Approximating 3D surfaces using generalized waterbomb tessellations

Abstract

Origami has received much attention in geometry, mathematics, and engineering due to its potential to construct 3D developable shapes from designed crease patterns on a flat sheet. Waterbomb tessellation, which is one type of traditional origami consisting of a set of waterbomb bases, has been used to create geometrically appealing 3D shapes and been widely studied. In this paper, we propose a method for approximating target surfaces, which are parametric surfaces of varying or constant curvatures, using generalized waterbomb tessellations. First, we generate a base mesh by tiling the target surface using waterbomb bases. Then, by applying a simple numerical optimization algorithm to the base mesh, we achieve a developable waterbomb tessellation, which can be developed onto a plane without stretching. We provide a prototype system using which the user can adjust the resolution of the tessellation and modify waterbomb bases. Our work could expand the exploration of building developable 3D structures using origami.

Highlights

• Generalizing waterbomb tessellations to fit target 3D parametric surfaces.

• Achieving developable tessellations by a simple numerical optimization algorithm.

• Non-axisymmetric or non-orientable resulting approximations are demonstrated.

Graphical Abstract

00-01, 99-00

Computational origami, Approximation, Developable constraint, Computer aided design

1. Introduction

Origami, also known as paper folding, has the potential to construct 3D shapes by folding thin sheets of paper along predefined creases without introducing cuts and distortions. An origami piece can be defined by its crease pattern, which contains a set of mountain and valley folded lines (shown in red and blue in this paper) appearing on a sheet of paper when the origami is opened flat. The crease pattern is scale independent and thus can be applied at the nanometric (Edwards & Yan, 2014; Nangreave, Han, Liu, & Yan, 2010; Rothemund, 2006; Tørring, Voigt, Nangreave, Yan, & Gothelf, 2011) or space level (Lang, 2009; Miura, 1989; Pohl & Wolpert, 2009; Wilson, Pellegrino, & Danner, 2013).

Among the types of origami, waterbomb tessellation is a traditional one that can be used to create geometrically appealing 3D shapes, such as that in Fig. 1 (a). As shown in Fig. 1, a 3D waterbomb origami (a) is defined by its crease pattern (b), which contains a set of waterbomb bases (c). Such origami pieces are developable, which is guaranteed by the fact that the sum of the sector angles around each interior vertex equals $360°$⁠. The waterbomb base, which is also referred to as a regular base, has a mirror-symmetric property. The base has the geometric feature containing four valley and two mountain folded lines meeting at the center vertex. Here, we introduce a generalized waterbomb base (Fig. 1(d)) that inherits this geometric feature but could omits the mirror-symmetric property. Furthermore, we introduce a generalized waterbomb tessellation that contains generalized waterbomb bases to approximate target 3D surfaces.

Fig. 1.

Geometry of generalized waterbomb origami.

Waterbomb tessellation has also been widely studied. Tachi, Masubuchi, and Iwamoto (2012) analyzed the kinematics of waterbomb tessellations to achieve adaptive freeform surfaces. They generated a model based on the multiple degree of freedom inherent in waterbomb tessellation containing regular bases. Additionally, Kuribayashi et al. (2006) made the first origami stent to achieve a large deployable ratio. Onal, Wood, and Rus (2013) demonstrated a worm robot, and Lee, Kim, Kim, Koh, and Cho (2013) fabricated a deformable wheel robot. Chen, Feng, Ma, Peng, and You (2016) proposed a comprehensive kinematic analysis on a waterbomb origami with one degree of freedom motion under symmetric folding.

In this paper, we approximate target surfaces, which are parametric surfaces of varying or constant curvatures, using generalized waterbomb tessellations. An overview of our method is shown in Fig. 2. We take a 3D parametric surface, e.g., Fig. 2(a), as input. Then, we sample u and v coordinates in the parametric uv-plane to achieve a quad approximation (Fig. 2(b)). Next, we generate a base mesh (Fig. 2(c)) by creating waterbomb bases in the quads. Here, our prototype system enables us to generate base meshes with variable resolutions and modify waterbomb bases interactively. Then, by applying a simple numerical optimization algorithm to the base mesh, we achieve a developable waterbomb tessellation (Fig. 2(d)), which can be developed onto a plane without stretching. Finally, the user can fold the crease pattern (Fig. 2(e)) to achieve the origami piece (Fig. 2(f)). We demonstrate several resulting approximations, which expands the exploration of building developable structures using origami.

Fig. 2.

An overview of our method.

The rest of this paper is organized as follows. Related work is discussed in Section 2. Section 3 presents a detailed description of our method. In Section 4, we demonstrate several developable approximations. We conclude this paper in Section 5 and discuss future work.

2. Related work

TreeMaker is software used to design flat-foldable origami (Lang). Its basic concept was first introduced by Meguro (1991) and fully described by Lang (1996). This software generates a crease pattern from a graph tree that represents the base structure of the object by using a circle/river packing technique. Tess is another computer program that makes crease patterns for origami tessellations and involves twist folds in a repeating pattern (Bateman). These approaches focus on flat-foldable origami, while we aim at approximating 3D surfaces using origami which may not be flat-foldable.

Mitani proposed methods for designing 3D origami on the basis of rotational sweep (Mitani, 2009, 2011). The methods generate a crease pattern for an axisymmetric structure by adding flaps outside of the target shape. Although the flaps might be considered obtrusive, his method succeeds in generating 3D curved origami. One of his methods (Mitani, 2012), which combines the advantages of the rotational sweep and mirror reflection approaches, has been used to build geometrically attractive origami pieces. Mitani & Igarashi (2011) also proposed an interactive system that allows the user to design 3D curved origami surfaces with mirror operations specified by selecting and moving vertices on the 3D origami while maintaining the developability of the resulting shape.

Zhao, Kanamori, and Mitani (2017) proposed a method for handling a family of axisymmetric 3D origami consisting of triangle facets. This method first designs a rotationally-symmetric crease pattern and then calculates an axisymmetric 3D origami piece on the basis of geometric constraints. By adding a cut in the crease pattern, such 3D origami can be axisymmetrically deployed or flat-folded by changing one parameter. More recently, Zhao, Kanamori, and Mitani (2018) proposed a method for designing axisymmetric 3D origami based on a mirror-symmetric crease pattern. The method explored the variations of the calculated 3D origami and presented a rigid folding motion that can axisymmetrically deploy or flatten the 3D shape.

For approximating 3D surfaces using origami, Tachi proposed the Origamizer algorithm (Tachi, 2009, 2010a), which generates crease patterns for arbitrary 3D triangle mesh models with a topological disc condition. Then, he proposed a system (Tachi, 2013) for approximating a target shape by using a subset of generalized Resch patterns. However, these approaches were based on the tucking technique, which hides unnecessary areas of a sheet of paper inside the shape. He also proposed a design system $\mathit{Freeform}\mathit{Origami}$ (Tachi, 2010b), which allows the user to vary a known origami in 3D while preserving the developability and other optional conditions inherent in the crease pattern. Through dragging the vertices in 3D, the system enables the user to edit a given pattern into a freeform. However, the method cannot fully support approximating target 3D surfaces.

In addition, several approximating approaches based on modified Miura-ori have been proposed. Zhou, Wang, and You (2015) developed a vertex method for generating developable 3D origami between two singly curved surfaces. Song, Zhou, Zang, Wang, and You (2017) proposed a mathematical framework for the generation of rigid-foldable 3D origami based on the crease pattern that can simultaneously fit two doubly curved surfaces with rotational symmetry about a common axis. Dudte, Vouga, Tachi, and Mahadevan (2016) used modified Miura cells to approximate orientable 3D surfaces with positive, zero, negative, and mixed Gauss curvatures. In this paper, we focus on another basic origami tessellation, waterbomb tessellation, to fit onto target surfaces.

3. Approximating target surfaces

We demonstrate the generation of a base mesh in Section 3.1. Optimizing the base mesh to achieve a developable approximation is discussed in Section 3.2.

3.1. Generation of base mesh

The generation of the base meshes on parametric surfaces is versatile; we can generate base meshes on axisymmetric or non-axisymmetric target surfaces and on orientable or non-orientable target surfaces. We tile a given surface using quads for the initial approximation. Parametric surfaces are taken as input in this work. Therefore, we can easily achieve a set of quads by isometrically sampling u and v coordinates, which vary within a certain domain D in the parametric $\mathit{uv}-$ plane, of the input parametric surface.

Hereafter, we explain the case of a catenoid surface as an example. A catenoid surface (Fig. 2(a)) is defined with $u\text{,}v$ parameters as:

$$P(x\text{,}y\text{,}z)=\left(\cosh \frac{v}{c}\cos u\text{,}\cosh \frac{v}{c}\sin u\text{,}v\right)\text{,}$$

(1)

where $u\in \left[0\text{,}2\pi \right]\text{,}v\in \left[-\pi \text{,}\pi \right]$⁠, and c is a non-zero real constant that is set as 2.5 in this case. As shown in Fig. 3 (a) and (b), we isometrically sample u and v coordinates to achieve sampling points. The steps of u and v for sampling are denoted as $\mathrm{\Delta }u$ and $\mathrm{\Delta }v$⁠, which equal $2\pi /N_u$ and $2\pi /N_v$⁠, respectively. $N_u$ indicates the number of quads in one strip, which is shown in red and green. $N_v$ means the number of strips used for constructing the approximation. Both $N_u$ and $N_v$ are integers and set as 10 in this case.

Fig. 3.

Initial approximation using quads.

As can be observed from waterbomb tessellations, adjacent strips are shifted against each other by $\mathrm{\Delta }u/2$ in the u direction in the $\mathit{uv}$ -plane. A naïve way of doing this is to shift only odd strips by $-\mathrm{\Delta }u/2$⁠. However, this works with axisymmetric shapes but fails with non-axisymmetric ones because quads along boundaries become jagged and cannot cover the target surfaces. To handle both axisymmetric and non-axisymmetric shapes, we first temporarily generate $N_u+1$ quads for odd strips. Suppose that parameter u in the given parameter surfaces ranges from $u_s$ to $u_e$⁠. Here, $N_u+1$ quads are achieved in the range from $u_s-\mathrm{\Delta }u/2$ to $u_e+\mathrm{\Delta }u/2$⁠. In particular, the first quad's u ranges from $u_s-\mathrm{\Delta }u/2$ to $u_s+\mathrm{\Delta }u/2$⁠, and the last quad's u ranges from $u_e-\mathrm{\Delta }u/2$ to $u_e+\mathrm{\Delta }u/2$ (Fig. 3(b)). In the case of axisymmetric shapes, the first and the last quads are identical because parameter u is periodic. We then generate a waterbomb base in each quad and select only a half of the first and the last waterbomb bases to ensure $N_u$ waterbomb bases in each strip (shown in Fig. 5 and discussed below).

Fig. 4.

Modification of waterbomb base

Fig. 5.

Merging waterbomb bases to achieve base mesh.

During the initial approximation using quads, we allow the user to adjust the density of quads by changing $N_u$ and $N_v$ interactively. Fig. 3(c) is an approximation created by double density sampling both in the u and v directions, and thus, it has four times more quads than that in Fig. 3(a) to represent the target surface. The more quads we use, the more accurately we can approximate the target surface. Considering fabrication by paper-folding, however, we also have to consider the increase of labor. Balancing the approximation accuracy and fabrication labor is an interesting problem, which is left as future work.

Next, we generate each waterbomb base by adding and moving three auxiliary vertices, $P_u\text{,}{P}_d$⁠, and $P_c$⁠, as shown in Fig. 4. At an initial state (Fig. 4(a)), $P_d$ and $P_u$ are the midpoints of segments $P_1{P}_2$ and $P_3{P}_4$⁠, respectively; $P_c$ is the midpoint of segment $P_d{P}_u$⁠. We connect boundary points $P_1\text{,}{P}_2\text{,}{P}_3\text{,}{P}_4\text{,}{P}_d$⁠, and $P_u$ to $P_c$⁠. The positions of $P_1\text{,}{P}_2\text{,}{P}_3$⁠, and $P_4$ are fixed. We then move $P_u\text{,}{P}_d$⁠, and $P_c$ to form a structure that “looks” like a waterbomb base, with four valley and two mountain folded lines. Specifically, $P_c$ can be moved along the normal of quad $P_1{P}_2{P}_3{P}_4$ (Fig. 4(b)). $P_u$ can be “dragged down” by rotating it in plane $P_u{P}_c{P}_d$ (Fig. 4(c)). Similarly, $P_d$ can be “dragged up” by rotating it in plane $P_u{P}_c{P}_d$ (Fig. 4(d)).

Finally, we merge adjacent bases to achieve a base mesh (Fig. 5 (c)). Fig. 5(a) is an approximation with gaps. By averaging the positions of adjacent vertices (b), we achieve a base mesh without gaps as shown in (c). Note that there are $N_u+1$ bases for odd strips. Here, we select only the right part of the first base and the left part of the last base to ensure $N_u$ bases in the odd strips.

3.2. Numerical optimization

In this section, we apply a simple numerical optimization to base meshes to produce developable surfaces. We use an angle constraint (Tachi, 2010b), which requires that the total angle around a developable vertex i be $360°$⁠:

$$D_i=360°-{\displaystyle \sum _{k=0}^{K_i-1}}{\alpha }_{i\text{,}k}=0\text{,}$$

(2)

where $K_i$ is the total number of sector angles around vertex i, and ${\alpha }_{i\text{,}k}$ is the k-th incident sector angle of vertex i. In our work, we classify vertices as interior vertices having six adjacent facets and boundary vertices having less than six adjacent facets. For a developable surface, all the interior vertices should satisfy the angle constraint.

We implemented the Levenberg-Marquardt algorithm to solve such an optimization problem. Boundary/interior vertices are viewed as fixed/free nodes, respectively. For each iteration of the Levenberg-Marquardt algorithm, we evaluate the maximum ${\alpha }_{\mathit{max}}$⁠, minimum ${\alpha }_{\mathit{min}}$⁠, and average ${\alpha }_{\mathit{ave}}$ of the sum of angles around each interior vertex for a termination criterion. Correspondingly, we introduce the errors of $e_{\mathit{max}}\text{,}{e}_{\mathit{min}}$⁠, and $e_{\mathit{ave}}$ represented as:

$$\begin{array}{cc}\hfill & e_{\mathit{max}}=|360°-{\alpha }_{\mathit{max}}|\text{.}\hfill \\ \hfill & e_{\mathit{min}}=|360°-{\alpha }_{\mathit{min}}|\text{.}\hfill \\ \hfill & e_{\mathit{ave}}=|360°-{\alpha }_{\mathit{ave}}|\text{.}\hfill \end{array}$$

(3)

Moreover, we introduce $e_{\mathit{total}}$ as the maximum among $e_{\mathit{max}}\text{,}{e}_{\mathit{min}}$⁠, and $e_{\mathit{ave}}$⁠. The procedure is terminated when $e_{\mathit{total}}$ is less than $e_d$⁠. In our experiment, we set $e_d$ as 1e−5 to produce a developable surface.

Fig. 6 shows graphs of convergence created during optimization on the base mesh (Fig. 2(c)), where Fig. 6(a) shows the relationship between the numbers of iterations and values of ${\alpha }_{\mathit{max}}\text{,}{\alpha }_{\mathit{min}}$⁠, and ${\alpha }_{\mathit{ave}}$⁠. Correspondingly, Fig. 6 (b) demonstrates the values of $e_{\mathit{max}}\text{,}{e}_{\mathit{min}}$⁠, and $e_{\mathit{ave}}$ that calculated during the iterations. In this case, $e_{\mathit{total}}$ becomes less than 1e−5 when the number of iterations is 198.

Fig. 6.

Graphs of convergence created during optimization for producing developable surface.

4. Results

We developed a prototype system using Java to implement our method. We ran our system on an Intel(R) Core(TM) i7-4770 CPU with an 8-GB-RAM PC. For a given target surface, our method allows the user to generate base meshes with variable resolutions and then produces developable approximations. As shown in Fig. 7, we show four results, each of which contains a base mesh, its corresponding approximation, and the approximation with the target surface, as shown in Fig. 2(a).

Fig. 7.

Approximations with variable resolutions for same target surface.

Fig. 7. $e_{\mathit{total}}$ of each approximation was less than 1e−5 after optimization, with which we consider the approximation to be developable. To evaluate the difference between the resultant approximation A and the target surface T, we define distance $d\left(A\text{,}T\right)$ as:

$$\begin{array}{cc}\hfill & d\left(A\text{,}T\right)=\text{mean}\left[d\left(x\text{,}T\right)\right]\text{,}x\in A\text{,}\hfill \\ \hfill & d\left(x\text{,}T\right)=\min \left[d\left(x\text{,}y\right)\right]\text{,}y\in T\text{,}\hfill \end{array}$$

(4)

where x and y denote vertices of the approximation A and the target surface T, respectively. $d\left(x\text{,}y\right)$ denotes the Euclid distance between x and y. $d\left(x\text{,}T\right)$ is the shortest distance between x and a set of y from T. $d(A\text{,}T)$ is similar in spirit to the Hausdorff distance, which is used to measure the difference between two surfaces. To compute $d\left(x\text{,}T\right)$⁠, we sample all vertices from the approximation A for x by considering A is a discrete tessellation. When the target surface T is continuous, Table 1, wenote that as the number of waterbomb bases increased, $d(A\text{,}T)$ decreased, which means that the result became closer to the target surface at the cost of more computational time.

Table 1

Parameters in detail and statistics of models shown in Fig. 7.

Approximations	$N_u$	$N_v$	Bases	$e_{\mathit{total}}$	$d(A\text{,}T)$	Time
(a)	8	6	48	9.68e−6	3.29e−2	0.31 min
(b)	10	7	70	6.54e−6	2.95e−2	0.85 min
(c)	13	9	117	9.80e−6	2.41e−2	9.35 min
(d)	20	14	280	9.94e−6	1.61e−2	158.60 min

We fabricated several approximations, shown in Fig. 8, where (a) shows a catenoid and (b) shows a cylinder. Both approximations contained 48 waterbomb bases. (c) shows a sphere containing 75 waterbomb bases, and (d) shows a vase containing 112 waterbomb bases. For each result shown in Fig. 8, we demonstrate a 3D model of the approximation, a crease pattern, and an origami piece.

Fig. 8.

Fabricated origami pieces.

We also approximated several 3D surfaces and show its crease pattern and rendered 3D model in Fig. 9, where (a) shows an approximation of a catenoid, (b) a sphere, (c) a cylinder, (d) a vase, (e) a torus, (f) a hyperbolic paraboloid, (g) a möbius strip. Details of the target surfaces are demonstrated in Fig. 9(e)). Additionally, we show an approximation of a hyperbolic paraboloid, which is the non-axisymmetric surface in Fig. 9(f), and an approximation of a Möbius strip, which is the non-orientable surface in Fig. 9(g). The Möbius strip approximation is not connected because the waterbomb bases at the start and end parts of the approximation had different orientations. Meanwhile, we demonstrate the detail results of the approximations (Fig. 9) in Table 3 correspondingly.

Fig. 9.

Developable approximations consisting of generalized waterbomb tessellations.

Table 2

Target surfaces used for generating developable approximations shown in Fig. 9.

Targets	Equations $(x\text{,}y\text{,}z)$
(a)	$(\cosh \frac{v}{2.5}\cos u\text{,}\cosh \frac{v}{2.5}\sin u\text{,}v)\text{,}u\in \left[0\text{,}2\pi \right]\text{,}v\in \left[-\pi \text{,}\pi \right]$
(b)	$(\cos v\cos u\text{,}\cos v\sin u\text{,}\sin v)\text{,}u\in \left[0\text{,}2\pi \right]\text{,}v\in \left[\frac{-\pi }{2.2}\text{,}\frac{\pi }{2.2}\right]$
(c)	$(\cos u\text{,}\sin u\text{,}v)\text{,}u\in \left[0\text{,}2\pi \right]\text{,}v\in \left[-\pi \text{,}\pi \right]$
(d)	$((2+\sin v)\cos u\text{,}(2+\sin v)\sin u\text{,}-v)\text{,}u\in \left[0\text{,}2\pi \right]\text{,}v\in \left[-3\text{,}4\right]$
(e)	$((3+\cos v)\sin u\text{,}(3+\cos v)\cos u\text{,}\sin v)\text{,}u\in \left[0\text{,}2\pi \right]\text{,}v\in \left[-\pi \text{,}\pi \right]$
(f)	$(u\text{,}v\text{,}\mathit{uv})\text{,}u\in \left[-1\text{,}1\right]\text{,}v\in \left[-1\text{,}1\right]$
(g)	$((1+\frac{v}{2}\cos \frac{u}{2})\cos u\text{,}(1+\frac{v}{2}\cos \frac{u}{2})\sin u\text{,}\frac{v}{2}\sin \frac{u}{2})\text{,}u\in \left[0\text{,}2\pi \right]\text{,}v\in \left[-1\text{,}1\right]$

Table 3

Details of statistics of models shown in Fig. 9.

Approximations	$N_u$	$N_v$	Bases	$e_{\mathit{total}}$	$d(A\text{,}T)$	Time
(a)	10	10	100	9.43e−6	2.60e−2	4.79 min
(b)	25	10	250	9.96e−6	1.44e−2	152.27 min
(c)	10	10	100	9.53e−6	1.59e−2	2.58 min
(d)	21	10	210	9.80e−6	1.59e−2	74.09 min
(e)	55	10	550	9.96e−6	8.97e−3	853.18 min
(f)	10	10	100	6.22e−6	1.35e−2	2.35 min
(g)	22	3	66	9.29e−6	1.40e−2	0.44 min

In terms of fabrication, folding a waterbomb tessellation is not an easy task because it requires multi-fold simultaneous actuation. The folding process becomes more difficult when the waterbomb tessellation contains more waterbomb bases. Pre-folding crease lines on a sheet of paper can alleviate this problem. However, the crease lines on paper become fuzzy after several pre-foldings. Therefore, we showed only crease patterns and rendered 3D models (Fig. 9) instead of results with folded paper. A more effective way for fabricating complex approximations with many waterbomb bases would be printing the crease patterns on a textile using polymers because a textile can be folded many times without obvious fatigue.

5. Conclusion and future work

We proposed a method for approximating target surfaces, which are parametric surfaces of varying or constant curvatures, using generalized waterbomb tessellations. First, we described the generation of a base mesh by tiling the target surface using waterbomb bases. Then, we applied a simple numerical optimization algorithm to the base mesh to produce a developable approximation. Several developable approximations were presented to demonstrate the validity of our method. We provided a prototype system which enables us to interactively generate base meshes with variable resolutions and modify waterbomb bases.

Our work is different from Origamizer (Tachi, 2009, 2010a) and the system (Tachi, 2013), because ours is not based on the tucking technique, which hides unnecessary areas of a sheet of paper inside the shape. Our method is also differs from $\mathit{Freeform}\mathit{Origami}$ (Tachi, 2010b), which generates a freeform surface by dragging the vertices of an origami in 3D. In addition, several existing approximating works were based on modified Miura-ori (Dudte et al., 2016; Song et al., 2017; Zhou et al., 2015), while we focus on the waterbomb tessellation, another basic origami tessellation, to fit on target surfaces. We have demonstrated that our method can tile waterbomb bases on target surfaces, which can be axisymmetric or non-axisymmetric as well as orientable or non-orientable.

As future work, three aspects of our study can be improved: (i) finding an optimal density to balance the approximation accuracy and amount of fabrication labor when generating a base mesh, (ii) achieving a developable approximation while restricting $d(A\text{,}T)$⁠, that is, the distance between the resultant approximation A and target surface T, and (iii) generating flat-foldable and self-intersection-free approximations. Furthermore, we hope this work can be extended to approximate complex 3D models which can be parameterized into $\mathit{uv}$ -plane and pave the way of fully solving the inverse-origami-design problem.

Footnotes

Peer review under responsibility of Society for Computational Design and Engineering.

Author notes

We wish to confirm that there are no known conflicts of interest associated with this publication.