Flattening single-vertex origami: The non-expansive case

A single-vertex origami is a piece of paper with straight-line rays called creases emanating from a fold vertex placed in its interior or on its boundary. The Single-Vertex Origami Flattening problem asks whether it is always possible to reconfigure the creased paper from any configuration compatible with the metric, to a flat, non-overlapping position, in such a way that the paper is not torn, stretched and, for rigid origami, not bent anywhere except along the given creases.Streinu and Whiteley showed how to reduce the problem to the carpenter's rule problem for spherical polygons. Using spherical expansive motions, they solved the cases of open <π and closed ⩽2π spherical polygons. Here, we solve the case of open polygons with total length between [π,2π), which requires non-expansive motions. Our motion planning algorithm works in a finite number of discrete steps, for which we give precise bounds depending on both the number of links and the angle deficit.