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°.
