Algorithms for detecting dependencies and rigid subsystems for CAD

Abstract

Geometric constraint systems underly popular Computer Aided Design software. 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.

Rigidity through a Projective Lens

“…Another important area of application for geometric constraints is the analysis of CAD constraints and the design of algorithms for detecting when the constraints being applied are dependent [72,175]. Two things happen in CAD when the next constraint is dependent: (i) there are more degrees of freedom than anticipated; and (ii) the numerical value assigned to the dependent constraint is not a free choice and is unlikely to be correct.…”