Parametric solutions to the kinematics of developable degree-4 rigid origami vertices

IF 2.9 3区 综合性期刊 Q1 MULTIDISCIPLINARY SCIENCES Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences Pub Date : 2023-11-01 DOI:10.1098/rspa.2023.0319
Yucai Hu, Changjun Zheng, Chuanxing Bi, Haiyi Liang
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Abstract

Developable degree-4 (DD4) vertices have four facets and four creases and can be unfolded flat. The rigid-folding kinematics of DD4 vertices is rich in that it generally has two folding modes and can get stuck when two facets bind together. To study the full spectrum of the kinematics of DD4 vertices, parametric solutions for fold angles in terms of the cotangents of half-angles are derived from the opposite and adjacent fold angle relationships. It is shown that any two fold angles of a general DD4 vertex are related by the equation of a hyperbola. When the vertex has collinear creases or is flat-foldable, the pertinent hyperbola equations degenerate into linear relationships. Meanwhile, when DD4 vertices are classified into three categories according to Grashof’s criterion, both unique and binding folds can be readily located from the facet with the largest or smallest sector angle. The rigid-folding kinematics of typical vertices is then investigated. In addition to the flat state, the two folding modes can also be switched at the binding states if self-intersection is permitted. The results provide new formulae and clear geometric views on the rigid-folding kinematics of DD4 vertices, which are fundamental for constructing larger origami patterns.
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可展次4刚性折纸顶点运动学的参数解
可展开度为4 (DD4)的顶点有四个切面和四个折痕,可以平展开。DD4顶点的刚性折叠运动学是丰富的,因为它通常有两种折叠模式,当两个面结合在一起时可能会卡住。为了研究DD4顶点的全谱运动学,从对角和邻角的对折角关系中导出了半角的等距的对折角的参数解。证明了一般DD4顶点的任意两个夹角是由双曲线方程联系起来的。当顶点有共线折痕或可平折时,相关双曲线方程退化为线性关系。同时,根据Grashof准则将DD4顶点分为三类时,可以很容易地从扇形角最大或最小的面定位到唯一褶皱和绑定褶皱。然后研究了典型顶点的刚性折叠运动学。除了平面状态外,如果允许自交,两种折叠模式也可以在结合状态下进行切换。这些结果为DD4顶点的刚性折叠运动学提供了新的公式和清晰的几何视图,为构造更大的折纸图案奠定了基础。
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来源期刊
CiteScore
6.40
自引率
5.70%
发文量
227
审稿时长
3.0 months
期刊介绍: Proceedings A has an illustrious history of publishing pioneering and influential research articles across the entire range of the physical and mathematical sciences. These have included Maxwell"s electromagnetic theory, the Braggs" first account of X-ray crystallography, Dirac"s relativistic theory of the electron, and Watson and Crick"s detailed description of the structure of DNA.
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