Properties of nonlinear systems and convergence of the Newton-Raphson method in geometric constraint solving

The paper describes an application of a variant of the Newton-Raphson method to solution of geometric constraint problems. Sparsity and rank deficiency of the corresponding nonlinear systems are emphasized and statistical data are presented. Several ways of handling underdeterminancy and overdeterminancy in solving the Newton linear systems are considered. The behavior of Newton’s method is shown on some examples of nonlinear systems. Two algorithms for solving linear systems are proposed based on rank-revealing LU and QR factorizations. The paper is concluded with a numerical comparison of the proposed linear solvers. 1. Geometric constraint problem A geometric constraint problem (GCP) consists of a finite set of objects and a finite set of constraints on them. Either two-dimensional or threedimensional space is considered. Each object has a type, for example: point, line, circle, plane, sphere, cylinder, etc. An “engineering variable” is an important special type of objects that has no geometrical meaning. Each constraint is imposed on a subset of objects of the problem. Constraints also have certain types: incidence, tangency, distance, angle, etc. Each type can have a different meaning depending on the types of objects involved. An “Engineering equation” is a special type of constraints that allows us to impose an arbitrary algebraic equation as a constraint on any set of engineering variables. A GCP is called underconstrained if it has continuum many solutions. If the problem is not underconstrained itself but becomes underconstrained after removing any single constraint, it is called well-constrained. In other cases, a GCP is considered overconstrained. It should be noted that the number of solutions is found up to the movement and rotation of the whole model. CAD users are generally advised to make their geometric models wellconstrained. However, it is often difficult to avoid overconstraining. Moreover, if a user wants to see the intermediate result of modelling, an underconstrained problem is likely to appear since not all the necessary constraints have been created by that time. Therefore, a geometric solver should not neglect the underconstrained and overconstrained problems.