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

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.

References in corpus (5)

• Programming Curvature using Origami Tessellations
• Inverse design of deployable origami structures that approximate a general surface
• The designs and deformations of rigidly and flat-foldable quadrilateral mesh origami
• Real and complexified configuration spaces for spherical 4-bar linkages
• Lagrangian approach to origami vertex analysis: Kinematics

Cited by in corpus (1)

• Quasi-symmetric nets: A constructive approach to the equimodular elliptic type of Kokotsakis polyhedra