Folding Flat Crease Patterns with Thick Materials

Abstract: Modeling folding surfaces with nonzero thickness is of practical interest for mechanical engineering. There are many existing approaches that account for material thickness in folding applications. We propose a new systematic and broadly applicable algorithm to transform certain flat-foldable crease patterns into new crease patterns with similar folded structure but with a facet-separated folded state. We provide conditions on input crease patterns for the algorithm to produce a thickened crease pattern avoiding local self intersection, and provide bounds for the maximum thickness that the algorithm can produce for a given input. We demonstrate these results in parameterized numerical simulations and physical models.

• The paper presents a novel algorithm that transforms flat-foldable crease patterns into thick-material designs using parallel crease pair offsets.
• The methodology includes crease width assignment, vertex polygon construction, and scale factor determination to prevent self-intersections.
• Numerical simulations and physical prototypes validate the approach, ensuring consistent folding behavior and robust design integration.

Folding Flat Crease Patterns with Thick Materials

Introduction

The paper "Folding Flat Crease Patterns with Thick Materials" (1601.05747) addresses the fundamental challenge in the field of mechanical engineering and computational origami: adapting traditional flat-foldable crease patterns to accommodate materials with nonzero thickness. With applications ranging from robotics to kinetic architecture, the need for techniques that account for material thickness is pressing. This paper proposes a novel algorithm to transform standard crease patterns into ones that support thicker materials, while preventing self-intersection and maintaining desired geometric properties in the folded state.

Algorithm Overview

The authors introduce a method that converts flat-foldable crease patterns into modified patterns, replacing purely angular creases with parallel crease pairs offset from the original. Each pair is strategically placed to prevent local self-intersections by utilizing a parameterized distance proportional to the width assigned to each original crease. The algorithm systematically discards material near vertex intersections, generating facets with increased degrees of freedom. Key steps include defining a crease width based on layer ordering graphs, constructing vertex polygons, refining intersections, and calculating scale factors for implementation.

Crease Width Assignment

The core of the algorithm involves specifying an offset distance between adjacent faces within the crease pattern, thus defining a crease width. The authors leverage a weight assignment to each direction in a layer ordering graph ensuring consistent layer separation across the folded state. Rather than optimizing crease width, which could be NP-hard, the assignment strategically accommodates local topology variations and global geometrical constraints.

Vertex Polygon Construction and Refinement

For each vertex, a polygon is constructed, scaling according to the desired thickness, evolving from trigonometric associations to ensure that adjacent faces avoid intersections. Refinement involves clipping intersecting regions, thereby maintaining structural integrity and manufacturability, particularly for complex vertex angles.

Scale Factor Determination

The paper determines an upper bound for polygon scaling that maintains the non-intersecting nature of the construction. Calculations ensure vertex polygons remain separated, accounting for complex interactions between widened crease edges and polygonal vertices, offering a safe range for material addition.

Adding Thickness and Model Implementation

The algorithm facilitates adding physical thickness to facets, optimizing for uniformity across panels. The authors propose maintaining thickness on widened crease regions while selectively thinning adjacent facets to prevent local self-intersections. They highlight potential for full range folding motion alongside the construction of robust physical models such as foam-core prototypes, affirming alignment with numerical simulations and theoretical predictions.

Numerical and Physical Validation

The paper showcases successful adaptation of rigid-foldable origami models, employing simulation tools like Freeform Origami and Mathematica. The models validate folding behavior predictions, demonstrating reassuringly consistent motions across states, both folded and unfolded. The underlying conjecture that single-vertex patterns possess spherical configuration space is supported by numerical evidence and preliminary experimentation.

Conclusion

This work presents a comprehensive approach to thickening crease patterns without sacrificing the intricate geometry and motion required for advanced engineering designs. The proposed method stands out against existing techniques by enabling planar facet construction, parallel alignment during fold, and accommodating a substantial range of panel thicknesses. While further exploration is necessary to assess the algorithm’s applicability to non-flat foldings, the theoretical groundwork laid here holds promise for expanding the boundaries of what is possible in origami-based engineering structures.