Rigidity of frameworks

Definition

Let $d \ge 1$ and let $G = (V, E)$ be a finite graph. A framework in $\R^d$ is a pair $(G, p)$ with $p \colon V \to \R^d$ . Write $\norm{\cdot}$ for the Euclidean distance. The framework $(G, p)$ is rigid if there is $\varepsilon > 0$ with the following property. Every $q \colon V \to \R^d$ that satisfies both conditions below also satisfies $\norm{q(u) - q(v)} = \norm{p(u) - p(v)}$ for all $u, v \in V$ .

$\norm{q(v) - p(v)} < \varepsilon$ for every $v \in V$ .
$\norm{q(u) - q(v)} = \norm{p(u) - p(v)}$ for every edge $uv \in E$ .

A framework that is not rigid is called flexible.

Remarks

So a framework is rigid when every nearby placement with the same bar lengths is congruent to it. Example: a triangle $K_3$ placed at three non-collinear points of $\R^2$ is rigid. Its three bar lengths are all the pairwise distances, so condition 2 already gives the conclusion.

Non-example: the cycle on four vertices placed at the corners of a unit square in $\R^2$ is flexible. For small $t > 0$ , the rhombus with corners $(0, 0)$ , $(1, 0)$ , $(1 + \sin t, \cos t)$ and $(\sin t, \cos t)$ keeps all four bar lengths equal to $1$ . Its diagonal from $(0, 0)$ to $(1 + \sin t, \cos t)$ has length different from $\sqrt{2}$ .

An equivalent form: every continuous motion of the vertices that keeps all bar lengths is the restriction of a family of isometries of $\R^d$ .

Reading path

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Euclidean space object
Graph definition
Length of a vector concept
Euclidean distance definition
Rigidity of frameworks definition

Tag	`3D1J3AXS69`
URI	`m2://3D1J3AXS69`
Kind	definition
Writing	W0 stub
MSC	52C25, 51K99
Defines	Rigidity of frameworks
Uses	Graph, Euclidean distance

Rigidity of frameworks

Definition

Remarks

Reading path

Further reading

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