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Computer-Aided Design and Applications© 2009 CAD Solutions, LLC
http://www.cadanda.com

A Design Method for 3D Origami Based on Rotational Sweep
Jun Mitani
University of Tsukuba/JST, mitani@cs.tsukuba.ac.jp
ABSTRACTIn this paper, a method for designing origami in which shape is based on the 3D
geometry generated by the rotational sweep that is commonly used in CG and CAD
systems is proposed. The target shape is designed by rotating a two-dimensional
polyline around an axis by a unit of 360/N degrees (N > 2). To make a shape with asingle sheet of paper, appropriate flaps are placed in between the constituent
polygonal faces that constitute the elements of the target shape. The proposed
method generates a crease pattern, i.e. a specification document consisting ofmountain/valley fold lines, for folding the origami. Because the shape becomessmoother if we set the input polyline to be finer and N larger, this method can also
produce approximate designs for three-dimensional curved origami. This paper
describes two principal types of rotational shape origami construction; cylindrical andconical. The origami are folded from a rectangle and a regular N-gon sheet of paper,
respectively. The proposed method was implemented on a PC and its validity wasdemonstrated by constructing several 3D origami.
Keywords: origami, crease pattern, rotational shape.
DOI: 10.3722/cadaps.2009.69-79
1. INTRODUCTION
Origami is a form of amusement and art that involves folding paper. In origami, a sheet of paper canbe folded to form a variety of shapes. Recently, because it has become common to use computerswhen designing origami, sophisticated artworks that were previously impossible have been created.
While there are various genres in origami, this paper explains a method for designing origami withshapes based on three-dimensional (3D) geometry by means of a rotational sweep that is commonlyused in CG and CAD systems. The target shape is designed by rotating a two-dimensional (2D) polyline
around an axis in 360/N degree (N > 2) intervals. Two typical forms of rotationally symmetric origami(cylindrical and conical) are discussed. Fig. 1 and Fig. 2 show simple examples. Fig. 1 shows acylindrical-type origami; Fig. 2 shows a conical-type origami. In each figure, (a) is the photo of theorigami work made by folding an actual sheet of paper while (b) describes the crease pattern. In thispaper, the red and blue lines in the various crease patterns denote mountain and valley folds,
respectively. The shape of the sheet of paper for a cylindrical origami is a rectangle (Fig. 1(b)), and a
regular N-gon for a conical origami (Fig. 2(b)). While it is normally agreed that an origami shape should
be designed using just a single square, this restriction was not applied. Instead, in this paper, it was
assumed that the most important property of origami making is creating a shape from a sheet ofpaper without tears and/or distortions.

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(a)

(b)
Fig. 1: (a) Photo of the cylindrical-type origami. (b) The crease pattern.

(a)(b)
Fig. 2: (a) Photo of the conical-type origami. (b) The crease pattern.
Although some conical-type origami works have been created in the past, and photos of such work canbe seen on the Internet [7], no documents that describe a method of designing the shapes, and/or
algorithms explaining how to generate their crease patterns, could be identified. This paper presents
algorithms for generating the crease pattern of the target shape from an input polyline in two ways,
after which examples that were designed using this system will be presented.
Basically, the surfaces that can be made from a single sheet of paper without tears and distortions arelimited only to developable surfaces such as a cone, cylinder, plane, and tangent developable surfaces.
These surfaces contain straight-line elements, and have zero Gaussian curvature. On the other hand, it
is possible for surfaces that are generated by rotational sweep to have positive or negative Gaussian
curvature. Hence, the method required to make an arbitrary shape that is generated by rotational
sweep, using a single sheet of paper, is not straightforward.
In the proposed method, appropriate flaps are placed in between the constituent polygonal faces tomake it possible to represent surfaces of non-zero Gaussian curvature using a single sheet of paper.
Although the target shapes are represented by a set of polygons, the shape approximates a set of
curved surfaces by making the generator polyline (input by a user) smoother and the divisor N larger.
In this paper, the concept of making crease patterns used for generalized rotational shape origami is
described, and methods for making two standard types of crease patterns (cylindrical and conical), by
adopting the general idea behind the method, are proposed. The proposed method enables users todesign new 3D origami with a simple interface, such as specifying a generator polyline, by clicking ona screen. This method is applicable for designing paper art objects or industrially useful items such as
gift boxes and lampshades. By using the system to visualize the resultant shape as a 3D CG image
before folding an actual a sheet of paper, the time used for trial and error can be considerably reduced.
In section 2, related studies are introduced. In section 3, the generation of a crease pattern for arotational sweep object as a preliminary is described. The details of the proposed method aredescribed in section 4. The results are shown in section 5. The proposed system and some extensions
are presented in section 6. Finally, conclusions are presented in section 7.

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2. RELATED WORK
By folding a piece of paper, various shapes can be produced. Many studies have investigated thegeometry of origami. Recently, origami has been recognized as a topic for scientific study since it has
wide applications in such fields as robotics, architecture, and industrial packaging (see the recent bookby Demaine and O'Rourke [2] for a comprehensive survey).
As techniques for designing crease patterns of complicated shapes using computers are being studied,some useful applications have appeared. For example, an application “Tess” [1], which generates crease
patterns for an Origami Tessellation that involves a twist fold in a repeating pattern, is available on theInternet. An application “Tree Maker” [6], which is also available on the Internet, generates crease
patterns of origami bases from a stick figure (the “tree”). This application aids designers in the creationof a crease pattern for the basic structure of the target shape. The target shape can be complicated (e.g.,shapes of animals, including insects). In contrast with most studies (including the two abovementioned applications), which focus on flat foldable origami, the target of this present study is 3D
structured origami. While designing 3D origami is a difficult feat, designing 3D origami with curved
surfaces is even more difficult. According to Demaine and O'Rourke [2], “little is known in the curved
origami”. Although there are few studies about curved origami, Huffman [4], Fuchs and Tabachnikov[3] studied the basic theory of curved folding. Recently, Killan et. al. [5] have proposed a method forconstructing a curved origami shape using a computer. The method utilized in this paper makes it
possible to generate a 3D shape as a set of polygons, which in turn makes possible curved origami withdesigns created using small polygons.
The “Origamizer”, a software program developed by Tachi [9], can generate a crease pattern for an
arbitrary polygonal model that has the topology of a disk. The outstanding advantage of his software is
its wide applicability as most 3D shapes can be modified so that they have the topology of a disk,
simply by adding some cuts. Using this software, Tachi managed to generate crease patterns for very
complicated 3D shapes [8]. The basic idea of his approach is to place clearances between each polygonof the unfolded pattern and to construct flaps from it behind the target shape; he called this “tucking”.
Although the Origamizer works for rotational shapes as well, the generated patterns tend to be overly
complicated for a simple model. For example, Fig. 3 shows a simple polygonal model and the crease
pattern generated by Origamizer. Though the appearance of the folded origami becomes equivalent tothat of the target shape by hiding flaps behind the shape (inside the model), it might be thought thatthe pattern is too complicated. The method described herein, on the other hand, generates a simple
pattern by making flaps placed in the foreground of the target shape as shown in Fig. 1 and Fig. 2. Even
though the flaps are essentially unnecessary for representing the target shape, and might beconsidered obtrusive, this approach results in the construction of 3D curved origami from simple
crease patterns.

Fig. 3: 3D polygonal model (left) and the crease pattern (right) generated with Origamizer [9].
3. PRELIMINARY
Before describing the details of the proposed method, the basic idea of making crease patterns for a
simple rotational shape will be explained. Since the target model is generated by rotating a polyline
around an axis in 360/N degree intervals, the model consists of a set of trapezoids. Fig. 4(a) shows a
simple example which is generated with the simplest polyline, which has one line segment, and N=6.
The unfolded pattern of this model, which has 6 trapezoids, becomes Fig. 4(b). The two dots indicatethat the two edges are identified when the design is folded.
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(a)
(b)
Fig. 4: (a) An example of a simple target shape, (b) The unfolded pattern for (a).
In this paper, two different approaches to construct rotational shapes from a single sheet of paper areproposed. The first involves a rectangle, resulting in origami of cylindrical type. The second involves an
N-gon disk, resulting in origami of conical type. We consider that an origami of cylindrical type is madefrom a cylinder by filling the gap between the target shape and a cylinder by adding adequate flaps. A
conical type origami is made using a cone such that the sum of the angles around the apex is 360
degrees (expressed more precisely, it is a “flat disk”). For example, when we consider making the shapeof Fig. 4(a) from a rectangle (cylindrical type) and a hexagon (conical type), the pattern becomes asshown in Fig. 5(a) and Fig. 5(b), respectively. The unnecessary area colored in light gray interleaved
between two trapezoids is folded as flaps. The folded shape for the cylindrical version is indicated inFig. 6. The shape for the conical version is almost the same, except for the shape of the flaps.

(a)(b)
Fig. 5: Crease patterns of Fig. 4(a): (a) cylindrical type, (b) equivalent conical type.

Fig. 6: Folded shape with flaps.
As shown in Fig. 7(a), a flap that contains a point P in the folded origami can rotate freely, even to theinside of the shape, around the edge AB located between neighboring two trapezoids, F0 and F1.
Although the vertex P can move freely when the model has only a single stage, this is not the casewhen the model has multiple stages, as is shown in Fig. 7(b). The vertex P is restricted so that it lies on
the plane that contains one of the trapezoids. This is because there are only two common points
between the loci generated by rotating P around the axes AB and BC. Although there are two possible
shapes as shown in Fig. 7(b), no effort is made to distinguish between these two in the following
explanations because one is simply a mirror symmetric version of the other.

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P

F0

C

C

B

B

F1

F0

A

P
F1

A

B

P
F0

F1

A

(a)
(b)
Fig. 7: (a) a flap can rotate when there is only a single stage, (b) a flap is restrained to one of two
positions when there are two stages.
When we take the axis of rotational sweep to be vertical, the horizontal cross section of the folded
shape becomes as shown in Fig. 8. When we express the total length of the girth of the cross sectionw
including flaps as Nw, the size of a flap b at the cross section can be expressed as b   r sin , where2Nr is the radius of the circumcircle, since we have a  2b  w and a  2r sin. When the target is ofN
cylindrical type, w is constant.

a

b

b

r

Fig. 8: Cross section of a target shape with flaps. r is the radius of the circumcircle. When the total
length of the girth of the cross section including flaps is Nw, a and b are defined with N, w, and r.
4. METHOD
Based on the idea described in the previous section, the algorithms used to generate crease patterns
from an input polyline are introduced for a cylindrical-type origami and a conical-type origami.
4.1 Input
We assume that the input is a 2D polyline, i.e., a sequence of n vertices, P0 ,  , Pn1 , that has n-1 line
segments as shown in Fig. 9. For the convenience of later explanation, we define l n  Pn1  Pn . The
target shape is generated by rotating the polyline around the y axis by a unit of 360/N degrees. For aconical-type origami, P0 should lie on the rotational axis.
4.2 Coordinates of Vertices on the Crease Pattern
4.2.1 Cylindrical-type
Since the same pattern appears repeatedly N times (see Fig. 1(b)) in the cylindrical crease pattern, weonly have to consider the crease pattern inside a single unit rectangle, a rectangle of width w. Then we
can obtain the complete crease pattern simply by repeating the unit rectangle N times in the directionof the x axis. P'n , which is the position in the unit rectangle corresponding to Pn (see Fig. 10), andwhich is defined using the following equations.
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y

Pn+1

Pn

Pn-1

P1
l0

P0

x

Fig. 9: Input polyline on 2D is defined as a sequence of n vertices, P0 ,  , Pn1 .
w
 Pn. x sin
2N0 (n  0)
P'n, y  22
 P' n1, y  l n1  {(a n  bn )  (a n1  bn1 )} (n  0)
P'n, x and P'n, y denote the x coordinate and the y coordinate of the vertex P'n in the crease pattern
P 'n, x  an  bn coordinates respectively. Pn, x and Pn, y denote the x coordinate and the y coordinate of vertex Pn in the
input coordinate system, respectively. an and bn correspond to a and b in Fig. 8 on the cross section at
Pn . As described by the above equations, P'n, x can be identified simply with Pn, x . On the other hand,
for all n because the size ofN
the flap bn cannot be negative. This means that the smallest value for w is max( 2 Pn, x sin ) .
N
P'n, y is identified with P' n1, y and l n1 . w has to be larger than a n  2 Pn, x sin

ya n+1 +b n+1 b n+1
P’n+1
a n+bnbn
P’n

a n-1+b n-1

b n-1

P’n-1

w

x

Fig. 9: Coordinates of each vertex in the unit of a crease pattern of cylindrical type.

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4.2.2 Conical-type
For a conical-type origami, the shape of a unit becomes an isosceles triangle with an apex angle of360/N degrees. Similar to the cylindrical-type, the whole crease pattern can be obtained by repeating
the unit N times around the apex. When we consider the coordinate system shown in Fig. 11, thecoordinates of each point P'n are represented by the following equations.
0 (n  0)
P' n, x  a n a n1 2
2)
 P' n 1, x  l n 1  ( 
2
2

a

P ' n, y  n  Pn. x sin
2N( n  0)yl0
P’1 a1/2

P’0

P’2

l1

a2 /2

x

Fig. 11: Coordinates of vertices in the unit of a crease pattern of conical type.
Since the x coordinate of the first vertex of an input polyline P'n is zero as described in Section 4.1,
P ' 0 is placed at the origin. P' n , y can be identified simply with Pn, x . On the other hand, P' n, x is identified

with P' n 1, x and l n1 .
4.3 Mountain/Valley Flags of Folding Lines
To fold origami correctly, the mountain/valley flag must be set for each folding line on the crease
pattern. In Fig. 12, which shows a part of the folded shape, mountain lines and valley lines are coloredin red and blue, respectively. The folding lines used for making ridges of the flaps (AB and BC in Fig.
12) are flagged as mountain. On the other hand, the folding lines between a flap and a trapezoid (ADand DC in Fig. 12) are flagged as valley. As a result, the flags of these lines are automatically defined.
Contrariwise, the flags of horizontal folding lines (EB and DF in Fig. 12) alter depending on the angle
between vertically connected trapezoids. This can be defined by seeing the angle between twoconnected line segments in the input polyline.

C
C
E

D

F0

F
B
P
F1

A

Fig. 12: Flags of folding lines. Mountain lines and valley lines are colored in red and blue, respectively.
4.4 3D Geometric Model
The appearance of the folded origami is not actually the same as that generated by rotating the input
polyline because additional flaps are placed outside it. Thus, previewing the folded shape is useful in
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reducing trial and error. To do this, it is necessary to construct a 3D geometric model on a PC. This isachieved by a simple program. After the rotated shape is generated with a common rotational sweep
algorithm used in standard CG software, flaps are added so that they lie on planes containing theneighboring trapezoid. The size of a flap is determined by the value bn described in the Preliminary
section.
5. RESULTWe implemented the proposed methods using a PC and found that users can design origami shapes byspecifying a polyline. The polyline is specified by simply clicking points in the application window. The
system automatically generates a crease pattern and a corresponding 3D model based on thealgorithms introduced in the previous section. When the user is satisfied with the result, the crease
pattern can then be printed and folded. A number of cylindrical- and conical-type are presented in Fig.
13 and Fig. 14, respectively. In the figures, the input polyline, the constructed 3D geometry, the crease
pattern, and the photo of the folded origami are shown from left to right. N is set to 8 for all examples
except Fig. 14(a), where N is set to 12. The input polylines for Fig. 13(b) and Fig. 13(c) are generated by
calculation so that the target shape becomes a mathematically appropriate hyperboloid and sphere.
Others are input manually.

(a)

(b)

(c)
Fig. 13: Cylindrical-type examples: (a) bell, (b) hyperboloid, (c) sphere. From left to right: input, theconstructed 3D geometry, the crease pattern, and the photo of folded origami.

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(a)

(b)
Fig. 14: Conical-type examples: (a) vase. (b) chestnut. From left to right: input, the constructed 3D
geometry, the crease pattern, and the photo of folded origami.
6. DISCUSSION
6.1 Approximation to Curved Origami
The shapes generated with the proposed method consist of a set of polygons. By making N larger andthe input polyline finer, the shape becomes smoother. Although it is difficult to fold the origami byhand if N is large, making the input polyline finer does not change the difficulty significantly. In fact,
we can omit the horizontal folding lines if the folding angles are close to 180 degrees. Hence, a smooth
input polyline simplifies the folding task. For example, Fig. 13(b) and Fig. 13(c) can be folded without
horizontal lines by assuming that the model consists of curved surfaces.
6.2 Turning Points
As indicated in Fig. 7(b), the flaps can be positioned in two ways. A particular choice can be switched at
the point where b (in Fig. 8) is zero. This is called the turning point. If we see the flaps as part of thedesign, this choice can add variations to the shapes obtained from the same input polyline. For
example, Fig. 15(b) and 15(d) shows variations of Fig. 15(a) and 15(c), respectively. The position of theflaps located on the upper half of Fig. 15(a), and the position of flaps located on the middle two stagesof Fig. 15(c) are flipped.
Basically, the coordinates of the y axis (the axis of rotation) of the vertices of the input polyline have to
be monotonically increasing or decreasing, i.e. Pn1, y  Pn, y (1  n  N ) or Pn1, y  Pn, y (1  n  N ) , in order toavoid penetration of the paper. Fig. 16 shows an example in which there are penetrations. The
polygonal faces of generated shapes are colored so that the penetrations can be seen clearly. It can beseen that the red face penetrates the blue face. However, the direction of the input polyline along the yaxis can be flipped at the turning point without penetrations because the width of the flap at that point
becomes zero. This fact increases the number of possible variations in the design. For example, ashape such as Fig. 17 is possible.

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(a)
(b)
(c)
(d)
Fig. 15: Flaps on the upper half of (a) are flipped in (b). Flaps in the middle two stages of (c) are flippedin (d).

P0
P2

P1
(a)
(b)
Fig. 16: (a) Input polyline, (b) Generated shape from (a) with N=6.

P0
P1
P3

P2
(a)
(b)
Fig. 17: (a) Input polyline. The direction of the line segment (along the y axis) is changed at P2 , (b)Generated shape from (a) with N=8.
6.3 Crossing the Axis of Rotation
Generally speaking, the input polyline should be located in one side against the axis of rotation, i.e. thepolyline should not cross the axis, when the rotational shape is designed. However, this becomes
possible in the proposed system. As shown in Fig. 18, the cylinder closes where the polyline crosses
the axis because the value of a n (in Fig. 11) becomes zero. This makes it possible to design interesting
shapes. Furthermore, a shape with this closing makes the model more stable. This scheme is used in
the examples shown in Fig. 13(a) and 13(c).

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(a)
(b)
Fig. 18: (a) input polyline that crosses the axis, (b) generated shape from (a) with N=8.
7. CONCLUSION
In this paper, a method for rotational origami has been proposed and details regarding how to
generate the crease pattern from an input polyline have been described. Additionally, a system that
implements the proposed method was developed and the validity of the proposed method wasevaluated by presenting some origami models designed with the system. Furthermore, it was shownthat approximated curved origami could be designed by making the input polyline smooth. It was also
shown that, by taking account of the turning points, additional variations could be generated. Thismethod will be applicable to the design of paper art objects or industrially useful items such as giftboxes, paper cups, lampshades, and so on.
8. REFERENCES[1]
Bateman, A.: Tess, http://www.papermosaics.co.uk/software.html.
[2]
Demaine, E.; O’Rourke, J.: Geometric folding algorithms, Cambridge university press, 2007.

[3]
[4]
[5]
[6]
[7]
[8]
[9]

Fuchs, D.; Tabachnikov, S.: More on Paperfolding, The American Mathematical Monthly, 106(1), 1999,
27-35.
Huffman, D. A.: Curvature and Creases: A Primer on Paper, Transactions on Computers, C-25(10), 1976,
1010 – 1019.
Kilian, M.; Flöry, S.: Chen, Z.; Mitra, N. J.; Sheffer, A.: Pottmann, H.: Curved Folding, ACM
Transactions on Graphics, 27(3), #75, 1-9, 2008.
Lang, R. J.:TreeMaker, http://www.langorigami.com/science/treemaker/treemaker5.php4.
Flickr: curved fold, http://www.flickr.com/groups/783444@N25/.
Tachi, T.: 3D Origami Design based on Tucking Molecule, The Fourth International Conference on
Origami in Science, Mathematics, and Education, to appear.
Tachi, T.: Origamizer, http://www.tsg.ne.jp/TT/software/index.html#origamizer.

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