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Projective rigidity of point-line configurations in the plane

Leah Berman, Signe Lundqvist, Bernd Schulze, Brigitte Servatius, Herman Servatius, Klara Stokes, Walter Whiteley

TL;DR

This work develops a rigorous framework for studying incidence-preserving motions of point-line configurations in the real projective plane through a projective rigidity matrix, whose kernel encodes infinitesimal motions and whose co-kernel captures incidence dependencies (self-stresses). It introduces a symmetry-adapted projective orbit rigidity matrix to analyze symmetry-preserving motions, and formalizes how orbits under a symmetry group constrain admissible deformations. The authors connect rigidity to classical incidence theorems, provide illustrative examples such as the Desargues configuration, complete quadrilateral, and autopolar configurations, and relate the projective rigidity perspective to matroid realization and slack realization spaces. The framework lays groundwork for probing realizability questions, inductive constructions, and higher-dimensional generalizations, with potential impact on understanding symmetric configurations and their realization spaces.

Abstract

In this paper, we establish a general setup for studying incidence-preserving motions of projective geometric configurations of points and lines via a "projective rigidity matrix". The spaces of infinitesimal motions of a point-line configuration and dependencies amongst the point-line incidences can be interpreted as the kernel and co-kernel of this projective rigidity matrix, respectively. We also introduce a symmetry-adapted projective rigidity matrix for analysing symmetric configurations and their symmetry-preserving motions. The symmetry may be a point group or a more general symmetry, such as an autopolarity.

Projective rigidity of point-line configurations in the plane

TL;DR

This work develops a rigorous framework for studying incidence-preserving motions of point-line configurations in the real projective plane through a projective rigidity matrix, whose kernel encodes infinitesimal motions and whose co-kernel captures incidence dependencies (self-stresses). It introduces a symmetry-adapted projective orbit rigidity matrix to analyze symmetry-preserving motions, and formalizes how orbits under a symmetry group constrain admissible deformations. The authors connect rigidity to classical incidence theorems, provide illustrative examples such as the Desargues configuration, complete quadrilateral, and autopolar configurations, and relate the projective rigidity perspective to matroid realization and slack realization spaces. The framework lays groundwork for probing realizability questions, inductive constructions, and higher-dimensional generalizations, with potential impact on understanding symmetric configurations and their realization spaces.

Abstract

In this paper, we establish a general setup for studying incidence-preserving motions of projective geometric configurations of points and lines via a "projective rigidity matrix". The spaces of infinitesimal motions of a point-line configuration and dependencies amongst the point-line incidences can be interpreted as the kernel and co-kernel of this projective rigidity matrix, respectively. We also introduce a symmetry-adapted projective rigidity matrix for analysing symmetric configurations and their symmetry-preserving motions. The symmetry may be a point group or a more general symmetry, such as an autopolarity.
Paper Structure (10 sections, 5 theorems, 32 equations, 7 figures)

This paper contains 10 sections, 5 theorems, 32 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Rigidity and flexibility of projective configurations
  4. Self-stresses of projective configurations
  5. Symmetric projective configurations
  6. Symmetries, dualities and polarities of projective space
  7. Projective orbit rigidity matrix
  8. Free actions
  9. Actions with fixed points, lines, and incidences
  10. Conclusion and future work

Key Result

Theorem 1

Let $(S, \mathbf{p}, \mathbf{l})$ be a projective configuration. Then the following are equivalent:

Figures (7)

  • Figure 1: The $10_3$ Desargues configuration.
  • Figure 2: A cyclic $10_3$-configuration.
  • Figure 3: A cyclic $20_4$-configuration.
  • Figure 4: The complete quadrilateral.
  • Figure 5: An autopolar configuration: the polarity $\pi_C$ sends the configuration to itself, i.e. it sends the points of the configuration to the lines of the configuration and vice versa.
  • ...and 2 more figures

Theorems & Definitions (13)

  • Theorem 1
  • proof
  • Remark 1
  • Theorem 2: fields
  • Example 1: The Desargues configuration
  • Example 2
  • Proposition 3
  • proof
  • Example 3: The complete quadrilateral
  • Theorem 4
  • ...and 3 more