Projective background of the infinitesimal rigidity of frameworks

TL;DR

This paper proves that infinitesimal rigidity of frameworks is preserved under projective transformations and explores the relationships between Euclidean, hyperbolic, and spherical frameworks using a geometric approach.

Contribution

It provides new proofs of classical theorems showing the projective invariance of infinitesimal rigidity and introduces infinitesimal Pogorelov maps linking different geometric frameworks.

Findings

Infinitesimal rigidity is a projective invariant.

Established relations between Euclidean, hyperbolic, and spherical frameworks.

Provided a geometric derivation of key formulas for framework transformations.

Abstract

We present proofs of two classical theorems. The first one, due to Darboux and Sauer, states that infinitesimal rigidity is a projective invariant; the other one establishes relations (infinitesimal Pogorelov maps) between the infinitesimal motions of a Euclidean framework and of its hyperbolic and spherical images. The arguments use the static formulation of infinitesimal rigidity. The duality between statics and kinematics is established through the principles of virtual work. A geometric approach to statics, due essentially to Grassmann, makes both theorems straightforward. Besides, it provides a simple derivation of the formulas both for the Darboux-Sauer correspondence and for the infinitesimal Pogorelov maps.