A 2D geometric constraint solver using a graph reduction method

Keywords: Geometric constraints Modeling by constraints Graph-constructive solver Decomposition–recombination (DR) planning Graph algorithms Under-constrained problems a b s t r a c t 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 decomposi-tion–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 subsystems and recombines the solutions of these subsystems to derive the solution of the entire system. 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(n 3) 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 …