4.4 Continuous rigidity for countable frameworks
4.4.2 Continuous rigidity for countable algebraically generic frame-
Let p be a finite placement in Rd, and define pv(i) to be the i-th coordinate of pv.
We define p to be algebraically generic if the set {pv(i) : v ∈ V, i = 1, . . . , d} is
algebraically independent over Q. If p is a countably infinite placement we define p
to be algebraically generic if every finite subplacement is algebraically independent.
We likewise define a framework (G, p) to be algebraically generic if pis algebraically
generic.
Lemma 4.4.8. Let|V|<∞, then the set of algebraically generic placements in Rd is
Proof. Fix k = |V|. Let F be the set of all polynomials f ∈ R[X1, . . . , Xdk] with
integer coefficients. We note that the set of algebraic placements is exactly
V(F) := {x∈Rdk :f(x) = 0 for all f ∈F},
thus the result holds as V(F) is an algebraic set (see Corollary B.3.7).
Corollary 4.4.9. Let V be a countable set. Then the set of algebraically generic
placements of V are dense in (Rd)V with respect to the box topology.
Proof. This follows Lemma 4.4.8 and Proposition 4.1.2
Proposition 4.4.10. LetX :=Rd have the standard Euclidean norm and (G, p) be
an algebraically generic framework in X. Then (G, p) is generic. Further, if (G, p) is
finite, then every placement q∈fG−1[fG(p)] is regular, and fG−1[fG(p)] is a C1-manifold
with dimension dimF(G, p).
Proof. To see that (G, p) is generic we apply [15, Proposition 3.1] to any finite subframe-
work. Suppose (G, p) is finite. By [15, Proposition 3.3], every point q ∈fG−1[fG(p)] is
regular. As X is Euclidean, every regular placement is well-positioned and constant,
thus by Proposition 1.2.7 and Lemma 1.2.8, fG is C1-differentiable on W(G) and
has constant rank at each point q ∈ fG−1[fG(p)]. By Corollary 2.1.3, fG−1[fG(p)] is a
C1-manifold with dimension dimF(G, p) as required.
For a framework (G, p) we shall denote by fG−1[fG(p)]Γ the path-connected compo-
nent of the configuration space of (G, p) that contains p.
Lemma 4.4.11. Let X :=Rd have the standard Euclidean norm. Suppose (G, p) is a
finite algebraically generic framework, then
4.4 Continuous rigidity for countable frameworks 179 Proof. By Proposition 4.4.10 and Lemma 4.1.16, both fG−1[fG(p)]Γ andf⟨−G1⟩[f⟨G⟩(p)]Γ
are smooth manifolds of equal dimension. We note immediately that f⟨−G1⟩[f⟨G⟩(p)]Γ is a
closed C1-submanifold of f−1
G [fG(p)]Γ. Since both have the same dimension we also
have that f⟨−G1⟩[f⟨G⟩(p)]Γ is an open smooth submanifold offG−1[fG(p)]Γ. As fG−1[fG(p)]Γ
is connected then its only clopen non-empty subset is itself, hence the result holds.
Theorem 4.4.12. Suppose (G, p) is infinitesimally rigid and algebraically generic in X :=Rd with the standard Euclidean norm, then (G, p) is continuously rigid.
Proof. Since (G, p) is infinitesimally rigid, by Proposition 1.3.24, (G, p) is spanning.
Let α : (−δ, δ) → XV(G) be a finite flex of (G, p), then as (G, p) is spanning in a
Euclidean space, α is trivial if and only if
∥αv(t)−αw(t)∥=∥pv−pw∥
for all v, w ∈ V(G) and t ∈ (−δ, δ) by Proposition 1.1.33. By Theorem 4.1.7, there
exists a vertex-complete relatively infinitesimally rigid tower ((Gn, pn))n∈N of (G, p),
where we may assume (G1, p1) is spanning. Then for each n∈N, F(Gn+1, pn+1) = F(Gn+1∪KV(Gn), pn+1).
As (G, p) is algebraically generic it is also completely well-positioned, thus by Lemma
4.1.19
D
Gn+1E=DGn+1∪KV(Gn)
E
for all n ∈N.
By Lemma 4.4.11, for each n∈N,
fG−n1+1[fGn+1(pn+1)]Γ = f⟨−1
= f−1 ⟨Gn+1∪K V(Gn)⟩[ f⟨Gn+1∪K V(Gn)⟩(p n+1)]Γ = fG−n1+1∪K V(Gn)[fG n+1∪K V(Gn)(p n+1)]Γ.
Choose any v, w ∈V(G), then there exists n ∈ N such that v, w ∈V(Gn). We now
note that α(t)|V(Gn+1) ∈f−1
Gn+1∪K
V(Gn)[fGn+1∪KV(Gn)(p
n+1)]Γ, thus
∥αv(t)−αw(t)∥=∥pv−pw∥
Chapter 5
Further research and open
problems
The natural avenue of research would be to expand geometric rigidity theory in normed spaces to other classical topics such as redundant and global rigidity [26], universal rigidity [22], formation control [55] and periodic symmetry-forced rigidity [9]. We have below a list of some other areas of research and open problems that stem from our previous results.
5.1
Expanding Theorem 2.1.5 to a larger class of
frameworks
While Theorem 2.1.5 requires that a framework is constant, Theorem 1.3.19 only requires that the framework is regular. For a large class of normed spaces, regular implies constant - i.e. those with an open set of smooth points, see Proposition 2.1.1 - however there do exist normed spaces without an open set of smooth points such as Example 1.1.19.
Some hope in how to remedy this can be seen in a paper by F. H. Clarke [14]. Given a Lipschitz function f : Rn →
Rn he notes that by Rademacher’s theorem, f
is differentiable on a set U where Uc is negligible. It follows we can define for each x0 ∈ Rn the set Df(x) to be the convex hull of all linear operators T where there
exists a sequence of differentiable points (xn)n∈N such that xn → x as n → ∞ and
T = limn→∞df(xn), i.e. Df(x) := conv lim n→∞df(xn) :xn∈U, xn→x asn → ∞ .
As noted in [14, Proposition 1], each Df(x) is a compact convex set, and if the map df : U →Rn is continuous then Df(x) = {df(x)}. Using this generalisation, Clarke
forms a version of the Inverse Function Theorem for Lipschitz functions.
Theorem 5.1.1. [14, Theorem 1] Suppose Df(x0) has maximal rank i.e. for all
T ∈ Df(x0), T is bijective. Then there exists open neighbourhoods U and V of x0
and f(x0) respectively and Lipschitz function g :V →U such that g(f(u)) =u and
f(g(v)) = v for all u∈U and v ∈V.
To be able to utilise Theorem 5.1.1 with the rigidity map we would need to natu- rally extend the definition of regular and constant frameworks to non-well-positioned frameworks. Some viable definitions would be as follows:
(i) A finite framework (G, p) is regular if DfG(p) has maximal rank i.e. for all
T ∈DfG(p) and S ∈DfG(q) for some q∈XV(G), rankT ≥rankS
(ii) A finite framework (G, p) is constant if there exists an open neighbourhood U of p and k ∈N such that for all q∈U and T ∈DfG(q), rankT =k.
We would then also define a framework to be infinitesimally rigid if for all T ∈DfG(p),
kerT = T(p) and independent if for all T ∈ DfG(p), T is surjective. Using these
5.2 Infinitesimal rigidity in linear metric spaces 183