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rigidity theory and some applications

This presentation will probably involve audience discussion, which will create action items. Use PowerPoint to keep track of these action items during your presentation

  • In Slide Show, click on the right mouse button
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Rigidity Theory andSome Applications

Brigitte Servatius

WPI

ingredients
Ingredients:

Universal (ball) Joint

V: vertices

Rigid Rod (bar)

E: edges

Framework:

embedding

graph

basic definitions
Basic Definitions:

Deformation:

Continuous 1-parameter family

of frameworks.

Length of the bars is preserved

trivial deformations
Trivial Deformations
  • Deformations from isometries
  • Trivial degrees of freedom
non trivial deformations
Non-trivial Deformations

Watt Engine

Peaucellier Engine

non trivial deformations1
Non-trivial Deformations

Watt Engine

Peaucellier Engine

non trivial deformations2
Non-trivial Deformations

Watt Engine

Peaucellier Engine

non trivial deformations3
Non-trivial Deformations

Watt Engine

Peaucellier Engine

rigidity
Rigidity:
  • Rigid: Every deformation is locally trivial.
  • Globally Rigid:
ambient dimension

Rigidity in

dimension 0

is pointless

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.
infinitesimal analysis
Infinitesimal Analysis

Quadratic:

Linear:

|E| equations in m|V| unknowns:

Evaluate at 0:

There is always a subspace of trivial solutions

Rank depends only on the dimesnion: m(m+1)/2

Infinitesimal Motion (Flex): Non-Trivial Solution

Infinitesimally Rigid: Only trivial Solutions

infinitesimal rigidity

(It may be rigid anyway…)

Infinitesimally Rigid Rigid

Infinitesimal Rigidity

Unknowns

Infinitesimally Rigid: Only trivial Solutions

Infinitesimally Rigid Rigid

visual linear algebra

CONTRADICTION!!!

The last red bar insists on an infinitesimal rotation centered on its pinned vertex.

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 top

triangle force the third vertex velocity

to match the infinitesimal rotation.

Visual Linear Algebra

Is the Framework Infinitesimally rigid?

parallel redrawings

The three connecting edges happen

to be concurrent.

Dilate the larger triangle.

The blue displacement vectors satisfy

the equation at the left.

Displacing the points results in a

PARALLEL REDRAWING

of the original framework.

The vector condition is familiar…

The blue redrawing displacements

correspond a red flex.

Conclusion:

The original framework did have

an infinitesimal motion.

Parallel Redrawings

Is the Framework Infinitesimally rigid?

the rigidity matrix
The Rigidity Matrix

A framework is infinitesimally rigid in m-space

if and only if

its rigidity matrix has rank

euler conjecture
Euler Conjecture

“A closed spacial

figure allows no

changes as long as

it is not ripped apart”

1766.

cauchy s theorem 1813
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!

bricard octahedra 1897

Animation by Franco Saliola,

York University using STRUCK.

Bricard Octahedra - 1897

By Cauchy’s Theorem,

an octahedron is rigid.

If the 1-skeleton is knotted ...

more euler spin offs
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.
more euler spin offs1
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.
more euler spin offs2
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

combinatorial rigidity
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.
generic embeddings
Generic Embeddings

Generic embedding –

think random embedding.

Theorem: If some generic framework is rigid, then ALL generic embeddings of the graph are also rigid.

A graph is generically rigid (in dimension

m) if it has any infinitesimally rigid

embedding.

the rigid world
The Rigid World

Generically

Rigid

Rigid

Infinitesimally

Rigid

generic rigidity in dimension 1
Generic Rigidity in Dimension 1:
  • All embeddings on the line are generic.
  • Rigidity is equivalent to connectivity
generic rigidity in dimension 2
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.
generic rigidity in the plane
Generic Rigidity in the Plane:

The following are equivalent:

  • Generic Rigidity
  • Laman’s Condition
  • 3T2: 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.
b mat 1
B Mat 1

A

b

a

B

C

c

A

c

C

a

b

B

henneberg moves
Henneberg Moves
  • Zero Extension:
  • One Extension:
henneberg moves1

No vertices of degree 2

  • NO TRIANGLES!
  • No vertices of degree 2
  • NO TRIANGLES!
Henneberg Moves
  • Zero Extension:
  • One Extension:
applications
Applications
  • Computer Modeling
    • Cad
    • Geodesy (mapping)
  • Robotics
    • Navigation
  • Molecular Structures
    • Glasses
    • DNA
  • Structural Engineering
    • Tensegrities
applications cad
Applications: CAD
  • Combinatorial (discrete) results preferred
  • Generic results not sufficient
glass model
Glass Model
  • Edge length ratio at most 3:1
  • No small rigid subgraphs
    • 1st order phase transition
cycle decompositions
Cycle Decompositions
  • The graph decomposes into disjoint Hamiltonian cycles
  • The are many “different” ones:
applications1
Applications

Molecular Structures

Ribbon Model

applications2
Applications

Molecular Structures

Ball and Joint Model

PROTASE

applications3
Applications

Molecular Structures

Ball and Joint Model

HIV

applications4
Applications

Tensegrities

Bob Connelly

Kenneth Snelson

applications5
Applications

Tensegrities

Tensegrities

Photo by Kenneth Snelson

open problems
Open Problems
  • 3D – characterize generic rigidity
  • 2D
    • Find a “good” algorithm to detect rigid subgraphs of a large graph.
    • Find good recursive constructions of 3-connected dependent graphs.
    • Rigidity of random regular graphs
    • CAD: How do you properly mix length, direction, and angle constraints.