Folded developables

Abstract

A plane, inextensible sheet may be folded or creased along a curved line to produce two connected but distinct developable surfaces. Various theorems applying to this folding process are identified and two special cases investigated. In one, the fold line remains a plane curve during deformation and in the other the dihedral angle at the fold is constant along the curve. Curved-line folding occurs naturally in the collapse of thin-sheet-metal structures composed of developable surfaces. The theorems presented identify the kinematic constraint existing between pairs of developable surfaces connected by curved-line folds and permit the design of sheet-metal products that use these surfaces. This expands considerably the range of engineering products that can be made by folding and bending a single inextensible sheet.