Elastica surface generation of curved-crease origami

Curved-crease origami are studied for many novel applications across engineering and architecture, as they are developable but possess a non-zero principal curvature and a corresponding energy storage capability when folded. However, geometric modelling techniques are limited, with most methods requiring numerical discretisation of a target curved surface to allow developability constraints to be enforced at vertices. The discretised surface can approximate a physical surface through relaxation for minimum bending energy, however such methods are cumbersome and their accuracy is largely unknown. This paper presents an analytical geometric construction method for curved-crease origami that avoids the need for surface discretisation. The new method combines a 1D elastica solution for large elastic bending deformation with a straight-crease origami projection and reflection process; it can thus concisely and accurately capture the principal surface curvature and developability characteristics of elastically-bent curved-crease origami. A surface error analysis of 3D scanned physical prototypes is used to validate the model, which is shown to be accurate to within  ± 50% of the sheet thickness for a 2 mm thick model for a range of elastica surface profiles. Limitations of the model are also explored including the derivation of a maximum compressibility limit; investigation of accuracy of numerical folding motion simulation; and an investigation of a free edge distortion behaviour which occurs in certain origami forms.