{"title":"可展次4刚性折纸顶点运动学的参数解","authors":"Yucai Hu, Changjun Zheng, Chuanxing Bi, Haiyi Liang","doi":"10.1098/rspa.2023.0319","DOIUrl":null,"url":null,"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.","PeriodicalId":20716,"journal":{"name":"Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences","volume":"70 4","pages":"0"},"PeriodicalIF":2.9000,"publicationDate":"2023-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Parametric solutions to the kinematics of developable degree-4 rigid origami vertices\",\"authors\":\"Yucai Hu, Changjun Zheng, Chuanxing Bi, Haiyi Liang\",\"doi\":\"10.1098/rspa.2023.0319\",\"DOIUrl\":null,\"url\":null,\"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.\",\"PeriodicalId\":20716,\"journal\":{\"name\":\"Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences\",\"volume\":\"70 4\",\"pages\":\"0\"},\"PeriodicalIF\":2.9000,\"publicationDate\":\"2023-11-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1098/rspa.2023.0319\",\"RegionNum\":3,\"RegionCategory\":\"综合性期刊\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MULTIDISCIPLINARY SCIENCES\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1098/rspa.2023.0319","RegionNum":3,"RegionCategory":"综合性期刊","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MULTIDISCIPLINARY SCIENCES","Score":null,"Total":0}
Parametric solutions to the kinematics of developable degree-4 rigid origami vertices
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.
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.