Infinitesimal Rigidity in Normed Planes

Abstract

A bar-joint framework (G, p) is the combination of a graph G and a map p assigning positions, in some space, to the vertices of G. The framework is rigid if every edge-lengthpreserving continuous motion of the vertices arises from an isometry of the space. We will analyse rigidity when the space is a (non-Euclidean) normed plane and two designated vertices are mapped to the same position. This non-genericity assumption leads us to a count matroid first introduced by Jackson, Kaszanitsky and the third author. We show that independence in this matroid is equivalent to independence as a suitably regular bar-joint framework in a normed plane with two coincident points; this characterises when a regular normed plane coincident-point framework is rigid and allows us to deduce a delete-contract characterisation. We then apply this result to show that an important construction operation (generalised vertex splitting) preserves the stronger property of global rigidity in normed planes and use this to construct rich families of globally rigid graphs when the normed plane is analytic.

Generalised rigid body motions in non-Euclidean planes with applications to global rigidity

“…Note that this generalised the result for the Euclidean case obtained by Pollaczek-Geiringer [25] and popularised by Laman [19]. Kitson and Power's work led, for example, to Dewar's analysis of more general normed planes [7,6,8] and Levene and Kitson's extension to matrix norms [15].…”

“…Global rigidity implies local rigidity, however the converse is not always true (simply consider a tree when X is 1-dimensional). It is not too hard to show that a framework in X is minimally rigid if and only if it is both infinitesimally rigid and independent (see, for example, [7]). We can also link infinitesimal and local rigidity.…”

Non-Euclidean braced grids

“…In particular for the non-Euclidean norms • p , 1 ≤ q ≤ ∞, we obtained analogues of the Laman/Pollaczek-Geiringer combinatorial characterisation of generic rigidity for the Euclidean plane. This has recently been generalised to arbitrary norms by Dewar [7]. See also Remark 3.7.…”

“…The characterisations in Dewar [7], and Kitson and Power [11], show that the existence of a (2, 2)-tight spanning subgraph of G is necessary and sufficient for the infinitesimal rigidity of a bar-joint framework (G, p) which is "sufficiently generic". Under our assumptions for the underlying norm "sufficiently generic" corresponds to the non-Euclidean variant of the Euclidean rigidity matrix R(G, p) having maximum rank, in which case the framework (G, p) is said to be regular.…”

“…For some further discussions of non-Euclidean frameworks see also Dewar [7], Kitson [9], Kitson, Nixon and Schulze [10], and Nixon and Power [14].…”