Rigidity of Frameworks Supported on Surfaces

Abstract

Abstract. A theorem of Laman gives a combinatorial characterisation of the graphs that admit a realisation as a minimally rigid generic bar-joint framework in R 2 . A more general theory is developed for frameworks in R 3 whose vertices are constrained to move on a two-dimensional smooth submanifold M. Furthermore, when M is a union of concentric spheres, or a union of parallel planes or a union of concentric cylinders, necessary and sufficient combinatorial conditions are obtained for the minimal rigidity of generic frameworks.

Distinct Distances on Curves Via Rigidity

“…We will introduce some definitions from the theory of structural rigidity; we have adapted them from the standard ones to be more suited to our particular application of algebraic curves embedded in an ambient Euclidean space. The reader may consult [1] and [16] for some background, although we will not assume the reader has knowledge of this area and we define the terminology used in the paper below. Definition 2.18 (Framework).…”

A constructive characterisation of circuits in the simple (2,2)-sparsity matroid

“…The rigidity of frameworks on surfaces [11,16] (particularly on a cylinder) provides geometric motivation for the study of M * (2, 2). In particular the question of global rigidity-when a geometric realisation of a graph on a cylinder is unique (up to ambient motions).…”

“…When the (k, l)-tight graph is simple, they still induce a matroid and we denote it as M * (k, l). Recursive constructions for the bases of M * (2, l) (l = 2, 1) can be found in [16,17,15]. In this paper we study circuits in M * (2,2).…”

Double-Distance Frameworks and Mixed Sparsity Graphs

“…The proofs make use of inductive characterisations of bi-coloured structure graphs, of the appropriate mixed sparsity type, together with determinations of minimal rigidity preservation for a range of coloured graph Henneberg extension moves. These new contexts in geometric rigidity theory and their combinatorial characterisations extend the analyses in: Nixon, Owen and Power [10,11] of 3-dimensional frameworks which are vertex-constrained to surfaces; Kitson and Power [7] of 2-dimensional frameworks with non-Euclidean distances; and Servatius and Whiteley [14] of 2-dimensional (Euclidean) direction-length frameworks.…”