This paper addresses the problem of discretizing the curved developable surfaces that are satisfying the equivalent surface curvature change discretizations. Solving basic folding units occurs in such tasks as simulating the behavior of Gauss mapping. The Gauss spherical curves of different developable surfaces are setup under the Gauss map. Gauss map is utilized to investigate the normal curvature change of the curved surface. In this way, spatial curved surfaces are mapped to spherical curves. Each point on the spherical curve represents a normal direction of a ruling line on the curved surface. This leads to the curvature discretization of curved surface being transferred to the normal direction discretization of spherical curves. These developable curved surfaces are then discretized into planar patches to acquire the geometric properties of curved folding such as fold angle, folding direction, folding shape, foldability, and geometric constraints of adjacent ruling lines. It acts as a connection of curved and straight folding knowledge. The approach is illustrated in the context of the Gauss map strategy and the utility of the technique is demonstrated with the proposed principles of Gauss spherical curves. It is applicable to any generic developable surfaces.

References

References
1.
Dai
,
J. S.
, and
Jones
,
J. R.
,
2002
, “
Kinematics and Mobility Analysis of Carton Folds in Packing Manipulation Based on the Mechanism Equivalent
,”
J. Mech. Eng. Sci.
,
216
(
10
), pp.
959
970
.
2.
Liu
,
H.
, and
Dai
,
J. S.
,
2002
, “
Carton Manipulation Analysis Using Configuration Transformation
,”
J. Mech. Eng. Sci.
,
216
(
5
), pp.
543
555
.
3.
Balkcom
,
D. J.
, and
Mason
,
M. T.
,
2008
, “
Robotic Origami Folding
,”
Int. J. Rob. Res.
,
27
(
5
), pp.
613
627
.
4.
Resch
,
R.
, and
Christiansen
,
H.
,
1970
, “
The Design and Analysis of Kinematic Folded Plate Systems
,”
IASS Symposium on Folded Plates and Prismatic Structures
, Vienna, Sept.–Oct.
5.
Miura
,
K.
,
1970
, “
Proposition of Pseudo-Cylindrical Concave Polyhedral Shells
,”
IASS Symposium on Folded Plates and Prismatic Structures
, Vienna, Sept.–Oct., pp. 141–163.
6.
Yao
,
W.
, and
Dai
,
J. S.
,
2008
, “
Dexterous Manipulation of Origami Cartons With Robotic Fingers Based on the Interactive Configuration Space
,”
ASME J. Mech. Des.
,
130
(
2
), p.
22303
.
7.
Lang
,
R. J.
, 2018, “
Crease Pattern Gallery
,” LANG ORIGAMI, California, accessed Oct. 12, 2018, https://langorigami.com/crease-patterns
8.
Chen
,
Y.
,
Peng
,
R.
, and
You
,
Z.
,
2015
, “
Origami of Thick Panels
,”
Science
,
349
(
6246
), pp.
396
400
.
9.
Resch
,
R.
,
Barnhill
,
R. E.
, and
Riesenfeld
,
R. F.
,
1974
, “
The Space Curve as a Folded Edge
,”
Computer-Aided Geometric Design
,
Academic Press
,
Dordrecht, The Netherlands
, pp.
255
258
.
10.
Xu
,
H. Y.
, and
Dai
,
J. S.
,
2002
, “
Three-Dimensional Implicit Curve Interpolation
,”
Int. J. Adv. Manuf. Technol.
,
19
(
5
), pp.
325
329
.
11.
Huffman
,
D. A.
,
1976
, “
Curvature and Creases: A Primer on Paper
,”
IEEE Trans. Comput.
,
C-25
(
10
), pp.
1010
1019
.
12.
Demaine
,
E.
,
Demaine
,
M.
,
Koschitz
,
D.
, and
Tachi
,
T.
,
2015
, “
A Review on Curved Creases in Art, Design and Mathematics
,”
Symmetry: Culture Sci.
,
26
(
2
), pp.
145
161
.http://martindemaine.org/papers/CurvedCrease_IASS2011/paper.pdf
13.
Fuchs
,
D.
, and
Tabachnikov
,
S.
,
1999
, “
More on Paperfolding
,”
Am. Math. Mon.
,
106
(
1
), pp.
27
35
.
14.
Redont
,
P.
,
1989
, “
Representation and Deformation of Developable Surfaces
,”
Comput. Aided Des.
,
21
(
1
), pp.
13
20
.
15.
Bo
,
P.
, and
Wang
,
W.
,
2007
, “
Geodesic-Controlled Developable Surfaces for Modeling Paper Bending
,”
Comp. Graph. Forum
,
26
(
3
), pp.
365
374
.
16.
Demaine
,
E.
,
Demaine
,
M.
,
Huffman
,
D.
,
Koschitz
,
D.
, and
Tachi
,
T.
,
2016
, “
Characterization of Curved Creases and Rulings: Design and Analysis of Lens Tessellations
,”
Origami6
,
American Mathematical Society
, Robert J. Lang, and Patsy Wang-Iverson, eds.,
Tokyo, Japan
, pp.
209
230
.
17.
Koschitz
,
D.
,
2014
, “
Computational Design With Curved Creases: David Huffman's Approach to Paperfolding
,”
Ph.D, dissertation
, MIT, Boston, MA.https://dspace.mit.edu/handle/1721.1/93013
18.
Geretschlaeger
,
R.
,
2009
, “
Folding Curves
,”
Origami4
,
A K Peters
,
Natick, MA
, pp.
151
164
.
19.
Kilian
,
M.
,
Floery
,
S.
,
Mitra
,
N. J.
, and
Pottmann
,
H.
,
2008
, “
Curved Folding
,”
ACM Trans. Graph.
,
27
(
3
), pp.
1
9
.
20.
Kergosien
,
Y.
,
Gotoda
,
H.
, and
Kunii
,
T.
,
1994
, “
Bending and Creasing Virtual Paper
,”
IEEE Comput. Graph. Appl.
,
14
(
1
), pp.
40
48
.
21.
Wang
,
F.
,
Gong
,
H.
,
Chen
,
X.
, and
Chena
,
C. Q.
,
2016
, “
Folding to Curved Surfaces: A Generalized Design Method and Mechanics of Origami-Based Cylindrical Structures
,”
Sci. Rep.
,
6
, p.
33312
.
22.
Tachi
,
T.
,
2013
, “
Composite Rigid-Foldable Curved Origami Structure
,”
The First Conference Transformables
, Seville, Spain, Sept. 18–20, pp. 1–6.https://pdfs.semanticscholar.org/cab7/997c3bd3cb503d2e6b83960eb990b727603b.pdf
23.
Tachi
,
T.
, and
Epps
,
G.
,
2011
, “
Designing One-DOF Mechanisms for Architecture by Rationalizing Curved Folding
,”
ALGODE
, pp. 1–14.http://www.tsg.ne.jp/TT/cg/RigidOrigamiCurvedFoldingTachiEppsALGODE2011.pdf
24.
Dudte
,
L. H.
,
Vouga
,
E.
,
Tachi
,
T.
, and
Mahadevan
,
L.
,
2016
, “
Programming Curvature Using Origami Tessellations
,”
Nat. Mater.
,
15
(
5
), pp.
583
588
.
25.
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
.
26.
Nelson
,
T. G.
,
Lang
,
R. J.
,
Magleby
,
S. P.
, and
Howell
,
L. L.
,
2016
, “
Curved-Folding-Inspired Deployable Compliant Rolling-Contact Element (D-CORE)
,”
Mech. Mach. Theory
,
96
, pp.
225
238
.
27.
Mentrasti
,
L.
,
Cannella
,
F.
,
Pupilli
,
M.
, and
Dai
,
J. S.
,
2013
, “
Large Bending Behaviour of Creased Paperboard—I: Experimental Investigations
,”
Int. J. Solids Struct.
,
50
, pp.
20
21
.
28.
Mentrasti
,
L.
,
Cannella
,
F.
,
Pupilli
,
M.
, and
Dai
,
J. S.
,
2013
, “
Large Bending Behaviour of Creased Paperboard—II: Structural Analysis
,”
Int. J. Solids Struct.
,
50
, pp.
20
21
.
29.
Bobenko
,
A.
, and
Suris
,
Y.
,
2005
, “
Discrete Differential Geometry Consistency as Integrability
,” e-print
arXiv:math/0504358
.https://arxiv.org/abs/math/0504358
30.
Desbrun
,
M.
,
Polthier
,
K.
, and
Schöder
,
P.
,
2005
, “
Discrete Differential Geometry
,” ACM SIGGRAPH' 05, Los Angeles, pp. 263–324.
31.
Bobenko
,
A.
, and
Pinkall
,
U.
,
1990
, “
Discrete Surfaces With Constant Negative Gaussian Curvature and the Hirota Equation
,”
J. Differ. Geom.
,
43
(
3
), pp.
527
611
.
32.
Carroll
,
D.
,
Hankins
,
E.
,
Kose
,
E.
, and
Sterling
,
I.
,
2014
,
A Survey of the Differential Geometry of Discrete Curves
, Vol.
36
,
Springer Science & Business Media
,
New York
, pp. 28–35.
33.
Hoffman
,
T.
,
2000
, “
Discrete Curves and Surfaces
,” Ph.D. thesis, Technische Universität Berlin, Berlin.
34.
Doliwa
,
A.
, and
Santini
,
P.
,
1995
, “
Integrable Dynamics of a Discrete Curve and the Ablowitz-Ladik Hierarchy
,”
J. Math. Phys.
,
36
(
3
), pp.
1259
1273
.
35.
Bodduluri
,
R. M. C.
, and
Ravani
,
B.
,
1993
, “
Design of Developable Surfaces Using Duality Between Plane and Point Geometries
,”
Comput.-Aided Des.
,
25
(
10
), pp.
621
632
.
36.
Pottmann
,
H.
, and
Farin
,
G.
,
1995
, “
Developable Rational Bézier and B-Spline Surfaces
,”
Comput. Aided Geom. Des.
,
12
(
5
), pp.
513
531
.
37.
Bennis
,
C.
,
Vezien
,
J. M.
, and
Iglesias
,
G.
,
1991
, “
Piecewise Flattening for Non-Distorted Texture Mapping
,” 18th Annual Conference on Computer Graphics and Interactive Techniques (SIGGRAPH '91), New York, July, pp. 237–246.
38.
Wang
,
D. L.
, and
Wang
,
W.
,
2015
,
Kinematic Differential Geometry and Saddle Synthesis of Linkages
,
Wiley
,
Singapore
.
39.
Lang
,
R. J.
,
2018
,
Twists, Tilings, and Tessellations: Mathematical Methods for Geometric Origami
,
A K Peters/CRC Press
,
Natick, MA
.
40.
Evans
,
T. A.
,
Lang
,
R. J.
,
Magleby
,
S. P.
, and
Howell
,
L. L.
,
2015
, “
Rigidly Foldable Origami Gadgets and Tessellations
,”
R. Soc. Open Sci.
,
2
(
9
), p.
150067
.
41.
Tachi
,
T.
,
2009
, “
Generalization of Rigid Foldable Quadrilateral Mesh Origami
,”
Evolution and Trends in Design, Analysis and Construction of Shell and Spatial Structures
, International Association for Shell and Spatial Structures (IASS) Symposium, Valencia, Spain, Sept. 28–Oct. 2, pp.
173
179
.
42.
Guven
,
J.
, and
Müller
,
M. M.
,
2008
, “
How Paper Folds: Bending With Local Constraints
,”
J. Phys. A: Math. Theor.
,
41
(
5
), p.
055203
.
43.
Zhang
,
K. T.
,
Qiu
,
C.
, and
Dai
,
J. S.
,
2016
, “
An Extensible Continuum Robot With Integrated Origami Parallel Modules
,”
ASME J. Mech. Rob.
,
8
(
3
), p.
031010
.
44.
You
,
Z.
, and
Pellegrino
,
S.
,
1997
, “
Foldable Bar Structures
,”
Int. J. Solid Struct.
,
34
(
15
), pp.
1825
1847
.
45.
Zhang
,
K. T.
, and
Dai
,
J. S.
,
2013
, “
Classification of Origami-Enabled Foldable Linkages and Emerging Applications
,”
ASME
Paper No. DETC2013-12227.
You do not currently have access to this content.