Well-formed Systems of Point Incidences for Resolving Collections of Rigid Bodies

For tractability, many modern geometric constraint solvers recursively decompose an input geometric constraint system into standard collections of smaller, generically rigid subsystems or clusters. These are recursively solved and their solutions or realizations are recombined to give the solution or realization of the input constraint system. The recombination of a standard collection of solved clusters typically reduces to positioning and orienting the rigid realizations of the clusters with respect to each other, subject to incidence constraints representing primitive, shared objects between the clusters and other external constraints relating objects in different clusters. Even for generically wellconstrained systems in 3D, and even when the shared objects are restricted to be points, finding a system of incidence constraints that extends to a wellconstrained system for recombining a cluster decomposition is a significant hurdle faced by geometric constraint solvers. In general, we would like a wellformed system of incidences that generically preserves the classification of the original, undecomposed system as a well, under or overconstrained system. Here we motivate, formally state and give an efficient, greedy algorithm to find such a wellformed system for a general constraint system, when the shared objects in the cluster decomposition are restricted to be points. Our solution relies on isolating an interesting new matroid structure underlying collections of rigid clusters with shared point objects.