Algorithms for detecting dependencies and rigid subsystems for CAD

TL;DR

This paper introduces algorithms to detect dependencies and special positions in geometric constraint systems used in CAD, improving robustness and user feedback by identifying when constraints become dependent.

Contribution

It presents a pebble game algorithm for generic dependency detection and combinatorial algorithms for identifying special positions causing dependencies in CAD systems.

Findings

Pebble game algorithm effectively detects generic dependencies.

Combinatorial algorithms identify subgraphs linked to polynomial factors indicating dependencies.

Potential for geometric interpretation through Grassmann-Cayley algebra.

Abstract

Geometric constraint systems underly popular Computer Aided Design soft- ware. Automated approaches for detecting dependencies in a design are critical for developing robust solvers and providing informative user feedback, and we provide algorithms for two types of dependencies. First, we give a pebble game algorithm for detecting generic dependencies. Then, we focus on identifying the "special positions" of a design in which generically independent constraints become dependent. We present combinatorial algorithms for identifying subgraphs associated to factors of a particular polynomial, whose vanishing indicates a special position and resulting dependency. Further factoring in the Grassmann- Cayley algebra may allow a geometric interpretation giving conditions (e.g., "these two lines being parallel cause a dependency") determining the special position.