An Curved-folding Algorithm to Adapt to The Freeform Surface: Design andFabrication○ Hong-Cing Tung＊ 1, Pei-hsien Hsu＊ 2*1Graduate Student,Graduate Institute of Architecture, National Chiao Tung University*2 Professor, Graduate Institute of Architecture, National Chiao Tung University, Ph.D..
Keywords: Curved-folding; Developable surface; Freeform; Grasshopper3D; fabrication.

1. Introduction

this paper is a proposed design and fabrication process forIn the field of architecture ,design and art, curved-folding has thearchitects and designers. They can easily used in not only early

exciting potential of a particular technique. It can provide

stage of design thinking but also fabrication.

architects with the generation of complex geometries that

The article presents the detail of Grasshopper settings fordesigning spatial and performative capacities. Most of thecurved folding simulation by explaining how the proposed

researches have used curved-folded tessellations that applied to aalgorithmic framework and the parameter of Grasshopper

predefined global geometry with a top-down method.

1)2)3)(Figure1.) However, the pavilions are made of numerous self-similarity

components. Lastly, it discusses the shortcomings and thepotentials of the design and fabrication process.

of sheet material, They used a method of discrete differential
geometry to produce developable strips of triangles. The triangles
producing fabrication data that ensures consistent assembly.

2.Geometry of paper :Developable surface
Developable surface have been researched widely inConsequently, the constructing process is demanding due tomathematics

complex self-similarity of triangle.

The challenge of curved-

mathematical researches attempted to understand and describe

andcomputergraphics

fields. Theearlyfolding is to understand and measure the three-dimensional

the behavior of the paper sheet along the curves. The sheet paper

configuration of curved folded geometry.

is one of specific class of surface in mathematics, which is known
as a developable surface. 4)
A developable surface is curved surface and is isometricallytransformable, capable of changing to a plane withoutcompressing or expanding. A developable surface consists ofnumerous line elements, which means that it is ruled surface. In
mathematical field, the developable surface is expressed byEquationp"(s) × p"'(s)X (s,t) = p(s) + t．
｜p"(s)｜2(1)Figure 1.Applications of curved folding in architecture.
Epps and Verma,2013(left); Bhooshan et al.2015(middle);
Chandra et al.,2015(right)

It should be point out that the pavilion we studied describe
three-dimensional geometry of developable surface based on apriori knowledge of developable surfaces such as “the sum of theThe objective of this research is to developing an curved-angle between edges around a point must be 360 degree.” 5)

folding technique of design and fabrication. The technique canpreserve the continuity of developable surface with curved-

3. Algorithm and Design process

folded sheet fitting on free-form surface. In this study, weThe algorithms developed whitn the Grasshopper are visual

generated and unfolded curved-folding paper in the digitalprogramming environment and Using the plugin, Kangaroo, toenvironment by using Kangaroo, a plugin of Grasshopper withset mathematical rules of developable surface. In order tophysical engine, has to balance the effects of multiple rules,

generate the mesh of developable surfaces, we converted thethrough Rhinoceros environment. The essential contribution ofmathematical rule into physical method and existing components

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第 42 回情報･システム･利用･技術シンポジウム論文集， 146-149 ， 2019 年 12 月，東京
Proceedings of the 42nd Symposium on Computer Technology of Information ，Systems and Applications ， AIJ ， 146-149 ， Dec. ， 2019 ， Tokyo－ 146 －

of kangaroo through defining the combination of components inGrasshopper.
3.1 DESIGN CURVED-CREASES PATTERN
We designed curved-crease pattern on the freeform surface inRhinoceros environment. By operating the command in
Rhino,”FlowalongSurf”, the curved-crease pattern can easily toadapt to the freeform surface. Also, the curved-crease patternmust be smooth appearance. In other words, the degree ofNURBS must be more than 2.
3.2 GENERATING MESH OF DEVELOPABLE SURFACES
In order to generate developable surfaces between curved-

Figure 3. Simulating the developable surfaces between curved-

creases pattern, the numerous effects of forces in Kangaroocreases from 10 iterations (left) ,50 iterations (middle) and 90(Figureiterations (right).

2.)

mustbebalanced.

forces,”Planarize”,”OnCurve”,”EqualLength”

andThe”Anchor”

calculate and generate developable surfaces between curves in
the meantime.

The main goal of this simulation is to generate the geometries
which are relatively similar to developable surfaces as same as

In this part, the most vital force is “Planarize” because it makes

possible. Based on the results of testing combinations of force

the line elements between each curves describe the developablestrength values, it can be concluded that there must be one

surface. Furthermore, ”OnCurve” comfirms the line elements

governing force. For example, if the strength of “OnCurve” given

closely attaches to the curved-crease pattern.”EqualLength” tryto the mesh is high, the strength given to the “Planarize” should

to verify that length of each pair of adjacent segments along the

be relatively low, and then we start with planarization strength

curves equal, and to avoid it degenerating into triangles.”Anchor”

low, and gently slide it up to simulate the developable surfaces.

keep end points of each curves moveless, so the simulation of

If the strength of “planarize” start high, the line elements can’t

meshes can describe developable surface completely between

smoothly move between curved-creases pattern to form thecurved-crease pattern.

meshes of developable surface.
3.3.UNFOLDINGTHEMESHOF

DEVELOPABLE

SURFACES
A key difference between our method and the precedents westudied is that we unfold the constructive geometry of meshinstead of unfolding the underlying triangulated mesh. It help
fabricate the developable surface and construct easily.
Figure 2. For generating developable surfaces, the proposed
definition of Grasshopper to simulate using Kangaroo.

In order to unfold the meshes ofbetween curved-creases pattern, the numerous effects of forces in
Kangaroo(FigureIt should be noted that the difficulty in balancing the effect offorces on the Planarize system occurs because we converted thedevelopable surfaces4.)

must

bebalanced.

The

forces ”OnPlane”, ”Length”, ”Hinge” and ”OnCurve” calculate
and unfold the developable surfaces in the meantime.

mathematical principles of developable surface into physicalmethod. Namely, It is different from precise mathematical model.
Therefore, we didn’t run the simulation to convergences , instead,simulation for 50-100 iterations. Figure 3. shows that variation of
the line elements during the simulation. The iterations need to beunder control due to generating developable surfaces better.
During the iteration of kangaroo, checking the sum ofneighbouring sector angles is approximate to 2π in order toredurce the inaccuracy of unfolding the mesh to be flat.

報告 H14－ 147 －4. Paper Model Experiment
The final stage is to realize the result, we test on amathematical surface,hyperbolic paraboloid,and then use a sheet
paper with curved-creases pattern we designed and folded them
manually to testify the feasibility of the algorithm.
After folding paper along the mountain and valley creases, thepaper can automatically form the expected shape(Figure 6.).

Figure 4. For unfolding the mesh of developable surface, theproposed definition of Grasshopper to simulate using Kangaroo.
In this part, the most crucial is “OnPlane”for the reason that it

Figure 6. Paper Model Experimentcan force the each vertex coplanar. Inaddition, “Length” try tokeep the each edge length of mesh same. Further,“Hinge” unfold
the angle between each triangulated mesh to π. In other word, it

5.Result of Simulation on Freeform Surface
The positive and negative Gaussian curvature can coexist on

can relax the folded shape to be flat. It is important to mention

freeform surface due to the mathematical rule. Figure 8. showsthat because each quadrilateral mesh divide into triangular mesh,an example of this testing. We simulate the developable surface

so ”Hinge” works successfully. Finally,”OnCurve”pull every

between curved-crease that we designed on the freeform

vertex of boundary of folded shape outward.

surface(Figure 7.).

Figure 5. Unfolding the curved-folding paper to be flat
It should be noted that using “zombie” , one of Kangaroo
solver, to simulating due to conveniently unfolding the meshes tobe completely flat.(Figure 5.) In addition, the strength of“Length” should be relatively high in order to keep the mesh ofdevelopable surfaces same. However, balancing the effect offorces have difficulty keeping every edge changeless during thesimulation. Importantly, the geometry simulated by this method
is a visual approximation and not precise curved folded geometry.
Nonetheless, it enable unfolded two-dimensional shapes can be

Figure 7. Implementation on the freeform surface.

folded and fit on the freeform surfaces.

報告 H14－ 148 －6.Results and Discussion

7.Conclusion

In our design process of approximating geometry as mentioned

In this research, we demonstrated the inverse technique toin 3.2, we maked a fundamental assumption that inaccuracy ofdesign curved-folding paper and a design process to achieve togeometry can be tolerated due to using Kangaroo engine toadapt to the free form surface. Further, the approach presented theunfold the geometry.

fabrication of constructive developable surfaces and is applicable toIn the design process of generating developable surfaces, it

any freeform surface.

should be pointed out that increasing line elements facilitate

The domain of curved-folding has a strong potential for

better generating mesh of developable surfaces. Further, the sumarchitecture and manufacture due to aesthetic and flexibility ofof the angle between edges around a point have difficulty

structure. In recent decades, the development of computer graphics

summing up to 2π (Figure 8.) However, In the design process ofand software of Curved-folding tool facilitate architects andunfolding the meshes, the inaccuracy angle of developable

designers with multiple constraints and advance algorithms. The

surface will be eliminated due to edge of meshes variation(Figuresoftware of designing curved-folding have specific function for9.). Namely, we set the strength of “Length” extremely high indesign requirements. However, Grasshopper platform provide

order to keep every edge of mesh same, but it turn out to be only

friendly design environment for architects and designers.

approximate same for the reason that unfolding the meshes to be

The method presented offer various chances to inform

flat is more critical. Finally, the folded model only approximatesparameters such as line elements of developable surface. Also, it

the visual appearance of the object, which is critical to a designer.

successfully works for intergrating with generating of developablesurface and fabrication. We hope to further this approach byenriching it with relevant conmtemporary studies in related origami
amd architecture domain. Bhooshan et al.2) presented a method
applicable to any freeform surface that can be tessellated into
convex polygons with curved-folding. Our method has highpotential of curved-folding tessellation to adapt to the freeform
surface.

Figure 8. Simulation of generating mesh of developable surfaces(left), the yellow panel(right) shows that sum of the angles
between edges around a point are only closely 2π.

Figure 9. Simulation of unfolding the mesh of developable

References (sample)
1） Epp, G.,&Verma. S.: 2013, Curved Folding: Design to fabrication
process of RoboFold, Shape Modeling Internation 2013, 75.
2） Bhooshan, S., Bhooshan, V., ElSayed, M., Chandra, S., Richens,P., & Shepherd, P: 2015, Applying dynamic relaxation techniques
to form-find and manufacture curve-crease folded panels,Simulation, 91(9), 773-786.
3） Chandra, S., Körner, A., Koronaki, A., Spiteri, R., Amin, R., Kowli,
S., & Weinstock, M.: 2015, Computing curved-folded tessellations
through straight-folding approximation, In Proceedings of theSymposium on Simulation for Architecture & Urban Design,Society for Computer Simulation International, 152-159.
4） Pottmann, H.: 2007, Architectural Geometry, Bentley Institute
Press5） Bacinoglu, Z.,Piskorec, L.,&Kotnik, T.: 2019, CURVED. IT: A
design tool to intergrate making with curved folding into digital
design process, ITU journal of Faculty of Architecture, 16, 11-27.
6） Mitani, J.: 2019, Curved-Folding Origami Design, Boca Raton , C
Press7） Web: 2019, “To FlatFold a Curved-folding paper”. Available from:Open Source Re-pository < https://discourse.mcneel.com/t/toflatfold-a-curved-folding-paper/85542 > (accessed 4 Octorber2019).
8）Web: 2017, “Developable Surface Optimization with Kangaroo 2”.
Available
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https://www.grasshopper3d.com/forum/topics/developablesurface-optimization-with-kangaroo-2?groupUrl=kangaroo>(accessed 4 Octorber 2019).

surfaces (left), the yellow panel(right) shows that sum of theangle between edges around a point are 2π.

報告 H14－ 149 －