Algorithms for Designing Pop-Up Cards

We prove that every simple polygon can be made as a (2D) pop-up card/book that opens to any desired angle between 0 and 360◦. More precisely, given a simple polygon attached to the two walls of the open pop-up, our polynomial-time algorithm subdivides the polygon into a single-degree-of-freedom linkage structure, such that closing the pop-up flattens the linkage without collision. This result solves an open problem of Hara and Sugihara from 2009. We also show how to obtain a more efficient construction for the special case of orthogonal polygons, and how to make 3D orthogonal polyhedra, from pop-ups that open to 90◦, 180◦, 270◦, or 360◦.