Projective rigidity of point-line configurations in the plane

TL;DR

This paper introduces a framework using a projective rigidity matrix to analyze the infinitesimal motions and symmetries of point-line configurations in the plane, advancing understanding of their geometric rigidity properties.

Contribution

It develops a general setup for studying incidence-preserving motions via a rigidity matrix, including symmetry-adapted versions for symmetric configurations.

Findings

Characterization of infinitesimal motions as the kernel of the rigidity matrix

Identification of dependencies among incidences as the co-kernel

Introduction of a symmetry-adapted rigidity matrix for symmetric configurations

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.