Related Work Glassner described a stable analytical solution for the location

Glassner described a stable analytical solution for the location of important points in the three main pop-up techniques (single-slit, asymmetric single-slit, and V-fold mechanisms) when the Quilling cards folds and unfolds in interactive pop-up card design [4] [5] [6]. He also implemented his solution in a small design program. However, he did not describe the interactive behavior of the program in detail, nor did he report any user experience. Mitani and Suzuki proposed a method for creating a 180◦ flat fold Origamic Architecture with lattice-type cross sections [7]. This system creates pieces from a 3D model. Those pieces can be used in our system because the base mechanism is same as the angle fold open box mechanism described in the following section. However, a 3D model is not always available. Furthermore, the structure is set only on the center fold line and cannot be combined with other pieces. Lee et al. described calculations and geometric constraints for modeling and simulating multiple V-fold pieces [8]. However, that system is limited to V-fold mechanisms and is not designed as an interactive system. Several interactive interfaces have been proposed for 90◦ pop-up cards. Mitani et al. proposed a method to design Origamic Architecture [9] models with a computer using a voxel model representation [10] [11].

Using this method, the system can store 90◦ pop-up card models and display them using computer graphics. User operations are the interactive addition and deletion of voxels. This system was used for graphics science education [12]. Mitani et al. also proposed a method for designing Origamic Architecture models with a computer using a model based on a set of planar polygons [13]. That system computes and imposes constraints to guarantee that the model can be constructed with a single sheet of paper. Thus, it enables the user to make more complex 90◦ pop-up cards interactively from the beginning through to the pattern printing stage. Hendrix and Eisenberg proposed a pop-up workshop system [14] [15] that enables the user to design pop-up cards by making two-dimensional (2D) cuts. The result in 3D can be opened and closed on the Viewer window. They also showed that children could design pop-up cards using their system [16]. However, their interfaces are not available for 180◦ pop-up cards. 90◦ pop-up cards are made with single sheet of paper and their system works well given this constraint. However, it is difficult to design 180◦ pop-up cards using a voxel model or planar polygon model. Moreover, it is difficult for people to edit 3D structures in 2D because they can not imagine the resulting 3D shape. 3 Assisted Pop-up Card Design