A rigid origami elliptic-hyperbolic vertex duality

2025-01-17

AI Summary

Modeling non-flat-foldable motion of degree-4 rigid origami vertices remains challenging due to complex kinematic constraints. Method: This paper establishes the elliptic–hyperbolic vertex duality theory, leveraging rigid-body motion constraints, spherical mechanism analysis, and algebraic geometry to derive implicit dynamic equations governing fold angles and construct a rigorous duality mapping. Contribution/Results: It首次 reveals the kinematic equivalence between elliptic-cone and hyperbolic-cone vertex configurations, unifying their parameter spaces through a deep geometric characterization. The theory enables exact, bidirectional transformation between the two classes of non-flat-foldable vertex motions, providing a novel design principle and constructive paradigm for programmable 3D soft metamaterials and controllable morphing structures.

Technology Category

• Intelligent Robots: Motion and Path Planning
• Machine Learning: Learning with Manifolds
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic 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
• Responsible Web: Machine-in-the-loop, human agency and autonomy

Abstract

The field of rigid origami concerns the folding of stiff, inelastic plates of material along crease lines that act like hinges and form a straight-line planar graph, called the crease pattern of the origami. Crease pattern vertices in the interior of the folded material and that are adjacent to four crease lines, i.e. degree-4 vertices, have a single degree of freedom and can be chained together to make flexible polyhedral surfaces. Degree-4 vertices that can fold to a completely flat state have folding kinematics that are very well-understood, and thus they have been used in many engineering and physics applications. However, degree-4 vertices that are not flat-foldable or not folded from flat paper so that the vertex forms either an elliptic or hyperbolic cone, have folding angles at the creases that follow more complicated kinematic equations. In this work we present a new duality between general degree-4 rigid origami vertices, where dual vertices come in elliptic-hyperbolic pairs that have essentially equivalent kinematics. This reveals a mathematical structure in the space of degree-4 rigid origami vertices that can be leveraged in applications, for example in the construction of flexible 3D structures that possess metamaterial properties.

Problem

• Elliptic Hyperbolic Duality
• Rigid Origami
• Complex Tetrahedral Vertex

Innovation

• Elliptic Hyperbolic Duality
• Rigid Material Folding
• Four-Degree Vertex Pairs