Infinitesimal rigidity and prestress stability for frameworks in normed spaces

Abstract

A (bar-and-joint) framework is a set of points in a normed space with a set of fixed distance constraints between them. Determining whether a framework is locally rigid - i.e. whether every other suitably close framework with the same distance constraints is an isometric copy - is NP-hard when the normed space has dimension 2 or greater. We can reduce the complexity by instead considering derivatives of the constraints, which linearises the problem. By applying methods from non-smooth analysis, we shall strengthen previous sufficient conditions for framework rigidity that utilise first-order derivatives. We shall also introduce the notions of prestress stability and second-order rigidity to the topic of normed space rigidity, two weaker sufficient conditions for framework rigidity previously only considered for Euclidean spaces.

Figures (4)

• Figure 1: (i) A framework in the Euclidean plane that is locally and continuously rigid but infinitesimally flexible; possible non-trivial infinitesimal flexes is indicated by the red arrows. (ii) A framework in the $\ell_\infty$ normed plane (see \ref{['sec:ex']} for a definition) that is locally and continuously flexible but infinitesimally rigid; the dashed line represents other possible positions of the outer vertex.
• Figure 2: (i) A graph $H$ that has rigid placements in the $\ell_\infty$ normed plane. (ii) A badly-positioned placement $p$ of $H$ in the $\ell_\infty$ normed plane as described in described in \ref{['ex1']} that is locally and continuously flexible. An edge $v_iv_j$ is red edge (respectively, blue) if the Jacobian of the norm at $p_{v_i}-p_{v_j}$ is either $[1 ~ 0]$ or $[- 1 ~ 0]$ (respectively, either $[0 ~ 1]$ or $[0~ -1]$).
• Figure 3: An example of the framework $(G',q')$ described in \ref{['sec:ex2']}. If the vertex $v_0$ is removed, then we obtain the infinitesimally rigid framework $(G,q)$. The point $q'_{v_0}$ is chosen to lie on the line between $q_{v_1}$ and $q_{v_2}$.
• Figure 4: (i) A continuously flexible doubly-braced grid in the $\ell_p$ normed plane. A non-trivial continuous flex can be formed by rotating the red edges in unison whilst maintaing the orientation of all other edges. (ii) A prestress stable doubly-braced grid in the $\ell_p$ normed plane for $p>2$. The grid is not prestress stable for $1 \leq p < 2$ since it is not second-order well-positioned, and is in fact not even continuously rigid for $p=2$.

Theorems & Definitions (51)

• Definition 2.1
• Theorem 2.2: Rademacher's theorem
• Lemma 2.3: see, for example, clarke
• Lemma 2.4: see, for example, clarke
• Theorem 2.5: btv
• Definition 2.6
• Lemma 2.7: D21
• Lemma 2.8
• proof
• Theorem 2.9: Alexandrov's theorem