Why directions are good, distances are OK, and angles are bad'

1
Why directions are good, distances are OK, and
angles are bad.

• Walter Whiteley
• York University
• Work supported, in part, by grants from NIH(USA)
  and NSERC (Canada).

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Overview

• Exploring geometric constraints, linear algebra,
  counting, and pebble games.
• Claim there is a class of counting properties
  which define nice independent sets, which
  correspond to significant geometry, and which
  lead to solvable problems and fast algorithms.
• Class includes plane bar and joint, 3-space body
  and bar,
• Issue of extension to borderline cases 3-space
  rigidity
• Introduce topics, methods for latter sessions

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Rigidity of frameworks
Examples of good and not good counts

• Framework Graph G(V,E) and configuration p
  V-gt Rn

pi - pj qi - qj if (i,j)Î E
G(p) is rigid if p(t) continuous path of
configurations with same bar lengths, 0tlt1
then p(t) congruent to p(0) p.
Otherwise G(p) is flexible.
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Test the possible paths, by testing derivatives
pi(t)- pj(t)2 pi(0)- pj(0)2 if (i,j)Î E
Derivative
pi(t)- pj(t).pi(t)- pj(t) 0 if (i,j)Î
E
t0
pi- pj.pi- pj 0 if (i,j)Î E
System of linear equations for unknown velocities
pi
first-order flexible find solution p not from
a congruence.
first-order rigid not first-order flexible.
Connections first-order rigid -gt rigid.
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Work with first-order rigidity easier, good
enough.
pi- pj.pi- pj 0 if (i,j)Î E
pi- pj.pipj- pi.pj 0
Rigidity Matrix R(G,p) E rows, dV
columns want to know rank, dimension of kernal,
independent rows,
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3 6 matrix rank 3 3 dim space of flexes
Space of trivial motions - from
congruences translations and rotations - 3 dim
space
Triangle is rigid!
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Plane Frameworks - counting to predict rank
(generically).
Necessary E2V-3, E2V-3
Lamans Theorem G(p) is minimal first-order
rigid for almost all configurations p if and
only if E2V-3, and for all
non-empty subsets E, E2V-3.
Almost all - the rank is polynomial in the
coordinates. Either polynomial is always 0, or
almost always ?0.
E2V-3 2(V-1) - 1
two trees - then truncation
2(V-1)
- 1
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Other nice properties

• Minimal dependent sets (circuits) are rigid and
  2-connected (vertex sense).
• Rigid Subgraphs are 2-connected.
• Gluing two rigid subgraphs on two vertices gives
  rigid subgraph.
• Any 6 connected graph (vertex-wise) is rigid.
• And minimal dependent set is essentially 4
  edge- connected
• Essentially means single vertices may be only 3
  edge connected to rest, but two larger subsets
  are at least 4-connected.

• Have fast algorithms for independence,
  dependence, etc. Pebble games - O(VE)

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Other nice properties

• Gluing two rigid subgraphs on two vertices gives
  rigid subgraph.
• Have inductive constructions (Hennebergs
  methods) from single edge.

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Not so nice example 3-space Bar and Joint
Frameworks.
Necessary E3V-6, E3V-6
Not sufficient.
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Second example not so nice

• Find that minimal dependent sets need not be
  rigid!
• Do not have any fast algorithm,
• Do not anticipate any fast algorithm.
• Very important problem - this afternoon a
  workshop session on what we can do to find
  special types of graphs for which we can prove an
  algorithm, and dependent sets as rigid.
• E.g. graphs of the form G2

• Molecular Conjecture this class has good
  algorithmic properties.

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Second example not so nice

• Would like inductive constructions (very useful
  for proofs)
• Have partial results, and conjectures

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Second example not so nice.

• Other inductive techniques, known to work for
  rigidity
• Gluing identify two rigid pieces across at least
  three common vertices.
• vertex splitting

Generates all triangulated surfaces (and more)
(Fogelsanger, Moshe and Whiteley )
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Conjecture (relevant to global rigidity)

• Vertex splitting takes a rigid circuit to a rigid
  circuit.
• Know it can take non-rigid circuit to independent
  set (split from double banana to a rigid
  framework).

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Third example bad

• Angles in the plane (constraints on pairs of
  edges)
• Conditions for independence
• No more than 2V-4 angles on V points
• No cycle in the graph of edges, angles
• Do not have nice constructions, or counting, or
  algorithms, or hope of such

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When will a system of constraints give nice
properties?

• Have geometric constraints and associated linear
  algebra
• Have necessary counting properties of the form
• of the form E nV - k for independence
• If working on a graph, a single edge is
  independent
• n(2) - k gt 0, or klt2n
• Works also for hypergraphs (constraints on
  triples etc.)
• This count, alone, defines a very nice
  independence structure (matroid - behaves like
  the rows of a matrix).
• Submodular functions
• From the counts, get nice pebble game algorithm
  for this independence structure
• Need theorem that independence in linear algebra
  of constraints matches independence from
  counting.

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Nice properties - continued

• From the counts, get nice pebble game algorithm
• Implementing this general pebble game, plan to
  put it onto flexweb.asu.edu
• Can hope for a nice inductive construction.
• Minimal dependent sets are connected and full
  rank.
• If k gt 0, then minimal dependent sets, and full
  rigid sets (full rank) sets are connected.
• If k gt m then bases (maximal independent sets
  (or rigid subgraphs) are at least 2-connected.
• Implementing this general pebble game, plan to
  put it onto flexweb

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Other constraints that match this paradigm

• Bar and Body frameworks in 3-space
• Have necessary counting properties of the form
  E 6V - 6 for independence
• This count, alone, defines a matroid.
• From the counts, get nice pebble game algorithm
  for this independence structure
• Rigid is equivalent to 6 spanning trees
• Tays Theorem independence in linear algebra of
  constraints matches independence from counting.

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Nice properties - continued

• Can get a nice inductive construction.
• Minimal dependent sets are connected and rigid.
• Can glue two rigid objects across a single vertex
  (or more) to get larger rigid objects.
• 12 edge-connectivity is sufficient for rigidity.
• Any minimal dependent set is at least 7
  edge-connected.

• Have model of generic bar and body frameworks in
  3-space.
• Have geometric specialization to groups of five
  edges as hinges - and all nice properties carry
  over.

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Molecular Conjecture

• This nice independence structure, with all its
  nice properties and algorithms matches
  independence of linear algebra for bond graphs of
  molecules treats as hinges.
• That is, restricting to a special class of
  realizations, with hinges of each body concurrent
  in a central atom does not reduce the rank.
• This was a major topic at the Banff workshop on
  protein flexibility and motions.
• Two molecular conjectures, and related problems,
  will be subject of a session this afternoon.

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Other constraints that match this paradigm

• Directions of line segments in the plane
  parallel drawing
• Have necessary counting properties
• of the form E 2V - 3 for independence
• Defines the same independence structure as
  framework (distance) constraints.
• Trivial deformations are translation and scaling.

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Other constraints that match this paradigm

• Old engineering technique
• Isomorphism of parallel drawing constraints and
  distance constraints in the plane.
• Isomorphism of matrices for any given realization

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Other constraints that match this paradigm

• Direction constraints in 3-space
• Note a single edge in graph is two constraints,
  similar in appearance to body and hinge in
  3-space, call set of constraints C
• Have necessary counting properties for
  independent constraints of the form C
  3V - 4
• This count, alone, defines a very nice
  independence structure.
• From the counts, get nice pebble game algorithm
  for this independence structure.
• Theorem (Whiteley) independence in linear
  algebra of constraints matches independence from
  counting.

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Nice properties - continued

• Can get a nice inductive construction.
• Minimal dependent sets are connected and rigid.
• Can glue two rigid objects across a two vertices
  (or more) to get larger rigid objects.
• 4 edge-connectivity is sufficient for
  rigidity(edges as pairs of constraints).

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Nice properties - continued

• Has multiple applications in geometry CAD,
  Minkowski decomposition of polytopes,
• Appears in the dipole alignment data for NMR
  structures.
• Are implementing pebble game algorithm for this.
• No longer have simple correspondence to 3-space
  rigidity.
• Have one way implication
• If G is rigid for 3-space distance constraints,
  then G is rigid for parallel drawing.
• Converse is not true.

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Other constraints that match this paradigm

• Combine distance and direction in the plane
• Call sets D, L
• Have necessary counting properties for
  independent constraints of the form
• DU L 2V - 2
• D 2V - 3
• L 2V - 3
• This count, alone, defines a nice independence
  structure.
• From the counts, get sequential pebble game
  algorithm for this independence structure.
• Use tight as word for maximal rank subsets
• Theorem (Servatius, Whiteley) independence in
  linear algebra of constraints matches
  independence from counting.

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Other constraints that match this paradigm

• Combines distance and direction designs in the
  plane
• Sample inductive construction

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Interesting mix of good and bad

• Combine distance and direction in 3-space
• Call sets D, L
• Have necessary counting properties for
  independent constraints of the form
• DU L 3V - 3
• D 3V - 4
• L 3V - 6
• Kinds of data structures in NMR data
• Unexplored for classes of partial results.

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Other constraints that match this paradigm

• Body and pin frameworks in plane
• Each pin removes 2 degrees of freedom.
• Have necessary counting properties for subsets of
  rows of the form E 3V - 3 for
  independence
• From the counts, get nice pebble game algorithm
  for this independence structure
• Corollary to Lamans Theorem independence in
  linear algebra of constraints matches
  independence from counting.
• Rigidity is equivalent to finding 3 spanning
  trees in the graph of bodies and pins, with
  duplicate edges for each pin.

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Plane Analog of Molecular Conjecture

• Conjecture (Whiteley)
• Given a set of bodies and pins G, which is
  rigid for generic realizations of pins (contains
  three edge disjoint spanning trees in
  multi-graph), then this is also rigid for generic
  rod structures in which all pins of each body are
  realized as collinear.
• Partial result (Whiteley)
• this holds for independent sets of pins.

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Conclusions

• There is a class of counting properties which
  define nice independent sets, which correspond to
  significant geometry, and which lead to solvable
  problems and fast algorithms.
• Issue of extension to borderline cases (3-space
  rigidity)
• Does symmetry rigidity produce a good algorithm?
• Issue of restricting selection to special
  subclass realizations and proving rank function,
  algorithms etc still apply.
• Nice underlying geometry - topic for tomorrow
• Pseudo triangulations for plane isostatic graphs
  and plan circuits.