Numerical decomposition of geometric constraints

Abstract

Geometric constraint solving is a key issue in CAD/CAM. Since Owen's seminal paper, solvers typically use graph based decomposition methods. However, these methods become difficult to implement in 3D and are misled by geometric theorems. We extend the Numerical Probabilistic Method (NPM), well known in rigidity theory, to more general kinds of constraints and show that NPM can also decompose a system into rigid subsystems. Classical NPM studies the structure of the Jacobian at a random (or generic) configuration. The variant we are proposing does not consider a random configuration, but a configuration similar to the unknown one. Similar means the configuration fulfills the same set of incidence constraints, such as collinearities and coplanarities. Jurzak's prover is used to find a similar configuration.

Decomposition of Geometric Constraint Systems: A Survey

“…72,73 We show that this principle can be integrated in existing decomposition methods or be the base of new ones.…”

Geometric constraint solving: The witness configuration method

“…Moreover, the WC makes possible the decomposition of the complete system into rigid subsystems (Section 6.3). It can also be interrogated; for instance, a WC of the Pappus hypothesis PH (of the Desargues hypothesis D2H in 2D, or of the Desargues hypothesis D3H in 3D), will satisfy the Pappus conclusion (the Desargues conclusion in 2D, in 3D): this is a (probabilistic) proof of the theorem; it is the principle of Jurzak's prover [4]. If the WC has not the conjectured incidence property, then the conjecture is wrong -with no doubt at all: thus it is useless to repeat the test with another WC.…”

“…Moreover, it can also decompose systems into well-constrained subsystems (Section 6.3). It is also possible to interrogate the WC: if it has some property, then this property holds generically and a theorem has been (probabilistically) proved [4][5][6].…”

A 2D geometric constraint solver using a graph reduction method

“…Other approaches optimize the resolution process by pruning the solutions space (see [25]). Foufou et al [7] used efficient numerical methods to solve systems of constraints.…”