Rigidity Theory and

Some Applications

Brigitte ServatiusWPI

Ingredients:

Universal (ball) JointV: vertices

Rigid Rod (bar)E: edges

Framework:

nRVp : embedding

),,( pEVF

),( EVG graph

Basic Definitions:

Continuous 1-parameter familyof frameworks.

))(,,( tpEVF

Deformation:

2))()(())()(( ijjiji ctptptptp

Length of the bars is preserved

Trivial Deformations

Deformations from isometriesTrivial degrees of freedom

Non-trivial Deformations

Watt Engine

Peaucellier Engine

Non-trivial Deformations

Watt Engine

Peaucellier Engine

Non-trivial Deformations

Watt Engine

Peaucellier Engine

Non-trivial Deformations

Watt Engine

Peaucellier Engine

Rigidity:

Rigid: Every deformation is locally trivial.

Globally Rigid:

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

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

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/

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.

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.

The Rigidity Matrix

nmq

qq

qq

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

Euler Conjecture

“A closed spacial figure allows no changes as long as it is not ripped apart”

1766.

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.

By Cauchy’s Theorem, an octahedron is rigid.

If the 1-skeleton is knotted ...

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-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-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

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

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.

The Rigid World

RigidGenericallyRigid

InfinitesimallyRigid

Generic Rigidity in Dimension 1:

All embeddings on the line are generic.

Rigidity is equivalent to connectivity

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 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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Henneberg Moves

Zero Extension:

One Extension:

Henneberg Moves

Zero Extension:

One Extension:

1. No vertices of degree 22. NO TRIANGLES!1. No vertices of degree 22. NO TRIANGLES!

Applications

Computer Modeling– Cad– Geodesy (mapping)

Robotics– Navigation

Molecular Structures– Glasses– DNA

Structural Engineering– Tensegrities

Applications: CAD

Combinatorial (discrete) results preferred

Generic results not sufficient

Glass Model

Edge length ratio at most 3:1No small rigid subgraphs – 1st order phase transition

Cycle Decompositions

The graph decomposes into disjoint Hamiltonian cycles

The are many “different” ones:

ApplicationsMolecular Structures

Ribbon Model

ApplicationsMolecular Structures

PROTASE

Ball and Joint Model

ApplicationsMolecular Structures

HIV

Ball and Joint Model

ApplicationsTensegrities

Bob ConnellyKenneth Snelson

Applications

Photo by Kenneth Snelson

Tensegrities

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.