Peraza Hernandez E A, Hartl D J, Lagoudas D C
Department of Aerospace Engineering, Texas A&M University, College Station, TX 77843, USA.
Department of Materials Science and Engineering, Texas A&M University, College Station, TX 77843, USA.
Proc Math Phys Eng Sci. 2017 Apr;473(2200):20160716. doi: 10.1098/rspa.2016.0716. Epub 2017 Apr 26.
Origami has enabled new approaches to the fabrication and functionality of multiple structures. Current methods for origami design are restricted to the idealization of folds as creases of zeroth-order geometric continuity. Such an idealization is not proper for origami structures of non-negligible fold thickness or maximum curvature at the folds restricted by material limitations. For such structures, folds are not properly represented as creases but rather as bent regions of higher-order geometric continuity. Such fold regions of arbitrary order of continuity are termed as . This paper presents a method for solving the following origami design problem: given a goal shape represented as a polygonal mesh (termed as the ), find the geometry of a single planar sheet, its pattern of smooth folds, and the history of folding motion allowing the sheet to approximate the goal mesh. The parametrization of the planar sheet and the constraints that allow for a valid pattern of smooth folds are presented. The method is tested against various goal meshes having diverse geometries. The results show that every determined sheet approximates its corresponding goal mesh in a known folded configuration having fold angles obtained from the geometry of the goal mesh.
折纸技术为多种结构的制造和功能实现带来了新方法。当前的折纸设计方法局限于将折痕理想化地视为零阶几何连续性的褶皱。对于那些折痕厚度不可忽略或受材料限制在折痕处有最大曲率的折纸结构而言，这种理想化并不恰当。对于此类结构，折痕不能恰当地表示为褶皱，而应表示为具有高阶几何连续性的弯曲区域。这种具有任意连续性阶数的折痕区域被称为 。本文提出一种方法来解决以下折纸设计问题：给定一个表示为多边形网格（称为 ）的目标形状，找到单个平面片材的几何形状、其平滑折痕的图案以及允许该片材逼近目标网格的折叠运动过程。文中给出了平面片材的参数化以及允许有效平滑折痕图案的约束条件。该方法针对具有不同几何形状的各种目标网格进行了测试。结果表明，每个确定的片材在已知的折叠构型中逼近其相应的目标网格，该折叠构型具有从目标网格几何形状得出的折叠角度。
