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

Sean Dewar, Anthony Nixon

Abstract

A bar-joint framework $(G,p)$ in a (non-Euclidean) real normed plane $X$ is the combination of a finite, simple graph $G$ and a placement $p$ of the vertices in $X$. A framework $(G,p)$ is globally rigid in $X$ if every other framework $(G,q)$ in $X$ with the same edge lengths as $(G,p)$ arises from an isometry of $X$. The weaker property of local rigidity in normed planes (where only $(G,q)$ within a neighbourhood of $(G,p)$ are considered) has been studied by several researchers over the last 5 years after being introduced by Kitson and Power for $\ell_p$-norms. However global rigidity is an unexplored area for general normed spaces, despite being intensely studied in the Euclidean context by many groups over the last 40 years. In order to understand global rigidity in $X$, we introduce new generalised rigid body motions in normed planes where the norm is determined by an analytic function. This theory allows us to deduce several geometric and combinatorial results concerning the global rigidity of bar-joint frameworks in $X$.

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

Abstract

A bar-joint framework in a (non-Euclidean) real normed plane is the combination of a finite, simple graph and a placement of the vertices in . A framework is globally rigid in if every other framework in with the same edge lengths as arises from an isometry of . The weaker property of local rigidity in normed planes (where only within a neighbourhood of are considered) has been studied by several researchers over the last 5 years after being introduced by Kitson and Power for -norms. However global rigidity is an unexplored area for general normed spaces, despite being intensely studied in the Euclidean context by many groups over the last 40 years. In order to understand global rigidity in , we introduce new generalised rigid body motions in normed planes where the norm is determined by an analytic function. This theory allows us to deduce several geometric and combinatorial results concerning the global rigidity of bar-joint frameworks in .

Paper Structure

This paper contains 21 sections, 51 theorems, 72 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Negligible sets
  4. Real analytic functions
  5. Rigidity theoretic background
  6. Rigidity in normed spaces
  7. Analytic normed spaces
  8. Global rigidity in smooth normed spaces with finitely many linear isometries
  9. Generalising rigid body motions
  10. Reflections and rotations in analytic normed planes
  11. Normal points of reflections and rotations
  12. Global rigidity in analytic normed planes
  13. $K_5^-$ and $H$ are globally rigid
  14. Some classes of globally rigid graphs
  15. Proof of Theorems \ref{['l:sisom2']} and \ref{['l:srefl']}
  16. ...and 6 more sections

Key Result

Theorem 2.2

analytic Let $U,V$ be open subsets of $\mathbb{R}^d$. If $f:U \rightarrow V$ is an analytic map and $\operatorname{rank} df(x) = d$ for some $x \in U$, then there exists a neighbourhood $U' \subset U$ of $x$ so that $f|_{U'}$ is a local analytic diffeomorphism. Furthermore, if $f$ is injective and $

Figures (6)

  • Figure 1: Two frameworks in the Euclidean plane. The framework on the left is strongly regular but not completely regular as it has a colinear triple. The framework on the right is completely regular but not strongly regular as we can flatten the framework into colinear non-regular framework.
  • Figure 2: (Left) An independent placement in $\ell_4^2$ of the complete graph on 4 vertices minus an edge. (Right) An equivalent independent framework that can be reached by continuous motion that preserves edge lengths. As the distance between the two non-adjacent vertices is altered during the motion, the framework is not locally rigid. The framework can be seen to be infinitesimally flexible by applying Theorem \ref{['t:asimowroth']}.
  • Figure 3: (Left) A graph that is minimally rigid in any normed plane. Although the graph has a separating vertex, the normed plane itself does not have a continuous family of rotational isometries to exploit so as to flex the framework. The graph is also not globally rigid as we can translate the framework so that the separating vertex is at the origin and then map every vertex in the right copy of $K_4$ to its negative. (Right) The graph $H$ is globally rigid in all analytic normed planes (Theorem \ref{['thm:H']}). It is not globally rigid in the Euclidean plane; for almost all placements, the left two vertices may be reflected across the dashed line to obtain an equivalent but non-congruent framework. However, the required reflection does not exist in most normed planes.
  • Figure 4: (Left) A $z$-reflection in $\ell_4^2$ of a point $x$; the grey point represents its new position $R_z(x)$. (Right) A $z$-rotation in $\ell_4^2$ of a point $x$; the grey triangle represents the new positions $r_z(x,t)$ and $r_z(z,t)$ of $x$ and $z$ respectively.
  • Figure 5: A $z$-rotation of two points $x,y$ in $\ell_4^2$. The distance between the points $x$ and $y$ will (for most choices of $x,y$) change during the $z$-rotation.
  • ...and 1 more figures

Theorems & Definitions (96)

  • Definition 2.1
  • Theorem 2.2
  • Proposition 2.3
  • Proposition 2.4
  • Proposition 2.5
  • proof
  • Theorem 2.6
  • Remark 2.7
  • Theorem 2.8
  • Remark 2.9
  • ...and 86 more