Polycubes via Dual Loops

2024-10-22

arXiv.org

AI Summary

This work addresses the complete characterization and constructive generation of axis-aligned polycubes, aiming for exact, verifiable modeling from given quadrilateral mesh surfaces to valid polycubes. We introduce a novel representation framework based on tri-directional oriented loop dual structures: each polycube corresponds uniquely to a nested set of closed loops aligned with the x-, y-, and z-axes, whose overlapping patterns fully encode both topology (of arbitrary genus) and geometry. We establish, for the first time, a bijective correspondence between polycubes and their dual loop systems, and provide combinatorial validity criteria for loop insertion and deletion, enabling verifiable iterative construction. Integrating dual graph theory, combinatorial topology, and axis-aligned geometric modeling, we design loop encoding and validation algorithms that automatically map input quad meshes to topologically and geometrically valid polycubes, with theoretical guarantees of correctness and completeness.

Technology Category

• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Computer Vision: 3D Computer Vision
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Web Mining and Content Analysis: Models for Web evolution

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.

Problem

• Multicubes Representation
• Quad Surface
• Model Construction

Innovation

• Cubic Structures
• Loop Analysis
• Modeling Algorithm