---
tag: 3D1J3AXS69
kind: definition
name: Rigidity of frameworks
msc: [52C25, 51K99]
defines: V5YS9BZX0N
uses: [0JD5C82QMB, 0C95TG69SG]
sources:
  - url: https://en.wikipedia.org/wiki/Structural_rigidity
---

## Definition

Let $d \ge 1$ and let $G = (V, E)$ be a finite [[graph|0JD5C82QMB]].
A *framework* in $\R^d$ is a pair $(G, p)$ with $p \colon V \to \R^d$.
Write $\norm{\cdot}$ for the [[Euclidean distance|0C95TG69SG]].
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 $u, v \in V$.

1. $\norm{q(v) - p(v)} < \varepsilon$ for every $v \in V$.
2. $\norm{q(u) - q(v)} = \norm{p(u) - p(v)}$ 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 $\R^d$.