Polycubes via Dual Loops

TL;DR

This paper introduces a complete characterization of polycubes using dual loop structures, enabling systematic construction and analysis of polycubes of any genus based on their intersection patterns.

Contribution

It presents a novel dual loop framework that uniquely represents polycubes and provides algorithms for their construction and modification.

Findings

Characterization of polycubes via dual oriented loops

Algorithms for adding and removing loops while maintaining validity

Iterative method for constructing polycube maps from surfaces

Abstract

In this paper we study polycubes: orthogonal polyhedra with axis-aligned quadrilateral faces. 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.