Gruber P. Convex and Discrete Geometry

298 Convex Polytopes
of V. Fori ∈ V let a(i) ={j ∈ V :{i, j} ∈E} be the set of vertices adjacent to the
vertex i.Aframework F(p) in E d is an abstract framework F =〈V, E〉 together with
a point p = (p1,...,pv) ∈ E d ×···×E d = E dv . A point pi, i ∈ V, isavertex
and a line segment [pi, p j ], {i, j} ∈E, isanedge of F(p). The framework F(p) is
called a realization of the abstract framework F. Lete be the number of edges of F.
Consider the e functions f{ij} : E dv → R, {i, j} ∈E, defined by:
Let
f{ij}(x) =||xi − x j|| 2 for x = (x1,...,xv) ∈ E dv .
R(p) = � x : f{ij}(x) = f{ij}(p) for all {i, j} ∈E �
= � �
x : f{ij}(x) = f{ij}(p) � ⊆ E dv .
{i, j}∈E
Then � F(x) : x ∈ R(p) �
is the set of all realizations F(x) of F with edge-lengths equal to the corresponding
edge-lengths of F(p). Callx = (x1,...,xv) congruent to p = (p1,...,pv) if there
is a rigid motion m : E d → E d such that x1 = mp1,...,xv = mpv.Let
C(p) = � x : x congruent to p � ⊆ R(p).
If aff{p1,...,pv} =E d , then it is not too difficult to show that C(p) is a smooth
manifold of dimension 1 2d(d + 1) in Edv , where by smooth we mean of class C∞ .
Here 1 2d(d − 1) dimensions come from the rotations and d from the translations.
Then {F(x) : x ∈ C(p)} is the set of all realizations F(x) of F which are congruent
to F(p) in the ordinary sense. The framework F(p) is flexible if there is a continuous
function x :[0, 1] →E dv such that
(1) x(0) = p ∈ C(p) and x(t) ∈ R(p)\C(p) for 0 < t ≤ 1.
F(p) is rigid if it is not flexible.
A Rigidity Criterion
The above makes it clear that flexibility of a framework F(p) is E d is determined by
the way in which C(p) is included in R(p) in a neighbourhood of p. The following
result is a simple version of the rigidity predictor theorem of Gluck [381].
Theorem 17.2. Let F(p) be a framework in E d , where p = (p1,...,pv) and
aff{p1,...,pd} =E d . Assume that the e vectors
grad f{ij}(p), {i, j} ∈E,
are linearly independent in E dv . Then e ≤ dv − 1 2 d(d + 1) and F(p) is rigid if and
only if equality holds.