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Infinitesimal rigidity in normed planes

Sean Dewar

TL;DR

It is proved that a graph has an infinitesimally rigid placement in a non-Euclidean normed plane if and only if it contains a (2,2)-tight spanning subgraph.

Abstract

We prove that a graph has an infinitesimally rigid placement in a non-Euclidean normed plane if and only if it contains a $(2,2)$-tight spanning subgraph. The method uses an inductive construction based on generalised Henneberg moves and the geometric properties of the normed plane. As a key step, rigid placements are constructed for the complete graph $K_4$ by considering smoothness and strict convexity properties of the unit ball.

Infinitesimal rigidity in normed planes

TL;DR

It is proved that a graph has an infinitesimally rigid placement in a non-Euclidean normed plane if and only if it contains a (2,2)-tight spanning subgraph.

Abstract

We prove that a graph has an infinitesimally rigid placement in a non-Euclidean normed plane if and only if it contains a -tight spanning subgraph. The method uses an inductive construction based on generalised Henneberg moves and the geometric properties of the normed plane. As a key step, rigid placements are constructed for the complete graph by considering smoothness and strict convexity properties of the unit ball.

Paper Structure

This paper contains 22 sections, 56 theorems, 65 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Support functionals, smoothness and strict convexity
  4. Isometries of Euclidean and non-Euclidean planes
  5. Framework and graph rigidity
  6. Frameworks
  7. The rigidity map and rigidity matrix
  8. Infinitesimal rigidity and independence of frameworks
  9. Rigidity and independence of graphs in the plane
  10. Pseudo-rigidity matrices and approximating not well-positioned frameworks
  11. Rigidity of K4 in all normed planes
  12. The rigidity of K4 in not strictly convex normed planes
  13. The rigidity of K4 in strictly convex but not smooth normed planes
  14. The rigidity of K4 in strictly convex and smooth normed planes
  15. Graph operations for the normed plane
  16. ...and 7 more sections

Key Result

Proposition 1.1

laman Henneberg moves preserve the $(2,3)$-tightness and $(2,3)$-sparsity of graphs. Further, any $(2,3)$-tight graph on $2$ or more vertices can be constructed from $K_2$ by a finite sequence of Henneberg moves.

Figures (6)

  • Figure 1: A diagram to illustrate Lemma \ref{['keylemma1']} applied to a not strictly convex normed plane $X$. (Left): The constructed infinitesimally rigid framework $(K_4,p^r)$. (Right): The unit ball of $X$. The edge directions from our placement have been added as their corresponding colour lines, $x_1,x_2$ have been added as blue dashed lines and $\operatorname{cone}[x_1, x_2]$ is shown as the blue area indicated.
  • Figure 2: From left to right: $(K_4,p^{-n_i})$, $(K_4,p)$ and $(K_4,p^{n_i})$ for $i \in \mathbb{N}$. The red dashed edge indicates the edge $v_1 v_4$ of $(K_4,p)$ is not well-positioned. We note that the support functional of the green edge will approximate $g$ while the support functional of the blue edge will approximate $f$.
  • Figure 3: The frameworks $(K_4-e ,q^1)$ and $(K_4-e, q^2)$ in some strictly convex and smooth normed plane $X$, as described in Lemma \ref{['keylemma3']}. The inner dotted shape represents the unit sphere of $X$ and the outer dotted shape represents the sphere of $X$ with radius $\|q^2_{v_4}\|$. As the framework follows the differentiable path $\alpha(t)$ the distance $\| \alpha_{v_1}(t) - \alpha_{v_4}(t) \|$ is non-constant; when the derivative of $t \mapsto \| \alpha_{v_1}(t) - \alpha_{v_4}(t) \|$ is non-zero at point $s$ we add the edge $v_1 v_4$ and note $(K_4, \alpha(s))$ will be infinitesimally rigid.
  • Figure 4: A $0$-extension (left) and a $1$-extension (right).
  • Figure 5: A vertex split (left) and a vertex-to-$K_4$ extension (right).
  • ...and 1 more figures

Theorems & Definitions (101)

  • Proposition 1.1
  • Proposition 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Proposition 1.5
  • Remark 2.1
  • Proposition 2.2
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • ...and 91 more