Lorentz transformation for the kinematics of degree-4 rigid origami vertices and compatibility of rigid-foldable polygons
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866911441482678272 |
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| author | Hu, Yucai Lin, Licheng Zheng, Changjun Bi, Chuanxing |
| author_facet | Hu, Yucai Lin, Licheng Zheng, Changjun Bi, Chuanxing |
| contents | We offer new insight into the folding kinematics of degree-4 rigid origami vertices by drawing an analogy to spacetime in special relativity. Specifically, folded states of the vertex, described by pairs of fold angles in terms of cotangent of half-angles, are related through Lorentz transformations in $1+1$ dimensions. Linear ordinary differential equations are derived for the tangent vectors on two-dimensional fold-angle planes, with the coefficient matrix depending exclusively on the sector angles. By taking the limit to the flat state, we generalize the fold-angle multipliers previously defined for flat-foldable vertices to general and collinear developable degree-4 vertices, and obtain a compatibility theorem on the rigid-foldability of polygons with $n$ developable degree-4 vertices. We further explore the rigid-foldable polygons of equimodular type and compose tangent vectors involving fold angles at the creases of the central polygon. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_08816 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Lorentz transformation for the kinematics of degree-4 rigid origami vertices and compatibility of rigid-foldable polygons Hu, Yucai Lin, Licheng Zheng, Changjun Bi, Chuanxing Soft Condensed Matter We offer new insight into the folding kinematics of degree-4 rigid origami vertices by drawing an analogy to spacetime in special relativity. Specifically, folded states of the vertex, described by pairs of fold angles in terms of cotangent of half-angles, are related through Lorentz transformations in $1+1$ dimensions. Linear ordinary differential equations are derived for the tangent vectors on two-dimensional fold-angle planes, with the coefficient matrix depending exclusively on the sector angles. By taking the limit to the flat state, we generalize the fold-angle multipliers previously defined for flat-foldable vertices to general and collinear developable degree-4 vertices, and obtain a compatibility theorem on the rigid-foldability of polygons with $n$ developable degree-4 vertices. We further explore the rigid-foldable polygons of equimodular type and compose tangent vectors involving fold angles at the creases of the central polygon. |
| title | Lorentz transformation for the kinematics of degree-4 rigid origami vertices and compatibility of rigid-foldable polygons |
| topic | Soft Condensed Matter |
| url | https://arxiv.org/abs/2408.08816 |