On rigid origami I: Piecewise-planar paper with straight-line creases

Origami (paper folding) is an effective tool for transforming two-dimensional materials into threedimensional structures, and has been widely applied to robots, deployable structures, metamaterials, etc. Rigid origami is an important branch of origami where the facets are rigid, focusing on the kinematics of a panel-hinge model. Here we develop a theoretical framework for rigid origami, and show how this framework can be used to connect rigid origami and its cognate areas, such as the rigidity theory, graph theory, linkage folding and computer science. First, we give definitions regarding fundamental aspects of rigid origami, then focus on how to describe the configuration space of a creased paper. The shape and 0-connectedness of the configuration space are analyzed using algebraic, geometric and numeric methods. In the algebraic part we study the tangent space and generic rigid-foldability based on the polynomial nature of constraints for a panelhinge system. In the geometric part we analyze corresponding spherical linkage folding and discuss the special case when there is no cycle in the interior of a crease pattern. In the numeric part we review methods to trace folding motion and avoid self-intersection. Our results will be instructive for the mathematical and engineering design of origami structures