Body-and-cad Geometric Constraint Systems

TL;DR

This paper develops a comprehensive rigidity theory for 3D body-and-cad structures with various geometric constraints, extending classical models and introducing a new counting condition for minimal rigidity.

Contribution

It introduces a general framework for body-and-cad rigidity, identifies 21 geometric constraints, and proposes the nested sparsity condition as a necessary criterion for minimal rigidity.

Findings

Identified 21 relevant geometric constraints for body-and-cad structures.

Developed the infinitesimal rigidity theory for these structures.

Proposed nested sparsity as a necessary condition for minimal rigidity.

Abstract

Motivated by constraint-based CAD software, we develop the foundation for the rigidity theory of a very general model: the body-and-cad structure, composed of rigid bodies in 3D constrained by pairwise coincidence, angular and distance constraints. We identify 21 relevant geometric constraints and develop the corresponding infinitesimal rigidity theory for these structures. The classical body-and-bar rigidity model can be viewed as a body-and-cad structure that uses only one constraint from this new class. As a consequence, we identify a new, necessary, but not sufficient, counting condition for minimal rigidity of body-and-cad structures: nested sparsity. This is a slight generalization of the well-known sparsity condition of Maxwell.