Spherical Image Analysis of Origami and Kirigami

Analysing origami structures typically consists of computing the dihedral relationships between facets, for which bar-and-hinge models and spatial linkages are widely used. An alternative geometric viewpoint is provided by the spherical image: the map of surface points to the unit sphere via their normal vectors. Although classical in differential geometry, its application in origami has mostly been confined to four-crease vertices, leaving its broader potential largely unexplored. In this study we provide an overview of the mathematical foundations of the spherical image and its application to general polyhedral vertices. We then specialise this approach to the kinematic analysis of origami. We demonstrate how the discrete Gaussian curvature encodes a key geometric constraint independent of the dihedral angles. The spherical image approach is then applied to the analysis of the four-crease and symmetric waterbomb vertices. We extend spherical image analysis to the case of non-rigid origami, including stretching deformations. Finally, we consider the consequences of a cut on the spherical image to apply the approach to kirigami and clarify the definitions of the e-cone and k-cone. The spherical image offers a versatile, and geometrically intuitive, means to analyse the kinematics of origami and kirigami that also provides additional insight into the geometry.