Locked and Unlocked Chains of Planar Shapes

We extend linkage unfolding results from the well-studied case of polygonal linkages to the more general case of linkages of polygons. More precisely, we consider chains of nonoverlapping rigid planar shapes (Jordan regions) that are hinged together sequentially at rotatable joints. Our goal is to characterize the families of planar shapes that admit locked chains, where some configurations cannot be reached by continuous reconfiguration without self-intersection, and which families of planar shapes guarantee universal foldability, where every chain is guaranteed to have a connected configuration space. Previously, only obtuse triangles were known to admit locked shapes, and only line segments were known to guarantee universal foldability. We show that a surprisingly general family of planar shapes, called slender adornments, guarantees universal foldability: roughly, the distance from each edge along the path along the boundary of the slender adornment to each hinge should be monotone. In contrast, we show that isosceles triangles with any desired apex angle less than 90 degrees admit locked chains, which is precisely the threshold beyond which the inward-normal property no longer holds.Comment: 23 pages, 25 figures, Latex; full journal version with all proof details. (Fixed crash-induced bugs in the abstract.

Convex Geometry and its Applications

The past 30 years have not only seen substantial progress and lively activity in various areas within convex geometry, e.g., in asymptotic geometric analysis, valuation theory, the $L_p$-Brunn-Minkowski theory and stochastic geometry, but also an increasing amount and variety of applications of convex geometry to other branches of mathematics (and beyond), e.g. to PDEs, statistics, discrete geometry, optimization, or geometric algorithms in computer science. Thus convex geometry is a flourishing and attractive field, which is also reflected by the considerable number of talented young mathematicians at this meeting

Rigidity of frameworks with coordinated constraint relaxations

This thesis is concerned with the rigidity of coordinated frameworks. These are considered to be bar-joint frameworks for which the requirement that the lengths of bars be kept fixed is relaxed on some collection of bars, with the caveat that all bars within a coordination class must change length by the same amount. We begin by formulating the conditions for a framework to be continuously coordinated rigid, infinitesimally coordinated rigid, and statically coordinated rigid. We prove that static and infinitesimal rigidity are equivalent for coordinated frameworks, and that for regular coordinated frameworks, continuous rigidity and infinitesimal rigidity are equivalent. We give a characterisation of the rigidity of frameworks in d-dimensional Euclidean space with k coordination classes, based on the rigidity of the structure graph of such a framework. Since minimal infinitesimal rigidity of bar-joint frameworks is characterised in 1- and 2-dimensions, we extend the standard characterisations to a combinatorial characterisation of minimally infinitesimally rigid frameworks with one class of coordinated bars, and with two classes of coordinated bars, in both dimension 1 and dimension 2. We also obtain an inductive characterisation of such minimally infinitesimally rigid frameworks using coordinated analogues to standard inductive constructions. We conclude by considering coordinated frameworks with symmetric realisations, and give some initial results in this area

LIPIcs, Volume 261, ICALP 2023, Complete Volume

LIPIcs, Volume 261, ICALP 2023, Complete Volum