One-DOF Rigid Foldable Structures from Space Curves
Tomohiro TACHIAssistant Professor
Graduate School of Arts andSciences, The University ofTokyotachi@idea.c.u-tokyo.ac.jp

Tomohiro Tachi received his
doctoral degree in engineeringfrom the University of Tokyo,
Japan. His research interests
include origami, deployablestructures, computational
design, and digital fabrication.

SummaryWe show a novel design method of one-DOF deployable mechanism based on a space curve,
through creating a curved folding and discretizing the folding into rigid origami. By interpreting
constant angle curved folding as a flat-foldable quadrilateral mesh origami, we design novel
irregular tessellated, cylindrical, and cellular flatly collapsible structures, whose behavior is easily
controlled by space curves.
Keywords: deployable structure; structural morphology; origami, rigid foldable structure; curved
folding.

1. Introduction
Rigid origami structures, i.e., panels and hinges polyhedral surface mechanisms, can be used fordesigning deployable structures such as retractable roof, temporary shelters, space structures, andmanipulators, or ordinary building elements such as solar shading, doors, and kinetic openings. Our
objective is to morphologically investigate such structures to add more design freedom. In thiscontext, quadrilateral panel structures is a powerful tool for designing rigid foldable structures since
this yields one-DOF synchronized and robust mechanisms resulting from redundant mechanicalconstraints. This implies that the whole structure easily transforms by asserting force to a part of thestructure. Examples of such mechanism are Miura-ori [1], Hoberman’s designs [2], and rigidfoldable cylindrical polyhedra by Miura and Tachi [3]. Author’s recent studies generalize theses
structures to achieve further design freedom through perturbation-based computational methods toobtain variations of Miura-ori-like structures [4][5]. However, since these methods use continuoustransformation from one pattern to another for the design, the achievable forms are restricted by theinitial patterns. Finding the initial figures with controlled global shape and behavior is still an
important yet unsolved problem.
In order to find such globally controllable form of one-DOF rigid origami structure, we propose amodeling method based on discretizing curved folding which can be constructed from a free single
space curve.
The following shows the contribution of this work.
1. We show the geometry of discretized version of the curved folding constructed from a spacecurve.
2. We interpret the discretized version of the curved folding as a one-DOF mechanism andpropose a novel method to construct a foldable structure that follows a given space curve or
polyline.
3. We show that the method can be extended to tessellated, cylindrical, and cellular structures
because of the flat-foldability originating from constant-angle folding.
We first start from the continuous curved folding, its construction from a space curve, and thendiscuss the discretized version. Next, we extend the method into tessellated, cylindrical, and cellularstructures.

2. Curved Folding
There have been geometrical analysis and design proposals of curved folding. One of the earlyexamples are studies by Huffman [6] and Resch [7]. A curved folded surface comprises uncreased(G1 continuous) developable patches and the creases (curve on which surface is G1 discontinuous)
between them. An uncreased developable patch is represented as a collection of ruled surfaces inwhich the rulings have non-twisting nature.
2.1

Curved Folding from a Space Curve

It is known that a “nice” generic space curve, i.e., a ribbon curve, is a geodesic line of a developablesurface. Such a curve can be assigned with local coordinates called Frenet–Serret frame. From adifferential geometric analysis, it is known that such a curve can be interpreted as the folded curved
crease of a developable surface as shown by computer graphics example created by Resch [7]. We
can obtain different curved folded surfaces from one space curve by changing the folding angle.
Fixing the folding angle function along the curve denoted by 2 (t ) (where t parameterizes thespace curve by its arc length) uniquely determines the folded surface around the curve. The
relationship between the folding angle and the ruling configuration has been investigated in Fuchs
and Tabachnikov [8]. Here is a brief summary of their results with a more common notation.
First, we parameterize a given curve as x(t ) . We assume there exists a Frenet-Serret frame
comprising three orthonormal vectors: tangent vector T(t ) , normal vector N(t ) , and bi-normalvector B(t ) . The curvature  (t ) and torsion  (t ) relate to the frame in the following well-known
formula: (t )   0T t 0   T(t )    t  N(t )
0
N(t )    t  B (t )   0  t  0   B(t ) (1)The surface is represented by the crease and ruling pattern in 2D and the half of the foldingangle  (t ) at the crease. The local configuration of the former is characterized by curvature of thecurve in 2D  2 D (t ) and the ruling angles  L (t ) and  R (t ) defined as the angles between therulings and the tangent vector at the point (Figure 1). The left and right tangent planes of the surface
at t form angle  (t ) with the osculating plane, i.e., a plane defined by T(t ) and N(t ) , and thesevariables are related as follows.

 2 D t    (t ) cos  t (2)cot  L t   t    t  t    t 
 2 D t  tan  t   t sin  t (3)cot  R t    t    t    t    t 
 2 D t  tan  t   t sin  t (4)Equation (2) indicates that the ratio between curvatures in 3D and in 2D is high if the folding angle
is big, and equations (3) and (4) imply that the torsion and the change in the folding angle affect therulings configuration. We can use these equations to build up the curve-folded surface, crease
pattern, and rulings from a space curve and the folding angle function  (t ) .

Fig.1 : Parameter of curved folding.
2.2

Constant Angle Curved Folding

Here we focus on the case of the constant-angle folding, where  t     const. . In this case,Equations (3) and (4) are combined intocot  L t   cot  R t    t   t .(5) 2 D t  tan   t  sin 

Therefore, in the crease pattern, the left and right ruling lines emanate at the same angle from thetangent vector of the curve, meaning that the rulings are drawn such that they reflect at the curve.
2.3

RibbonWe can further build the surface from a given space curve (with finite  t  ) and the foldingfunction  t  at every point, then the two developable surfaces lie onto each other. Then2
 2 D (t )  0 , indicating that the curve is a straight line in 2D, i.e., a geodesic line of the surface. Thisis the shape of a straight strip of paper (or a ribbon) curved along the given space curve. The surface
is locally on the plane defined by Bt  and Tt  .

3. Discrete Curved Folding
Developable surfaces and curved folding surface can be represented computationally by discretely
sampling the rulings as a planar-quadrilateral mesh as shown in Kergosien et al. [9], Bo and Wang[10], and Kilian et al. [11]. Here, the creases and rulings are discretized as the edges of thepolyhedral mesh, and the non-twisting nature of the rulings is represented by the planarity of eachquadrilateral facet. Here we show the process of constructing a discretized version of curved folding
from a discretized space curve i.e., polyline.
3.1

Discrete Ribbon

We can construct a discrete ribbon from a given polyline so that the center line of the ribbon is thegiven polyline. The construction is performed sequentially starting from one edge and propagate
from there. Suppose that an edge Ei is assigned with a frame comprising Ti , N i , and B i so that theincident quadrangle patch Qi is perpendicular to N i . Our objective is to derive the rulings vector

ri 1 emanating from the incident point of edges i and i  1 from Ti , N i , B i and Ti 1 (note that
T0 , ..., Tn are given as the directions of the edges) (see Fig. 2). We can identify ri 1 by threeconstraints: 1) since ri 1 is the intersection of the Qi and Qi 1 , ri 1 is perpendicular to N i ; 2) sincepolyline Ei Ei 1 is a geodesic line of the quadrangle strip, the angle between Ti and ri 1 equals tothat of Ti 1 and ri 1 ; 3) the distance from the end point of ri1 from the edge should be half of thewidth of the strip w / 2 . Therefore, the ruling vector is given by solving the following matrixequation.
 0 
Νi

T  T T r   0 
 i 1 i  i 1  
w 
Bi
 2

(6)

Once we define ri 1 , then the next frame is calculated by N i 1  ri 1  Ti 1 and B i 1  Ti 1  N i 1 . In
the discrete case, the frame of the initial edge is freely determined. However, in order to beconsistent with a smooth curve case with infinite number of vertices, we approximate the initialframe using Ti 1 , Ti , and Ti 1 .

Fig. 2: Discrete Ribbon.
3.2

Constant Angle Discrete Curved Folding

Once we assign consistent frame for every edge and obtain the ribbon, we can construct a discrete
curved folding by replacing the strip with V-shaped ridge as shown in Figure 3. The process is asfollows.
1. Construct a pair of symmetric half-planes through each edge i to form angle

2  withplane Qi .
2. Trim corresponding adjacent planes at the intersection line and join them to form acontinuous ridge.
This can be processed in the same manner as in constructing ruling vectors of a ribbon by justrotating each frame before the calculation. This construction procedure yields a developable andflat-foldable pattern for a certain range of  . Note that the process relies on the special condition
under constant-angle folding and does not apply to the case where each edge has different folding
angle.

Fig. 3: Discrete curved folding with constant folding angle..Note that every vertex is flat-foldable in
the isometric sense, i.e., Kawasaki’s theorem is satisfied (alternating sum of incident angles is 0).

4. Rigid Origami Structures
Here, we interpret such a polyhedral mesh as a rigid origami structure and propose a novel methodfor designing rigid-foldable structure from a space curve. Even though a physical curved folding ofreal material is floppy and flexible, the polyhedral structure constructed from a curved folding hasless flexibility than the original material. This is because the change in the position and orientationof a ruling is not allowed in the rigid origami structures.
The mechanism of the structure is characterized by the number of degrees of freedom (DOF). Thisis generically calculated by the number of variables (fold angle of each edge) and constraints (eachvertex producing three constraints). Therefore an interior vertex composed of four edges produces aone-DOF mechanism, i.e., one of the folding angles determines the entire configuration of the fourfoldlines. Therefore, a planar quadrangle mesh composed of degree-4 vertices yields the followingstructure.
1. A 2×n quadrangle array (a single crease curved folding) yields one-DOF mechanism (Fig. 4).
2. A m×n (where m>2) quadrangle array (a pattern with more than two creases) yields overconstrained static structure or a redundant one-DOF mechanism because fold angles are
multiply defined (Fig. 4).
The necessary and sufficient condition of the pattern for having a one-DOF mechanism is not
perfectly investigated. However, we can use special cases that the planar quad mesh produces
redundant constraints: when the crease pattern is flat-foldable and there is at least one 3D state [3];and when the panels are constructed from parallelograms to form identity constraints around eachfacet. Here, we will start from a single crease curved folding and then extend to general patterns
using similarity.

Fig. 4: One-DOF foldable mechanism (left) vs. over constrained mechanism (right).
4.1

Single Curve

As described before, a single curved folding that can be constructed from a space polyline can bediscretized to form a one-DOF mechanism (Fig. 5 ). We can observe that the curvature and torsionof the form increase by folding. This can be understood as the change in parameters of Equation (2)and (5) when ruling and crease pattern parameters  and  2 D are fixed.
Since the curvature only increases as the surface folds, structure from a smooth curve generally tries
to roll up as shown in Fig. 5 left. This often causes collision between facets for a general curve, andthe folding cannot go up to actual flat-folding even though flat-foldability of the pattern in isometricsense is satisfied.

Fig. 5: Folding sequence of the structure from single curve. Solid model based on thick panel
construction technique [5].
One variational form that allows actual flat-folding and thus compact packaging is to use a zigzagged line (alternating curvature) instead of polyline that directly follows a given curve (Fig. 6).
The zig-zagged line is constructed from a directly sampled polyline by 1) constructing the frames
for the polyline and 2) moving vertices alternately along the curvature vector.

Fig. 7: Folding sequence of the zig-zagged structure. The folded form approximates the original
space curve drawn in red.
4.2

Planar Tessellation

We can extend the structure from 2×n to m×n (where m>2) by constructing patterns that avoids
contradiction between the mechanisms of adjacent vertices. Such a pattern can be constructed byusing parallel lines to copy and duplicate adjacent vertex structures. Fig. 8 shows how the array canbe extended in width direction. 1) First, trim the developable surface so that every perimeter edge isparallel to adjacent crease edge. 2) Then, construct quadrangle strips so that every other strip has
the parallel sequence of rulings vectors. Because of the similarity of vertices, we can assign
consistent folding angle function to each crease. The resulting structure is a generalized Miura-ori
surface that are bent and twisted along the curve (Fig. 9).

Fig. 8: Construction of planar tessellation with rigid-foldability.
4.3

Cylindrical and Cellular Structures

Our method can also extend to cylindrical and cellular structures since the folded strips satisfy flatfoldability condition necessary for the construction of cylindrical structures [5]. A cylinder can beconstructed similarly to the planar tessellation using parallel lines. We chose the inverse direction
for attaching a strip then we can loop back to the original strip. Since the section is a parallelogram,we can obtain a transformable cylindrical structure. Additionally, if we construct two surfaces inboth directions at each time, we can obtain a cellular structure that fills three-dimensional volume(Fig. 10). Fig. 11 shows an example design of deployable cylindrical structure.

Fig. 9: Tessellated Rigid-Foldable Structure.

Fig. 10: Constructing Cylindrical and Cellular Structures.

Fig. 11: Example Cylindrical Structure.

5. Conclusion
We have shown a family of design methods of deployable linkages with a one-DOF motion
constructed from single space curve. We enabled a compactly packageable geometric structure, byusing the constant angle curved folding that can be interpreted as a flat-foldable pattern. The basic
method can extend to corrugated structures based on zigzagged space curves, planar tessellation
exhibiting Miura-ori like form, and cylindrical and cellular collapsible structures whose shapes arefreely controlled by space curves.

Acknowledgement
This research is supported by KAKENHI (22800009), Grant-in-Aid for Research Activity Start-up
by Japan Society for the Promotion of Science. The geometric construction and the animation areprogrammed using Grasshopper [12].

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