Statics and kinematics of frameworks in Euclidean and non-Euclidean geometry

TL;DR

This survey explores the infinitesimal rigidity of frameworks across Euclidean, hyperbolic, and spherical geometries, highlighting static and kinematic equivalences, projective force interpretations, and mappings between frameworks and their geodesic images.

Contribution

It provides a comprehensive overview of the static and kinematic aspects of framework rigidity in various geometries, including new insights into projective force representations and infinitesimal Pogorelov maps.

Findings

Equivalence of static and kinematic formulations of rigidity.

Projective interpretation of forces as bivectors.

Extension of Maxwell-Cremona correspondence to non-Euclidean frameworks.

Abstract

This is a survey article on the infinitesimal rigidity of frameworks in Euclidean, hyperbolic, and spherical geometry. We discuss the equivalence of the static and kinematic formulations of the infinitesimal rigidity, the projective interpretation of statics (representing forces as bivectors), and the infinitesimal Pogorelov maps that establish correspondence between infinitesimal motions of a framework and of its geodesic image. Also we describe the Maxwell-Cremona correspondence between equilibrium loads and polyhedral lifts, both for Euclidean and for non-Euclidean frameworks.