Pascal Mathis

Decomposition of Geometric Constraint Systems: a Survey

Signiflcant progress has been accomplished during the past decades about geometric constraint solving, in particular thanks to its applications in industrial flelds like CAD and robotics. In order to tackle problems of industrial size, many solving methods use, as a preprocessing, decomposition techniques that transform a large geometric constraint system into a set of smaller ones. In this paper, we propose a survey of the decomposition techniques for geometric constraint problems a . We classify them into four categories according to their modus operandi, establishing some similarities between methods that are traditionally separated. We summarize the advantages and limitations of the difierent approaches, and point out key issues for meeting industrial requirements such as generality and reliability.

Geometric construction by assembling solved subfigures

Abstract Among the expected contributions of Artificial Intelligence to Computer-Aided Design is the possibility of constructing a geometric object, the description of which is given by a system of topological and dimensional constraints. This paper presents the theoretical foundations of an original approach to formal geometric construction of rigid bodies in the Euclidian plane, based on invariance under displacements and relaxation of positional constraints. This general idea allows to explain in greater detail several methods proposed in the literature. One of the advantages of this approach is its ability to efficiently generalize and join together different methods for local solving. The paper also describes the main features of a powerful and extensible operational prototype based on these ideas, which can be viewed as a simple multi-agent system with a blackboard. Finally, some significant examples solved by this prototype are presented.

Formal resolution of geometrical constraint systems by assembling

Handling geometric objects described declaratively by a system of geometric constraints is an important issue in CAD. But until now, this requires the effective geometric construction of the objects. This paper presents an original approach to formal geometric constructions in the Euclidian plane, based on invariance under displacements and relaxation of positional constraints. This approach allows to efficiently generalize and join different methods for local solving. The paper also describes the main features of a powerful and extensible operational prototype based on these ideas, which can be viewed as a simple multi-agent system with a blackboard.

A formalization of geometric constraint systems and their decomposition

For more than a decade, the trend in geometric constraint systems solving has been to use a geometric decomposition/recombination approach. These methods are generally grounded on the invariance of systems under rigid motions. In order to decompose further, other invariance groups (e.g., scalings) have recently been considered. Geometric decomposition is grounded on the possibility to replace a solved subsystem with a smaller system called boundary. This article shows the central property that justifies decomposition, without assuming specific types of constraints or invariance groups. The exact nature of the boundary system is given. This formalization brings out the elements of a general and modular implementation.

GEOMETRICAL CONSTRAINT SYSTEM DECOMPOSITION: A MULTI-GROUP APPROACH

Since they help to specify the shape of real objects, geometric constraint systems encountered in CAD domain are often invariant by isometries. But other transformation groups can be considered to improve the solving process. More precisely, using different transformation groups leads to a new approach of decomposition which generalizes in some sense the classical approaches. This paper presents a method able to perform such a multi-group decomposition.

Decomposition of geometrical constraint systems with reparameterization

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.

Tracking Method for Reparametrized Geometrical Constraint Systems

In CAD, constraint solvers allow a user to describe a figure or an object with a set of constraints like distances, angles, tangencies, incidences and so on. Geometric solvers proceed in two stages. First, a symbolic construction plan is provided from the set of constraints. Then, the dimensions of constraints are used in a numerical stage to evaluate the construction plan. However, construction plans can not be easily provided for many problems in 3D. A classic idea consists in removing and adding some constraints in order to make the problem solvable by a geometric method. This leads to a numerical problem in which numerical values for the added constraints have to be computed in order to find the values of the added dimensions that validate the removed dimensions. Finding these values is usually done by sampling which is very time-consuming when there are more than 2 variables to sample. In this paper we address the numerical stage by adapting a path-tracking method. This allows to find several solutions and this method is efficient when the number of values is greater than 2.

Leading a continuation method by geometry for solving geometric constraints

Geometric constraint problems arise in domains such as CAD, Robotics, Molecular Chemistry, whenever one expects 2D or 3D configurations of some geometric primitives fulfilling some geometric constraints. Most well-constrained 3D problems are resistant to geometric knowledge based systems. They are often solved by purely numerical methods that are efficient but provide only one solution. Finding all the solutions can be achieved by using, among others, generic homotopy methods, that become costly when the number of constraints grows. This paper focuses on using geometric knowledges to specialize a so-called coefficient parameter continuation to 3D geometric constraint systems. Even if the proposed method does not ensure obtaining all the solutions, it provides several real ones. Geometric knowledges are used to justify it and lead the search of new solutions.

A framework for geometric constraint satisfaction problem

This article presents a metalanguage called GCML which allows the description of geometric constraint problems. In the spirit of algebraic specifications, it constitutes a framework to accompany a geometric problem from its expression to its solution. Its originality is to provide a problem with its framework called geometric universe, a tuple (syntax, semantic) which allows to get rid of ambiguities and limitations concerning description which is then freed from any software restriction. Moreover, distinction between syntax and semantic allows pre and post treatments in order to generate tools while adopting different semantic points of view in the following fields: modeling, visualization, resolution and documentation. Pragmatically, this metalanguage is based upon XML which is a language of terminology description, and allows to embed other terminologies to express different semantics.

Interactive handling of a construction plan in CAD

Solving geometric metric constraints is a topical issue in CAD. An original way to solve a constraint system is to use geometric methods, providing a symbolic construction plan. Then, this plan can be numerically interpreted to generate the required figure. If multiple solutions are produced, most solvers propose to scan the entire space of the solutions found, that is generally tedious. We show how the inner properties of a symbolic solver allow to deal more efficiently with this case. After briefly recalling our sketch-based selection method, that enables to easily eliminate most of the solutions and to keep the only, or at worst the few solutions that have the best likeness with the original drawing, we introduce a new step by step interpretation mechanism implemented as a debugger-like tool, that allows to browse the remaining solutions tree in order to help the user choosing the required solution.