Point-hyperplane frameworks, slider joints, and rigidity preserving transformations

TL;DR

This paper extends classical rigidity theory to frameworks of points and hyperplanes, establishing new correspondences and combinatorial characterizations that enhance understanding of rigidity across different geometric models.

Contribution

It introduces a framework for analyzing infinitesimal rigidity of point-hyperplane structures and generalizes existing results on collinearity in rigidity graphs.

Findings

Derived a combinatorial characterization for infinitesimal rigidity with collinear points in the plane.

Extended classical rigidity correspondences to include point-hyperplane frameworks.

Connected rigidity models across different geometric spaces.

Abstract

A one-to-one correspondence between the infinitesimal motions of bar-joint frameworks in $\mathbb{R}^d$ and those in $\mathbb{S}^d$ is a classical observation by Pogorelov, and further connections among different rigidity models in various different spaces have been extensively studied. In this paper, we shall extend this line of research to include the infinitesimal rigidity of frameworks consisting of points and hyperplanes. This enables us to understand correspondences between point-hyperplane rigidity, classical bar-joint rigidity, and scene analysis. Among other results, we derive a combinatorial characterization of graphs that can be realized as infinitesimally rigid frameworks in the plane with a given set of points collinear. This extends a result by Jackson and Jord\'{a}n, which deals with the case when three points are collinear.