e present sufficient conditions for a class of 3D surfaces

Paper architectures, also called origamic architectures, are paper buildings created by folding combined with paper cutting. Originated in Japan by Masahiro Chatani [1987] in the 1980’s, the craft has been popularized by artists around the world, in particular Bianchini, Siliakus and Aysta [2009].
Paper architecture appears in many forms, such as greeting cards and desktop decorations, and
We present a formal, geometric formulation of planar layouts that can rigidly and stably pops-up to a paper architecture (Section 3). • We present sufficient conditions for a class of 3D surfaces, consisting of patches oriented in either one of two directions, to be realizable by popping-up a planar layout in a rigid and stable way (Section 4). • We present an automatic algorithm that generates paper architectures and their planar layouts that approximate any given 3D models with guaranteed realizability. (Section 5).

can be “startling realistic” [Chatani et al. 1987]. Some examples created by artists are shown in Figure 2. Further exhibits could be found from the online galleries of Ingrid Siliakus and Gerry Stormer. A paper architecture is made from cutting and folding from a single piece of paper, and is stored by folding the two halves of the paper close. As the paper is opened, the 3D building “stands-up” or “pops-up”. While similar to pop-up books, a paper architecture is made with no gluing or splicing, which puts additional constraints to the design of cut and fold patterns on the paper (called a planar layout). What is even more challenging is to create layouts that would pop-up into a desired 3D look. Numerous books exist on the mechanism of designing pop-up crafts [Birmingham 1997; Carter 1999; Cheong et al. 2009], and a number of computer-aided tools have been developed to provide virtual design environments [Lee et al. 1996; Glassner 2002; Hendrix and Eisenberg 2006; Mitani and Suzuki 2004a]. However, the user is ultimately responsible for deciding where and how the cuts and folds should be placed on the 2D paper, and it remains a labor-intensive and highly skilldemanding task to generate 2D layouts that pop-up into realistically looking 3D buildings. In this paper, we develop a completely automatic algorithm that produces paper architectures approximating user-given 3D models, which enables novice users to create realistic and complex crafts in an effortless way (see the example on Figure 1 right). Our algorithm is grounded on novel geometric formulations of planar layouts that can physically pop-up  cards to paper architectures. In particular, regions in the layout should maintain rigid and non-intersecting when popping-up, and the architecture should be able to stably erect with no additional help from the user other than holding the two halves of the paper. Based on the formulation, we present suf- ficient conditions for a class of 3D surfaces, consisting of planar patches oriented in two directions, to be physically realizable by popping-up a planar layout. Guided by the conditions, we design a grid-based algorithm that produces 3D realizable paper architectures automatically from any input model given by the user, while requiring only the users to specify the paper location with respect to the model. An example is shown in figure 1. Contributions To the best of our knowledge, our algorithm is one of the first automated methods for creating paper architecture that mimics a given 3D input. To achieve this goal, we make the following contributions: (a) Amsterdam Central Station (b) Statue of Liberty (c) Bellesguard Tower (d) Himeji Castle