Revisiting decomposition analysis of geometric constraint graphs

Abstract

Geometric problems defined by constraints can be represented by geometric constraint graphs whose nodes are geometric elements and whose arcs represent geometric constraints. Reduction and decomposition are techniques commonly used to analyze geometric constraint graphs in geometric constraint solving.In this paper we first introduce the concept of deficit of a constraint graph. Then we give a new formalization of the decomposition algorithm due to Owen. This new formalization is based on preserving the deficit rather than on computing triconnected components of the graph and is simpler. Finally we apply tree decompositions to prove that the class of problems solved by the formalizations studied here and other formalizations reported in the literature is the same.

Well-constrained completion for under-constrained geometric constraint problem based on connectivity analysis of graph

“…Joan-Arinyo et al define a deficit function deficit(G) = (2|V | − 3) − |E| in 2D to compute the difference between the number of edges for a constraint graph G = (V, E) to be structurally well-constrained and its actual number of |E| in 2D [7]. Here we give the general expression to compute the difference between the sum of weight of edges for a constraint graph G = (V (G), E(G), ω) to be structurally well-constrained and its actual weight ω(E(G)) in 2D as follows.…”

Towards an homogeneous handling of under-constrained and well-constrained systems of geometric constraints

“…An important work on this domain was done by Joan-Arinyo et al [8], introducing the notion of decit and explaining a decomposition method, based on a modication of Owen's algorithm, that extracts under-constrained subsystems and nds which parameterized constraints should be added to deal with the under-constriction. Based on this method, Gao et al [3] proposed ecient completion algorithms.…”

“…Otherwise, the propagation method does only consider the new constraint as a closure condition. There are dierent ways to solve this problem: one can compute the decit [8] of the closed chain, and if it is zero, rewrite the term to show that the closed chain actually is rigid; one may also use a more complex geometric reasoning, such as aloci method [2].…”

“…One could for instance prefer the use of a classical decomposition method to extract all greatest rigid subsystems, before nally using the Assemble function to assemble them if there is more than one. Should the user be unsatised and ask for a rigid solution, this approach enables us to add relative references or to use another completion algorithm, like [8].…”

Transforming an under-constrained geometric constraint problem into a well-constrained one

“…These solvers are incomplete, and, therefore, it makes sense to ask for their domain. We have characterized the domain of these methods in [10,12,13].…”