a given pop-up craft can be opened or closed

Computational paper architecture Unlike other paper art forms, algorithmic solutions for paper architecture have been scarce. Most computational work revolves around creating computer-aided environment for designing pop-up crafts. Glassner introduced a system in [Glassner 2002] where users can interactively design single-slit and V-fold, two basic skills of pop-up card. A similar system is the Popup Workshop for children [Hendrix and Eisenberg 2006]. However, self-intersections may happen in these systems and they have to be resolved by users. 

Mitani and Suzuki [2004a] proposed a CAD system for paper architecture design, which ensures the geometric validity of the output and the foldability of the planar layout by their construction mechanism. In addition, a pop-up condition is proposed that checks whether a layout can be erected when the paper opens. However, this condition is not automatically guaranteed by their system, and hence needs to be resolved by the user in a trial-and-failure manner. Note that, in general, deciding whether a given pop-up craft can be opened or closed is a NP-hard problem [Uehara and Teramoto 2006]. In our work,
we proposed a sufficient condition that the layout can erect in a stable manner as the paper opens, and further provides an automatic algorithm that guarantees the satisfaction of the condition in the output. There are few automated methods for creating pop-up crafts or paper architectures. Hara and Sugihara [2009] considered a 2D version of the pop-up problem (given a polygon as the desired popupped shape) and proposed an automated solution involving polygonal subdivision. However, the method requires gluing multiple paper pieces, which is not allowed in paper architectures. The only previous work we know of that produces paper architecture from 3D models is by Mitani et al. [2003] (in Japanese).
Like our method, their work considers voxel grid to construct the pop-up surfaces. However, their algorithm creates 3D buildings with simple, stair-stepping appearances and lacking guarantees of pop-up or stability. In contrast, we propose a robust algorithm, grounded on geometric formulations of foldability and stability, that produces results closely resembling the input models. Note that stability of architectural models has also been considered by Whiting et al. [2009] in procedural modeling, although the modeling primitives there and hence the stability conditions are very different from ours. Shape abstraction Approximating a 3D model by a paper architecture is a stylistic way of abstracting the shape. Previous work on shape abstraction is mostly based on segmentation of surface patches [Lai et al. 2006; Shamir 2008; Lai et al. 2009] and approximating them with simpler primitives such as quasi-developable or nearly-flat patches [Julius et al. 2005; Yamauchi et al. 2005; Wang 2008; Mehra et al. 2009]. The result of our method can be considered as a special type of abstraction using planar patches in two directions, parallel to either one half of the paper, that have additional physical properties (e.g., being able to pop-up from the plane).