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Applied Geometry: Foldings and Unfoldings
Hellmuth Stachel
[email protected] — http://www.geometrie.tuwien.ac.at/stachel
ISTEC 2016, International Science and Technology Conference
July 13–15, Vienna University of Technology
Table of contents
1. Unfolding and folding
2. Flexible quad-meshes
3. Curved folding, Example 1
4. Curved folding, Example 2
Funding source:
Partly supported by the Austrian Academy of Sciences
July 13: ISTEC 2016, Vienna University of Technology 1/57
1. Unfolding and folding
unfold
fold
There are standard procedures provided for theconstruction of the unfolding (development,net) of polyhedra or developable surfaces.
The result is unique, apart from the placementof the different components, and it shows theintrinsic metric of the spatial structure.
July 13: ISTEC 2016, Vienna University of Technology 2/57
E.g., the unfolding of the Oloid
Institut fuer GeometrieInstitut fuer Geometrie
Technische Universitaet WienTechnische Universitaet WienTechnische Universitaet Wien
Oloid
Oloid
Oloid
Oloid
1
2 3
3
Developable surfaces are ruled surfaces withvanishing Gaussian curvature
July 13: ISTEC 2016, Vienna University of Technology 3/57
Institut fuer GeometrieInstitut fuer Geometrie
Technische Universitaet WienTechnische Universitaet WienTechnische Universitaet Wien
Oloid
Oloid
Oloid
Oloid
1
2 3
3
Oloid:
arc-length parametrization of theunfolding of the circles:
x(s) =2√3
9
[
arccos
√2 cos s√1 + cos s
−√
2(1− cos s)(1 + 2 cos s)(1 + cos s)
]
y(s) =
√3
9
[
ln2
1 + cos s+11 + 7 cos s
1 + cos s
]
.
July 13: ISTEC 2016, Vienna University of Technology 4/57
1. Unfolding and folding
unfold
fold
The inverse problem, i.e., the determination ofa folded structure from a given unfolding ismore complex. In the smooth case we obtaina continuum of bent poses.
In the polyhedral case the computation leads toa system of algebraic equations. Also here thecorresponding spatial object needs not be unique.
July 13: ISTEC 2016, Vienna University of Technology 5/57
1. Unfolding and folding
Only if the polyhedron bounds a convex solid then the result is unique, due to AleksandrDanilovich Alexandrov (1941).
In this case, for each vertex the sum of intrinsic angles for all adjacent surfaces is< 360◦ (= convex intrinsic metric).
Theorem: [Uniqueness Theorem]For any convex intrinsic metric there is a unique convex polyhedron.
July 13: ISTEC 2016, Vienna University of Technology 6/57
1. Unfolding and folding
If convexity is not required the unfoldingof a polyhedron needs not define itsspatial shape uniquely !
Definition 1: A polyhedron is calledglobally rigid if its intrinsic metricdefines its spatial form uniquely — up tomovements in space.
e.g., a tetrahedron
f1
f2
f4
f5
f 6
f1
f2
f4
f6
f7A flipping (or snapping) polyhedronadmits two sufficiently close realizations– by applying a slight force.
July 13: ISTEC 2016, Vienna University of Technology 7/57
1. Unfolding and folding
Definition 2: A polyhedron is called (continuously) flexible if there is a continuousfamily of mutually incongruent polyhedra sharing the intrinsic metric. Each member ofthis family is called a flexion.
f8
f 1
f5
f 6
f8
f 1
f6
f7
f 1
f2
Even a regular octahedron is flexible — after being re-assembled. The regular pose onthe left hand side is called locally rigid.
July 13: ISTEC 2016, Vienna University of Technology 8/57
1. Unfolding and folding4
B
C
B′
C ′
A
A′
B
C
B′
C ′
A
A′Two particular examplesof flexible octahedrawhere two faces areomitted. Both have anaxial symmetry (types 1and 2)
Below: Nets of the twooctahedra.
B C
B′ C ′C ′ A
A′A′ BC
B′ C ′C ′ A
A′A′
July 13: ISTEC 2016, Vienna University of Technology 9/57
1. Unfolding and folding8
N
A
A′
B
B′
C
C ′ t1
t2
P
S
According to R. Bricard (1897)there are three types of flexibleoctahedra (four-sided double-pyramids).
The first two have axis or plane ofsymmetry. Those of type 3 admittwo flat poses. In each such pose,the pairs (A,A′), (B,B′), and(C, C′) of opposite vertices areassociated points of a strophoid S.
July 13: ISTEC 2016, Vienna University of Technology 10/57
1. Unfolding and folding
NA
A′B
B′
C
C′
A′′
B′′
C′′
According to Bricard’s construc-tion, all bisectors must passthrough the midpoint N of theconcentric circles.
The two flat poses of a type-3flexible octahedron, when ABCremains fixed.
July 13: ISTEC 2016, Vienna University of Technology 11/57
1. Unfolding and folding
16
13
10
09
06
05
03 02
16 12
09
08 05
02
01
16
15
12
09
08 05
02
01
This polyhedron called“Vierhorn” is locallyrigid, but can flipbetween its spatialshape and two flatrealizations in theplanes of symmetry(W. Wunderlich, C.Schwabe).
At the science exposition “Phänomena” 1984 in Zürich this polyhedron was exposedand falsely stated that this polyhedron is flexible.
July 13: ISTEC 2016, Vienna University of Technology 12/57
1. Unfolding and folding
16 12
09
08 05
02
01 0102
0607
1415
0910
0805
0304
1112
1613
the “Vierhorn” and its unfolding
Wolfram MathWorld: A flexible polyhedron which flexes from one totally flat configuration to another, passing through
intermediate configurations of positive volume.
July 13: ISTEC 2016, Vienna University of Technology 13/57
2. Flexible quad-meshes
A polygonal mesh is a simply connected subsetof a polyhedral surface (sphere-like or withboundary) consisting of (not necessarily planar)polygons, edges and vertices in the Euclidean 3-space.
The edges are either internal when they areshared by two faces, or they belong to theboundary of the mesh.
When all polygons are quadrangles, then it iscalled a quadrilateral surface.
July 13: ISTEC 2016, Vienna University of Technology 14/57
2. Flexible quad-meshes
In modern architecture, most freeform surfaces are designed as polyhedral surfaces –like the Capital Gate in Abu Dhabi, built by the Austrian company Waagner Biro
(160 m high, 18◦ inclination)
July 13: ISTEC 2016, Vienna University of Technology 15/57
2. Flexible quad-meshes
V1
V2V3
V4
f0
f1
f2
f3
f4
a1
a2
a3
a4 Under which conditions is this 3 × 3mesh continuously flexible ?
July 13: ISTEC 2016, Vienna University of Technology 16/57
2. Flexible quad-meshes
V3
V4
f0
f1
f2
f3
I10
I20
I30
α1
β1
γ1
δ1
α2
β2 γ2
δ2
Transmission from f1 to f3via the quadrangle f2
I10I20
I30
A1B1
A2
B 2
ϕ1
ϕ2ϕ3
α1 β1
γ1
δ1
α2β2
γ2
δ2
Composition of two spherical four-barlinkages
July 13: ISTEC 2016, Vienna University of Technology 17/57
2. Flexible quad-meshes
A Kokotsakis mesh is continuously flexible⇐⇒the transmission from f1 to f3 can in twoways be decomposed into two sphericalfour-bar mechanisms, one via V1 and V2,the other via V4 and V3.
The internal edges can be arranged in two‘horizontal’ (blue) and two ‘vertical’ edgefolds (red).
V1
V2V3
V4
f0
f1
f2
f3
f4
I10
I20
I30
I40
July 13: ISTEC 2016, Vienna University of Technology 18/57
2. Flexible quad-meshes
I. Planar-symmetric type (Kokotsakis 1932):
The reflection in the plane of symmetry of V1 and V4maps each horizontal fold onto itself while the twovertical folds are exchanged.
V1
V2V3
V4
II. Translational type:
There is a translation V1 7→ V4 and V2 7→ V3 mappingthe three faces on the right hand side onto the tripleon the left hand side.
V1
V2
V3
V4
a4
July 13: ISTEC 2016, Vienna University of Technology 19/57
2. Flexible quad-meshes6
Vi
αi
βiγi
δi
III: Isogonal type (Kokotsakis 1932):
A Kokotsakis mesh is flexible when at eachvertex Vi opposite angles are either equal orcomplementary, i.e.,
αi = βi , γi = δi or
αi = π − βi , γi = π − δi and (n = 4)
sinα1 ± sin γ1sin(α1 − γ1)
· sinα2 ± sin γ2sin(α2 − γ2)
=sinβ3 ± sin γ3sin(β3 − γ3)
· sinβ4 ± sin γ4sin(β4 − γ4)
A quad mesh where all 3 × 3 complexes are of this type is continuously flexible andcalled Voss surface (Kokotsakis, Graf, Sauer)
July 13: ISTEC 2016, Vienna University of Technology 20/57
2. Flexible quad-meshes
These are Voss surfaces:
Nadja Posselt:Synthese von zwangläufig beweglichen 9-gliedrigen VierecksflachenDiploma thesis, TU Dresden 2010
July 13: ISTEC 2016, Vienna University of Technology 21/57
2. Flexible quad-meshes
IIIa. Generalized isogonal type:
A. Kokotsakis (1932): At all vertices oppositeangles are congruent or complementary.
G. Nawratil (2010): At least at two of the fourpyramids opposite angles are congruent.
IV. Orthogonal type (Graf, Sauer 1931):
Here the horizontal folds are located in parallel (say:horizontal) planes, the vertical folds in vertical planes(T-flat).
V1 V2
V3V4
July 13: ISTEC 2016, Vienna University of Technology 22/57
2. Flexible quad-meshes
V. Line-symmetric type (H.S. 2009):
A line-reflection maps the pyramid at V1onto that of V4; another one exchanges thepyramids at V2 and V3.
This includes Kokotsakis’ example of aflexible tessellation.
V1
V2
V3
V4
f0
f1
f2
f3
f4
a1
a2
a3
a4
ρ1
ρ2
July 13: ISTEC 2016, Vienna University of Technology 23/57
2. Flexible quad-meshes
fold
ing
Miu
ra-O
ri
Miu
ra-O
rifo
ldin
g
mou
ntai
n fo
lds
full
valle
y fo
lds
dash
ed
Unfolded miura-ori;dashs are valley folds,
full lines are mountain folds
1) Miura-ori is a Japanese foldingtechnique (1970?) named afterProf. Koryo Miura, The University ofTokyo (military secret in Russia).
It is used for solar panels because itcan be unfolded into its rectangularshape by pulling on one corner only.
On the other hand it is used as kernelto stiffen sandwich structures.
July 13: ISTEC 2016, Vienna University of Technology 24/57
2. Flexible quad-meshesreplacements
180◦
we start with twoparallelograms sharingone edge . . .
July 13: ISTEC 2016, Vienna University of Technology 25/57
2. Flexible quad-meshesreplacements
2δε1
and rotate the right one againstthe left one through the angle2δ.
The lower sides span a plane ε1,the upper sides a plane ε2 parallelε1 .
July 13: ISTEC 2016, Vienna University of Technology 26/57
2. Flexible quad-meshes
ε1
ε2
By translations we generate a zig-zag strip of parallelogramsbetween the two parallel planes ε1 and ε2 .
July 13: ISTEC 2016, Vienna University of Technology 27/57
2. Flexible quad-meshes
ε1
ε2
By reflection in ε2 we generate a second zig-zag strip ofparallelograms sharing the border line in ε2 with the initial strip
— and we iterate . . .
July 13: ISTEC 2016, Vienna University of Technology 28/57
2. Flexible quad-meshes
Miura-ori, a Japanese folding technique
Miura-ori, a Japanese folding technique
Miura-ori, a Japanese folding technique
Miura-ori, a Japanese folding technique
Miura-ori, a Japanese folding technique
Miura-ori, a Japanese folding technique
Miura-ori, a Japanese folding technique
Miura-ori, a Japanese folding technique
Miura-ori, a Japanese folding technique
Miura-ori, a Japanese folding technique
Miura-ori, a Japanese folding technique
Miura-ori, a Japanese folding technique
Miura-ori, a Japanese folding technique
Miura-ori, a Japanese folding technique
Miura-ori, a Japanese folding technique
Miura-ori, a Japanese folding technique
Miura-ori, a Japanese folding technique
Miura-ori, a Japanese folding technique
Miura-ori, a Japanese folding technique
Miura-ori, a Japanese folding technique
Miura-ori, a Japanese folding techniqueMiura-ori, a Japanese folding technique
Miura-ori, a Japanese folding techniqueMiura-ori, a Japanese folding technique
Miura-ori, a Japanese folding techniqueMiura-ori, a Japanese folding technique
Miura-ori, a Japanese folding techniqueMiura-ori, a Japanese folding technique
Miura-ori, a Japanese folding techniqueMiura-ori, a Japanese folding technique
Miura-ori, a Japanese folding techniqueMiura-ori, a Japanese folding technique
July 13: ISTEC 2016, Vienna University of Technology 29/57
2. Flexible quad-meshes
V
x
y
z
ϕϕ
ε2ψ
ψ
α
P1
P2
P3
P4
P∗3
P∗4
There is a hidden localsymmetry at eachvertex V :
The parallelogramsP1,P2 with angle αand the elogationsP∗3,P
∗3 of those with
angle 180◦ − α forma pyramid symmetricwith respect to thefixed planes.
July 13: ISTEC 2016, Vienna University of Technology 30/57
2. Flexible quad-meshes
Cf4
f3 f2
f1r
l
A. Kokotsakis, 1932Athens
2) Any arbitrary plane quadrangle isa tile for a regular tessellation of theplane. It is obtained from the initialquadrangle
• by iterated 180◦-rotations about themidpoints of the sides — or
• by iterated translations of centrallysymmetric hexagon.
For a convex f1 this polyhedral surfaceis continuously flexible.
July 13: ISTEC 2016, Vienna University of Technology 31/57
2. Flexible quad-meshes
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*Kokotsakis*tessellation*Kokotsakis*
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*Kokotsakis*tessellation*Kokotsakis*
*Kokotsakis*tessellation*Kokotsakis*
*Kokotsakis*tessellation*Kokotsakis*
*Kokotsakis*tessellation*Kokotsakis*
*Kokotsakis*tessellation*Kokotsakis*
*Kokotsakis*tessellation*Kokotsakis*
*Kokotsakis*tessellation*Kokotsakis*
*Kokotsakis*tessellation*Kokotsakis*
*Kokotsakis*tessellation*Kokotsakis*
*Kokotsakis*tessellation*Kokotsakis*
*Kokotsakis*tessellation*Kokotsakis*
*Kokotsakis*tessellation*Kokotsakis*
*Kokotsakis*tessellation*Kokotsakis*
*Kokotsakis*tessellation*Kokotsakis*
*Kokotsakis*tessellation*Kokotsakis*
*Kokotsakis*tessellation*Kokotsakis*
*Kokotsakis*tessellation*Kokotsakis*
*Kokotsakis*tessellation*Kokotsakis*
*Kokotsakis*tessellation*Kokotsakis*
*Kokotsakis*tessellation*Kokotsakis*
*Kokotsakis*tessellation*Kokotsakis*
*Kokotsakis*tessellation*Kokotsakis*
*Kokotsakis*tessellation*Kokotsakis*
*Kokotsakis*tessellation*Kokotsakis*
*Kokotsakis*tessellation*Kokotsakis*
*Kokotsakis*tessellation*Kokotsakis*
*Kokotsakis*tessellation*Kokotsakis*
*Kokotsakis*tessellation*Kokotsakis*
*Kokotsakis*tessellation*Kokotsakis*
*Kokotsakis*tessellation*Kokotsakis*
*Kokotsakis*tessellation*Kokotsakis*
*Kokotsakis*tessellation*Kokotsakis*
*Kokotsakis*tessellation*Kokotsakis*
*Kokotsakis*tessellation*Kokotsakis*
*Kokotsakis*tessellation*Kokotsakis*
*Kokotsakis*tessellation*Kokotsakis*
*Kokotsakis*tessellation*Kokotsakis*
*Kokotsakis*tessellation*Kokotsakis*
*Kokotsakis*tessellation*Kokotsakis*
*Kokotsakis*tessellation*Kokotsakis*
*Kokotsakis*tessellation*Kokotsakis*
*Kokotsakis*tessellation*Kokotsakis*
*Kokotsakis*tessellation*Kokotsakis*
*Kokotsakis*tessellation*Kokotsakis*
*Kokotsakis*tessellation*Kokotsakis*
*Kokotsakis*tessellation*Kokotsakis*
*Kokotsakis*tessellation*Kokotsakis*
*Kokotsakis*tessellation*Kokotsakis*
*Kokotsakis*tessellation*Kokotsakis*
*Kokotsakis*tessellation*Kokotsakis*
*Kokotsakis*tessellation*Kokotsakis*
*Kokotsakis*tessellation*Kokotsakis*
*Kokotsakis*tessellation*Kokotsakis*
*Kokotsakis*tessellation*Kokotsakis*
*Kokotsakis*tessellation*Kokotsakis*
*Kokotsakis*tessellation*Kokotsakis*
*Kokotsakis*tessellation*Kokotsakis*
*Kokotsakis*tessellation*Kokotsakis*
At each flexion all vertices are located on a right circular cylinder.
July 13: ISTEC 2016, Vienna University of Technology 32/57
2. Flexible quad-meshes
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otsa
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otsa
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otsa
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otsa
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otsa
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otsa
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*Kokotsakis*tessellation*K
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*Kokotsakis*tessellation*K
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n*K
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kis*
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n*K
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kis*
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is*tessellatio
n*K
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is*tessellatio
n*K
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*Kokotsakis*tessellation*K
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*Kokotsakis*tessellation*K
okotsakis*
*Kokotsakis*tessellation*K
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*Kokotsakis*tessellation*K
okotsakis*
*Kokotsakis*tessellation*K
okotsakis*
*Kokotsakis*tessellation*K
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*Kokotsakis*tessellation*K
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*Kokotsakis*tessellation*K
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n*K
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n*K
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*Kokotsakis*tessellation*Kokotsakis*
*Kokotsakis*tessellation*Kokotsakis*
*Kokotsakis*tessellation*Kokotsakis*
*Kokotsakis*tessellation*Kokotsakis*
*Kokotsakis*tessellation*Kokotsakis*
*Kokotsakis*tessellation*Kokotsakis*
*Kokotsakis*tessellation*K
okotsakis*
*Kokotsakis*tessellation*K
okotsakis*
*Kokotsakis*tessellation*K
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*Kokotsakis*tessellation*K
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*Kokotsakis*tessellation*K
okotsakis*
*Kokotsakis*tessellation*K
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*Kokotsakis*tessellation*Kokotsakis*
*Kokotsakis*tessellation*Kokotsakis*
*Kokotsakis*tessellation*Kokotsakis*
*Kokotsakis*tessellation*Kokotsakis*
*Kokotsakis*tessellation*Kokotsakis*
*Kokotsakis*tessellation*Kokotsakis*
*Kokotsakis*tessellation*Kokotsakis*
*Kokotsakis*tessellation*Kokotsakis*
*Kokotsakis*tessellation*Kokotsakis*
*Kokotsakis*tessellation*Kokotsakis*
*Kokotsakis*tessellation*Kokotsakis*
*Kokotsakis*tessellation*Kokotsakis*
1
*Kokotsakis*
tessellatio
n*Kokotsakis*
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tessellatio
n*Kokotsakis*
*Kokotsakis*tessellation*Kokotsakis*
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*Kokotsakis*
tessella
tion*Kokotsa
kis*
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tessella
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kis*
*Kokotsakis*
tessellatio
n*Kokotsakis*
*Kokotsakis*
tessellatio
n*Kokotsakis*
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akis*te
ssellatio
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tessella
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tessella
tion*Kokotsa
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*Kokotsakis*tessellation*Kokotsakis*
*Kokotsakis*tessellation*Kokotsakis*
*Kokotsakis*tessellation*Kokotsakis*
*Kokotsakis*tessellation*Kokotsakis*
*Kokotsakis*
tessellation*Kokotsa
kis*
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tessellation*Kokotsa
kis*
*Kokotsakis*tessellation*Kokotsakis*
*Kokotsakis*tessellation*Kokotsakis*
*Kokotsakis*tessellation*Kokotsakis*
*Kokotsakis*tessellation*Kokotsakis*
*Kokotsakis*tessellation*Kokotsakis*
*Kokotsakis*tessellation*Kokotsakis*
*Kokotsakis*tessellation*Kokotsakis*
*Kokotsakis*tessellation*Kokotsakis*
*Kokotsakis*tessellation*Kokotsakis*
*Kokotsakis*tessellation*Kokotsakis*
*Kokotsakis*tessellation*Kokotsakis*
*Kokotsakis*tessellation*Kokotsakis*
*Kokotsakis*tessellation*K
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*Kokotsakis*tessellation*K
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*Kokotsakis*tessellation*Kokotsakis*
*Kokotsakis*tessellation*Kokotsakis*
*Kokotsakis*tessellation*Kokotsakis*
*Kokotsakis*tessellation*Kokotsakis*
*Kokotsakis*tessellation*Kokotsakis*
*Kokotsakis*tessellation*Kokotsakis*
*K
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otsak
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otsak
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otsa
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otsa
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otsa
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otsa
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otsa
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otsak
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otsa
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otsa
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otsa
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*Kokotsakis*tessellation*Kokotsakis*
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*Kokotsakis*tessellation*Kokotsakis*
*Kokotsakis*tessellation*Kokotsakis*
*Kokotsakis*tessellation*Kokotsakis*
*Kokotsakis*tessellation*Kokotsakis*
*Kokotsakis*tessellation*Kokotsakis*
*Kokotsakis*tessellation*Kokotsakis*
*Kokotsakis*tessellation*Kokotsakis*
*Kokotsakis*tessellation*Kokotsakis*
*Kokotsakis*tessellation*Kokotsakis*
*Kokotsakis*tessellation*Kokotsakis*
*Kokotsakis*tessellation*K
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4
2
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*Kokotsakis*tesse
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*Kokotsakis*tessellation*Kokotsakis*
*Kokotsakis*tessellation*Kokotsakis*
*Kokotsakis*tessellation*Kokotsakis*
*Kokotsakis*tessellation*Kokotsakis*
*Kokotsakis*tessellation*Kokotsakis*
*Kokotsakis*tessellation*Kokotsakis*
*Kokotsakis*tessellation*Kokotsakis*
*Kokotsakis*tessellation*Kokotsakis*
*Kokotsakis*tessellation*Kokotsakis*
*Kokotsakis*tessellation*Kokotsakis*
*Kokotsakis*tessellation*Kokotsakis*
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otsak
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*Kokotsakis*tessellation*Kokotsakis*
*Kokotsakis*tessellation*K
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otsa
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*Kokotsakis*tessellation*Kokotsakis*
*Kokotsakis*tessellation*Kokotsakis*
*Kokotsakis*tessellation*Kokotsakis*
*Kokotsakis*tessellation*Kokotsakis*
*Kokotsakis*tessellation*Kokotsakis*
*Kokotsakis*tessellation*Kokotsakis*
*Kokotsakis*tessellation*Kokotsakis*
*Kokotsakis*tessellation*K
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3
Different flexions of a 9×6tessellation mesh (dashesindicate valley folds).
Under which conditions isthere a flexion where theright border zig-zag fitsexactly to the left border —apart from a vertical shift ?
July 13: ISTEC 2016, Vienna University of Technology 33/57
2. Flexible quad-meshes
f11
ρ4
l−1r
l−1rρ4
l−2r2
l−2r2ρ4
l−3r3
l−3r3ρ4
l r
l rρ4
r2
r2ρ4
l−1r3
l−1r3ρ4
l−2r4
l−2r4ρ4
l2r2
l2r2ρ4
l r3
l r3ρ4
r4
r4ρ4
l−1r5
l−1r5ρ4
l3r3
l2r4 l r
5r6
l−1ρ4 l
−2rρ4 l
−3r2ρ4
r
rρ4
l−1r2
l−1r2ρ4
l−2r3
l−2r3ρ4
l r2
l r2ρ4
r3
r3ρ4
l−1r4
l−1r4ρ4
l2r3
l2r3ρ4
l r4
l r4ρ4
r5
r5ρ4
This scheme of a 7×7 tesselation mesh indicateswhich product of helical motions l , r and 180◦-rotations ρ4 maps f11 onto f i j .
Theorem:
f i j=
li−j2 r
i+j2−1(f11)
for i + j ≡ 0 (mod 2)
li−j−12 r
i+j−32 ρ4(f11)
for i + j ≡ 1 (mod 2)
( r = ρ2 ◦ ρ1 and l = ρ4 ◦ ρ1)
July 13: ISTEC 2016, Vienna University of Technology 34/57
2. Flexible quad-meshes
f11
ρ4
l−1r
l−1rρ4
l−2r2
l−2r2ρ4
l−3r3
l−3r3ρ4
l r
l rρ4
r2
r2ρ4
l−1r3
l−1r3ρ4
l−2r4
l−2r4ρ4
l2r2
l2r2ρ4
l r3
l r3ρ4
r4
r4ρ4
l−1r5
l−1r5ρ4
l3r3
l2r4
l r5
r6
l−1ρ4 l
−2rρ4 l
−3r2ρ4
r
rρ4
l−1r2
l−1r2ρ4
l−2r3
l−2r3ρ4
l r2
l r2ρ4
r3
r3ρ4
l−1r4
l−1r4ρ4
l2r3
l2r3ρ4
l r4
l r4ρ4
r5
r5ρ4
Coaxial helical motions commute=⇒
Theorem:A flexion of a m × n tesselationmesh closes with a vertical shiftof k faces ⇐⇒there exist a, b ∈ Z with
lar b = d2π, k = −a − b.
July 13: ISTEC 2016, Vienna University of Technology 35/57
2. Flexible quad-meshes
Closing flexion of a 7× 9 tesselation mesh with l−6r = d2π, k = 5.
*Kokotsakis*tessellation*m
esh*
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h*
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otsa
kis*
tess
ella
tion
*mes
h*
*Kok
otsa
kis*
tess
ella
tion*
mes
h*
*K
okots
akis
*te
ssel
lati
on*m
esh*
*Kok
otsa
kis*
tess
ella
tion*
mes
h*
*K
okots
akis
*te
ssell
ati
on*m
esh
* *Kok
otsak
is*tes
sellat
ion*m
esh*
*K
okots
akis
*te
ssel
lation*m
esh*
*Kokotsakis*tessellation*mesh*
*Kok
otsa
kis*
tess
ella
tion*
mes
h*
*Kokotsakis*tessellation*mesh*
f 91
f 81
f 71
f 61
f 51
f 41
f 31 f 21
f 11
f 92
f 82 f 72
f 62 f 13
f 24
f 14
f 95
f25
f 96 f 8
6
f57 f37
f 27 f 17
How obtainable ?
• numerically:minimize a distance, or
• start with two coaxialhelical motions r, lsatisfying lar b = d2π.
July 13: ISTEC 2016, Vienna University of Technology 36/57
2. Flexible quad-meshes
The analogon of a Schwarz boot with trapezoids; a closing flexion of a 17 × 16tesselation mesh with l−10r 6 = d2π and k = 4.
*K
ok
otsak
is*tessellatio
n*
mesh
*K
ok
otsak
is*tessellatio
n*
6,0
*Kokotsakis*tessellation*m
esh*Kokotsakis*tessellation*
6,1
*K
ok
otsak
is*tessellatio
n*
mesh
*K
ok
otsak
is*tessellatio
n*
*Kokotsakis*tessellation*m
esh*Kokotsakis*tessellation*
*K
ok
otsak
is*tessellatio
n*
mesh
*K
ok
otsak
is*tessellatio
n*
*Kokotsakis*tessellation*m
esh*Kokotsakis*tessellation*
*K
ok
otsak
is*tessellatio
n*
mesh
*K
ok
otsak
is*tessellatio
n*
*Kokotsakis*tessellation*m
esh*Kokotsakis*tessellation*
*K
ok
otsak
is*tessellatio
n*
mesh
*K
ok
otsak
is*tessellatio
n*
*Kokotsakis*tessellation*m
esh*Kokotsakis*tessellation*
*K
ok
otsak
is*tessellatio
n*
mesh
*K
ok
otsak
is*tessellatio
n*
*Kokotsakis*tessellation*m
esh*Kokotsakis*tessellation*
*K
ok
otsak
is*tessellatio
n*
mesh
*K
ok
otsak
is*tessellatio
n*
*Kokotsakis*tessellation*m
esh*Kokotsakis*tessellation*
*K
ok
otsak
is*tessellatio
n*
mesh
*K
ok
otsak
is*tessellatio
n*
*Kokotsakis*tessellation*m
esh*Kokotsakis*tessellation*
*K
ok
otsak
is*tessellatio
n*
mesh
*K
ok
otsak
is*tessellatio
n*
*Kokotsakis*tessellation*m
esh*Kokotsakis*tessellation*
7,0*K
okotsakis*tessellation*mesh*K
okotsakis*tessellation*
7,1*K
okotsakis*tessellation*mesh*K
okotsakis*tessellation*
*Kokotsakis*tessellation*m
esh*Kokotsakis*tessellation*
*Kokotsakis*tessellation*m
esh*Kokotsakis*tessellation*
*Kokotsakis*tessellation*m
esh*Kokotsakis*tessellation*
*Kokotsakis*tessellation*m
esh*Kokotsakis*tessellation*
*Kokotsakis*tessellation*m
esh*Kokotsakis*tessellation*
*Kokotsakis*tessellation*m
esh*Kokotsakis*tessellation*
*Kokotsakis*tessellation*m
esh*Kokotsakis*tessellation*
*Kokotsakis*tessellation*m
esh*Kokotsakis*tessellation*
*Kokotsakis*tessellation*m
esh*Kokotsakis*tessellation*
*Kokotsakis*tessellation*m
esh*Kokotsakis*tessellation*
*Kokotsakis*tessellation*m
esh*Kokotsakis*tessellation*
*Kokotsakis*tessellation*m
esh*Kokotsakis*tessellation*
*Kokotsakis*tessellation*m
esh*Kokotsakis*tessellation*
*Kokotsakis*tessellation*m
esh*Kokotsakis*tessellation*
8,0 8,1
*Kokotsakis*tessellation*m
esh*Kokotsakis*tessellation*
*Kokotsakis*tessellation*m
esh*Kokotsakis*tessellation*
*Kokotsakis*tessellation*m
esh*Kokotsakis*tessellation*
*Kokotsakis*tessellation*m
esh*Kokotsakis*tessellation*
*Kokotsakis*tessellation*m
esh*Kokotsakis*tessellation*
*Kokotsakis*tessellation*m
esh*Kokotsakis*tessellation*
*Kokotsakis*tessellation*m
esh*Kokotsakis*tessellation*
*Kokotsakis*tessellation*m
esh*Kokotsakis*tessellation*
*Kokotsakis*tessellation*m
esh*Kokotsakis*tessellation*
*Kokotsakis*tessellation*m
esh*Kokotsakis*tessellation*
*Kokotsakis*tessellation*m
esh*Kokotsakis*tessellation*
9,0 9,1
*Kokotsakis*tessellation*mesh*Kokotsakis*tessellation*
*Kokotsakis*tessellation*mesh*Kokotsakis*tessellation*
*Kokotsakis*tessellation*mesh*Kokotsakis*tessellation*
*Kokotsakis*tessellation*mesh*Kokotsakis*tessellation*
*Kokotsakis*tessellation*mesh*Kokotsakis*tessellation*
*Kokotsakis*tessellation*mesh*Kokotsakis*tessellation*
*Kokotsakis*tessellation*mesh*Kokotsakis*tessellation*
10,0 10,1
*Kokotsakis*tessellation*mesh*Kokotsakis*tessellation**Kokotsakis*tessellation*mesh*Kokotsakis*tessellation*
*Kokotsakis*tessellation*mesh*Kokotsakis*tessellation**Kokotsakis*tessellation*mesh*Kokotsakis*tessellation*
11,0 11,1
*Kokotsakis*tessellation*mesh*Kokotsakis*tessellation* *Kokotsakis*tessellation*mesh*Kokotsakis*tessellation* 11,11
*Kokotsakis*tessellation*mesh*Kokotsakis*tessellation* *Kokotsakis*tessellation*mesh*Kokotsakis*tessellation* 11,13
*Kokotsakis*tessellation*mesh*Kokotsakis*tessellation*
12,0 12,1
*Kokotsakis*tesse
llation*mesh*Kok
otsakis*tessellatio
n**Ko
kotsakis*t
essellatio
n*mesh*
Kokotsak
is*tessell
ation*
*Kokotsakis*tesse
llation*mesh*Kok
otsakis*tessellatio
n**Ko
kotsakis*t
essellatio
n*mesh*
Kokotsak
is*tessell
ation*
*Kokotsakis*tesse
llation*mesh*Kok
otsakis*tessellatio
n*
*Kok
otsak
is*tes
sellat
ion*m
esh*K
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akis*
tessel
lation
**K
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is*tess
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h*K
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s*te
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*Kokotsakis*tessellation*mesh*Kokotsakis*tessellation*
5,0
*K
ok
otsak
is*tessellatio
n*
mesh
*K
ok
otsak
is*tessellatio
n*
5,1
*Kokotsakis*tessellation*mesh*Kokotsakis*tessellation*
*K
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otsak
is*tessellatio
n*
mesh
*K
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otsak
is*tessellatio
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*Kokotsakis*tessellation*mesh*Kokotsakis*tessellation*
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otsak
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mesh
*K
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otsak
is*tessellatio
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*Kokotsakis*tessellation*mesh*Kokotsakis*tessellation*
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otsak
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mesh
*K
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otsak
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*Kokotsakis*tessellation*mesh*Kokotsakis*tessellation*
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otsak
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*K
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otsak
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*Kokotsakis*tessellation*mesh*Kokotsakis*tessellation*
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otsak
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otsak
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otsak
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*K
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mesh
*K
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otsak
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n*
*Kokotsakis*tessellation*mesh*Kokotsakis*tessellation*
*Kokotsakis*tessellation*m
esh*Kokotsakis*tessellation*
4,0
*Kokotsakis*tessellation*mesh*Kokotsakis*tessellation*
4,1
*Kokotsakis*tessellation*m
esh*Kokotsakis*tessellation*
*Kokotsakis*tessellation*mesh*Kokotsakis*tessellation*
*Kokotsakis*tessellation*m
esh*Kokotsakis*tessellation*
*Kokotsakis*tessellation*mesh*Kokotsakis*tessellation*
*Kokotsakis*tessellation*m
esh*Kokotsakis*tessellation*
*Kokotsakis*tessellation*mesh*Kokotsakis*tessellation*
*Kokotsakis*tessellation*m
esh*Kokotsakis*tessellation*
*Kokotsakis*tessellation*mesh*Kokotsakis*tessellation*
*Kokotsakis*tessellation*m
esh*Kokotsakis*tessellation*
*Kokotsakis*tessellation*mesh*Kokotsakis*tessellation*
*Kokotsakis*tessellation*m
esh*Kokotsakis*tessellation*
*Kokotsakis*tessellation*mesh*Kokotsakis*tessellation*
*Kokotsakis*tessellation*m
esh*Kokotsakis*tessellation*
*Kokotsakis*tessellation*mesh*Kokotsakis*tessellation*
*Kokotsakis*tessellation*m
esh*Kokotsakis*tessellation*
*Kokotsakis*tessellation*mesh*Kokotsakis*tessellation*
*Kokotsakis*tessellation*mesh*Kokotsakis*tessellation*
*Kokotsakis*tessellation*mesh*Kokotsakis*tessellation*
*Kokotsakis*tessellation*mesh*Kokotsakis*tessellation*
*Kokotsakis*tessellation*mesh*Kokotsakis*tessellation*
*Kokotsakis*tessellation*mesh*Kokotsakis*tessellation*
*Kokotsakis*tessellation*mesh*Kokotsakis*tessellation*
*Kokotsakis*tessellation*mesh*Kokotsakis*tessellation*
*Kokotsakis*tessellation*mesh*Kokotsakis*tessellation*
*Kokotsakis*tessellation*mesh*Kokotsakis*tessellation*
*Kokotsakis*tessellation*mesh*Kokotsakis*tessellation*
*Kokotsakis*tessellation*mesh*Kokotsakis*tessellation*
*Kokotsakis*
tessellation*mesh*Kokotsa
kis*tesse
llation*
*Kokots
akis*te
ssell
ation*m
esh*K
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ation*
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tessellation*mesh*Kokotsa
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f 161 f 15
1
f 141 f 13
1f 121 f 11
1
f 101 f 9
1
f 81
f 71
f 61
f 51
f 41
f 31
f 21
f 11
f 172 f 14
2f 122
f 102
f 92f 82
f 72f 62
f 52f 42
f 32f 22
f 12
f 16
f 27 f 17
f 28
f 18
f 29 f 19
f 210
f 115
f 1616
f 1416
f 1216
f 1016
f 816
f 616
f 416
f 216
July 13: ISTEC 2016, Vienna University of Technology 37/57
2. Flexible quad-meshes
*Kokotsakis*tessellation*m
esh*Kokotsakis*tessellation*m
esh*Kokotsakis*tessellation
*Kokotsakis*tessellation*m
esh*Kokotsakis*tessellation*m
esh*Kokotsakis*tessellation
*Kokotsakis*tessellation*m
esh*Kokotsakis*tessellation*m
esh*Kokotsakis*tessellation
*Kokotsakis*tessellation*m
esh*Kokotsakis*tessellation*m
esh*Kokotsakis*tessellation
*Kokotsakis*tessellation*m
esh*Kokotsakis*tessellation*m
esh*Kokotsakis*tessellation
*Kokotsakis*tessellation*m
esh*Kokotsakis*tessellation*m
esh*Kokotsakis*tessellation
*Kokotsakis*tessellation*mesh*Kokotsakis*tessellation*mesh*Kokotsakis*tessellation
*Kokotsakis*tessellation*mesh*Kokotsakis*tessellation*mesh*Kokotsakis*tessellation
*Kokotsakis*tessellation*mesh*Kokotsakis*tessellation*mesh*Kokotsakis*tessellation
*Kokotsakis*tessellation*mesh*Kokotsakis*tessellation*mesh*Kokotsakis*tessellation*Kokotsakis*tessellation*mesh*Kokotsakis*tessellation*mesh*Kokotsakis*tessellation
*Kokotsakis*tessellation*mesh*Kokotsakis*tessellation*mesh*Kokotsakis*tessellation
*Kokotsakis*tessellation*mesh*Kokotsakis*tessellation*m
esh*Kokotsakis*tessellation
*K
okotsakis*tessellation*m
esh*K
okotsakis*tessellation*m
esh*K
okotsakis*tessellation
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f 1616
f31 f11
f32 f12
f39 f19f310
f110
f317 f117
f318
f118
Theorem:Each closing flexion of a m × n tesselation mesh is infinitesimally rigid.
July 13: ISTEC 2016, Vienna University of Technology 38/57
3. Curved folding, Example 1
A common way of producing small boxes is to push up appropriate planar cardbordforms Φ0 with prepared creases. Below the case of creases along circular arcs c0.
f
c0
Φ0
planar version with circular creases
Φ
c
corresponding box with planar creases
July 13: ISTEC 2016, Vienna University of Technology 39/57
3. Curved folding, Example 1
As proved by W. Wunderlich (1958), the spatial creases c are again planar andknown as meridians of surfaces of revolution with constant Gaussian curvature.
f
c0
Φ0
planar version with circular creases
Φ
c
corresponding box with planar creases
July 13: ISTEC 2016, Vienna University of Technology 40/57
3. Curved folding, Example 1
P
c
x
yc′
ρ1c′′
M1
M2ρ 1
ρ2
α
On surfaces of revolution the meridians andparallel circles are the principal curvature lines.Therefore, the signed principal curvatures are
κ1 = − y ′′
cosα, κ2 =
cosα
y.
The Gaussian curvature is definedas K = κ1κ2 . Hence,
K = const. ⇐⇒y ′′ +Ky = 0, x ′ =
√
1− y ′ 2.
provided that cosα 6= 0.
July 13: ISTEC 2016, Vienna University of Technology 41/57
3. Curved folding, Example 1
The general solution of y ′′ + Ky = 0with constant K 6= 0 is
for K > 0 :
y = a cos s√K + b sin s
√K ,
for K < 0 :
y = a cosh s√−K + b sinh s
√−K
with constants a, b ∈ R, and
x =∫
√
1− y ′ 2 ds.
we can restrict to six cases, up tosimilarities (Gauß, Minding).
Pseudosphere (tractroid)
July 13: ISTEC 2016, Vienna University of Technology 42/57
3. Curved folding, Example 1
a = 0.78 a = 1.00 a = 1.18
1. 2. 3.
xxx
yyy
b = 0.3
tractrix
a = 0.8
4.
5.
6.
M1
M2
xx
x
y
y
y
1
There are six types of meridians to distinguish at the surfaces of revolution withconstant Gaussian curvature K 6= 0.
July 13: ISTEC 2016, Vienna University of Technology 43/57
3. Curved folding, Example 12
x
yy0
z
a
P
T
c
β
Φ
ε
Let c0 satisfy y ′′0+Ky0 = 0 and bounda cylindrical patch Φ0 with generatorsorthogonal to the x-axis a0.
Theorem: If at a cylindrically bentpose Φ of Φ0 the boundary c lies ina plane ε, then it satisfies the samedifferential equation as c0.
Proof: y0(s) = y(s) cosβ withβ < π/2 being the (constant) angleof inclination of the cylinder.
The axis of c is the meet of ε and the plane of the orthogonal section a, which is thebent counterpart of the original axis a0 of c0.
July 13: ISTEC 2016, Vienna University of Technology 44/57
3. Curved folding, Example 1
Theorem:
Let Φ0 be a planar ‘ruled surface’ with a transversalcurve (crease) c0, which separates Φ0 into two patchesΦ10 and Φ20.
Suppose the generators of the ruling remain straight atthe bent pose Φ1, Φ2 with a curved edge c between.Then c must be a planar curve.
If all generators of Φ1 and Φ2 are extended to infinity,we obtain two torses, which are symmetric with respectto the plane of c .
Φ10
Φ20
c0α
1/κg
E.g., take a cone of revolution with a parabolic section c and reflect the part oppositeto the apex in the plane of c . In Origami this is called reflection operation.
July 13: ISTEC 2016, Vienna University of Technology 45/57
3. Curved folding, Example 1
Sketch of the Proof:
Let κ(s) and τ(s) denote the curvature and torsion of c . In terms of the angle γ1(s)between the osculating plane of c and the tangent plane of the torse Φ1, the geodesiccurvature of c w.r.t. Φ1 is
κg = κ cos γ1.
The geodesic curvature κg must be the same w.r.t. Φ2 =⇒ γ2 = −γ1.
The angle α between the tangent of c and the generator of Φ1 satisfies
cosα : sinα = (τ − γ ′1) : −κ sin γ1.The angle α must be the same w.r.t. Φ2 =⇒ (τ − γ ′1) = −(τ + γ ′1), hence τ = 0.
July 13: ISTEC 2016, Vienna University of Technology 46/57
4. Curved folding, Example 2
c0 P0
Q0
Unfolding and corresponding spatial form (photos: G. Glaeser)
The spatial form Φ is obtained by gluing together the semicircles with the straightsegments. How to model the resulting convex body ?
July 13: ISTEC 2016, Vienna University of Technology 47/57
4. Curved folding, Example 2
c
Φ
Unfolding and corresponding spatial form (photos: G. Glaeser)
The crucial point is here that the ruling is unknown.
M. Kilian, S. Flöry, Z. Chen, N.J. Mitra, A. Sheffer, H. Pottmann: Curved Folding.ACM Trans. Graphics 27/3 (2008), Proc. SIGGRAPH 2008.
July 13: ISTEC 2016, Vienna University of Technology 48/57
4. Curved folding, Example 2y
A0
B0C0
D0
M0Φ0
c10
c ′10
c20
c ′20
r
A physical model shows:
• The spatial body with its developable boundary Φis convex and uniquely defined.
• The helix-like curve c = c1 ∪c2 is a proper edge of Φ; theresulting solid is the convex hullof c .
• The semicircular disks arebent to cones with apices Aand C. Hence, Φ is a C1-compound of two cones and atorse between.
• The body has an axis a ofsymmetry which connects themidpoint M with the remainingtransition point B = D on c .
July 13: ISTEC 2016, Vienna University of Technology 49/57
4. Curved folding, Example 2
A0
B0C0
B′0=D0
M0
E10
E ′10
E ′20
E20
Φ0
c10
c ′10
c20
c ′20
g0
r
Consequences:
• Because of the straight segments of c10, thedevelopable surface on the left hand side of c1belongs to the rectifying torse of c1.
• At A and C the surface Φcan be approximated by a rightcone with apex angle 60◦.
• The tangent tA to c1 at Ais a generator, the osculatingplane of c1 a tangent plane ofthis cone; the rectifying planepasses through the cone’s axis.
•When g0 meets both straightsides of c0, then g meets c1and c2 at points with paralleltangents =⇒ coincidingtangent indicatrices.
July 13: ISTEC 2016, Vienna University of Technology 50/57
4. Curved folding, Example 2
A0
B0C0
B′0=D0
M0
E10
E ′10
E ′20
E20
Φ0
c10
c ′10
c20
c ′20
gM
rψ
• The tangent at the point E2 ∈ c2 of transitionbetween the cone with apex A and the torse mustbe parallel to tA.
• The tangent at the analoguepoint E1 ∈ c1 is parallel to thefinal tangent tC of c2.
• The subcurves AE1 ⊂ c1and E2C ⊂ c2 have concidingtangent indicatrices.
At a first approximation thecone with apex A is specified asright cone with apex angle 60◦;c1 is a geodesic circle on thiscone.
July 13: ISTEC 2016, Vienna University of Technology 51/57
4. Curved folding, Example 2
A′
t ′A
B′
E ′1
c ′1
A′′
t ′′A
B′′E ′′1
c ′′1
A′′′
t ′′′A
B′′′
C ′′′
E ′′′1
E ′′′2
c ′′′1
c ′′′2
a′′′
A0
c0
Approximation 1:
c1 is an algebraic curve.
tA is parallel to the tangent atE2 ∈ c2. Analogously, tC is parallelto the tangent at E1 ∈ c1. Thisdefines the axis a of symmetry.
We notice a contradiction sincethe osculating plane of c1 at B isnot orthogonal to BC.
July 13: ISTEC 2016, Vienna University of Technology 52/57
4. Curved folding, Example 22
tE2=tA
tB
tC=tE1
‖ a
tE2=tA
tB
tC=tE1
‖ a
Left: Tangent indicatrices of c1 and c2 for the firstapproximation; no coinciding subcurves!
Approximation 2 is defi-ned by alined side viewsof the tangent indicatrices(right) =⇒
• the subcurve AE1 ⊂ c1 isa curve of constant slope.
• the central torse is acylinder,
• a translation maps AE1onto the subcurve E2C ⊂c2.
July 13: ISTEC 2016, Vienna University of Technology 53/57
A′
B′
C ′
M ′
E ′1
E ′2
F ′1
F ′2
c ′1
c ′2
a′
A′′
B′′
C ′′
M ′′
E ′′1
E ′′2
F ′′1
F ′′2
c ′′1
c ′′2
A′′′
B′′′
C ′′′
M ′′′
E ′′′1
E ′′′2
F ′′′1
F ′′′2
c ′′′1
c ′′′2
a′′′
a′′′1
a′′′2
Approximation 2:
The product of the translation A 7→ E2and the half-rotation about a maps thesubcurve AE1 onto itself, but in reverseorder.
Therefore this portion AE1 has an axisa1 of symmetry passing through themidpoint F1.
July 13: ISTEC 2016, Vienna University of Technology 54/57
Approximation 2 shows an excellentaccordance with the physical model.
. . . but there remains a contradiction.
A
B
C
E1
E2
F1
F2
c1
c20
a
a1
a2
July 13: ISTEC 2016, Vienna University of Technology 55/57
4. Curved folding, Example 2
A0
B0
M0 E10
E ′10
F10
F ′10
E ′20
Nψ
A
N
E1
F1
c1
a1
Due to the symmetry w.r.t. a1, the midpoint N of AE1 lies on a1. The distancesA0F10 and A0E10 are preserved, the triangle ANF1 is congruent to its counterpartA0N0F10 in the unfolding. But NF1 is not (exactly) orthogonal to the tangent of c1at F1.
July 13: ISTEC 2016, Vienna University of Technology 56/57
References
[1] A.D. Alexandrov: Convex Polyhedra. Springer Monographs in Mathematics,Springer 2005, ISBN 9783540263401, (first Russian ed. 1950).
[2] A.I. Bobenko, I. Izmestiev: Alexandrov’s theorem, weighted Delaunay triangulations,and mixed volumes. Annales de l’Institut Fourier 58/(2), 447–505 (2008),arXiv:math.DG/0609447.
[3] R. Bricard: Mémoire sur la théorie de l’octaèdre articulé. J. math. pur. appl.,Liouville 3, 113–148 (1897).
[4] R. Bricard: Leçon de Cinématique, Tome II, de Cinématique appliqueé. Gauthier-Villars et Cie, Paris 1927.
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[5] E.D. Demaine, J. O’Rourke: Geometric folding algorithms: linkages, origami,polyhedra. Cambridge University Press 2007.
[6] H. Dirnboeck, H. Stachel: The Development of the Oloid. J. Geom. Graph. 1,105–118 (1997).
[7] M. Kilian, S. Flöry, Z. Chen, N.J. Mitra, A. Sheffer, H. Pottmann: Curved Folding.ACM Trans. Graphics 27/3 (2008), Proc. SIGGRAPH 2008.
[8] A. Kokotsakis: Über bewegliche Polyeder. Math. Ann. 107, 627–647, 1932.
[9] G. Nawratil, H. Stachel: Composition of spherical four-bar-mechanisms. In D. Pislaet al. (eds.): New Trends in Mechanism Science, Springer 2010, pp. 99–106.
[10] G. Nawratil: Flexible octahedra in the projective extension of the Euclidean 3-space.J. Geom. 14/2, 147–169 (2010).
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[11] G. Nawratil: Reducible compositions of spherical four-bar linkages with a sphericalcoupler component. Mech. Mach. Theory 46/5, 725–742 (2011).
[12] H. Stachel: What lies between the flexibility and rigidity of structures. SerbianArchitectural J. 3/2, 102–115 (2011).
[13] H. Stachel: On the Rigidity of Polygonal Meshes. South Bohemia Math. Letters19/1, 6–17 (2011).
[14] H. Stachel: A Flexible Planar Tessellation with a Flexion Tiling a Cylinder ofRevolution. J. Geometry Graphics 16/2, 153–170 (2012).
[15] H. Stachel: On the flexibility and symmetry of overconstrained mechanisms. Phil.Trans. R. Soc. A 372, num. 2008, 20120040 (2014).
[16] H. Stachel: Flexible Polyhedral Surfaces With Two Flat Poses. In Special Issue, B.Schulze (ed.): Rigidity and Symmetry, Symmetry 7(2), 774–787 (2015).
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[17] H. Stachel: Two examples of curved foldings. Proceedings of the 17th Internat.Conf. on Geometry and Graphics, Beijing 2016 (to appear).
[18] W. Whiteley: Rigidity and scene analysis. In J.E. Goodman, J. O’Rourke(eds.): Handbook of Discrete and Computational Geometry, 2nd ed., Chapman& Hall/CRC, Boca Raton 2004, pp. 1327–1354.
[19] W. Wunderlich: Aufgabe 300. El. Math. 22, p. 89 (1957); author’s solution: El.Math. 23, 113–115 (1958).
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