A 2D geometric constraint solver using a graph reduction method

Abstract

Modeling by constraints enables users to describe shapes by specifying relationships between geometric elements. These relationships are called constraints. A constraint solver derives then automatically the design intended by exploiting these constraints. The constraints solvers can be classified in four categories: symbolic, numerical, rule-oriented and graph-constructive solvers. The graph constructive approach is widely used in recent Computer Aided Design (CAD) systems. In this paper, we present a decomposition–recombination (DR) planning algorithm, called S-DR, that uses a graph reduction method to solve systems of 2D geometric constraints. Based on the key concept of skeletons, S-DR planner figures out a plan for decomposing a well constrained system into small sub-systems and recombines the solutions of these sub-systems to derive the solution of the entire system.

Introduction

Geometric constraint solving has applications in many different fields, such as Computer Aided Design (CAD), molecular modeling, tolerance analysis, and geometric theorem proving. Geometric modeling by constraints enables users to describe shapes by specifying a rough sketch and adding to it geometric constraints, i.e. a set of required relations between geometric elements. The constraint solver must derive automatically the correct shape needed. Typically, in 2D, geometric modeling by constraints specifies geometrical objects such as points, lines, circles, conics by a set of constraints: distances between points, points and lines, parallel lines, angles between lines, incidence relations between points and lines, points and circles, tangency relations between lines and circles or between circles.

Many resolution methods have been proposed for solving systems of geometric constraints. We classify the resolution methods in four broad categories: symbolic, numerical, rule-oriented and graph-constructive solvers.

In symbolic methods, the constraints are translated into a system of equations. Methods such as Gröbner bases or elimination with resultants are applied to find symbolic expressions for the solutions. These methods are “extremely” time consuming. They are typically exponential in time and space (see [4], [15]). They can be used only for small systems.

Numerical methods (Newton–Raphson’s iteration, homotopy, Gaussian elimination and so on) for solving systems of equations are O(n³) or worse (see [3], [17], [23], [26]). Most numerical methods have difficulties for handling over- and under-constrained schemes. Some systems use a preprocess treatment to decompose the system of equations before launching the numerical solver (see [1], [26]). We can also mention the use of Cayley–Menger determinants to yield simpler algebraic systems (see [21], [31], [32]). 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.

Rule-based solvers rely on the predicates formulation (see [28], [29], [30]). Although they provide a qualitative study of geometric constraints, the “huge” amount of computations needed (exhaustive searching and matching) makes them inappropriate for real world applications.

Graph-constructive solvers are stemming from graph theory. They are based on the analysis of the structure of the constraint graph. The graph constructive approach provides means for developing sound and efficient algorithms (see [2], [6], [9], [11], [13], [14], [18], [19], [20], [22], [24], [27], [33]).

A decomposition–recombination (DR) planner, as defined by Hoffman et al. [12], figures out a plan for decomposing a constraint system into small sub-systems and recombines the solutions of these sub-systems to derive the solution of the entire system. DR planners should have desirable properties such as solvability preserving, Church–Rosser property and completeness [12].

In this paper, we describe a decomposition–recombination planning method based on skeletons called S-DR algorithm that uses a graph based reduction method to solve systems of geometric constraints.

During the decomposition phase, the constraint solver develops a plan to decompose the constraint graph into small sub-systems. A sequence of constructions steps is derived. During the recombination phase, an evaluation of the constructions steps (recombining solutions of the sub-systems) is made to derive the shape needed.

This paper is organized as follows. The graph representation of the constraint problem is explained in Section 2. We present in Section 3, the core algorithm that handles structurally well constrained problems. In Section 4, we briefly compare our method with commonly used comparable methods. Section 5 gives conclusions.