Proceedings of the First Conference
Transformables 2013. In the Honor of Emilio Perez Piñero
18th-20th September, 2013, School of Architecture, Seville, SpainEDITORIAL STARBOOKS. Felix Escrig and Jose Sanchez (eds.)

Composite Rigid-Foldable Curved Origami Structure
Tomohiro Tachi
Assistant Professor, The University of Tokyo, Tokyo, Japan, tachi@idea.c.u-tokyo.ac.jp
Summary: In this study, we show a family of multilayered rigid-foldable and flat-foldable vault structures. A vault can be designed by constructing arigid-foldable curved folded tubular arch and assembling the arches to construct a multilayered surface. The resulting vault is also rigidly flat-foldable.
We first show the process involved in the geometric construction of the tubular structure from a given space curve defined by curvature and torsion
functions. Next, we show a single-DOF rigid-folding motion that is enabled by the curved folded tube. Finally, we show the parametric design of avault structure and discuss its design parameters and the resulting form.

Keywords: Deployable Structure, Rigid Origami, Curved Foldingof smooth curved folding, and (2) the geometricconsideration of the change in the proportion of the tube’ssection leading to the parametric design of the vault
structure with different section curves.

INTRODUCTION
Rigid-foldable origami, a mechanism comprising rigid
panels and hinges, can be used for designing deployable
shells. For large-scale implementation of such amechanism, it is important to consider the thickness, notonly that of a single panel but also the total depth ofcomposite structures to increase the structure’s stiffness
and insulate the interior from heat and sound. A general
method for adding thickness to panels of rigid origami [1]involves a tradeoff between the thickness of panels andthe maximum fold angles; thus, the structure cannot befolded compactly when the panels are thick. An
alternative approach is to construct a tessellation ofcellular structures comprising thin panels and thereby
acquire virtual thickness. This approach allows efficientcompact folding in a continuous one-DOF motion. To
achieve this, the overall structure must be carefully
designed such that it maintains the mechanism tocontinuously fold flat. Symmetric cylindrical rigidfoldable modules have been used to construct a cellular
structure in [2]. However, in the previous studies, there
were problems in the design and construction processes:(1) design variations relied on translational symmetry, and(2) the number of panels tends to be very large.

First, we review the geometric construction of the folding
structure generated from a space curve and discuss its
kinematics in a cylindrical form. Next, we show theparametric design of the assembled vault and discuss themorphological characteristics related to its design
parameters.
GEOMETRIC CONSTRUCTION
Single Curved Folding
First, we review the basic geometry of the curved folding
structure generated from a space curve. Consider ageneric space curve
defined by the boundary
0 and its curvature and torsion
conditions 0 andfunctions
and, respectively, whereparameterizes the space curve by its arc length. Next, weobtain a valid continuous Frenet–Serret frame composedoftangent,normal,andbinormal
vectors,,′ , ′′ / , ′′′ /by integrating the curvature and torsion
functions using the Frenet–Serret formula.

In this study, we solve these two problems by assemblingrigid-foldable flat-foldable cylindrical structures using
curved folding, a developable surface with smoothly
curved creases. Unlike ordinary polyhedral origami,curved folding uses the bending of the surface for form
development. As a result, the number of foldlines isreduced, making the folding-based fabrication easier
while keeping the resulting 3D form complex and rich.

0000.

(1)

0

It is known that if a continuous valid frame is defined, we
can construct a curved folding whose crease lies on thespace curve (Fig. 01). We can find design examples inearly computer graphics shading images by Resch [4].

The design process is based on the method proposed in[3]. We generate a family of tubular structures from acommon space curve so that they rigidly fold consistently
even as they are assembled. In this study, we extend themethod to smooth curved folding without tiny rigidpanels, to improve constructability. The contributions of
this study include (1) the use of a tube for the construction
module for avoiding any unwanted kinematic flexibility

1first (L+R+) or third (L−R−) quadrants, a curved folding,i.e., a developable creased surface, can be constructed.
From the second (L−R+) or fourth (L+R−) quadrants, acurved seam, i.e., a flat-foldable creased surface
constructed from a two-ply sheet welded at the curve, canbe obtained.
Modular Tube
The tubular and cellular structures are constructed in thefollowing manner. First, we calculate a curved folding in
the first quadrant (L+R+) with the constant folding angle(2. ∈ 0,180° ) from a given space curve.
At the constant folding angle, the crease pattern ischaracterized by the reflecting ruling lines forming angles).
with the curved crease (Fig. 01. Space curve and curved folding.

A variety of curved folded surfaces can be constructedfrom the same space curve. By defining the folding angle
along the curve denoted by 2, we can uniquelydetermine the folded surface attached to the curve. Here,is calculated as the angle from the osculating planefacing the binormal vector(Fig. 02) to the tangent
plane of the surface. The geometric relationship between
the curves and the angles are summarized by thefollowing equations, as investigated by Fuchs andTabachnikov [5].
cos,

cot

cot

cot

cot,

(3)
,

(4)

where the curvature of the crease in the crease pattern isdenoted byand the angles between the tangent
and the rulings of the two surfaces are denoted by, respectively.
andFig. 02. Parameters (Left: Crease pattern, Right: Folded form inperspective and orthogonal projection along vector T.

cos

sin

cos

sin

sin,

cos

sin

cos

sin

sin. (6)From the derived curve, we build a curved unfolding inthe second quadrant (L+R−) using the same folding angle. Because the two curves share the same frame, they also
share the same ruling vectors, ,and,.
This
ensuresthatthegeneratedsurfaces,,
overlap the right-hand side of the previous crease to form
a valid two-crease surface (Fig. 04).

(5)

The normal vectors of the surfaces are represented asfollows:
cos

sin

cos,.

(9)

Fig. 03. Construction of curves 2 and 4, indicating the change in theproportion.

Therefore, the ruling vectors are described as follows:sin.

Next, we draw another curve on one side (say, the righthand side) of the attached surfaces such that thecorresponding tangent vectors of curves 1 (the originalcurve) and 2 (the new curve) are parallel (Fig. 03). The
correspondence between the arc length parameters ofthese curves sharing the common ruling line is described
and, respectively, using a commonby≡
parameter ∈ 0,1 . The curves satisfy, where the tangent vector in curve is denoted
00 and1 equals the length ofby . Here,curve . This construction ensures that curves 1 and 2
have the same frame of , ,at the corresponding
and, respectively. The new curve ispositions
uniquely determined if its endpoint is fixed. Here, thewidth between the curves at
0 is denoted by|ℓ | sin , where ℓ is the length of theruling line between curves 1 and 2.

(2)

cotcot(7)(8)

Here, four types of geometrically equivalent creased
surfaces can be constructed from the same space curve
and the set of ruling vectors, by choosing one from thefour quadrants obtained by extending the ruling vectors inboth positive and negative directions (Fig. 02). From the2Proceedings of the First Conference
Transformables 2013. In the Honor of Emilio Perez Piñero
18th-20th September, 2013, School of Architecture, Seville, SpainEDITORIAL STARBOOKS. Felix Escrig and Jose Sanchez (eds.)

structures behave exactly in the same manner as the basic
quadrangle tube.

Fig. 06. Derived curve is defined by two parameters
sequence does not matter).

Fig. 04. Construction of the tube.

We continue constructing curves from the previously
derived surfaces in the same manner, choosing the third
quadrant (L− R−) and, thereafter, the fourth quadrant (L−R+). Here, we choose|||,|||,and(theWidth Proportion
From Fig. 03, we can see that the widths between thecurves (the distance between the corresponding tangent
sin ℓ ) change with respect to thelines, i.e., w
parameter t as a result of the construction method basedon parallel tangent vectors. The change in the width isdescribed as(10)

so that curves 5 and 1 lie on each other to form a valid
tubular surface. Equation (10) also ensures that theconstructed tube is flat-foldable. In other words, the tube
can be constructed by welding two sheets and folding it
up along the center creases (Fig. 05). There also exits acontinuous folding motion, which is discussed later.

cot,

,

(12)

cot,.

(13)

This implies that a curve with torsion of the same sign,e.g., a helix, yields a curved folding with the proportion ofto, increasing or decreasing monotonically withrespect to . An interesting aspect is that the areaof the section parallelogram is constant withrespect to .
Regression
The construction step of a new curve assumes that it lies
within the developable surface. However, because ageneric (non-cylindrical) developable surface has either amustcurve of regression or cone apex, the width |be set such that /1, whereis the width
between the crease and the curve of regression.
iscalculated as

Fig. 05. Flattening the tube.

Subsequently, we can construct a tube with arbitrarilyand ∗ | . Therefore, any
scaled widths of ∗ |of the derived three creases
with respect to theoriginal curve can be described by the linear combination
and ℓasof two vector functions ℓℓℓ,.

(14)

.

(15)

Similarly,(11)

Thus, the speed functions of the rulings can berepresented as follows:

wheredescribes the original curve and and arereal number parameters. This means that a set of anduniquely determines the curve, and the construction
sequence can be arbitrarily set as long as the lengths ofsegments along sum up to , and those along sumup to (Fig. 06). This makes sure that complex cellular

1.

(16)

The linearity can be exploited to obtain a global limit forthe cylindrical and cellular structures that can beconstructed from a given space curve. Assuming we get a3

curve from a sequence of construction steps, which arerepresented by two parameters and in Equation (11),then1.

In contrast, the drifting motion can be restricted in thetubular structure. We similarly parameterize theconfiguration of the tube by ruling alignment , : →0 (1, ⋯ ,4 . However, in this case, theandparameters cannot be independently determined because
if each crease is separated by adding a cut in the middle ofthe developable strips connecting the creases, the cut
boundary does not match with the adjacent one when
0 are arbitrarily given. To construct aand,consistent surface shared by adjacent creases and1,
the curvature of the surface relative to the speed function,,andmust be identical:,
which can be represented as(17)This produces a regression line for parameter .
1.

(18)

This forms a ruled surface boundary across upon which
the space curve cannot lie, thereby defining a domain in
thespace that results in a valid curve.

,,,

1,,

whereandrefer to,,
respectively, if is odd, and , andif is even.

,(19)

and, ,,respectively,,In addition, to maintain the homeomorphism to a cylinder,
the closure condition must be assigned. Assuming that theboundary curves of the tube are kept straight, theboundary follows a sheering motion keeping
0000 .

Fig. 07. Regression

To sum up, the motion is parameterized by a single
parameter
0 and four variations of the rulingalignment functions, which are constrained by four,
identities given by Equation (19). This implies that in ageneric case, there is sufficient number of constraints to
0 is fixed. We conjecture that if
stabilize the tube if
the boundary curves are straight quads and no cone apex
exists on the surface, rigid folding motion will be the only
motion available; however, this has not been proven. Here,the former assumption can be realized by reinforcementof the boundary whereas the latter is the result of thephysical behavior of a thin elastic sheet that stores aninfinite elastic energy at the cone apex.

KINEMATICS
Single Curved Folding vs. Tube Structure
In general, a curved folding can transform in two ways:
folding without a change in the rulings position (0), and twisting with a change in the rulings (Fig.
08). We call former motion “rigid folding” of the curved
fold because it is the limit of the rigid folding motion of adiscrete quad panel folding as the panels are infinitelysubdivided. The latter motion is also a valid motion
without stretching of the material and is unique to thesmooth curved folding. We can parameterize theconfiguration using the continuous rulings alignment
mappings→and→between the, :, :boundary curves 0 and 2, the crease curve 1, and thefolding angle at the start point
0 . ,uniquelydetermines
andand Equations (3) and (4) give a
differential equation, through which the folding angle
and torsion
are uniquely determined. Any ofandthe variations,, corresponds to a validtransformation with the change in ruling alignment.
Therefore, a curved folding with a single curved crease
has excess degrees of freedom and is hard to control its
form and motion.

Tubes Can Rigidly Fold
In a rigid folding motion, intrinsic parameters such as, ,and
do not change. Therefore, the,≡δ≡δ≡folding motion with δ≡ δ satisfies Equation (19). For a given foldingδ
angle 2 ∈ 0,180° , the curve defined by the curvatureand torsion≡is≡the curve in a folded state, where the original space curve∈ 0,180° is defined by theat folding angle 2and torsion.
curvature
The cellular structure folds rigidly and flatly in the samemanner. This can be visualized as the parallel tangent
vectors and tangent planes of the ruled surfaces at acommon parameter orthogonally projected from thecommon tangent vector, which form a consistentpantograph mechanism.
COMPOSITE DEPLOYABLE VAULT

Fig. 08. Folding motion with drifting rulings.

We show designs of deployable shells using the proposed
tubular structures. The basic strategy for the design is to4

Proceedings of the First Conference
Transformables 2013. In the Honor of Emilio Perez Piñero
18th-20th September, 2013, School of Architecture, Seville, SpainEDITORIAL STARBOOKS. Felix Escrig and Jose Sanchez (eds.)

first construct a tubed arch from a curve and thenassemble the arches to form a composite vault.

Fig. 10 shows the design variations obtained by changing
parametersand . By adjusting these parameters,

We design the arch tube such that its boundary quads lie
on a common xy plane in a deployed state of 0 2
180° , say 290° . If this condition is satisfied, theconsecutive curves follow the same property and theboundary of the vault is also coplanar. The planarity of90° and thus,
each boundary implies that 000, where the total length of the curve isdenoted by . The coplanarity of the boundaries, ingeneral, requires the coincidence of four parameters, i.e.,
and0 ⋅ 00 . In0general, the tangent vectors and the positions arecomputed using numerical integration. Here, we show asubset of the design space by assuming reflection,. Insymmetry, i.e.,
this case, the coplanarity is simplified tothe slope of the sections at the xy and zx planes can bevaried, as seen in Fig. 10.

0.5

⋅

00.

The structure continuously folds to a flat state. Once thefolding angle is varied, the boundary curves do not lie on
the ground plane. This is attributed to Bellow’s Theorem,
which forbids rigidly collapsible, closed polyhedra.
CONCLUSION
In this study, we showed the geometric construction
method of curved folded tubular and cellular structuresusing a space curve. We discussed the folding kinematicsof the structure and showed a family of design variationsof the deployable vault, by assembling multiple arches.
The shape of the assembled vault can be controlled bycontrolling the amount of total torsion and the proportionof the rectangular section.

(20)

Typical Design

ACKNOWLEDGMENT

We built a parametric design system using Grasshopper tocalculate the curve, and the tubular and cellular structures.
The following shows a typical parametric design exampleof an assembled vault using space curves of constantcurvature and harmonic torsion.

This work was supported by the JST Presto program.
References[1] Tachi, T., “Rigid-Foldable Thick Origami”, inOrigami5, pp. 253-263, 2010.
[2] Tachi, T., Miura, K., “Rigid-Foldable Cylinders andCells”, Journal of the International Association forShell and Spatial Structures (IASS), 53(4), pp. 217226, 2012.
[3] Tachi, T., “One-DOF Rigid Foldable Structures fromSpace Curves”, in Proceedings of the IABSE-IASS
Symposium 2011, London, UK, September 20-23,2011.
[4] Resch R. D., “Portfolio of Shaded Computer Images”,
Proc. IEEE, Vol. 62, No. 4, pp. 496-502, 1974.
[5] Fuchs D., Tabachnikov S., “More on Paperfolding”.
The American Mathematical Monthly, Vol. 106, No.
1, pp. 27-35, 1999.

First, the curve is set such that it has constant curvatureand harmonic torsion
sin 2.
We calculate the curve and its frame by discretizing theiscurve by converting it into a polyline. Then,numerically determined using a line search algorithm tosatisfy Equation (20). We continue the construction oftubes so that they share the developable creases, i.e.,curve 3 of a certain tube is curve 1 of the following tube.
Equations (12) and (13) calculates the change in the width
proportion. Owing to the symmetry of the harmonic
function, the two end sections share a common
proportion
s0 , and the change in the
proportion is the maximum at the middle of the curve as
0.5
0 . Here, the change in the
proportion at the middle of the curve is an importantdesign parameter for the resulting vault as the aspect ratio
0 determines theof the rectangular section / atsection feature curve on a horizontal plane (xy plane),determines thewhereas the proportion at
0.5
section feature on a vertical plane (zx plane).
As an example, Fig. 09 shows a vault structure from acurve with0.3, in which Equation (20) is satisfied at
3.305, and thus, the change in the proportion is.
1.37 . We assemble tubes with rectangularsections of aspect ratio pp.
.

0.75 and obtain

1.41 at xz plane.

5Fig. 09. Folding motion of an example vault constructed form foldable tubes.

Fig. 10. Variations of the derived form; the slope of the section in the zx plane is indicated in blue and the slope in the xy plane is indicated in red.

6