a polycube. We also describe all combinatorially different ways to add a loop to a loop structure while maintaining its validity; similarly, we show how to identify loops that can be removed from a polycube loop structure without invalidating it. Our characterization gives rise to an iterative algorithm to construct provably valid polycube-maps for a given input surface; polycube-maps are frequently used to solve texture mapping, spline fitting, and hexahedral meshing. We showcase some results of a proof-of-concept implementation of this iterative algorithm.