Combinatorial Rigidity Jack Graver, Brigitte Servatius, Herman Servatius.

1
Combinatorial RigidityJack Graver, Brigitte
Servatius, Herman Servatius.

• Jenny Stathopoulou
• December 2004

2
Infinitesimal Rigidity Introduction

• We say that a framework (V, E, p) is generic if
  all frameworks corresponding to points in a
  neighborhood of Pp(V) in are rigid or not
  rigid.
• A set of points P in m-space is said to be
  generic if each framework (V, E, p) with Pp(V)
  is generic.
• We define a graph (V, E) to be rigid (in
  dimension m) if the frameworks corresponding to
  the generic embeddings of V into m-space are
  rigid.
• We say that a framework (V, E, p) in m-space is
  generically rigid if the graph (V, E) is rigid in
  dimension m.

Rm
3
Infinitesimal Rigidity Introduction

• Is (V, E) generically rigid (generically
  independent) in dimension m?
• In m 3, the only technique is for all graphs
  (V, E) chose a generic embedding p of V into
  and check if the framework (V, E, p) is
  infinitesimal rigid or infinitesimal independent.

Rm
4
Infinitesimal Rigidity Basic Definitions

• V(E) i?j ?V with (i,j) ? E or (j,i)?E
• K(U) (i,j) ilt j and i,j?U
• p embedding of V into

Rm
p (p1, , pi, , pn)
p (p11, p12 , , p1m , ,pi1, pi2 ,, pim,,
pn1, pn2 ,, pnm )
5
Infinitesimal Rigidity Basic Definitions

• Let p be an embedding of V into and P
  denote set p(V). The set P is in general position
  and p is a general embedding if, for any
  q-element subset Q of P with qlt(m1) the affine
  space spanned by Q has dimension q-1. (e.g. if
  q2, Q spans a line if q3, Q spans a plane.)
• We may also measure the length of the edges by
  evaluating the rigidity function ?
  defined by
• (where the ijth coordinate of ?(p) is the
  square of the length of the edge ij in K.).
• We define the rigidity matrix for the embedding
  p, by ?(p)2R(p). R(p) is an n(n-1)/2 by nm
  matrix whose entries are functions of the
  coordinates of p as a point in .

Rm
Rmn ? RE(K)
?(p)ij (pi pj)2
Rmn
6
Infinitesimal Rigidity Basic Definitions

• e.g n4 and m2

p (p1, p2, p3, p4)
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Infinitesimal Rigidity Basic Definitions
Vector in . Corresponding to
edge(2,4) of K
Rmn

• R(p)

n
p1-p2 p2-p1 0 0 p1-p3 0
p3-p1 0 p1-p4 0 0
p4-p1 0 p2-p3 p3-p2 0 0 p2-p4
0 p4-p2 0 0 p3-p4
p4-p3
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Infinitesimal Rigidity Independence and the
Stress Space

• If E is independent with respect to one generic
  embedding into m-space, then it is independent
  with respect to all generic embeddings into
  m-space.
• If E is independent with respect to all generic
  embedding into m-space, then it is generically
  independent for dimension m.
• If E is dependent with respect to one
  m-dimensional embedding of V, then E is
  generically dependent for dimension m and is, in
  fact, dependent with respect to all embeddings of
  V into m-space.

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Infinitesimal Rigidity Independence and the
Stress Space

• An edge set E?K is independent with respect to p
  if the corresponding set of rows ( in R(p)) is
  independent as a set of vectors in
  
• An edge set will be dependent if and only if the
  corresponding set of rows of R(p) satisfies a
  non-trivial dependency relation.
• where is the row corresponding to the
  edge (i,j), and is a scalar and some
  ? 0 .
• For each i?V(E)
  , over K(V(E)). ()

Rmn
?(i,j)?E sij rij 0
sij
rij
sij
?i?j sij(pi - pj ) 0
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Infinitesimal Rigidity Independence and the
Stress Space
S341
P3 (0,1)
P4 (1,1)
S23 -1
For i1 1(-1,0) 1(0,-1) (-1)(-1,-1)
0 For i2 1(1,0) (-1)(1,-1) 1(0,-1)
0 We conclude that for this p the set K is
dependent !
S131
S14 -1
P1 (0,0)
P2 (1,0)
S121
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Infinitesimal Rigidity Independence and the
Stress Space
P3 (0,1)
P4 (1,1)
?i?j sij(pi - pj ) 0
In order that
is satisfied at we must have
p1

• s12(p1-p2) s13(p1-p3) s14(p1-p4) 0 ?
• 0 s13(0,-1) s14(-1,-1) 0
  ?
• s13 s14 0
• Similarly at s23 s24 0
• and it follows that s34 0

P1 (0,0)
P2 (1,0)
p2
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Infinitesimal Rigidity Independence and the
Stress Space

• Let E? K and consider the vector space of
  all functions from E to the reals. For s ?
  , s is called a set of stresses for E and
  is the stress on the edge (i,j). A set of
  stresses for E, not all zero, which satisfy the
  equations in () is called a non-trivial
  resolvable set of stresses for E , S(E).
• We consider the linear transformation
  where
  and S(E) ker( ),
  where 0 if (i,j) is not in E.
• Thus, E will be independent with respect to p if
  and only if the kernel of is trivial.

RE
RE
si j
TE
RE ?(Rm)V
(TE(s))i ?j?i sij (pi-pj)
TE
sij
TE
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Infinitesimal Rigidity Independence and the
Stress Space
e.g.
P3 (0,1)
S34
P4 (x,y)
S32
S13
T(s)1 ( -s12 - xs14 , -s13 ys14 ) T(s)2 (
s12 s23 (1- x)s24 , -s23 ys24 ) T(s)3 (
-s23 xs34 , s13 s23 (1- y)s34 ) T(s)4 (
xs14 ( x -1)s24 xs34 , ys14 ys24 (y-1)s34
)
S24
S14
S12
P1 (0,0)
P2 (1,0)
Where x?0, y?0 and xy ?1
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Infinitesimal Rigidity Independence and the
Stress Space

• dim (ker( )) dim(im( ))
  dim(domain ( )).
• But dim(domain( )) E.
• The space im( ) is a subspace of
  , so
• dim(im( )) dim(im( ) ) mn.
• E is independent if and only if
• dim(im( ) ) mn -E dim(S(E)).

TE
TE
TE
TE
TE
Rmn
TE
TE
TE
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Infinitesimal Rigidity Infinitesimal Motions
and Isometries

• Let V 1,,n, p mapping V into , E
  be an edge set of (V,K) and consider the
  framework (V, E, p). Then u? is an
  infinitesimal motion of (V, E, p) if
  .
• u? ? ,for
  all (i,j) in E.
• 
  0
• The set of infinitesimal motions of the
  framework is the orthogonal complement of the
  subspace spanned of the R(p) which correspond to
  the edges of E , or , equivalently the orthogonal
  complement of the image .
• Denote the space of infinitesimal motions of E
  by V(E).

Rm
(Rm)V
(ui uj) (pi pj) 0
(u1, , un)
(Rm)V
(u1, , un)
(0, , 0, pi-pj, 0, ..., 0, pj-pi, 0, ..., 0)
TE
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Infinitesimal Rigidity Infinitesimal Motions
and Isometries

• Denote V(K(V(E))) the space of infinitesimal
  isometries of V(E) by ,D(E).
• Since E?K(V(E)), the orthogonal complement of
  the space spanned by the rows of R(p)
  corresponding to E contains the orthogonal
  complement of the space spanned by the rows of
  R(p) corresponding to K(V(E)).
• A framework is rigid if V (E) D(E).

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Infinitesimal Rigidity Infinitesimal Motions
and Isometries

• By an isometry of , we mean a 1-1
  function from to which preserves
  the distance between pairs of points.
• e.g. a vector field U is
  an infinitesimal isometry of if,
  (U(x) U(y)) (x-y) 0 , for all x,y ?
  then it is clear that u? defined by
  , for i 1,..,n is an
  infinitesimal motion for (V, E, p) , for all
  E?K.
• For any subset S of , a function U S
  is a infinitesimal isometry of S if (U(x)
  U(y)) (x-y) 0 , for all x,y?S.
• Isometries direct ( translations, rotations),
  opposite (reflections).
  

Rm
Rm
Rm
Rm
Rm
Rm
Rm
ui
U(pi)
(Rm)V
Rm
Rm
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Infinitesimal Rigidity Infinitesimal and
Generic Rigidity

• Let X be the set of p for which the set of E
  is independent. If p?X then p is called generic
  embedding.
• We define the dependency number of E as the
  dim(S(E)), denoted by dn(E).
• We define the degree of freedom of E to be the
  dim(V(E)) dim(D(E)), denoted by df(E).
• If a framework (V(E), E, p) is infinitesimal
  rigid for some embedding p then E is generically
  rigid for dimension m.

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Infinitesimal Rigidity Infinitesimal and
Generic Rigidity
P1 (0,2)
P2 (1,1)
P6 (-1,1)
P5 (-1,0)
P3 (1,0)
P4 (0,0)
Let W be any infinitesimal motion of the
framework. The restriction of W to
is an infinitesimal isometry of that set. U(
) U( ) 0. Let (x, y)
then ((x,y) (0,0))((-1,0)-(0,0)) 0 ? x0
(0,a). Similarly (0,b).
From (x,y) ? ya, x-a , b3a. And
form ? a 3b . We conclude
that abc0. ? Infinitesimal rigid
p1 , p4
p1
p4
U(p5)
U(p5)
U(p3)
U(p6)
U(p2)
See Appendix for examples
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Infinitesimal Rigidity Rigidity Matroids

• Let S be any set an operator ?.? mapping the
  power set of S onto the power set of S is called
  a closure operator if the following conditions
  are satisfied
• C1 if T ?S then T ??T?.
• C2 if R?T ?S then ?R? ??T?.
• C3 if T ?S then ??T?? ??T?.
• A matroid consists of a finite set S.
• A matroid closure operation ?.? on S satisfies
  the additional condition
• C4 if T ?S and s,t ?(S-T) then s ? ?T U
  t? if and only if t ? ?T U s ?.

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Infinitesimal Rigidity Rigidity Matroids

• Let S be a finite set of vectors from some vector
  space and, for any T ?S let ?T? span(T) nS.
  Then ?.? is a matroid closure operator on S.
• The matroid closure operation on S is an
  associated concept of dependency. We say that a T
  ?S is independent ( with respect to the matroid)
  if, for every s?T, s??T -s?.
• Let a ?.? be a matroid closure operator on a
  finite set S. Let T ?S and s?S we say that s is
  independent of T if s is not in ?T? .

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Infinitesimal Rigidity Rigidity Matroids

• Let V and the embedding p of V . Let KK(V) and
  let ?.? to be the closure operator of this
  embedding. Then , for all E ?K , ?E??K(V(E))
  furthermore E is rigid if and only if ?E?
  K(V(E)) .
• Let V and the embedding p of V . Let KK(V) and
  let ?.? to be the closure operator of this
  embedding. Then ?.? satisfies
  C5 if E,F ?K and V(E)nV(F)ltm then ?E?F?
  ?K(V(E))?K(V(F)). C6 if E,F ?K are rigid and
  V(E)nV(F) m then E?F is rigid.
• A matroid Am on KK(V) whose closure operator
  satisfies the C5 and C6 is called m-dimensional
  abstract rigidity matroid of p.

23
Appendix (Figure 1)
24
Appendix (Figure 1)

• At ( a ) is rigid because the chain of the
  three rods is pulled taught. If the slightest
  slack is permitted, (b), the framework is no
  longer rigid.
• At ( a ), assigning the zero vector to all points
  except to one of the two points on the chain
  (to this points lets assign any non-zero vector
  perpendicular to the chain), there is an
  infinitesimal flex of that framework. So the
  framework ( a ) is rigid but not infinitesimal
  rigid and hence no generically rigid.
• At (c ), there is a crossing that has no
  significance and adds no constrains. We think of
  the rods as being able to slide across one
  another. The framework ( c ) is no rigid because
  the three vertical rods are equal in length and
  parallel and hence permit a horizontal shear. If
  the uniform length or parallelism is destroyed,
  as in framework becomes rigid.
• Framework (b) is neither rigid nor generically
  rigid framework (c ) is not rigid but
  generically rigid while (d) is both rigid and
  generically rigid.

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Appendix
We say that a framework is strongly rigid, if all
solutions to the corresponding system of
quadratic equations correspond to congruent
frameworks a framework is rigid if all solutions
to the corresponding system in some neighborhood
of the original solution come from congruent
solutions. Clearly, strongly rigid implies
rigid. The triangle is strongly rigid and hence
rigid, while the rectangle is rigid but not
strongly rigid.
We flip one of the triangles ( of the next
rectangle (a) ) over the diagonal . The framework
obtained, (b), is not congruent to the original
framework but is a solution to the system of
quadratic equations
(a)
(b)
26
Appendix

• Strong rigidity ? rigidity
• Infinitesimal rigidity ? rigidity
• Infinitesimal rigidity ? generic rigidity

Figure 2

• (a) Rigid , NO Strong Rigid , NO
  Infinitesimal Rigid , NO Generically Rigid
• (b) Rigid , NO Strong Rigid , NO
  Infinitesimal Rigid , Generically Rigid
• (c) Rigid , Strong Rigid , NO Infinitesimal
  Rigid , Generically Rigid
• (d) Rigid , Strong Rigid , NO Infinitesimal
  Rigid , NO Generically Rigid

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?he End