Of the many valid configurations that a curved fold may assume, it is of particular interest to identify natural—or lowest energy—configurations that physical models will preferentially assume. We present normalized coordinate equations—equations that relate fold surface properties to their edge of regression—to simplify curved-fold relationships. An energy method based on these normalized coordinate equations is developed to identify natural configurations of general curved folds. While it has been noted that natural configurations have nearly planar creases for curved folds, we show that nonplanar behavior near the crease ends substantially reduces the energy of a fold.

References

References
1.
Demaine
,
E. D.
, and
O’Rourke
,
J.
,
2008
,
Geometric Folding Algorithms: Linkages, Origami, Polyhedra
,
Cambridge University Press
,
Cambridge
.
2.
Zhao
,
Y.
,
Kanamori
,
Y.
, and
Mitani
,
J.
,
2018
, “
Design and Motion Analysis of Axisymmetric 3d Origami With Generic Six-Crease Bases
,”
Comput. Aided Geom. Des.
,
59
, pp.
86
97
.
3.
Hanna
,
B. H.
,
Magleby
,
S. P.
,
Lang
,
R. J.
, and
Howell
,
L. L.
,
2015
, “
Force–Deflection Modeling for Generalized Origami Waterbomb-Base Mechanisms
,”
ASME J. Appl. Mech.
,
82
(
8
), p.
081001
.
4.
Lee
,
T.-U.
, and
Gattas
,
J. M.
,
2016
, “
Geometric Design and Construction of Structurally Stabilized Accordion Shelters
,”
ASME J. Mech. Rob.
,
8
(
3
), p.
031009
.
5.
Feng
,
H.
,
Ma
,
J.
,
Chen
,
Y.
, and
You
,
Z.
,
2018
, “
Twist of Tubular Mechanical Metamaterials Based on Waterbomb Origami
,”
Sci. Rep.
,
8
(
1
), p.
9522
.
6.
Chen
,
Y.
,
Peng
,
R.
, and
You
,
Z.
,
2015
, “
Origami of Thick Panels
,”
Science
,
349
(
6246
), pp.
396
400
.
7.
Chen
,
Z.
,
Wu
,
T.
,
Nian
,
G.
,
Shan
,
Y.
,
Liang
,
X.
,
Jiang
,
H.
, and
Qu
,
S.
,
2019
, “
Ron Resch Origami Pattern Inspired Energy Absorption Structures
,”
ASME J. Appl. Mech.
,
86
(
1
), p.
011005
.
8.
Ma
,
J.
, and
You
,
Z.
,
2014
, “
Energy Absorption of Thin-Walled Square Tubes With a Prefolded Origami Pattern Part I: Geometry and Numerical Simulation
,”
ASME J. Appl. Mech.
,
81
(
1
), p.
011003
.
9.
Zhou
,
C.
,
Jiang
,
L.
,
Tian
,
K.
,
Bi
,
X.
, and
Wang
,
B.
,
2017
, “
Origami Crash Boxes Subjected to Dynamic Oblique Loading
,”
ASME J. Appl. Mech.
,
84
(
9
), p.
091006
.
10.
Zhou
,
C.
,
Ming
,
S.
,
Li
,
T.
,
Wang
,
B.
, and
Ren
,
M.
,
2018
, “
The Energy Absorption Behavior of Cruciforms Designed by Kirigami Approach
,”
ASME J. Appl. Mech.
,
85
(
12
), p.
121008
.
11.
Tang
,
C.
,
Bo
,
P.
,
Wallner
,
J.
, and
Pottmann
,
H.
,
2016
, “
Interactive Design of Developable Surfaces
,”
ACM Trans. Graph.
,
35
(
2
), p.
12
.
12.
Kilian
,
M.
,
Flöry
,
S.
,
Chen
,
Z.
,
Mitra
,
N. J.
,
Sheffer
,
A.
, and
Pottmann
,
H.
,
2008
, “
Curved Folding
,”
ACM Trans. Graph.
,
27
(
3
), p.
75
.
13.
Lee
,
T.-U.
,
You
,
Z.
, and
Gattas
,
J. M.
,
2018
, “
Elastica Surface Generation of Curved-Crease Origami
,”
Int. J. Solids Struct.
136-137
, pp.
13
27
.
14.
Fuchs
,
D.
, and
Tabachnikov
,
S.
,
1999
, “
More on Paperfolding
,”
Am. Math. Mon.
,
106
(
1
), pp.
27
35
.
15.
Duncan
,
J. P.
, and
Duncan
,
J.
,
1982
, “
Folded Developables
,”
Proc. R. Soc. Lond. Math. Phys. Sci.
,
383
(
1784
), pp.
191
205
.
16.
Struik
,
D. J.
,
1961
,
Lectures on Classical Differential Geometry
,
Courier Dover Publications
,
New York
.
17.
Lang
,
R.
,
Nelson
,
T.
,
Magleby
,
S.
, and
Howell
,
L.
,
2017
, “
Kinematics and Discretization of Curved Fold Mechanisms
,”
ASME
2017 IDETC/CIE, No. IDETC2017-59747, ASME.
18.
Aumann
,
G.
,
2003
, “
A Simple Algorithm for Designing Developable Bzier Surfaces
,”
Comput. Aided Geom. Des.
,
20
(
8
), pp.
601
619
(in memory of Professor J. Hoschek).
19.
Weisstein
,
E. W.
,
2002
,
CRC Concise Encyclopedia of Mathematics
,
CRC Press
,
Boca Raton, FL
.
20.
Mitani
,
J.
, and
Igarashi
,
T.
,
2011
, “
Interactive Design of Planar Curved Folding by Reflection
,”
Proceedings of PG 2011
,
Kaohsiung, Taiwan
,
Sept. 21–23
.
21.
Demaine
,
E. D.
,
Demaine
,
M. L.
,
Huffman
,
D. A.
,
Koschitz
,
D.
, and
Tachi
,
T.
,
2014
, “
Designing Curved-Crease Tessellations of Lenses: Qualitative Properties of Rulings
,”
6th International Meeting on Origami in Science, Mathematics and Education (OSME 2014)
,
Tokyo
,
Aug
., pp.
10
13
.
22.
Dias
,
M. A.
,
2012
, “
Swelling and Folding as Mechanisms of 3d Shape Formation in Thin Elastic Sheets
,” PhD thesis,
University of Massachusetts Amherst
,
Amherst, MA
.
23.
Nelson
,
T. G.
,
Lang
,
R. J.
,
Pehrson
,
N. A.
,
Magleby
,
S. P.
, and
Howell
,
L. L.
,
2016
, “
Facilitating Deployable Mechanisms and Structures Via Developable Lamina Emergent Arrays
,”
ASME J. Mech. Rob.
,
8
(
3
), p.
031006
.
24.
Abbena
,
E.
,
Salamon
,
S.
, and
Gray
,
A.
,
2006
,
Modern Differential Geometry of Curves and Surfaces With Mathematica
,
CRC Press
,
Boca Raton, FL
.
25.
Verbeek
,
P. W.
, and
Van Vliet
,
L. J.
,
1993
, “
Curvature and Bending Energy in Digitized 2D and 3D Images
,”
8th Scandinavian Conference on Image Analysis
,
Tromso, Norway
,
May 25
, pp.
1403
1410
.
26.
Wang
,
W.
, and
Qui
,
X.
,
2017
, “
Coupling of Creases and Shells
,”
ASME J. Appl. Mech.
,
85
(
1
), p.
011009
.
27.
Lee
,
T. U.
,
You
,
Z.
, and
Gattas
,
J. M.
,
2018
, “
Curved-Crease Origami With Multiple States
,”
Origami 7: Seventh International Meeting of Origami Science, Mathematics, and Education
,
Oxford
,
Sept. 5–7
.
28.
Woodruff
,
S. R.
, and
Filipov
,
E. T.
,
2018
, “
Structural Analysis of Curved Folded Deployables
,”
ASCE Earth and Space Conference
,
Cleveland, OH
,
Apr. 9–12
, p.
793
.
29.
Seffen
,
K. A.
, and
Guest
,
S. D.
,
2011
, “
Prestressed Morphing Bistable and Neutrally Stable Shells
,”
ASME J. Appl. Mech.
,
78
(
1
), p.
011002
.
30.
Nelson
,
T. G.
,
Zimmerman
,
T. K.
,
Magleby
,
S. P.
,
Lang
,
R. J.
, and
Howell
,
L. L.
,
2019
, “
Developable Mechanisms on Developable Surfaces
,”
Sci. Rob.
,
4
(
27
), p.
5171
.
31.
DeFigueiredo
,
B.
,
Pehrson
,
N. A.
,
Tolman
,
K.
,
Crampton
,
E.
,
Magleby
,
S. P.
, and
Howell
,
L. L.
,
2019
, “
Origami-Based Design of Conceal-and-Reveal Systems
,”
ASME J. Mech. Rob.
,
11
(
2
), p.
020904
.
32.
Cai
,
J.
,
Qian
,
Z.
,
Jiang
,
C.
,
Feng
,
J.
, and
Xu
,
Y.
,
2016
, “
Mobility and Kinematic Analysis of Foldable Plate Structures Based on Rigid Origami
,”
ASME J. Mech. Rob.
,
8
(
6
), p.
064502
.
33.
Cai
,
J.
,
Zhang
,
Q.
,
Feng
,
J.
, and
Xu
,
Y.
,
2018
, “
Modeling and Kinematic Path Selection of Retractable Kirigami Roof Structures
,”
Comput. Aided Civil Infrastruct. Eng.
,
34
(
4
), pp.
352
363
.
34.
Jianguo
,
C.
,
Yangqing
,
L.
,
Ruijun
,
M.
,
Jian
,
F.
, and
Ya
,
Z.
,
2017
, “
Nonrigidly Foldability Analysis of Kresling Cylindrical Origami
,”
ASME J. Mech. Rob.
,
9
(
4
), p.
041018
.
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