As a result, f is a valid pop-up transform from T via Sv,w, and Sv,w is popup-foldable

As a result, f is a valid pop-up transform from T via Sv,w, and Sv,w is popup-foldable. Note that the conditions in Proposition 1 can be checked computationally given a parallel PLS. In our algorithm, we will construct a specific class of parallel PLS that satisfies these conditions naturally by the construction process. Making one step further, we have the following sufficient condition for a parallel PLS that can be rigidly and stably popped-up: Proposition 2 A parallel PLS Sv,w satisfying Proposition 1 is popup-realizable if it further satisfies the following condition: there exists an ordering of patches in Sv,w = {p1, . . . , pn} such that p1, p2 are the backdrop and ground respectively, and for any k ∈ [3, n], either: 1. pk is connected to two parallel, non-coplanar patches pi, pj where i, j < k, or 2. pk is connected to pk+1, and pk, pk+1 are respectively connected to some pi, pj where i, j < k and pi, pk+1 are noncoplanar. Proof:

We will first show that a parallel PLS S meeting the two conditions above is stable with respect to p1, p2. We start with two observations. First, a rigid planar patch will be fixed if two non-colinear boundary edges are fixed. Second, consider two rigid planar patches pa, pb sharing a common edge e. Then pa is fixed if there is an edge ea of pa and an edge eb of pb such that e, ea are non-colinear, and ea, eb are fixed. With these observations, it is easy to show that the patches of Sv,w are stable by induction. By Proposition 1, Sv,w can be folded from a PLPA T by the popuptransform f defined as in Equation 2. Note that the two conditions here still hold during the transform, which preserves the connectivity and parallel relations among patches. Therefore, T is a realizable PLPA and Sv,w is popup-realizable. For example, the parallel PLS in Figure 4 (a) is popup-realizable according to the above conditions, using the ordering of patches marked in the figure. In contrast, no such ordering can be found on the PLS in Figure 4 (b). 5 The algorithm The input of our method is a 3D model with user assigned backdrop and ground plane locations. The output of the algorithm is a paper architecture represented as a popup-realizable, parallel PLS that approximates the input model. In this paper, we focus on parallel PLS with 90◦ fold angles, as such PLS encompasses a large range of models including urban architectures, though the algorithm can be easily adapted to other fold angles. To best capture the model geometry while fulfilling the conditions stated in the previous section, we proceed in three steps (illustrated in Figure 5): 1. Popup-foldable surface: First, we compute an initial, parallel PLS approximating the input model and consisting of orthogonal faces on a Cartesian grid. This initial PLS is referred to as the visible PLS. The construction process ensures the popup-foldability according to Proposition 1 (Figure 5 (b)). 2. Popup-realizable surface: Next, we modify the visible surface to further meet the realizability condition in Proposition 2 while maintaining foldability. This is achieved using a greedy, region-growing algorithm (Figure 5 (c)). 3. Geometric refinement: Finally, the zig-zag boundaries of the patches in the Cartesian surface computed above is improved while maintaining realizability (Figure 5 (d)).