Rigidity through a Projective Lens

Abstract

In this paper, we offer an overview of a number of results on the static rigidity and infinitesimal rigidity of discrete structures which are embedded in projective geometric reasoning, representations, and transformations. Part I considers the fundamental case of a bar–joint framework in projective d-space and places particular emphasis on the projective invariance of infinitesimal rigidity, coning between dimensions, transfer to the spherical metric, slide joints and pure conditions for singular configurations. Part II extends the results, tools and concepts from Part I to additional types of rigid structures including body-bar, body–hinge and rod-bar frameworks, all drawing on projective representations, transformations and insights. Part III widens the lens to include the closely related cofactor matroids arising from multivariate splines, which also exhibit the projective invariance. These are another fundamental example of abstract rigidity matroids with deep analogies to rigidity. We conclude in Part IV with commentary on some nearby areas.

Bar-and-joint rigidity on the moment curve coincides with cofactor rigidity on a conic

“…The matroids C d−2 d−1 and R d are invariant under projective transformation in RP d ⊃ R d , see e.g. [NSW21]. H d is, as far as we know, only invariant under linear transformation or, rather, under projective transformation in RP d−1 as a quotient space of R d \ {0}: Lemma 2.1.…”

“…Let us recall some of the needed rigidity concepts. Good comprehensive references for background are [NSW21,Whi96]; [CJT22] also contains everything we need. For hyperconnectivity see [Kal85] or [JT21].…”

Coincident-point rigidity in normed planes

“…The rigidity and global rigidity of bar-joint frameworks in Euclidean spaces has been intensely studied in recent years (e.g. [2,3,12,15,20,22]) and has a rich history going as far back as classical work of Euler and Cauchy on Euclidean polyhedra. In the last decade, work on rigidity has been generalised to various non-Euclidean normed spaces (e.g.…”

States of self-stress in symmetric frameworks and applications

“…Many gridshell structures possess symmetry (see Figure 1 for an example) so it is natural to try and utilise the symmetry-adapted counting rule as an analysis and design tool. As the states of self-stress of 2D frameworks are a projectively invariant property (Izmestiev, 2009;Nixon et al, 2021), it is possible to design highly symmetric frameworks with many states of selfstress and then project them to obtain a geometry which fits the construction requirements. Such an example is discussed in Section 3.5.…”

“…There are further methods -beyond symmetry -that can be used to create additional states of self-stress. These include subdivision methods and tools from projective geometry, such as 'pure conditions' (White and Whiteley, 1983;Nixon et al, 2021), to name but a few. Some of these methods are discussed in Section 4 by means of some basic examples.…”