Designing pop-up cards

Giovanni Viglietta

We will show 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, we present a polynomial-time algorithm that, given a simple polygon attached to the two walls of the open pop-up, subdivides it 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°.