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Generic Global Rigidity in $\ell_p$-Space and the Identifiability of the $p$-Cayley-Menger Varieties

Tomohiro Sugiyama, Shin-ichi Tanigawa

TL;DR

The paper settles the generic global rigidity problem in $\ell_p^2$ for even integers $p$, demonstrating that global rigidity is a generic graph property in the plane when $p$ is even. It achieves this by linking rigidity to the identifiability of the $p$-Cayley-Menger variety, leveraging tangential weak defectivity criteria, and deriving a Connelly-type sufficiency condition suited to $p$-norm measurements. A practical stress-based test is developed via coordinated self-stresses, enabling a graph-theoretic criterion that mirrors Euclidean results. Additionally, the authors provide a precise plane characterization: a generic $2$-D framework is globally rigid in $\ell_p^2$ if and only if $G$ is 2-connected and redundantly 2-tree-connected, with constructive bases and extension operations ensuring preservation of the property. The work bridges algebraic geometry and rigidity theory, offering both theoretical insight and a pathway to computational identification of global rigidity in non-Euclidean norms.

Abstract

The celebrated result of Gortler-Healy-Thurston (independently, Jackson-Jordán for $d=2$) shows that the global rigidity of graphs realised in the $d$-dimensional Euclidean space is a generic property. Extending this result to the global rigidity problem in $\ell_p$-spaces remains an open problem. In this paper we affirmatively solve this problem when $d=2$ and $p$ is an even positive integer. A key tool in our proof is a sufficient condition for the $d$-tangential weak non-defectivity of projective varieties due to Bocci, Chiantini, Ottaviani, and Vannieuwenhoven. By specialising the condition to the $p$-Cayley-Menger variety, which is the $\ell_p$-analogue of the Cayley-Menger variety for Euclidean distance, we provide an $\ell_p$-extension of the generic global rigidity theory of Connelly. As a by-product of our proof, we also offer a purely graph-theoretical characterisation of the $2$-identifiability of an orthogonal projection of the $p$-Cayley-Menger variety along a coordinate axis of the ambient affine space.

Generic Global Rigidity in $\ell_p$-Space and the Identifiability of the $p$-Cayley-Menger Varieties

TL;DR

The paper settles the generic global rigidity problem in for even integers , demonstrating that global rigidity is a generic graph property in the plane when is even. It achieves this by linking rigidity to the identifiability of the -Cayley-Menger variety, leveraging tangential weak defectivity criteria, and deriving a Connelly-type sufficiency condition suited to -norm measurements. A practical stress-based test is developed via coordinated self-stresses, enabling a graph-theoretic criterion that mirrors Euclidean results. Additionally, the authors provide a precise plane characterization: a generic -D framework is globally rigid in if and only if is 2-connected and redundantly 2-tree-connected, with constructive bases and extension operations ensuring preservation of the property. The work bridges algebraic geometry and rigidity theory, offering both theoretical insight and a pathway to computational identification of global rigidity in non-Euclidean norms.

Abstract

The celebrated result of Gortler-Healy-Thurston (independently, Jackson-Jordán for ) shows that the global rigidity of graphs realised in the -dimensional Euclidean space is a generic property. Extending this result to the global rigidity problem in -spaces remains an open problem. In this paper we affirmatively solve this problem when and is an even positive integer. A key tool in our proof is a sufficient condition for the -tangential weak non-defectivity of projective varieties due to Bocci, Chiantini, Ottaviani, and Vannieuwenhoven. By specialising the condition to the -Cayley-Menger variety, which is the -analogue of the Cayley-Menger variety for Euclidean distance, we provide an -extension of the generic global rigidity theory of Connelly. As a by-product of our proof, we also offer a purely graph-theoretical characterisation of the -identifiability of an orthogonal projection of the -Cayley-Menger variety along a coordinate axis of the ambient affine space.
Paper Structure (17 sections, 30 theorems, 61 equations, 1 figure)

This paper contains 17 sections, 30 theorems, 61 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Rigidity in $\ell_p$-space and main theorems
  3. Local and global rigidity
  4. Generic local rigidity
  5. Generic global rigidity in Euclidean space
  6. Global rigidity problem in $\ell_p$-space
  7. Main rigidity theorems
  8. Comments on the self-stress condition in Theorem \ref{['thm:main']}
  9. Identifiability and Global Rigidity
  10. Cayley-Menger varieties
  11. Defectivity
  12. Identifiability
  13. Tangential weak defectivity
  14. Proof of Proposition \ref{['prop:twnd_iden']}
  15. Sufficient condition in terms of local rigidity
  16. ...and 2 more sections

Key Result

Proposition 2.1

Let $p$ is an even positive integer with $p \neq 2$ and $(G,\bm p)$ be a framework in $\ell_p^d$. Then $(G,\bm p)$ is globally rigid if and only if any $\bm q$ in the fibre of $\pi_G\circ f_p^{\times d}(\bm p)$ under $\pi_G\circ f_p^{\times d}$ can be obtained from $\bm p$ by a translation, a permut

Figures (1)

  • Figure 1: (a) $K_5^-$ and (b) $B_1$.

Theorems & Definitions (55)

  • Proposition 2.1
  • Proposition 2.2
  • Theorem 2.3: Kitson and Power kitson2014infinitesimal
  • Theorem 2.4: Gortler-Healy-Thurston gortler2010characterizing
  • Proposition 2.5
  • Theorem 2.6: Dewar, Hewetson, and Nixon dewar2024uniquely
  • Theorem 2.7: Dewar, Hewetson, and Nixon dewar2024uniquely
  • Theorem 2.8
  • Conjecture 2.9
  • Example 2.10
  • ...and 45 more