Conditions for Rigid and Flat Foldability of Degree-n Vertices in Origami

Abstract

In rigid origami, the complex folding motion arises from the rotation of strictly rigid faces around crease lines that represent perfect revolute joints. The rigid folding motion of an origami crease pattern is collectively determined by the kinematics of its individual vertices. Establishing a kinematic model and determining the conditions for the rigid foldability of a single vertex is thus important to exploit rigid origami in engineering design tasks. Today, there exists neither an efficient kinematic model to determine the unknown dihedral angles nor an intrinsic condition for the rigid foldability of arbitrarily complex vertices of degree n. In this paper, we present the principle of three units (PTU) that provides an efficient approach to modeling the kinematics of single degree-n vertices. The PTU is based on the notion that the kinematics of a vertex is determined by the behavior of a single underlying spherical triangle. The condition for the existence of this triangle leads to the condition for the rigid and flat foldability of degree-n vertices. These findings are transferred from single vertices to crease patterns, resulting in a simple rule to generate kinematically determinate crease patterns that can be designed to fold rigidly. Finally, we discuss the limitations of the PTU with respect to the global rigid foldability of a crease pattern.

A Computational Design Synthesis Method for the Generation of Rigid Origami Crease Patterns

“…The Principle of Three Units. The PTU [52] predicates on the fact that every single degree-n vertex requires n − 3 inputs and 3 outputs to fold in a kinematically determinable manner [55]. The inputs are the dihedral angles that drive the motion of the vertex, here called the driving angles, which need to be prescribed in order to fold the vertex.…”

“…Independent of the chosen set of driving angles and the degree of the vertex (for n ≥ 4), virtually cutting the vertex at the locations of the unknown dihedral angles reveals three parts [55] called units u 1 , u 2 , and u 3 (Fig. 1(b)) [52]. What remains within these units are the sector and driving angles determined by the user, allowing for the entire kinematic behavior of each individual unit to be expressed analytically.…”

“…( 1), the work in Ref. [52] presents a kinematic model that derives the analytic expression for the unknown dihedral angles by applying the law of spherical cosines. Especially useful about this model is that the RBMs can be handled as an input, allowing for the design space to be enumerated based on the decision whether a vertex should fold up or downward.…”

“…In a recent work [52], whose findings will be briefly explained in Sec. 2, the authors addressed some of the above complexities by presenting the Principle of Three Units (PTU) that leads to an analytical, intrinsic condition for the rigid foldability of single degree-n vertices, where n is the number of crease lines incident to a vertex.…”

“…In addition to this condition, the PTU yields an analytical kinematic model for single vertices to which the RBMs are inputs rather than the results of simulation [53]. The PTU [52] further offers guidelines for the generation of crease patterns whose kinematic motion can be explicitly determined. The condition for the rigid foldability of a single vertex can then be extended to each vertex in the generated crease pattern to achieve global rigid foldability if the underlying crease pattern is acyclic.…”