The goal of our algorithm is to create realizable planar layouts

The goal of our algorithm is to create realizable planar layouts that approximate a given 3D model when folded. We therefore need to answer the inverse question: given a 3D surface (e.g., a PLS), is it pop-up realizable? In this paper, we consider a special type of PLS for which we are able to derive sufficient, computable conditions of realizability. These conditions will guide our algorithm design and ensure the correctness of the results. Definition 7 A parallel PLS Sv,w is a PLS where the normal direction of each patch is either v or w, which are unequal unit vectors. For example, the examples in Figure 3 (b) and Figure 4 (a,b) are all parallel PLSs where v, w are orthogonal. Note that most paper architecture falls into this class, such as the ones shown in Figure 2. We first show conditions for popup-foldable parallel PLSs: Proposition 1 Consider a parallel PLS Sv,w, and let U = v × w, V = w × U, and W = U × v (as shown in Fig 3 (b)). If parallel projection of Sv,w in the direction of V + W onto a plane with normal v results in a PLPA, then Sv,w is popup-foldable. Proof:

 Consider the parallel projection T on the plane containing the origin. We will show that T is a foldable PLPA by constructing a pop-up transform from T via Sv,w. We can express any point x on Sv,w as: x = (xU , xV , xW ) · (U, V, W) (1) Note that U, V, W are linearly independent, hence the decomposition is unique. Consider the following continuous mapping f where f(t) for t ∈ [0, 1] consists of points: x(t) = (xU , xV , xW ) · (U, V (tπ), W) (2) for all points x ∈ Sv,w. Here, V (α) = w(α) × U where w(α) is a vector rotated from v around U by α degrees. We show several properties of f: • f(0) = T. In fact, for any point x ∈ Sv,w, x(0) lies on the plane with normal v containing the origin, and x − x(0) = xV (V + W). • f(t) = Sv,w for t = θ/π where θ is the angle between v, w. • f(t) is intersection-free for any t ∈ [0, 1), due to the uniqueness of the decomposition (when t > 0) and the nonintersection property of a PLPA. • The transform f maintains the rigidity of the patches in S. In fact, for any two points x, y on a same patch in Sv,w with normal v (or w), x(t), y(t) will lie on a common plane orthogonal to v (or w(tπ)), and kx − yk = kx(t) − y(t)k for any t ∈ [0, 1].