Decomposition of geometrical constraint systems with reparameterization

Abstract

Decomposition of constraint systems is a key component of geometric constraint solving in CAD. On the other hand, some authors have introduced the notion of reparameterization which aims at helping the solving of indecomposable systems by replacing some geometric constraints by other ones. In previous works, the minimal change of the initial system is a main criterion. We propose to marry these two ingredients, decomposition and reparameterization, in a method able to reparameterize and to decompose a constraint system according to this reparameterization. As a result, we do not aim at minimizing the number of added constraints during the reparameterization, but we want to decompose the system such that each component owns a minimal number of such added constraints.

A robust and efficient method for solving point distance problems by homotopy

“…Constraints are added so that a CP is easy to establish. We do not detail here the way a RCP with a sole driving parameter in each instruction is obtained, see [7], [14], [4] for different approaches. This new CP is called a RCP and is completely characterized by: the CP itself, the removed constraints that must be satisfied by all solutions and the driving parameters that are the added dimensions.…”

Re-parameterization reduces irreducible geometric constraint systems

“…However, the reduction methods proposed in [9] have limitations. For instance, they do not apply to re-parameterized systems proposed in [8,10,11,12]. This inability to reduce re-parameterized systems is due to the fact that the methods of [9] are unable to reduce irreducible systems, and that re-parameterized systems are irreducible.…”

“…This inability to reduce re-parameterized systems is due to the fact that the methods of [9] are unable to reduce irreducible systems, and that re-parameterized systems are irreducible. Later on, after the locus intersection method of Gao et al [8] became popular, several techniques for the decomposition of geometric systems with re-parameterization have been proposed [12,10,11]. These methods decompose well-constrained 3D systems into re-parameterized subsystems with a small set of key unknowns per subsystem, perform in polynomial time, and provide suboptimal but good results.…”

“…Second, with basic configurations involving more than six geometric primitives or for systems of geometric constraints involving more complex geometric primitives (cylinders, spheres, cones, torii, etc. ), using one or two key unknowns is not enough, even when employing the best re-parameterization techniques known so far [12]. In other words, although LIM1 (Locus Intersection method with one key unknown) is very fast and simple, its variants LIMk (Locus Intersection methods involving k ≥ 2 key unknowns) become much less convenient.…”

“…Our work addresses the aforementioned major limitations of re-parameterization/decomposition techniques. It proposes a technique for efficiently reducing or unlocking irreducible re-parameterized systems of equations like those proposed in [8,10,11,12] and resulting from geometric constraint systems and geometric modeling applications, so that the decomposition methods proposed in [9], which were unable to reduce such re-parameterized irreducible systems, become applicable. Furthermore, this work shows that it is possible to benefit from these decomposition techniques even when the values of the key unknowns are not known and the number of these key unknowns k is greater than 2.…”