Some notes on the equivalence of first-order rigidity in various geometries

TL;DR

This paper discusses the equivalence of first-order rigidity theories across Euclidean, hyperbolic, and spherical geometries, providing foundational insights for constraint programming in CAD.

Contribution

It sketches the theoretical framework establishing the equivalence of rigidity concepts in different geometries, preparing for a comprehensive formal proof in a forthcoming paper.

Findings

Outline of the first-order rigidity theory in various geometries

Remarks on the implications for constraint programming in CAD

Preparation for a formal proof of equivalence

Abstract

These pages serve two purposes. First, they are notes to accompany the talk "Hyperbolic and projective geometry in constraint programming for CAD" by Walter Whiteley at the "Janos Bolyai Conference on Hyperbolic Geometry", 8--12 July 2002, in Budapest, Hungary. Second, they sketch results that will be included in a forthcoming paper that will present the equivalence of the first-order rigidity theories of bar-and-joint frameworks in various geometries, including Euclidean, hyperbolic and spherical geometry. The bulk of the theory is outlined here, with remarks and comments alluding to other results that will make the final version of the paper.