1 Introduction
We discuss here the inverse problem of rigid origami, that is, to approximate a target surface by rigid origami — usually starting from a planar creased paper. This problem has been preliminarily discussed in the review [Callens and Zadpoor 17]. Generally, we need to consider the following factors when designing a creased paper for approximation.

Which surface we can approximate.

The DOF (degree of freedom) during the rigid folding motion.

The utilization of materials.
Experience shows that, it is hard to make the creased paper behave well in all aspects. Generally, as we reduce the possible DOFs during the rigid folding motion, less surfaces can be approximated. For example, [Demaine and Tachi 17] give a universal algorithm to fold a planar creased paper to any piecewisepolygon orientable 2manifold, which can be used to approximate any orientable 2manifold. Although to archive the “watertight” property, they add small additional features at the vertices and along the edges, this algorithm is practical and guarantees a minimum number of seams. This result is undoubtedly successful, but will have many DOFs during its rigid folding motion, and a large proportion of materials are used in the connections and “walls” hidden in the “tuck” side. On the other hand, [Dudte et al. 16] and [Song et al. 17] use a flatfoldable creased paper. These algorithms guarantee one degreeoffreedom (1DOF) during the rigid folding motion and high utilization of materials, but only possible for a cylinderical developable surface or a surface of revolution, otherwise the creased paper will not be rigidfoldable. If we triangulate this creased paper, we can approximate more surfaces but without being 1DOF.
In this article we will focus on a branch of the inverse problem, to design a 1DOF rigid origami approximating some desired shapes. More formally, given a connected surface in , for any positive real number , find a creased paper , which is the union of a connected planar paper and a straightline crease pattern embedded on , such that

is rigidfoldable to its final rigidly folded state , where the rigid folding motion halts because some panels clash.

has one degree of freedom during the rigid folding motion.

the Hausdorff distance between and satisfies .
More details of the terminologies used here are given in [He and Guest 18b].
By using a family of developable quadrilateral creased papers that are rigidfoldable but not necessarily flatfoldable, we are able to give solutions for approximating some surfaces at the final rigidly folded state, but they cannot be completely arbitrary.
Furthermore, the approximation problem naturally induces an optimization problem, that is, find the “best” creased paper that fits the extra presupposed requirements. We will discuss it at the end of the article.
2 Choosing Design Example
To our best knowledge, the constraints on 1DOF rigidfoldability for a creased paper restrict the configurations it may have. Therefore it seems hard to approximate an arbitrary connected surface with 1DOF rigid origami. Our idea is, from several types of 1DOF developable and rigidfoldable quadrilateral creased papers we have known [He and Guest 18a, He and Guest 18c], we choose some of them as the design examples, and study the rigid folding motions of them. For each design example, the profile of its inner vertices can approximate certain types of surfaces. The more design examples we can use, the more surfaces we can approximate.
In this section we will analyze two design examples mentioned in [He and Guest 18c]. The first one is the developable case of the ”parallel repeating” type, which is generated among rows of parallel inner creases (Figure 1); and the second one is the developable case of the ”orthodiagonal” type, which is generated among several parallel straight line segments (Figure 6).
2.1 Parallel Repeating Type
Figure 1 shows the developable case of the parallel repeating type of rigidfoldable quadrilateral creased papers, which is generated among rows of parallel inner creases. In each column we can choose independent input sector angles , and , then construct the rest of the creased paper with both these angles and the supplement of these angles. should not form a cross. Here the length of creases does not affect the rigidfoldability, but will affect the profile of inner vertices. Based on that we start to analyze which surface the parallel repeating type can approximate.
Proposition 1.
The inner vertices on a column of the parallel repeating type are coplanar.
Proof.
Figure 2(a) shows a general column. We know at any rigidly folded state. Thus are coplanar. Because , this argument continues down the column, which means all the inner vertices are coplanar, and the angle between any two adjacent inner creases is , as illustrated in Figure 2(b). Figure 2(c) demonstrates a rigidly folded state of this column. ∎
To study the profile of inner vertices on a column of the parallel repeating type, we use
Definition 1.
A  coordinate system is built on the plane mentioned in Proposition 1. We say the inner vertices of a column can approximate a given planar curve if , there exists a column whose inner vertices make the Hausdorff distance between the set and the curve satisfy .
Proposition 2.
The inner vertices on a column of the parallel repeating type can only approximate a planar curve that satisfies the following condition: there exists a rotation and a shear transformation of magnitude , , s.t. after the affine transformation described below, is monotone decreasing.
(1) 
We name the curve approximated by the inner vertices of a column as a target curve.
Proof.
Sufficiency: An example of approximation is shown in Figure 3. For a given curve , after a rotation by and a shear transformation by , as in equation (1), (black curve in (b)) is mapped to (black curve in (c)), where the inner creases are parallel to the and axes. Then if is monotonous, we can construct an approximation corresponding to the partition when we apply the Darboux sum to describe the Darbouxintegrability (red line segments in (c)). Because is monotonous, it is Darbouxintegrable (if is unbounded, the limit of difference between lower and upper Darboux sum is zero when the partition is infinitesimally refined) and only has countable firstkind discontinuity points, so arbitrarily refining the partition will make the Hausdorff distance be arbitrarily small. Hence we can approximate by retransforming the approximation in the  coordinate system to the  coordinate system (red line segments in (b)). Here the requirement of monotone decreasing makes the angle between adjacent inner creases , not .
Necessity: If a curve can be approximated by the inner vertices of a column, we can always find corresponding and and do the affine transformation in equation (1). Because the approximation turns left and right alternatively, must be monotonous. To make the angle between adjacent inner creases , not , should be monotone decreasing. ∎
Corollary 2.1.
The inner vertices of a column cannot approximate a closed planar curve.
Remark 1.
We can require to be continuous and to be a closed interval in for the following reason. If is not connected, is the union of countable disjoint closed intervals. Each such subset can be approximated independently because the rigid folding motions of disconnected creased papers are independent. Therefore we can require to be connected, which means is continuous and is continuous. Then we can reparametrize , where is a closed interval in .
Next we will analyze the rigid folding motion of a row of the parallel repeating type.
Proposition 3.
The inner vertices on a row of the parallel repeating type can approximate a Darbouxintegrable curve at its final rigidly folded state, named the datum curve.
Proof.
An approximation can be generated following the steps mentioned below.

Assign a direction of , label each partition point () along this direction. From the coordinates of , calculate (), () and ().
is calculated by the rotation angle along the vector
. 
Then we continue to obtain other sector angles () in this row, as shown in Figure 4(b). Regarding , , , , , and as known variables, there are four equations for , , and . Two of them are related to , and .
(3) Note that the depends on the rigid folding motion we choose and the magnitude of sector angles. These equations can be directly derived from spherical trigonometry. Another equation is , but in order to simplify the steps we suppose the other vertices in this row are from column 2 in Figure 1. Hence there are two more equations:
(4) 
With and , , , (), draw the creased paper.
∎
Remark 2.
Generically, We can require to be continuous and to be a closed interval in for the following reason. Similar to our analysis in Remark 1, can be required as a connected and closed set. If has no secondkind discontinuity points, is continuous. Then we can reparametrize , where is a closed interval in . Besides, can be a closed curve.
Remark 3.
Here the final rigidly folded state is not special, we make it final by designing the rigid folding motion to be halted by clashing of panels at the first column of inner vertices from the left. If we release this condition we can make the datum curve be approximated by an intermediate rigidly folded state.
Corollary 3.1.
In step 4 of Proposition 3, If we choose other vertices in the first row from column 3 in Figure 1 , equation (3) will be simplified:
(5) 
Now there is only one branch of rigid folding motion and only one equation for and . However, it requires or , which means should be locally planar in a discrete sense. More essentially, [Fuchs and Tabachnikov 99] illustrate this in a continuous sense.
From Propositions 1, 2 and 3, we are now in a position of studying which surface the parallel repeating type can approximate.
Proposition 4.
The parallel repeating type can approximate a surface under a given Hausdorff distance . The creased paper is generated by the following steps, and is described below. (see Figure 5)

Choose a datum curve and approximate it in a row under a sufficiently small Hausdorff distance , as described in Proposition 3.

Choose a target curve and approximate it the first column from the left under a sufficiently small Hausdorff distance , as described in Proposition 2.

In other columns, the shape of the target curve () is an affine transformation of .
(6) where,
or vice versa. This depends on whether the line segments start in the or direction as shown in Figure 3(b). Additionally,
is the affine matrix we used in Proposition
2, and is the rotation angle in the approximation of . We then define the angle between the planes where and locate by (), which can be expressed by(7)
When the Hausdorff distance , the surface can be expressed as:
(8) 
where depends on (equation (6)) and locates on different planes determined by (equation (7)).
Corollary 4.1.
If is piecewisespiral, on each piece , , , and () will keep the same from the second column respectively. Then each piece induced by can be expressed analytically as scanning along . Some examples are demonstrated in [Song et al. 17] when is circular. Besides, if is piecewiselinear, remains piecewiselinear.
2.2 Orthodiagonal Type
The developable case of the orthodiagonal type of rigidfoldable quadrilateral creased papers is generated among several parallel line segments [He and Guest 18c], as shown in Figure 6. For all , The sector angles here should satisfy
(9) 
, . This equation guarantees each column and each row of inner vertices to be coplanar during the rigid folding motion.
Proposition 5.
The inner vertices on a column of the orthodiagonal type can approximate a Darbouxintegrable curve , named a datum curve.
Proof.
An approximation can be generated following the steps mentioned below.

Under a given Hausdorff distance , partition (black curve in Figure 7(a)) by points and connect adjacent partition points in sequence by line segments (orange line segments in Figure 7(a)). To make the angle between adjacent inner creases not too close to we choose partition points on an tube of (purple curves in Figure 7(a)). Note that here we use a method different from step 1 in Proposition 3, and there are many techniques to approximate a Darbouxintegrable curve by a series of line segments.

Assign a direction of , label each partition point () along this direction. From the coordinates of , calculate () and ().

Without loss of generality, we just consider the first column from the left. Calculate all the sector angles on the left () from equation (10), which is to make the rigid folding motion halt at the left side of the first column.
(10) Note that the depends on whether the line segments (coloured orange in Figure 7(a)) “turn left” or “turn right” along the direction assigned for . Only one of can be randomly chosen. Suppose it is , which should satisfy equation (11) to make sure the inner creases turn left or right properly, and the rigid folding motion halts at the left side.
(11) the other sector angles () can be calculated from equation (9).

With and , (), draw the creased paper.
∎
Remark 4.
As shown in Remark 2, generically, we can require to be continuous and to be a closed interval in . Besides, can be a closed curve. Another point is the approximation of does not need to turn left or right alternately, which means the mountainvalley assignment in each row not necessarily change alternately.
Then we will analyze which surface the orthodiagonal type can approximate. From equation (9), we know the sector angles in the first column and first row are independent, therefore we can also approximate a Darbouxintegrable curve with the inner vertices in a row. The problem of such approximation is, we cannot write a clear expression for the curve approximated by other rows of inner vertices in this creased paper, which makes it hard to grasp the feature of the target surface. (In step 3 of Proposition 4, we express the curve approximated by other columns of inner vertices as an affine transformation of the target curve.) Based on that we set (), and the result in Proposition 2 can be applied. For the orthodiagonal type, this simplification makes it possible to express the curve approximated by other rows of inner vertices as an affine transformation of the curve approximated by the first row of inner vertices. If we regard the surface approximated by such a simplified creased paper as a piece, the target surface can consist of these pieces stitched in the transverse direction, as shown in Proposition 6.
Proposition 6.
The orthodiagonal type can approximate a surface which is stitched by countable pieces in the transverse direction under a given Hausdorff distance . Each piece is generated by the following steps. (see Figure 7)

Choose a datum curve and approximate it in a column under a sufficiently small Hausdorff distance , as described in Proposition 5.

Choose a target curve and approximate it in the first row from the top under a sufficiently small Hausdorff distance , as described in Proposition 2.

Here (). With equation (9), calculate the other sector angles in the creased paper. In other columns, the shape of the target curve () is an affine transformation of .
(12) where,
is the affine matrix we used in Proposition 2, and for
is the rotation angle in the approximation of . locates on a plane perpendicular to .
When the Hausdorff distance , the piece can be expressed as:
(13) 
where depends on (equation (12)) and locates on the tangent plane of at that is also perpendicular to .
Corollary 6.1.
If is piecewisecircular, on each piece , and () will keep the same respectively. Then each piece induced by can be expressed analytically as scanning along . Besides, if is piecewiselinear, remains piecewiselinear, then each piece induced by will become a cylinderical developable surface.
3 Discussion
3.1 Comment on the Algorithms
This article presents some initial results for the problem we have set. We recognize there are flaws and limitations in the algorithms, some of which are discussed here.

In Proposition 2, for a given target curve and an angle , the condition we give to find an appropriate is not convenient. For the examples in this article we try scattered to find possible approximations.

In Proposition 3, the folding angle is a flexible parameter, that can be used to adjust the sector angles and . Actually, is the magnitude of all the folding angles on every row of inner creases. If we set too close to , will increase, but the width of approximation will be too small in the longitudinal direction, which may not be suitable for application. If we set too far from , the singularity of solutions will increase. For the approximation shown in Figures 4(c) and 4(d) we choose .

Even if we have obtained all the sector angles, when we plot the creased paper as described in Propositions 4 and 6, the inner creases in different columns may intersect at points other than vertices. We haven’t found a good way to control this. Sometimes scaling the datum or target curve will help.

In Proposition 5, the sector angle is also a flexible parameter, which can be used to control the width of the approximation. There is no singularity in the algorithm proposed here.
3.2 Comment on the Approximation
Apart from the algorithms, we want to mention a few points related to an approximation for a surface.

In Propositions 4 and 6, when the Hausdorff distance , we haven’t found a good way to express the target surface analytically, even though it is determined by the datum curve , target curve , and a folding angle (Proposition 4) or a sector angle (Proposition 6). We only give analytical expressions for special cases mentioned in Corollaries 4.1 and 6.1. However, an analytical solution for how to approximate a curve is given in [Tachi 13], which might be helpful in solving this problem.

It is not necessary to make the rigid folding motion halt at the first column from the left. Such halting columns can be inserted to the creased paper arbitrarily. Besides, the two examples shown in sections 2.1 and 2.2 design the halting column differently, and there are many other possible techniques.

The utilization of materials of our design depends on the proportion of halting columns with respected to the whole creased paper, which is at a relatively high level.

In this article we show that a series of developable surfaces can approximate a nondevelopable surface, which means the collection of all developable surfaces is not a closed set.

Another possible design example is a quadrilateral creased paper with no inner panel [He and Guest 18c], which can approximate more surfaces but will have some big “cracks” almost running through the final rigidly folded state. We will include the discussion on it in a future article.
3.3 Optimization
Optimizations can be applied to the algorithms mentioned in this article, which will lead to future work.

In Propositions 2, 3 and 5, the piecewiselinear approximation do not need to be uniformlyspaced. Furthermore, the partition points are not necessarily on the curve. Given the number of partition points, there are some classic methods to approximate the datum and target curves, which will improve the accuracy of approximation.

There are some other criteria for an approximation in Propositions 2, 3 and 5 under a given Hausdorff distance , such as minimizing the number of inner vertices, the total bending energy [Solomon et al. 12]; or restricting the minimum length of all the creases, the minimum of sector angles, etc, which will involve more complex calculations.
4 Conclusion
We have shown that it is possible to approximate some types of nondevelopable surfaces with a 1DOF rigid folding motion starting from a planar creased paper. This might have useful engineering applications, for instance as a way of forming a shell structure in 3dimensional space. However, given an arbitrary surface, the methods that can be used to generate an approximation with a planar creased paper that has limited DOFs are not fully understood.
5 Acknowledgment
We thank Tom Aldridge for some preliminary works on this topic, and Hanxiao Cui for helpful discussions on Proposition 2. This paper has been awarded the 7OSME Gabriella & Paul Rosenbaum Foundation Travel Award.
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