Analysis of the Rigid Foldability of Origami Patterns Based on Spatial Positions of Creases

Rigid foldability is a special property of rigid origami patterns, where each origami plane remains undeformed during continuous movement along the predetermined crease. Current research on the rigid foldability of origami patterns mainly focuses on kinematics, while less attention is paid to factors that cause deformation of the folding plane. Whether the relative spatial position of adjacent creases has been changed is a critical factor that influences the state (rigid or deformed) of the folding plane between the two adjacent creases during the folding process. This study considered two factors (linear relationship and Euclidean distance) to measure the changes in the spatial positions of creases, explored the relationship between the two factors and rigid folding, and identified deformation forms that affect rigid foldability. First, the origami pattern was regarded as a linkage mechanism, and the linear relationship between creases was determined from the single-vertex origami unit forming this origami structure. Then, the geometric parameters of the origami pattern were used to calculate the Euclidean distance between two points on adjacent creases during the folding process. If the linear relationship and Euclidean distance always remain the same, the origami pattern has rigid foldability. Based on changes in the Euclidean distance, this method can also help determine the main deformation of non-rigidly foldable origami patterns. In addition, it can be applied to origami patterns with four or five vertices and multiple loops, and it further provides a novel approach for determining the layout of the crease position and the judgment of rigid foldability during origami-inspired mechanism design.