Finite and infinitesimal rigidity with polyhedral norms

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by  D. Kitson

FINITE AND INFINITESIMAL RIGIDITY WITH
POLYHEDRAL NORMS
D. KITSON
Abstract. Wecharacterisefiniteandinfinitesimalrigidityforbar-joint
frameworks in Rd with respect to polyhedral norms (i.e. norms with
closed unit ball P a convex d-dimensional polytope). Infinitesimal and
4 continuous rigidity are shown to be equivalent for finite frameworks in
1 Rdwhicharewell-positionedwithrespecttoP. Anedge-labellingdeter-
0 mined by the facets of the unit ball and placement of the framework is
2 usedtocharacteriseinfinitesimalrigidityinRd intermsofmonochrome

n spanning trees. An analogue of Laman’s theorem is obtained for all
a polyhedral norms on R2.
J

7

]
G Contents
M
Introduction 1
h. 1. Preliminaries 3
t 2. Support functionals and rigidity 4
a
m 3. Edge-labellings and monochrome subgraphs 10
4. An analogue of Laman’s theorem 16
[
5. Infinite frameworks 23
1
References 25
v
6
3
3
1 Introduction
.
1
0 A bar-joint framework in Rd is a pair (G,p) consisting of a simple undi-
4 rected graph G = (V(G),E(G)) (i.e. no loops or multiple edges) and a
1 placement p : V(G) → Rd of the vertices such that p and p are distinct
: v w
v whenever vw is an edge of G. Given a norm on Rd we are interested in
i
X determining when a given framework can be continuously and nontrivially
r deformed without altering the lengths of the bars. A well-developed rigidity
a
theory exists in the Euclidean setting for finite bar-joint frameworks (and
their variants) which stems from classical results of A. Cauchy [6], J. C.
Maxwell [16], A. D. Alexandrov [3] and G. Laman [13]. Of particular rele-
vance is Laman’s landmark characterisation for generic minimally infinitesi-
mally rigid finite bar-joint frameworks in the Euclidean plane. Asimow and
Roth proved the equivalence of finite and infinitesimal rigidity for regular
bar-joint frameworks in two key papers [1], [2]. A modern treatment can be
2010 Mathematics Subject Classification. 52C25, 52A21, 52B12.
Key words and phrases. Bar-joint framework, infinitesimally rigid, Laman’s theorem,
polyhedral norm.
Supported by EPSRC grant EP/J008648/1.
1
2 D.KITSON
foundinworksofGraver,ServatiusandServatius[8]andWhiteley[23], [24].
More recently, significant progress has been made in topics such as global
rigidity ([5], [7], [10]) and the rigidity of periodic frameworks ([4], [15], [19],
[20]) in addition to newly emerging themes such as symmetric frameworks
[21] and frameworks supported on surfaces [17]. In this article we consider
rigidity properties of both finite and infinite bar-joint frameworks (G,p) in
Rd with respect to polyhedral norms. A norm on Rd is polyhedral (or a
block norm) if the closed unit ball {x ∈ Rd : (cid:107)x(cid:107) ≤ 1} is the convex hull
of a finite set of points. Such norms are important from a number of per-
spectives. Firstly, every norm on Rd may be approximated by a polyhedral
norm. Secondly, polyhedralnormsareusedindiverseareasofmathematical
modelling. Thirdly, the rigidity theory obtained with polyhedral norms is
distinctly different to the Euclidean setting in admitting edge-labelling and
spanning tree methods. A study of rigidity with respect to the classical
non-Euclidean (cid:96)p norms was initiated in [11] for finite bar-joint frameworks
andfurtherdevelopedforinfinitebar-jointframeworksin[12]. Amongthese
norms the (cid:96)1 and (cid:96)∞ norms are simple examples of polyhedral norms and
so the results obtained here extend some of the results of [11].
In Section 1 we provide the relevant background material on polyhedral
norms and finite and infinitesimal rigidity. In Section 2 we establish the role
of support functionals in determining the space of infinitesimal flexes of a
bar-joint framework (Theorem 2.5). We then distinguish between general
bar-joint frameworks and those which are well-positioned with respect to
the unit ball. The well-positioned placements of a finite graph are open and
dense in the set of all placements and we show that finite and infinitesimal
rigidity are equivalent for these bar-joint frameworks (Theorem 2.7). We
then introduce the rigidity matrix for a general finite bar-joint framework,
the non-zero entries of which are derived from extreme points of the polar
set of the unit ball. In Section 3 we apply an edge-labelling to G which
is induced by the placement of each bar in Rd relative to the facets of the
unit ball. With this edge-labelling we identify necessary conditions for infin-
itesimal rigidity and obtain a sufficient condition for a subframework to be
relatively infinitesimally rigid (Proposition 3.5). We then characterise the
infinitesimally rigid bar-joint frameworks with d induced framework colours
asthosewhichcontainmonochromespanningtreesofeachframeworkcolour
(Theorem 3.9). This result holds for both finite and infinite bar-joint frame-
worksanddoesnotrequiretheframeworktobewell-positioned. Forminimal
infinitesimal rigidity we must assume that the bar-joint framework is well-
positioned and an example is provided to demonstrate this. In Section 4 we
apply the spanning tree characterisation to show that certain graph moves
preserve minimal infinitesimal rigidity for any polyhedral norm on R2. We
then show that in two dimensions a finite graph has a well-positioned min-
imally infinitesimally rigid placement if and only if it satisfies the counting
conditions|E(G)| = 2|V(G)|−2and|E(H)| ≤ 2|V(H)|−2forallsubgraphs
H (Theorem 4.9). This is an analogue of Laman’s theorem [13] which char-
acterises the finite graphs with minimally infinitesimally rigid generic place-
ments in the Euclidean plane as those which satisfy the counting conditions
|E(G)| = 2|V(G)|−3 and |E(H)| ≤ 2|V(H)|−3 for subgraphs H with at
FINITE AND INFINITESIMAL RIGIDITY WITH POLYHEDRAL NORMS 3
least two vertices. Many of the results obtained hold equally well for both
finite and infinite bar-joint frameworks. We conclude in Section 5 with a
discussion of some aspects which are unique to the infinite case. Illustrative
examples are provided throughout.
1. Preliminaries
Let P be a convex symmetric d-dimensional polytope in Rd where d ≥ 2.
Following [9] we say that a proper face of P is a subset of the form P ∩H
whereH isasupportinghyperplaneforP. AfacetofP isaproperfacewhich
is maximal with respect to inclusion. The set of extreme points (vertices) of
P is denote ext(P). The polar set of P is denoted P(cid:52) and is also a convex
symmetric d-dimensional polytope in Rd,
(1) P(cid:52) = {y ∈ Rd : x·y ≤ 1, ∀ x ∈ P}
Moreover, there exists a bijective map which assigns to each facet F of P a
unique extreme point Fˆ of P(cid:52) such that
(2) F = {x ∈ P : x·Fˆ = 1}
The polar set of P(cid:52) is P.
The Minkowski functional (or gauge) for P defines a norm on Rd,
(cid:107)x(cid:107) = inf{λ ≥ 0 : x ∈ λP}
P
Thisiswhatisknownasapolyhedralnormorablocknorm. Thedualnorm
of (cid:107)·(cid:107) is also a polyhedral norm and is determined by the polar set P(cid:52),
P
(cid:107)y(cid:107)∗ = max x·y = inf{λ ≥ 0 : y ∈ λP(cid:52)} = (cid:107)y(cid:107)
P P(cid:52)
x∈P
Ingeneral,alinearfunctionalonaconvexpolytopewillachieveitsmaximum
valueatsomeextremepointofthepolytopeandsothepolyhedralnorm(cid:107)·(cid:107)
P
is characterised by,
(3) (cid:107)x(cid:107) = (cid:107)x(cid:107)∗∗ = (cid:107)x(cid:107)∗ = max x·y = max x·y
P P P(cid:52)
y∈P(cid:52) y∈ext(P(cid:52))
A point x ∈ Rd belongs to the conical hull cone(F) of a facet F if x =
(cid:80)n λ x for some non-negative scalars λ and some finite set of points
j=1 j j j
x ,x ...,x ∈ F. By formulas (1), (2) and (3) the following equivalence
1 2 n
holds,
(4) x ∈ cone(F) ⇔ (cid:107)x(cid:107) = x·Fˆ
P
Each isometry of the normed space (Rd,(cid:107)·(cid:107) ) is affine (by the Mazur-
P
Ulam theorem) and hence is a composition of a linear isometry and a trans-
lation. Alinearisometrymustleaveinvariantthefinitesetofextremepoints
of P and is completely determined by its action on any d linearly indepen-
dent extreme points. Thus there exist only finitely many linear isometries
on (Rd,(cid:107)·(cid:107) ).
P
A continuous rigid motion of (Rd,(cid:107)·(cid:107) ) is a family of continuous paths,
P
α : (−δ,δ) → Rd, x ∈ Rd
x
with the property that α (0) = x and for every pair x,y ∈ Rd the distance
x
(cid:107)α (t) − α (t)(cid:107) remains constant for all values of t. If δ is sufficiently
x y P
small then the isometries Γ : x (cid:55)→ α (t) are necessarily translational since
t x
4 D.KITSON
by continuity the linear part must equal the identity transformation. Thus
we may assume that a continuous rigid motion of (Rd,(cid:107)·(cid:107) ) is a family of
P
continuous paths of the form
α (t) = x+c(t), x ∈ Rd
x
for some continuous function c : (−δ,δ) → Rd (cf. [12, Lemma 6.2]).
An infinitesimal rigid motion of (Rd,(cid:107)·(cid:107) ) is a vector field on Rd which
P
arises from the velocity vectors of a continuous rigid motion. Since the
continuous rigid motions are initially of translational type, the infinitesimal
rigid motions of (Rd,(cid:107)·(cid:107) ) are precisely the constant maps
P
γ : Rd → Rd, x (cid:55)→ a
for some a ∈ Rd (cf. [11, Lemma 2.3]).
Let(G,p)beabar-jointframeworkin(Rd,(cid:107)·(cid:107) ). Acontinuous (or finite)
P
flex of (G,p) is a family of continuous paths
α : (−δ,δ) → Rd, v ∈ V(G)
v
such that α (0) = p for each vertex v ∈ V(G) and (cid:107)α (t) − α (t)(cid:107) =
v v v w P
(cid:107)p −p (cid:107) for all |t| < δ and each edge vw ∈ E(G). A continuous flex of
v w P
(G,p) is regarded as trivial if it arises as the restriction of a continuous rigid
motion of (Rd,(cid:107)·(cid:107) ) to p(V(G)). If every continuous flex of (G,p) is trivial
P
then we say that (G,p) is continuously rigid.
An infinitesimal flex of (G,p) is a map u : V(G) → Rd, v (cid:55)→ u which
v
satisfies,
(5) (cid:107)(p +tu )−(p +tu )(cid:107) −(cid:107)p −p (cid:107) = o(t), as t → 0
v v w w P v w P
for each edge vw ∈ E(G). We will denote the collection of infinitesimal
flexes of (G,p) by F(G,p). An infinitesimal flex of (G,p) is regarded as
trivial if it arises as the restriction of an infinitesimal rigid motion of (Rd,(cid:107)·
(cid:107) ) to p(V(G)). In other words, an infinitesimal flex of (G,p) is trivial
P
if and only if it is constant. A bar-joint framework is infinitesimally rigid
if every infinitesimal flex of (G,p) is trivial. Regarding F(G,p) as a real
vector space with component-wise addition and scalar multiplication, the
trivial infinitesimal flexes of (G,p) form a d-dimensional subspace T(G,p)
of F(G,p). The infinitesimal flex dimension of (G,p) is the vector space
dimension of the quotient space F(G,p)/T(G,p).
2. Support functionals and rigidity
In this section we begin by highlighting the connection between the infin-
itesimal flex condition (5) for a general norm on Rd and support functionals
on (Rd,(cid:107) · (cid:107)). We then characterise the space of infinitesimal flexes for a
general bar-joint framework in (Rd,(cid:107)·(cid:107) ) in terms of support functionals
P
and prove the equivalence of finite and infinitesimal rigidity for finite well-
positioned bar-joint frameworks in (Rd,(cid:107)·(cid:107) ). Following this we describe
P
the rigidity matrix for general finite bar-joint frameworks in (Rd,(cid:107)·(cid:107) ) and
P
compute some examples.
FINITE AND INFINITESIMAL RIGIDITY WITH POLYHEDRAL NORMS 5
2.1. Support functionals. Let(cid:107)·(cid:107)beanarbitrarynormonRd anddenote
by B the closed unit ball in (Rd,(cid:107)·(cid:107)). A linear functional f : Rd → R is a
support functional for a point x ∈ Rd if f(x ) = (cid:107)x (cid:107)2 and (cid:107)f(cid:107)∗ = (cid:107)x (cid:107).
0 0 0 0
Equivalently, f is a support functional for x if the hyperplane
0
H = {x ∈ Rd : f(x) = (cid:107)x (cid:107)}
0
is a supporting hyperplane for B which contains x0 .
(cid:107)x0(cid:107)
Lemma 2.1. Let (cid:107)·(cid:107) be a norm on Rd and let x ∈ Rd. If f : Rd → R is
0
a support functional for x then,
0
(cid:107)x +ty(cid:107)−(cid:107)x (cid:107)
0 0
f(y) ≤ (cid:107)x (cid:107) , ∀ t > 0
0
t
and
(cid:107)x +ty(cid:107)−(cid:107)x (cid:107)
0 0
f(y) ≥ (cid:107)x (cid:107) , ∀ t < 0
0
t
for all y ∈ Rd.
Proof. Since f is linear and f(x ) = (cid:107)x (cid:107)2 we have for all y ∈ Rd,
0 0
1
f(y) = (f(x +ty)−(cid:107)x (cid:107)2)
0 0
t
If t > 0 then since f(x) ≤ (cid:107)x (cid:107)(cid:107)x(cid:107) for all x ∈ Rd we have
0
(cid:107)x +ty(cid:107)−(cid:107)x (cid:107)
0 0
f(y) ≤ (cid:107)x (cid:107)
0
t
If t < 0 then applying the above inequality,
(cid:107)x −t(−y)(cid:107)−(cid:107)x (cid:107) (cid:107)x +ty(cid:107)−(cid:107)x (cid:107)
0 0 0 0
f(y) = −f(−y) ≥ −(cid:107)x (cid:107) = (cid:107)x (cid:107)
0 0
−t t
(cid:3)
Let (G,p) be a bar-joint framework in (Rd,(cid:107)·(cid:107)) and fix an orientation
for each edge vw ∈ E(G). We denote by supp(vw) the set of all support
functionals for p − p . (The choice of orientation on the edges of G is
v w
for convenience only and has no bearing on the results that follow. Alter-
natively, we could avoid choosing an orientation by defining supp(vw) to
be the set of all linear functionals which are support functionals for either
p −p or p −p .)
v w w v
Proposition 2.2. If (G,p) is a bar-joint framework in (Rd,(cid:107)·(cid:107)) and u :
V(G) → Rd is an infinitesimal flex of (G,p) then
(cid:92)
u −u ∈ kerf
v w
f∈supp(vw)
for each edge vw ∈ E(G).
Proof. Let vw ∈ E(G) and suppose f is a support functional for p −p .
v w
Applying Lemma 2.1 with x = p −p and y = u −u we have,
0 v w v w
(cid:107)x +ty(cid:107)−(cid:107)x (cid:107) f(y) (cid:107)x +ty(cid:107)−(cid:107)x (cid:107)
0 0 0 0
lim ≤ ≤ lim
t→0− t (cid:107)x0(cid:107) t→0+ t
Since u is an infinitesimal flex of (G,p), lim 1((cid:107)x +ty(cid:107)−(cid:107)x (cid:107)) = 0 and
t→0 t 0 0
so f(y) = 0.
6 D.KITSON
(cid:3)
Let (cid:107)·(cid:107) be a polyhedral norm on Rd. For each facet F of P denote by
P
ϕ the linear functional
F
ϕ : Rd → R, x (cid:55)→ x·Fˆ
F
Lemma 2.3. Let (cid:107)·(cid:107) be a polyhedral norm on Rd, let F be a facet of P
P
and let x ∈ Rd. Then x ∈ cone(F) if and only if the linear functional,
0 0
ϕ : Rd → R, x (cid:55)→ (cid:107)x (cid:107) ϕ (x)
F,x0 0 P F
is a support functional for x .
0
Proof. If x ∈ cone(F) then by formula (4), ϕ (x ) = (cid:107)x (cid:107)2. By (1) we
0 F,x0 0 0 P
have ϕ (x) ≤ (cid:107)x (cid:107) for each x ∈ P and it follows that ϕ is a support
F,x0 0 P F,x0
functional for x . Conversely, if x ∈/ cone(F) then by (4), ϕ (x ) <
0 0 F,x0 0
(cid:107)x (cid:107)2 and so ϕ is not a support functional for x . (cid:3)
0 P F,x0 0
For each oriented edge vw ∈ E(G) we denote by supp (vw) the set of all
Φ
linear functionals ϕ which are support functionals for pv−pw .
F (cid:107)pv−pw(cid:107)P
Proposition 2.4. Let (G,p) be a finite bar-joint framework in (Rd,(cid:107)·(cid:107) ).
P
If u : V(G) → Rd satisfies
(cid:92)
u −u ∈ kerϕ
v w F
ϕF∈suppΦ(vw)
for each edge vw ∈ E(G) then there exists δ > 0 such that the family
α : (−δ,δ) → Rd, α (t) = p +tu
v v v v
is a finite flex of (G,p).
Proof. Let vw ∈ E(G) and write x = p −p and u = u −u . If ϕ
0 v w 0 v w F
is a support functional for x0 then, by the hypothesis, ϕ (u ) = 0. By
(cid:107)x0(cid:107)P F 0
Lemma 2.3, x is contained in the conical hull of the facet F. Applying
0
formulas (3) and (4),
(cid:107)x (cid:107) = max x ·y = x ·Fˆ
0 P 0 0
y∈ext(P(cid:52))
By continuity there exists δ > 0 such that for all |t| < δ ,
vw vw
(cid:107)x +tu (cid:107) = max (x +tu )·y
0 0 P 0 0
y∈ext(P(cid:52))
= (x +tu )·Fˆ
0 0
= (cid:107)x (cid:107) +tϕ (u )
0 P F 0
= (cid:107)x (cid:107)
0 P
Since G is a finite graph the result holds with δ = min δ > 0. (cid:3)
vw∈E(G) vw
The following is a characterisation of the space of infinitesimal flexes of a
general bar-joint framework in (Rd,(cid:107)·(cid:107) ).
P
FINITE AND INFINITESIMAL RIGIDITY WITH POLYHEDRAL NORMS 7
Theorem 2.5. Let (G,p) be a bar-joint framework in (Rd,(cid:107)·(cid:107) ). Then a
P
mapping u : V(G) → Rd is an infinitesimal flex of (G,p) if and only if
(cid:92)
u −u ∈ kerϕ
v w F
ϕF∈suppΦ(vw)
for each edge vw ∈ E(G).
Proof. If u is an infinitesimal flex of (G,p) then the result follows from
Proposition 2.2. For the converse, let vw ∈ E(G) and write x = p −p
0 v w
and u = u −u . Applying the argument in the proof of Proposition 2.4,
0 v w
there exists δ > 0 with (cid:107)x +tu (cid:107) = (cid:107)x (cid:107) for all |t| < δ . Hence u is
vw 0 0 P 0 P vw
an infinitesimal flex of (G,p). (cid:3)
2.2. Equivalence of finite and infinitesimal rigidity. A placement of a
simple graph G in Rd is a map p : V(G) → Rd for which p (cid:54)= p whenever
v w
vw ∈ E(G). A placement p : V(G) → Rd is well-positioned with respect
to a polyhedral norm on Rd if p − p is contained in the conical hull of
v w
exactly one facet of the unit ball P for each edge vw ∈ E(G). We denote
this unique facet by F . In the following discussion G is a finite graph and
vw
eachplacementisidentifiedwithapointp = (p ) intheproductspace
v v∈V(G)
(cid:81) Rd which we regard as having the usual topology. The set of all
v∈V(G)
well-positionedplacementsofGin(Rd,(cid:107)·(cid:107) )isanopenanddensesubsetof
P
this product space. The configuration space for a bar-joint framework (G,p)
is defined as,
(cid:89)
V(G,p) = {x ∈ Rd : (cid:107)x −x (cid:107) = (cid:107)p −p (cid:107) , ∀ vw ∈ E(G)}
v w P v w P
v∈V(G)
Proposition 2.6. Let (G,p) be a finite well-positioned bar-joint framework
in (Rd,(cid:107)·(cid:107) ) with p −p ∈ cone(F ) for each vw ∈ E(G). Then there
P v w vw
exists a neighbourhood U of p in (cid:81) Rd such that,
v∈V(G)
(i) if x ∈ U then x −x ∈ cone(F ) for each edge vw ∈ E(G),
v w vw
(ii) (G,x) is a well-positioned bar-joint framework for each x ∈ U, and,
(iii) V(G,p)∩U = {x ∈ U : ϕ (x −x ) = ϕ (p −p ), ∀ vw ∈ E(G)}.
Fvw v w Fvw v w
In particular, V(G,p)∩U = (p+F(G,p))∩U.
Proof. Letvw ∈ E(G)beanorientededgeandconsiderthecontinuousmap,
(cid:89)
T : Rd → R, (x ) (cid:55)→ x −x
vw v(cid:48) v(cid:48)∈V(G) v w
v(cid:48)∈V(G)
Since (G,p) is well-positioned, p − p is an interior point of the conical
v w
hull of a unique facet F of P. The preimage T−1(cone(F )◦) is an open
vw vw vw
neighbourhood of p. Since G is a finite graph the intersection,
(cid:92)
U = T−1(cone(F )◦)
vw vw
vw∈E(G)
is an open neighbourhood of p which satisfies (i), (ii) and (iii).
Since(G,p)iswell-positioned,byLemma2.3,thereisexactlyonesupport
functional in supp (vw) for each edge vw and this functional is given by
Φ
8 D.KITSON
ϕ . If x ∈ U then define u = (u ) by setting u = x −p for each
Fvw v v∈V(G) v v v
v ∈ V(G). By (iii), x ∈ V(G,p)∩U if and only if x ∈ U and
ϕ (u −u ) = ϕ (x −x )−ϕ (p −p ) = 0
Fvw v w Fvw v w Fvw v w
for each edge vw ∈ E(G). By Theorem 2.5, the latter identity is equivalent
totheconditionthatuisaninfinitesimalflexof(G,p). Thusx ∈ V(G,p)∩U
if and only if x ∈ U and x−p ∈ F(G,p). (cid:3)
We now prove the equivalence of continuous rigidity and infinitesimal
rigidity for finite well-positioned bar-joint frameworks.
Theorem 2.7. Let (G,p) be a finite well-positioned bar-joint framework in
(Rd,(cid:107)·(cid:107) ). Then the following statements are equivalent.
P
(i) (G,p) is continuously rigid.
(ii) (G,p) is infinitesimally rigid.
Proof. (i) ⇒ (ii). If u = (u ) ∈ F(G,p) is an infinitesimal flex of
v v∈V(G)
(G,p) then by Theorem 2.5 and Proposition 2.4, the family
α : (−(cid:15),(cid:15)) → Rd, α (t) = p +tu , v ∈ V(G)
v v v v
is a finite flex of (G,p) for some (cid:15) > 0. Since (G,p) is continuously rigid this
finite flex must be trivial. Thus there exists δ > 0 and a continuous path
c : (−δ,δ) → Rd such that α (t) = p +c(t) for all |t| < δ and all v ∈ V(G).
v v
Now u = α(cid:48)(0) = c(cid:48)(0) for all v ∈ V(G) and so u is a constant, and hence
v v
trivial, infinitesimal flex of (G,p). We conclude that (G,p) is infinitesimally
rigid.
(ii) ⇒ (i). If (G,p) has a finite flex given by the family,
α : (−(cid:15),(cid:15)) → Rd, v ∈ V(G)
v
then consider the continuous path,
α : ((cid:15),(cid:15)) → V(G,p), t (cid:55)→ (α (t))
v v∈V(G)
ByProposition2.6, V(G,p)∩U = (p+F(G,p))∩U forsomeneighbourhood
U of p. Since α(0) = p, there exists δ > 0 such that α(t) ∈ V(G,p)∩U for
all |t| < δ. Choose t ∈ (−δ,δ) and define,
0
u : V(G) → Rd, u = α (t )−p
v v 0 v
Then u = α(t )−p ∈ F(G,p) is an infinitsimal flex of (G,p). Since (G,p) is
0
infinitesimally rigid, u must be a trivial infinitesimal flex. Hence u = c(t )
v 0
for all v ∈ V(G) and some c(t ) ∈ Rd. Apply this same argument to show
0
that for each |t| < δ there exists c(t) such that α (t) = p + c(t) for all
v v
v ∈ V(G). Note that c : (−δ,δ) → Rd is continuous and so {α : v ∈ V(G)}
v
is a trivial finite flex of (G,p). We conclude that (G,p) is continuously
rigid. (cid:3)
The non-equivalence of finite and infinitesimal rigidity for general finite
bar-joint frameworks in (Rd,(cid:107) · (cid:107) ) is demonstrated in Examples 2.9 and
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3.8.
FINITE AND INFINITESIMAL RIGIDITY WITH POLYHEDRAL NORMS 9
2.3. The rigidity matrix. We define the rigidity matrix R (G,p) for a
P
finite bar-joint framework (G,p) in (Rd,(cid:107)·(cid:107) ) as follows: Fix an ordering of
P
the vertices V(G) and edges E(G) and choose an orientation on the edges of
G. For each vertex v assign d columns in the rigidity matrix and label these
columns p ,...,p . For each directed edge vw ∈ E(G) and each facet F
v,1 v,d
with p −p ∈ cone(F) assign a row in the rigidity matrix and label this
v w
row by (vw,F). The entries for the row (vw,F) are given by
pv,1 ··· pv,d pw,1 ··· pw,d
(cid:2) 0 ··· 0 Fˆ ··· Fˆ 0 ··· 0 −Fˆ ··· −Fˆ 0 ··· 0 (cid:3)
1 d 1 d
where p −p ∈ cone(F) and Fˆ = (Fˆ ,...,Fˆ ) ∈ Rd. If (G,p) is well-positioned
v w 1 d
then the rigidity matrix has size |E(G)|×d|V(G)|.
Proposition 2.8. Let (G,p) be a finite bar-joint framework in (Rd,(cid:107)·(cid:107) ). Then
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(i) F(G,p)∼=kerR (G,p).
P
(ii) (G,p) is infinitesimally rigid if and only if rankR (G,p)=d|V(G)|−d.
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Proof. ThesystemofequationsinTheorem2.5isexpressedbythematrixequation
R (G,p)uT =0whereweidentifyu:V(G)→Rdwitharowvector(u ,...,u )∈
P v1 vn
Rd|V(G)|. Thus F(G,p) ∼= kerR (G,p). The space of trivial infinitesimal flexes of
P
(G,p) has dimension d and so in general we have
rankR (G,p)≤d|V(G)|−d
P
with equality if and only if (G,p) is infinitesimally rigid. (cid:3)
If F is a facet of P and y ,y ,...,y ∈ ext(P) are extreme points of P which
1 2 d
are contained in F then for each column vector y we compute [1···1]A−1y =1
k k
where A=[y ···y ]∈Md×d(R). Hence
1 d
(6) Fˆ = [1···1]A−1
(cid:104) (cid:105)T
Moreover, if y ,y ,...,y are pairwise orthogonal then A−1 = y1 ··· yd
1 2 d (cid:107)y1(cid:107)22 (cid:107)yd(cid:107)22
and so
d
(7) Fˆ =(cid:88) yj
(cid:107)y (cid:107)2
j=1 j 2
where (cid:107)·(cid:107) is the Euclidean norm on Rd.
2
Example 2.9. Let P be a crosspolytope in Rd with 2d many extreme points
ext(P) = {±e : k = 1,...,d} where e ,e ,...,e is the usual basis in Rd. Then
k 1 2 d
each facet F contains d pairwise orthogonal extreme points y ,y ,...,y each of
1 2 d
Euclidean norm 1. By (7), Fˆ = (cid:80)d y and the resulting polyhedral norm is the
j=1 j
1-norm
d
(cid:88)
(cid:107)x(cid:107) = max x·y = |x |=(cid:107)x(cid:107)
P i 1
y∈ext(P(cid:52))
i=1
Consider for example the placements of the complete graph K in (R2,(cid:107)·(cid:107) ) il-
2 1
lustrated in Figure 1. The polytope P is indicated on the left with facets labelled
F and F . The extreme points of the polar set P(cid:52) which correspond to these
1 2
facets are Fˆ =e +e =(1,1) and Fˆ =e −e =(1,−1). The first placement is
1 1 2 2 1 2
well-positioned with respect to P and the rigidity matrix is,
pv,1 pv,2 pw,1 pw,2
(cid:2) (cid:3)
(vw,F1) 1 1 −1 −1
10 D.KITSON
This bar-joint framework has infinitesimal flex dimension 1. The second placement
is not well-positioned and the rigidity matrix is,
pv,1 pv,2 pw,1 pw,2
(cid:20) (cid:21)
(vw,F1) 1 1 −1 −1
(vw,F2) 1 −1 −1 1
As the rigidity matrix has rank 2 this bar-joint framework is infinitesimally rigid
in (R2,(cid:107)·(cid:107) ), but continuously flexible.
1
1
F1 w
v w
−1 1 v
F2
−1
Figure 1. An infinitesimally flexible and an infinitesimally
rigid placement of K in (R2,(cid:107)·(cid:107) ).
2 1
3. Edge-labellings and monochrome subgraphs
In this section we describe an edge-labelling on G which depends on the place-
mentofthebar-jointframework(G,p)in(Rd,(cid:107)·(cid:107) )relativetothefacetsofP. We
P
provide methods for identifying infinitesimally flexible frameworks and subframe-
works which are relatively infinitesimally rigid. We then characterise infinitesimal
rigidity for bar-joint frameworks with d framework colours in terms of the mono-
chrome subgraphs induced by this edge-labelling.
3.1. Edge-labellings. Let (G,p) be a general bar-joint framework in (Rd,(cid:107)·(cid:107) ).
P
SinceP issymmetricinRd,ifF isafacetofP then−F isalsoafacetofP. Denote
by Φ(P) the collection of all pairs [F]={F,−F}. For each edge vw ∈E(G) define
Φ(vw)={[F]∈Φ(P):p −p ∈cone(F)∪cone(−F)}
v w
We refer to the elements of Φ(vw) as the framework colours of the edge vw. For
example, if p −p lies in the conical hull of exactly one facet of P then the edge
v w
vw has just one framework colour. If p −p lies along a ray through an extreme
v w
point of P then vw has at least d distinct framework colours. By Lemma 2.3, [F]
is a framework colour for an edge vw if and only if either ϕ or −ϕ is a support
F F
functional for pv−pw .
(cid:107)pv−pw(cid:107)P
For each vertex v ∈V(G) denote by Φ(v ) the collection of framework colours
0 0
of all edges which are incident with v ,
0
(cid:91)
Φ(v )= Φ(v w)
0 0
v0w∈E(G)
Proposition3.1. If(G,p)isaninfinitesimallyrigidbar-jointframeworkin(Rd,(cid:107)·
(cid:107) ) then |Φ(v)|≥d for each vertex v ∈V(G).
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Proof. If v ∈V(G) and |Φ(v )|<d then there exists non-zero
0 0
(cid:92)
x∈ kerϕ
F
[F]∈Φ(v0)