Combinatorics and the Rigidity of CAD Systems

TL;DR

This paper provides a combinatorial characterization of the generic minimal rigidity of 3D body-and-cad frameworks, addressing various geometric constraints used in CAD software, and lays the groundwork for deterministic rigidity algorithms.

Contribution

It introduces the first combinatorial characterization that simultaneously considers coincidence, angular, and distance constraints in body-and-cad frameworks.

Findings

Characterizes generic minimal rigidity using graph partitioning into spanning trees.

Addresses a broad class of constraints beyond classical point-distance constraints.

Establishes a theoretical basis for deterministic rigidity algorithms.

Abstract

We study the rigidity of body-and-cad frameworks which capture the majority of the geometric constraints used in 3D mechanical engineering CAD software. We present a combinatorial characterization of the generic minimal rigidity of a subset of body-and-cad frameworks in which we treat 20 of the 21 body-and-cad constraints, omitting only point-point coincidences. While the handful of classical combinatorial characterizations of rigidity focus on distance constraints between points, this is the first result simultaneously addressing coincidence, angular, and distance constraints. Our result is stated in terms of the partitioning of a graph into edge-disjoint spanning trees. This combinatorial approach provides the theoretical basis for the development of deterministic algorithms (that will not depend on numerical methods) for analyzing the rigidity of body-and-cad frameworks.