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arXiv · 2408.08816

Lorentz transformation for the kinematics of degree-4 rigid origami vertices and compatibility of rigid-foldable polygons

Yucai Hu·Licheng Lin·Changjun Zheng·Chuanxing Bi
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Abstract

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.

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Yucai Hu, Licheng Lin, Changjun Zheng, Chuanxing Bi. 2024-08-16. Lorentz transformation for the kinematics of degree-4 rigid origami vertices and compatibility of rigid-foldable polygons. https://doi.org/10.1016/j.ijsolstr.2025.113698

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