Geometry and design of popup structures

2026-03-07

AI Summary

This work proposes a programmable design method for three-dimensional pop-up structures that integrates origami and kirigami principles to approximate target curved surfaces. Addressing the inverse problem of deriving cutting-and-folding patterns from desired surface geometries, the study establishes, for the first time, a rigorous discrete differential geometry framework for pop-up structures. By analyzing the geometric configuration of individual units, the authors define their mean and Gaussian curvatures and combine four-bar linkage modeling with parametric design to enable a single cut-fold unit to evolve through multiple morphologies during deployment, including switching curvature signs. Experimental results demonstrate that the method efficiently generates a variety of prescribed surfaces and exhibits promising applications in drag reduction, smart packaging, and architectural facades.

Technology Category

Machine Learning: Structured LearningComputer Vision: 3D Computer VisionSearch and Optimization: Mixed Discrete/Continuous Search

Application Category

Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphsSemantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologiesSystems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

Origami and Kirigami, the famous Japanese art forms of paper folding and cutting, have inspired the design of novel materials&structures utilizing their geometry. In this article, we explore the geometry of the lesser known popup art, which uses the facilities of both origami and kirigami via appropriately positioned folds and cuts. The simplest popup-unit resembles a four-bar mechanism, whose cut-fold pattern can be arranged on a sheet of paper to produce different shapes upon deployment. Each unit has three parameters associated with the length and height of the cut, as well as the width of the fold. We define the mean and Gaussian curvature of the popup structure via the discrete surface connecting the fold vertices and develop a geometric description of the structure. Using these definitions, we arrive at a design pipeline that identifies the cut-fold pattern required to create popup structure of prescribed shape which we test in experiments. By introducing splay to the rectangular unit-cell, a single cut-fold pattern is shown to take multiple shapes along the trajectory of deployment, making possible transitions from negative to positive curvature surfaces in a single structure. We demonstrate application directions for these structures in drag-reduction, packaging, and architectural facades.

Problem

• popup structures
• origami
• kirigami
• geometric design
• curvature control

Innovation

• popup structures
• discrete differential geometry
• curvature design
• cut-fold patterns
• deployable mechanisms