Matlab code fig 10 Movie S5 & S6 from Design of rigid-foldable doubly curved origami tessellations based on trapezoidal crease patterns

This paper presents a mathematical framework for the design of rigid-foldable doubly curved origami tessellations based on trapezoidal crease patterns that can simultaneously fit two target surfaces with rotational symmetry about a common axis. The geometric parameters of the crease pattern and the folding angles of the target folded state are determined through a set of combined geometric and constraint equations. An algorithm to simulate the folding motion of the designed crease pattern is provided. Furthermore, the conditions and procedures to design folded ring structures that are both developable and flat-foldable and stacked folded structures consisting of two layers that can fold independently or compatibly are discussed. The proposed framework has potential applications in designing engineering doubly curved structures such as deployable domes and folded cores for doubly curved sandwich structures on the aircraft.

本论文提出了一种基于梯形折痕图案（trapezoidal crease patterns）的刚性可折叠（rigid-foldable）双曲面折纸镶嵌（doubly curved origami tessellations）结构的数学设计框架，该框架可同时适配两个绕公共轴具备旋转对称性的目标曲面。该框架通过一组联合几何方程与约束方程（constraint equations），确定折痕图案的几何参数与目标折叠状态的折叠角。同时提供了用于模拟所设计折痕图案折叠运动的算法。此外，本文还探讨了兼具可展性（developable）与可平坦折叠性（flat-foldable）的折叠环结构，以及由两层可独立折叠或协同折叠的结构组成的堆叠折叠结构的设计条件与流程。所提出的框架在工程双曲面结构设计中具备潜在应用价值，例如可展开穹顶（deployable domes），以及航空器用双曲面夹层结构的折叠芯层。