Geometric Constraint Problems Research Articles

A Systematic Framework for Solving Geometric Constraints Analytically

A systematic framework is presented for solving algebraic equations arising in geometric constraint solving. The framework has been used successfully to solve a family of spatial geometric constraint problems. The approach combines geometric reasoning, symbolic reduction, and homotopy continuation.

Geometric constraint satisfaction using optimization methods

The numerical approach to solving geometric constraint problems is indispensable for building a practical CAD system. The most commonly-used numerical method is the Newton–Raphson method. It is fast, but has the instability problem: the method requires good initial values. To overcome this problem, recently the homotopy method has been proposed and experimented with. According to the report, the homotopy method generally works much better in terms of stability. In this paper we use the numerical optimization method to deal with the geometric constraint solving problem. The experimental results based on our implementation of the method show that this method is also much less sensitive to the initial value. Further, a distinctive advantage of the method is that under- and over-constrained problems can be handled naturally and efficiently. We also give many instructive examples to illustrate the above advantages.

A 2-D geometric constraint solver using DOF-based graph reduction

In this paper, we propose a new DOF-based graph reduction approach to geometric constraint solving. In the proposed approach, we incrementally solve a geometric constraint problem that is not ruler-and-compass constructible by incrementally identifying a set of constrained geometric entities with 3 DOF (degree of freedom) as a rigid body and determining the geometric entities in the rigid body using one of two solving procedures: algebraic procedure and numerical procedure, instead of solving it simultaneously using a numerical method. By solving a complex geometric constraint problem incrementally, the proposed method can maximize the efficiency and robustness of a geometric constraint solver.

A geometric constraint engine

This paper describes a geometric constraint engine for finding the configurations of a collection of geometric entities that satisfy a set of geometric constraints. This task is traditionally performed by reformulating the geometry and constraints as algebraic equations which are then solved symbolically or numerically. Symbolic algebraic solution is NP-complete. Numerical solution methods are characterized by slow runtimes, numerical instabilities, and difficulty in handling redundant constraints. Many geometric constraint problems can be solved by reasoning symbolically about the geometric entities themselves using a new technique called degrees of freedom analysis. In this approach, a plan of measurements and actions is devised to satisfy each constraint incrementally, thus monotonically decreasing the system's remaining degrees of freedom. This plan is used to solve, in a maximally decoupled form, the equations resulting from an algebraic representation of the problem. Degrees of freedom analysis results in a polynomial-time, numerically stable algorithm for geometric constraint satisfaction. Empirical comparison with a state-of-the-art numerical solver in the domain of kinematic simulation shows degrees of freedom analysis to be more robust and substantially more efficient.