Structural rigidity

Definition

Let $d \ge 1$ and let $G = (V, E)$ be a finite graph. A framework in $\R^d$ is a pair $(G, p)$ where $p \colon V \to \R^d$ is a map. The framework $(G, p)$ is rigid if there is $\varepsilon > 0$ with the following property: every map $q \colon V \to \R^d$ such that

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

also satisfies $\norm{q(u) - q(v)} = \norm{p(u) - p(v)}$ for all $u, v \in V$ . A framework that is not rigid is flexible.

Remarks

Example: the triangle $K_3$ placed at three non-collinear points of $\R^2$ is rigid. Clause 1 already fixes all three pairwise distances, so the conclusion holds for every $\varepsilon$ .

Non-example: the $4$ -cycle placed at the corners of the unit square in $\R^2$ is flexible. For every $\varepsilon > 0$ there is a rhombus with sides $1$ close to the square, which satisfies clauses 1 and 2, but its diagonals have lengths different from $\sqrt{2}$ .

Rigidity is equivalent to the absence of a nontrivial flex. A flex is a continuous motion of the points, starting at $p$ , that keeps all bar lengths; it is nontrivial if it changes some distance between two vertices.

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Euclidean space object
Graph concept
Structural rigidity definition

Tag	`1RZV6TRTPE`
URI	`m2://1RZV6TRTPE`
Kind	definition
Writing	W0 stub
MSC	52C25, 05C62, 70B15
Defines	Structural rigidity
Uses	Graph, Euclidean space

Structural rigidity

Definition

Remarks

Reading path

Further reading

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