rigidity theory and some applications brigitte servatius wpi this presentation will probably involve...
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Rigidity Theory and
Some Applications
Brigitte ServatiusWPI
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Ingredients:
Universal (ball) JointV: vertices
Rigid Rod (bar)E: edges
Framework:
nRVp : embedding
),,( pEVF
),( EVG graph
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Basic Definitions:
Continuous 1-parameter familyof frameworks.
))(,,( tpEVF
Deformation:
2))()(())()(( ijjiji ctptptptp
Length of the bars is preserved
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Trivial Deformations
Deformations from isometriesTrivial degrees of freedom
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Non-trivial Deformations
Watt Engine
Peaucellier Engine
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Non-trivial Deformations
Watt Engine
Peaucellier Engine
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Non-trivial Deformations
Watt Engine
Peaucellier Engine
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Non-trivial Deformations
Watt Engine
Peaucellier Engine
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Rigidity:
Rigid: Every deformation is locally trivial.
Globally Rigid:
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Ambient Dimension
Tendency: the rigidity of a framework decreases as the dimension of the ambient space increases.
The complete graph is rigid in all dimensions.
Rigidity in dimension 0 is pointless
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Infinitesimal Analysis
2))()(())()(( ijjiji ctptptptp Quadratic:
0))(')('())()(( tptptptp jijiLinear:
|E| equations in m|V| unknowns: ip'
0))0(')0('())0()0(( jiji ppppEvaluate at 0:
Infinitesimal Motion (Flex): Non-Trivial Solution
There is always a subspace of trivial solutions Rank depends only on the dimesnion: m(m+1)/2
Infinitesimally Rigid: Only trivial Solutions
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Infinitesimal Rigidity
ip'0)''()( jiji pppp
Unknowns
Infinitesimally Rigid: Only trivial Solutions
)0(ii pp
Infinitesimally Rigid Rigid
(It may be rigid anyway…)
Infinitesimally Rigid Rigid/
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Visual Linear Algebra
Is the Framework Infinitesimally rigid? First let’s eliminate the trivial solutions by pinning the bottom vertices.
The equation at the left vertical rod forces the velocity at the top corner to lie along the horizontal direction.
The equation at the right vertical rod forces the velocity at the top left corner to also lie along the horizontal direction.
The top bar forces the two horizontal vectors to be equal in magnitude and direction.
The remaining vertices of the toptriangle force the third vertex velocity to match the infinitesimal rotation.
CONTRADICTION!!! The last red bar insists on an infinitesimal rotation centered on its pinned vertex.
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Parallel Redrawings
Is the Framework Infinitesimally rigid?
The three connecting edges happento be concurrent. Dilate the larger triangle.The blue displacement vectors satisfy the equation at the left.
0)()( jiji ddpp
Displacing the points results in a PARALLEL REDRAWING of the original framework.
0)''()( jiji pppp
The vector condition is familiar…The blue redrawing displacements correspond a red flex.
Conclusion: The original framework did have an infinitesimal motion.
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The Rigidity Matrix
nmq
pEVR
...000
00
0...0
),,( ...3113
2112
jiij ppq
A framework is infinitesimally rigid in m-space if and only if
its rigidity matrix has rank 2
)1(
mmmn
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Euler Conjecture
“A closed spacial figure allows no changes as long as it is not ripped apart”
1766.
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Cauchy’s Theorem - 1813
“If there is an isometry between the surfaces
of two strictly convex polyhedra which is
an isometry on each of the
faces, then the polyhedra
are congruent”.
The 2-skeleton of a strictly
Convex 3D polyhedron is rigid.
Like Me!
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Bricard Octahedra - 1897
Animation by Franco Saliola,York University using STRUCK.
By Cauchy’s Theorem, an octahedron is rigid.
If the 1-skeleton is knotted ...
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More Euler Spin-offs…
Alexandrov – 1950– If the faces of a strictly convex polyhedron are
triangulated, the resulting 1-skeleton is rigid.
Gluck – 1975– Every closed simply connected ployhedral
surface in 3-space is rigid.
Connelly – 1975– Non-convex counterexample to Euler’s
Conjecture.
Asimov & Roth - 1978
– The 1-skelelton of any convex 3D polyhedron with a non-triangular face is non-rigid.
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More Euler Spin-offs…
Alexandrov – 1950– If the faces of a strictly convex polyhedron are
triangulated, the resulting 1-skeleton is rigid.
Gluck – 1975– Every closed simply connected ployhedral
surface in 3-space is rigid.
Connelly – 1975– Non-convex counterexample to Euler’s
Conjecture.
Asimov & Roth - 1978
– The 1-skelelton of any convex 3D polyhedron with a non-triangular face is non-rigid.
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More Euler Spin-offs…
Alexandrov – 1950– If the faces of a strictly convex polyhedron are
triangulated, the resulting 1-skeleton is rigid.
Gluck – 1975– Every closed simply connected ployhedral
surface in 3-space is rigid.
Connelly – 1975– Non-convex counterexample to Euler’s
Conjecture.
Asimov & Roth - 1978
– The 1-skelelton of any convex 3D polyhedron with a non-triangular face is non-rigid.
“Jitterbug”Photo: Richard Hawkins
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Combinatorial Rigidity
Infinitesimal rigidity of a framework depends on the embedding.
An embedding is generic if small perturbations of the vertices do not change the rigidity properties.
Generic embeddings are an open dense subset of all embeddings.
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Generic Embeddings
Theorem: If some generic framework is rigid, then ALL generic embeddings of the graph are also rigid.
Generic embedding – think random embedding.
A graph is generically rigid (in dimensionm) if it has any infinitesimally rigidembedding.
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The Rigid World
RigidGenericallyRigid
InfinitesimallyRigid
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Generic Rigidity in Dimension 1:
All embeddings on the line are generic.
Rigidity is equivalent to connectivity
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Generic Rigidity in Dimension 2:
Laman’s Theorem– G = (V,E) is rigid iff G has a subset F
of edges satisfying • |F| = 2|V| - 3 and• |F’| < 2|V(F’) - 3 for subsets F’ of F
This condition says that:– G has enough edges to be rigid– G has no overbraced subgraph.
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Generic Rigidity in the Plane:
Generic RigidityLaman’s Condition3T2: The edge set contains the union
three trees such that– Each vertex belongs to two trees
– No two subtrees span the same vertex set G has as subgraph with a Henneberg
construction.
The following are equivalent:
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Mat1 )0,0(a)0,3(b)1,2(c
)3,0(d)3,3(e)2,1(f
ef
df
de
cf
be
bc
ad
ac
ab
ffeeddccbbaa
R
yxyxyxyxyxyx
121200000000
110011000000
000303000000
113000100000
000000003000
000000111100
000030000030
000000120012
000000000303
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Mat2 )0,0(a)0,3(b)1,2(c
)3,0(d)3,3(e)2,1(f
ef
df
de
cf
be
bc
ad
ab
ab
eddccbbaa
R
xyxyxyxyx
200000000
011000000
303000000
000100000
000003000
000111100
030000030
000120012
000000303
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Mat3 )0,0(a)0,3(b)1,2(c
)3,0(d)3,3(e)2,1(f
ef
df
de
cf
be
bc
ad
ab
ab
dcbaedcba
R
yyyyxxxxx
000020000
100001000
000033000
010000100
003000000
011000110
300300000
010100202
000000033
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ef
df
de
cf
be
bc
ad
ab
ab
dcbaedcba
R
yyyyxxxxx
000020000
100001000
000033000
010000100
003000000
011000110
300300000
010100202
000000033
Mat4 )0,0(a)0,3(b)1,2(c
)3,0(d)3,3(e)2,1(f
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ef
df
de
cf
be
bc
ad
ab
ab
dcbaedcba
R
yyyyxxxxx
000020000
100001000
000033000
010000100
003000000
011000110
300300000
010100202
000000033
Mat4 )0,0(a)0,3(b)1,2(c
)3,0(d)3,3(e)2,1(f
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B Mat 1Ba
A
cC
b
},{01 CbV },{02 AcV },{12 BaV
12V 01V
02V
0T
1T
2T
011000000000
001100000000
100100000000
000010010000
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000100100110
000000110011
001000011001
2
1
0
Aa
cA
BcT
Bb
aB
CaT
Cc
bC
AbT
BaAcBaCbAcCb yyyyxxxxxxyy
Ba
A
c
C
b
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Henneberg Moves
Zero Extension:
One Extension:
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Henneberg Moves
Zero Extension:
One Extension:
1. No vertices of degree 22. NO TRIANGLES!1. No vertices of degree 22. NO TRIANGLES!
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Applications
Computer Modeling– Cad– Geodesy (mapping)
Robotics– Navigation
Molecular Structures– Glasses– DNA
Structural Engineering– Tensegrities
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Applications: CAD
Combinatorial (discrete) results preferred
Generic results not sufficient
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Glass Model
Edge length ratio at most 3:1No small rigid subgraphs – 1st order phase transition
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Cycle Decompositions
The graph decomposes into disjoint Hamiltonian cycles
The are many “different” ones:
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ApplicationsMolecular Structures
Ribbon Model
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ApplicationsMolecular Structures
PROTASE
Ball and Joint Model
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ApplicationsMolecular Structures
HIV
Ball and Joint Model
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ApplicationsTensegrities
Bob ConnellyKenneth Snelson
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Applications
Photo by Kenneth Snelson
Tensegrities
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Open Problems
3D – characterize generic rigidity 2D
1. Find a “good” algorithm to detect rigid subgraphs of a large graph.
2. Find good recursive constructions of 3-connected dependent graphs.
3. Rigidity of random regular graphs4. CAD: How do you properly mix
length, direction, and angle constraints.