Papers and card can be bought in a wide range of colours

Card mechanics Here you will find techniques to help you make card and paper mechanisms for pop-ups. You can adapt these examples to create imaginative designs of your own. Materials Papers and card can be bought in a wide range of colours, textures, thicknesses and weights. This brief guide will help you decide on a suitable thickness of material. You will find more detailed information in the Paper and Card Chooser Chart. The backing sheet on which the pop-up is built should be strong enough to open without buckling. For pop-ups that involve a lot of stresses and strains, use a rigid, heavy weight of card, such as mounting board. For the pop-up design, you can use thick paper, such as cartridge paper, or thin card. Card that is too thick will be too bulky to fold flat and thin paper will not keep its shape when the page is opened. Decoration Take care when using water-based paints to colour your design. Too much moisture will wrinkle the surface of paper and cause thin card to warp. You can achieve effective results with markers, pencil crayon and pastels, or use a colour photocopy of your original artwork.

Scoring and folding Scoring makes it easier to produce a neat fold. Using a metal ruler as a guide, run the tip of a scalpel along the line of the fold. Only the surface of the card should be cut (do not cut too deeply or the card will break). The card is then folded away from the score line so that it can open slightly along the cut. You will find that the card bends and folds accurately. Remember that paper and card will fold more smoothly with the direction of the grain. 1 A pop-up design with scoring lines (- - - - -) and cutting lines (_____) 2 Just fold and the castle ‘pops up’ 3 Adding extra detail. Can you explain why the tower can’t be included in the original piece?

This activity should be restricted to those who can safely use and manipulate

The art form of quilling answers to many names: paper rolling, paper scrolling, paper filigree or mosaic. Whether quilling originated in ancient Egypt or China in 105 AD when paper was invented, the art form has a rich and fascinating history. In the 300’s and 400’s, silver and gold wire (known as metal filigree) was used to adorn pillars, vases and jewelry using the quilling technique. When metal became scarce, paper was substituted as the materials used for quilling which made the art form affordable and accessible to the masses. Quilling really came into its own in the 1500s and 1600s when French and Italian nuns used the torn edges from gilt-edged bibles and goose feathers to quill and decorate religious articles and pictures. In the 1700s and 1800s, quilling, along with needle point, were popular hobbies taught to young ladies. Although it was not typically practiced by working-class women, quilling eventually found its way to America where it was embraced and enjoyed by settlers. Today, with the advent of scrapbooking and card making, quilling has found a resurgence in popularity.

The projects in this article feature the teardrop coil, but there are many other intriguing shapes to try — marquises, arrowheads, holly leaves, and all sorts of beautiful scrolls, just to name a few.

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

Department of Computer Science, The University of Tokyo

Abstract. This paper describes an interface for assisting the design and production of pop-up cards by using a computer. A pop-up card is a piece of a folded paper from which a three-dimensional structure pops up when it is opened; it can be folded flat again afterward. Many people enjoy this interesting mechanism in pop-up books and greeting cards. However, nonprofessionals find it difficult to design pop-up cards because of the various geometric constraints required to make the card fold flat. We therefore propose an assistant interface to help people easily design and construct pop-up cards. In this paper, we deal with pop-up cards that open fully to 180◦ . We have designed a prototype that allows the user to design a pop-up card by setting new parts on the fold lines and editing their position and shape afterward. At the same time, the system examines whether the parts protrude from the card or whether the parts collide with one another when the card is closed. Users can concentrate on the design activity because the results are continuously fed back to them. We created several pop-up cards using our system and performed an informal preliminary user study to demonstrate its usability. 1 Introduction A pop-up card is a piece of a folded paper from which a three-dimensional (3D) paper structure pops up when it is opened. The card can be folded flat again afterward.

Many people enjoy this interesting mechanism in pop-up books [1] [2] [3] and greeting cards, and receiving and viewing pop-up cards appeals to people of all ages. Fig. 1 shows an example of pop-up book. Constructing a pop-up card is relatively easy; anyone can simply cut out the pieces and glue them together if a template is available. Unfortunately, it is much more difficult for nonprofessionals to design a pop-up card from scratch. The first problem is correctly understanding the pop-up card mechanism. The second problem is determining the positions of objects so that pop-up parts do not collide. This usually requires repetitive trial and error during design: cutting out component parts out of paper, pasting them on the card, and checking Fig. 1. “Alice’s Adventures in Wonderland”, a typical pop-up book [1]. whether they collide. If an error is found, re-thinking the design and starting over from the beginning. This process requires a lot of time, energy, and paper. Design and simulation in a computer eliminate the boring repetition and save time. Glassner proposed methods for designing pop-up cards [4] [5] [6]. He introduced several simple pop-up mechanisms and described how to use these mechanisms, how to simulate the position of vertices as an intersecting point of three spheres, how to check whether the structure sticks out beyond the cover or if a collision occurs during opening, and how to generate templates. His work is quite useful in designing simple pop-up cards. Our work builds on Glassner’s pioneering work and introduces several innovative aspects. Our system has two new mechanisms based on the V-fold: the box and the cube. We provide a detailed description of the user interface, which Glassner did not describe in any detail. In addition, our system provides realtime feedback to the user during editing operations by examining whether parts protrude from the card when closed or whether they collide with one another during opening and closing. Finally, we report on an informal preliminary user study of our system involving two inexperienced users.

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)).

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].

We are interested in PLPAs that can open and close “all the way”. Let fold angle be the outer angle of the central fold in a PLPA or any PLS folded from a PLPA (e.g., the fold angle of a PLPA is zero). Note that, strictly speaking, a PLPA cannot close completely (e.g., at fold angle 180◦ ) by our definition of foldability, as the backdrop and ground are not allowed to overlap. We therefore formulate a foldable layout as: Definition 4 A PLPA T is said to be foldable if there is a fold transform f from T during which the fold angle increases monotonically from 0 to 180◦ − with arbitrarily small . 

This transform f is called the pop-up transform of T, and any f(t) for t ∈ [0, 1] is called popup-foldable. For example, the PLPA shown in Figure 3 (a) is foldable, and a popup-foldable surface at 90◦ fold angle is shown in (b). 3.3 Realizable layouts A foldable PLPA as defined above may not be able to “pop-up” as the user opens or closes the ground and backdrop. For example, the PLPA may contain pieces disconnected from the ground or backdrop that needs extra support to hold them in place.

 Also, the pop-up transform may require external forces along the fold lines in order to crease them in practice. Ideally, we would to create stable paper architecture where the pop-up transform does not require additional forces other than opening and closing the backdrop and ground by the user, and the paper at each stage during the transform is able to hold steady when the backdrop and ground are fixed. We can define realizability based on the same rigidity assumption we made above. Intuitively, a PLS that can hold steady when some patches are fixed is one that cannot be folded to another state while keeping the fixed patches unchanged. Also, if the PLPA at each time point of a continuous folding process is stable with respect to the backdrop and ground, then no external forces would be required to get to the next time from a previous time other than moving the backdrop and ground. More formally, we define: Definition 5 Given a PLS S and a subset of patches P ⊂ S, S is said to be stable with respect to P if there is no other PLS S 0 6= S that is foldable from S via a fold transform f, where P keeps stationary during f.