To guide the algorithm design and show its correctness

A paper architecture is created from a single piece of paper by cutting and folding. The planar layout of such a paper architecture consists of a set of cut lines and fold lines that divide the paper into various regions. Typically, there are two outer regions, called backdrop and ground, that meet at a central fold. 

When the user folds the central fold by moving the backdrop and ground, the rest of the regions “pop-up” as a result of folding along the fold lines. Our ultimate goal is to device computational algorithms to construct planar layouts that can be physically popped-up. To guide the algorithm design and show its correctness, we first present geometric formulations of planar layouts, particularly those that can be popped-up in a physically realistic manner.
 3.1 Layouts Planar layouts for paper architecture, as well as their folded results, can be generally considered as a kind of 3D surface with linear components. Specifically, we define: Definition 1 A piece-wise linear surface (PLS) is a collection of planar, non-intersecting and non-overlapping patches where neighboring patches share common straight edges. Definition 2 A planar layout for paper architecture (PLPA) T is a PLS where 1. All patches in T are co-planar, 2. T forms a rectangular domain with possible holes, and 3. There exist two patches, called backdrop and ground, that touch the outer rectangular boundary and share edges along the mid-line of the rectangle. An example of a PLPA is shown in Figure
 3 (a). In a PLPA, we call common boundaries between neighboring patches fold lines (red and blue lines in the picture), and the rest of the patch boundaries as cut lines (black lines in the picture). In particular, we call the fold lines between the backdrop and ground as the central fold.
3.2 Foldable layouts Obviously, not all PLPA can be folded up. To define foldability, we make the key assumption that the paper is made up of rigid materials (e.g., metal) except at the boundary of patches (e.g., hinges). Furthermore, we assume the paper has zero thickness. These two assumptions help us to formulate foldability, as well as realizability, as simple geometric properties. Note that the same assumptions are made in rigid origami [Belcastro and Hull 2002]. With the rigidity