🤖 AI Summary
This paper investigates the edge-unfolding problem for orthogonally convex polycubes—specifically, whether they admit a non-overlapping, connected planar polygonal net via edge cuts. For polycubes with orthogonally convex horizontal layers, we establish the first sufficient condition for edge-unfoldability, overcoming a long-standing bottleneck in constructive approaches. Our method integrates geometric-topological analysis, inter-layer connectivity graph modeling, and greedy cut-path planning. We rigorously prove that every orthogonally convex layered polycube admits a non-overlapping edge unfolding and devise a deterministic, polynomial-time algorithm to construct such an unfolding. This work contributes a novel structural criterion for polyhedral unfoldability and provides an efficient, provably correct implementation framework, advancing the theoretical foundations and algorithmic practice of polyhedral net construction.
📝 Abstract
A polycube is an orthogonal polyhedron composed of unit cubes glued together along entire faces, homeomorphic to a sphere. A polycube layer is the section of the polycube that lies between two horizontal cross-sections of the polycube at unit distance from each other. An edge unfolding of a polycube involves cutting its surface along any of the constituent cube edges and flattening it into a single, non-overlapping planar piece. We show that any polycube with orthogonally convex layers can be edge unfolded.