International Journal of CAD/CAM Vol. 13, No. 2, pp. 23~29 (2013)

Interactive Design of 3D Geometry Made by Bending Inextensible Sheet Material
Containing Slits
Naoya Tsuruta1, Jun Mitani1,2, Yoshihiro Kanamori1 and Yukio Fukui1
1

Department of Computer Science, University of Tsukuba, JapanJST ERATO, Japan

2

Abstract − Most manufacturing materials, such as wooden boards, metal sheets, and paper, are planar and nearly inextensible.
Bending such materials not only allows us to form objects, but also avoids wastes and reduces gluing costs. However, theresulting shape is limited to developable surfaces. Due to this limitation, this method lacks flexibility in design. This problem
can be addressed by adding slits to the materials, thereby allowing the conformation of a single sheet into expressive surfaces
with high flexibility. In this paper, we propose an interactive design system for flexible geometry. A rectangular sheet material
containing parallel slits is used in this work. The regions between the slits are called strips, and by bending them, we cansimulate the desired shape. Both flexibility and geometric constraints are satisfied by a numerical optimization approach. The
proposed system optimizes a bending shape of the strips by considering the smoothness of the surface and the user input. We
show the effectiveness of our system using evaluation experiments.
Keywords: Geometric modeling, Developable surface, Constrained optimization, Sheet material, Containing slits

1. Introduction
Most well-known materials, such as wooden boards,
metal sheet, and paper, are planar and nearly inextensible. Forming an object from these materials bybending a single part avoids waste of the material andreduces the costs for joining parts. However, the shapeproduced by the bending is limited to the developablesurface. The shape thus lacks flexibility in the design.
Adding slits to the material is one approach to address
this problem. As an actual example, Naruse et al.
manually designed and created a chair from a wooden
board having parallel slits (Fig. 1).
It is remarkable that an expressive surface can beobtained from a single sheet of material by simply
adding slits. However, designing objects made with “asheet material with slits” has not been performed on acomputer. It is because the constraint, which is theunfolded pattern of the shape matching the predefined
one without any tearing and wrinkles, is difficult tosatisfy during the design process. Naruse et al. had toperform a number of trial and error processes with asheet of paper before obtaining the chair shown in Fig.
1. In this paper, we present a CAD system for designing
an object that can be reconstructed from a sheet
material with parallel slits without tearing and gluing.
*Corresponding author:
Tel: +81-29-853-5388
Fax: +81-29-853-5388
E-mail: tsuruta@npal.cs.tsukuba.ac.jp

The seating face of the chair consists of a set of strips.
We refer to the regions in the sheet material between
slits as ‘strips’ hereafter. The whole shape design
consists of problems on how each strip of the nearly
inextensible material can be shaped while maintaining
the constancy of the unfolded pattern. Thus, the lengthof the strip must be constant and the position of bothends of the strip must be constrained by other parts ofthe sheet material. Both ends of our target strip are
fixed so that it cannot move freely. Therefore, we haveto shape the strips under this constraint. Here, we
assume this problem as a constrained optimization problem and present an optimization-based shape modeling system. In Section 2, we introduce related work. In
Section 3, we show our proposed system. Then, we describe details of the optimization (3.1) and the modelingprocesses (3.2). We discuss our results in Section 4 byproviding several design examples and a fabricatedphysical object. We finally conclude in Section 5.

Fig. 1. Chair, designed by Naruse Inokuma Architects, made of arectangular wooden board having parallel slits [1].

24International Journal of CAD/CAM Vol.13, No. 2, pp. 23~29

2. Related work
Developability is important for products such as
papercrafts [2], stuffed toys [3] and garments [4], which
are cut out from sheets of paper or cloths. Developablesurfaces have valuable characteristics for industrial andarchitectural designs. Using developable surfaces inarchitectural design reduces the costs for construction[5], [6]. Another example can be found in the design ofa ship hull [7]. Methods that generate a set of developable patches from a closed space curve [8] and froma parametric curved surface [9] are available. These methods construct a whole shape with a set of developable
patches. In other words, the developable patches are
glued together along their boundaries. Our target shape,
which is a chair, is made from a single developable
sheet, and adding slits is the only process used.
Peternell proposed a method for reconstructing adevelopable surface that is fitted to 3D points measured
by a 3D scanner [10]. Kilian et al. reconstructed surfaces having folded creases from scanned 3D data using
an optimization-based approach [11]. Rohmer et al. proposed a method for reconstructing a folded paper geometry from a 2D pattern and a 3D contour curve [12].
In this paper, we propose an interactive design system.
Moreover, we extend the system so that a shape (achair) is automatically reconstructed from 3D scanned
data.
Mitani and Igarashi proposed an interactive design
system for origami pieces having curved folds, basedon mirror reflection [13]. Bo et al. proposed a systemwith which a user can interactively bend a sheet ofpaper [14]. These methods fix one end of the sheet
paper in 3D space and allow the other end to be moved.
This approach cannot be used for our target shapeformed from a sheet material with slits.

3. Proposed system
First, we define the target material as a planar rectangle containing one or more slits. The slits are paralleland of equal length. The shape of the material is symmetric and defined by four parameters, as illustrated inFig. 2. The designed 3D shape is also symmetric. Therefore, we only consider the half regions of the material in
the following. As illustrated in Fig. 3, a strip is divided
into M quads and has (M+1) ridge lines. A ridge line Ei
has three parameters, xL,i, xR,i, and θ i, where xL,i, xR,i are
the x coordinates of the upper and lower end points of
Ei, respectively, when the strip is oriented along the xaxis, and θ i is the dihedral angle of Ei. We name the two
end points of Ei as υL,i and vR,i. We assume that the firstedge E0 is fixed. Therefore, the shape of the strip isdetermined by the parameters associated with the ridge
lines. The proposed system calculates the values ofthese parameters to obtain the intended shape of the
strip while keeping the shape of the unfolded sheet

Fig. 2. Developed pattern defined by four numbers from L1 to L4
and number of strips N.

Fig. 3. Ridge lines before after optimization.

Fig. 4. Calculation of 3D and positions.

constant. We add two more constraints here so that the
strip becomes smooth and the user can control its shape.
To obtain the final shape of each strip as a solution ofan optimization problem, we define a penalty function.
We will describe details of the penalty function in thefollowing section. The proposed system solves anoptimization problem to minimize the value of thefunction.

3.1 Optimization
As the constraints, one end of the strip is located on
the symmetry plane, and the other end is fixed where
the user specifies. The constraint on the symmetry planeis given because we defined the target shape to besymmetric. The positions of υL,0 and υR,0 are defined by
the user. The positions of υL,i and υR,i (1 ≤ i ≤ M) in 3D
space are calculated by rotating the quad between Ei-1
and Ei by angle θ i around Ei-1. The positions υL,M andυR,M, which are the last to be obtained, have to be on thesymmetry plane, as illustrated in Fig. 4. We use an optimization approach to meet this constraint. Here, we
define the following penalty function for each strip.
Then, we solve the optimization problem that minimizes
the value of the function.
Fpenalty = w1Fno_gap + w2Fsmooth + w3Ffair + w4Fuser_control(1)

Naoya Tsuruta et al.

Interactive Design of 3D Geometry Made by Bending INextensible Sheet Material Containing Slits 25

We add terms in the function that makes the resulting
shape smooth and controllable. The components in
Fpenalty measure the connectivity at the symmetry planeof the strip, smoothness of the strip, fairness of thequads in the strip, and closeness to the user-specified
anchor points of the strip, respectively. To prevent theoccurrence of a gap at the symmetry plane, we set alarger value of ω1 in comparison with those of the otherweights.
To solve the optimization problem, we use a penalty
method with the Levenberg-Marquardt algorithm. The
derivative of the above equation is calculated numerically. Since the number of variable end points of theridge lines in a strip is 2(M−1), and the number ofvariable angles of the ridge lines in a strip is M, thenumber of whole variables becomes D = 2(M − 1) + M.
The coefficient matrix used in the optimization ishighly sparse, with a size of D × D. We describe thedetails of each component of the penalty function
below.
Position constraint for an end of a strip. The term
Fno_gap is the sum of the squared distances of υL,M andυR,M from the symmetry plane, as illustrated in Fig. 5.
Since the whole shape of the designed model issymmetric and obtained by mirroring the half model,
the value of this term indicates the size of the gap on a
strip at the center of the whole model. Ideally, this value
should be zero. We set the symmetry plane on the yz
coordinate plane. Then, the term is calculated by thefollowing equation.
Fno_gap: = (υL,M)x2 + (υR,M)x2(2)

Smoothness. The term Fsmooth is for smoothing a strip.
We measure the differences of the angles between
neighboring ridge lines. When the differences are small,the strip is smooth. Additionally, we introduce a term,
Fend_angle, which makes the strip touch the symmetry
plane perpendicularly, as illustrated in Fig. 6. This
makes the strip smooth at the plane of symmetry (thecenter of the whole shape). In the following equation,
υ p_o_s is a vector parallel to the symmetry plane. Since
the symmetry plane is on the yz coordinate plane, υ p_o_s= (0,1,0).
2

Fsmooth := ∑ (–θ i – 1 + 2θ i –θ i + 1) + Fend_anglei

Fend_angle = ((υL, M – 1 – υL, M) ⋅ υp_o_s)2

Elimination of degeneracy of quads. We measure
the distances between end points of neighboring ridge
lines to prevent crossing or degenerating of a quad into
a triangle (Fig. 7). The penalty value of the following
equation becomes larger when the distance between theend points of neighboring ridge lines becomes smaller.
1
1
Ffair: = ∑⎛⎝----------------------- + -----------------------⎞⎠
–
x
x
–
x
i
L, i
L, i – 1
R, i x R, i – 1(4)

User-specified anchors. The user controls the shapeby specifying two types of anchors: one directly controls the position of each vertex vL,i or vR,i, and the otherone controls the cross section on the symmetry plane(Fig. 8). The “position-control” anchor measures thesquared distance from vertex v to the user-specified
position υ'. This controls the shape of a strip locally.
The “cross-section” anchor measures the shortest distance from υL,M and υR,M to csymmetry, which is the cross-Fig. 5. Evaluation of constraint at end of strip.

Fig. 8. Evaluation of distances to user-specified anchors.

Fig. 6. Evaluation of smoothness.

Fig. 7. Evaluation of degeneracy of quads.

(3)

Fig. 9. Initial state.

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International Journal of CAD/CAM Vol.13, No. 2, pp. 23~29

sectional curve drawn on the symmetry plane by theuser. The curve controls the global shape of a strip. The
calculation of the distance from a point to a curve
requires iterative computation. Therefore, the curve isapproximated by small segments and the distance ismeasured from the nearest segment to reduce the computational cost.
2

2

Fuser_control: = ∑ ((υ′L, i – υL, i) + (υ′R, i – υR, i) )(5)i+ dist(υL,M, csymmetry)2 + dist(υR,M, csymmetry)2
In our implementation, we set (ω1,ω2,ω3,ω4) =(0.1,
0.04,0.01,0.03) as the weights of the terms in Eq. (1).
These values are determined by experience. The basic
idea is that ω1 should be a higher value to make gaps at
the symmetry plane small. Weight ω4 affects the controllability.
Before running the optimization, the ridge lines areequally spaced and their angles are zero except for θ 0.
As shown in Fig. 9, θ 0 is set to 90 degrees. The part towhich the strips are connected (the side panel of thechair) is perpendicular to the ground plane (xz plane)
and is placed at a distance of L3 from the symmetry
plane.

3.2 Workflow
Fig. 10 shows the GUI of the proposed system. First,
the user inputs the parameters L1 to L4, which define thedeveloped pattern (Fig. 2). The 3D positions of the endsof the strips are specified by two 2D Bézier curves, i.e.,
the leg curve and the seat curve, as shown in Fig. 11.
The positions of υL,0 and υR,0 are defined by the leg
curve and the parameter L2. The seat curve defines the
cross section on the symmetry plane. Both curves areedited by the user. The system runs an optimization tocompute a shape that satisfies the constraints. This
optimization runs during interactive modeling by theuser. The system updates the shape of the designed
model just after each operation by the user. The user
can add an anchor to a vertex to control the shape (Fig.
12). For precise control, the user can also add multiple
anchors. Until the user deletes the anchor, the anchor isFig. 11. General design by two curves.

used in the optimization process. The optimization runs
only on the affected strip when the user edits an anchor
point. The computational time for the optimization isreasonably fast. In addition, the user can interactively
edit the shape while seeing the effect of the anchor
points.
Another user operation can subdivide the strips tomake them smoother (Fig. 13). The new ridge lines areinserted with a zero angle between neighboring ridge
lines on the strips. The user starts by editing a coarse
quad mesh, then divides the quads to make the strips
smooth. This process effectively reduces the total
computational costs for the optimization. By applying
subdivision and optimization alternately, the strips

Fig. 12. Adding an anchor point (orange sphere) to a vertex.

Fig. 10. Screenshot of our system.

Fig. 13. Subdivision of a strip. From the left: before and after
subdividing, and after optimization.

Naoya Tsuruta et al.

Interactive Design of 3D Geometry Made by Bending INextensible Sheet Material Containing Slits 27

Fig. 14. Examples. Left two: 6-strip chair. Center: optimized chair model with reference to scanned points. Right two: monuments.

become smooth.

4. Results
The prototype system is implemented using Java on a
2.66 GHz PC. Fig. 14 shows some example models
designed using the system. Designing each model required approximately 20 minutes. Although composing
the rough shape with the coarse quad was easy, precise
control of the strip was difficult and took a long time.
For designing a chair, a long time was required to makeits seat flat since the surface tended to become wavy.
Changing the weights of the penalty function would
make it easier.
All models except the center model in Fig. 14 were
designed interactively. The legs of the rightmost model
were set parallel to the ground. The center model was
designed with the other approach. We first made achair-shaped model with a sheet of paper by hand toobtain a shape as shown in Fig. 1. Then, we obtained
the 3D coordinates of the points on the paper modelusing a 3D range scanner. By slightly modifying thepenalty function so that the optimized shape came close
to the 3D scanned points (details are described in
Section 4.1), we obtained the result.
With our approach, the distance from the symmetry
plane to the end of a strip was useful as an indicator of
the validity of the obtained shape. We calculated the
maximum gap distance G in all strips. By dividing G by
the length of the developed pattern, E = G/(L2 + L3) gives
the normalized value of error. The largest error in theexamples in Fig. 14 was 0.0015. This means that when
we use a 1768 mm long material (the length used forthe chair shown in Fig. 1), the gap is approximately
2.65 mm, which is acceptable for manufacturing.
The computational time for the optimization was lessthan 100 ms per strip. It would be possible for the user
to edit the shape with less distraction when the systemruns on a more powerful PC. Although subdividing
quads in a strip increases the computational cost, theoptimization converges fast because the starting shapeis already near its solution. The system was fast enougheven after the strips were subdivided twice, which madethem smooth enough. However, the performance was
not good, especially just after the initial state when the
shape was far from the final convergence. It tookapproximately 500 ms per strip in which ten quads are

Fig. 15. Unfolded pattern of the center model in Fig. ?.

Fig. 16. 3D scanned data and extracted scanned data (orange)
overlaid on the edited shape (blue).

contained. We limited the number of iterations forsolving an equation in the optimization process. The
user can avoid a slow convergence by changing theanchor positions.

4.1 Approximation to scanned data
We perform the following steps to make the strips
close to the scanned 3D model shown in Fig. 16 left.
1. Clean up the scanned data by eliminating noise.
2. Manually extract the points corresponding to each
strip and output them individually.

Fig. 17. 3D printed mold and the formed polyvinyl chloride board.
Size(W/H/D): 247×233×179 mm.

28International Journal of CAD/CAM Vol.13, No. 2, pp. 23~29

3. Display the strips in the system (Fig. 16 right).
4. Roughly adjust the position of the strips manually.
5. Optimize with an additional constraint.
We combine an additional penalty function shown in
Eq. (6), such that nearest(υi) is the closest point to vi inthe scanned points, into Eq. (1) to take consider thecloseness of a strip to point clouds. We could obtain ashape resembling the original chair by adding this term.
However, the chair shown in Fig. 1 seems to contain
deformation of the material, and the points clouds may
contain an error created while scanning. It was impossible to match them completely because our systemmodels an ideal shape.
2

Fprox: = ∑((nearest(υL, i) – υL, i) +i

2(nearest(υR, i) – υR, i ) )(6)

4.2 Fabrication
We verified the proposed system by making a physicalmodel. First we created a solid mold model from theside panels and strips by adding faces in the front, backand bottom. We printed out the mold on a 3D printer,then we pressed a 1 mm thick polyvinyl chloride sheetwith slits onto the mold. We heated the sheet with a hairdryer to bend it. Fig. 17 shows the 3D printed mold andthe formed sheet. The unfolded pattern to cover themold was slightly shorter. It seems that the shortage
was due to the fact that we did not consider the
thickness of the material when we printed out the mold.
However, we confirmed that we could reconstruct themodel designed on our system with a flat sheet
material.

5. Conclusion and Limitations
In this paper, we have proposed a method for designing a shape that can be made from a planar material
containing slits. The positions and angles of the ridge
lines on the strips are calculated by an optimization.
The user can control the final shape that satisfies thegeometric constraints. To evaluate our system, we
printed out a mold using a 3D printer, cut out theunfolded pattern, and formed the designed shape bycompressing the material onto the mold. We confirmedthat the shape could be reconstructed with a sheet
material.
The domain of the shapes that we can design on ourcurrent system is very narrow. Changing the shape andposition of slits is a way to allow more variation.
Asymmetric shapes could be obtained without mirroring. The user interface could be improved so that theuser can control the final shape more precisely. Optimizing all strips together with considering the gap-sizes
could generate smooth changes between strips. Combining a physical simulation of strength or balance
makes a practical system for designing self-supporting
and robust shapes.

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Interactive Design of 3D Geometry Made by Bending INextensible Sheet Material Containing Slits 29

Naoya Tsuruta is currently pursuing his Ph.D. in Engineering
at the University of Tsukuba, Japan. He received his B.S. in
Computer Science and M.S. in Engineering from University ofTsukuba in 2010 and 2012, respectively. His research interests
include geometric modeling, optimization, and interactive CAD
systems for the geometry of origami.

Jun Mitani received his Ph.D. in Engineering from the
University of Tokyo in 2004. He had been working at RIKEN
and University of Tsukuba since 2004 and 2005, respectively.
He has been an associate professor in the Department of Computer Science at University of Tsukuba since 2009. He has been
studying geometric modeling in the field of computer graphics.

Yoshihiro Kanamori received his B.S., M.S., and Ph.D. degrees in computer science from the University of Tokyo, Japan,
in 2003, 2005, and 2009, respectively. Since April 2009, he has
been an assistant professor at the University of Tsukuba. His research interests center in computer graphics including renderingand visual simulations. He now studies image editing techniques
for reproducing real-world phenomena.

Yukio Fukui received his Ph.D. in Engineering from the University of Tokyo in 1993. He had been working at the National
Institute of Bioscience and Human-Technology, M.I.T.I. and theInstitute of Information Sciences and Electronics at Universityof Tsukuba since 1993 and 1998, respectively. He has been aprofessor in the Department of Computer Science at Universityof Tsukuba since 2004.

Naoya Tsuruta

Jun Mitani

Yoshihiro Kanamori

Yukio Fukui